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Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change2", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ \(TraditionalForm\`log(1 + X) = 1 + X + \[ScriptCapitalO](X\^2)\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`log(1 + X) = X + \[ScriptCapitalO](X\^2)\)]], "\"." }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ A Bayesian Derivation of an Iterative Autofocus/Super-Resolution Algorithm\ \>", "Title"], Cell["\<\ Stephen P Luttrell RSRE, Malvern, WR14 3PS, UK\ \>", "Author"], Cell["\<\ This paper appeared in Inverse Problems, 1990, vol. 6, pp. 975-996. Received 5 February 1990, in final form 8 June 1990.\ \>", "Text"], Cell["\[Copyright] Controller, Her Majesty's Stationery Office, 1990", "Text"], Cell[TextData[{ StyleBox["Abstract.", FontWeight->"Bold"], " We derive an estimate-maximise formulation of a Bayesian super-resolution \ algorithm for reconstructing scattering cross sections from coherent images. \ We generalise this result to obtain an 'autofocus/super-resolution' method, \ which simultaneously autofocuses an imaging system and super-resolves its \ image data. We present an explanatory numerical example to illustrate the \ implementation of our method on images of single and double point targets \ that are defocused by \[ScriptCapitalO](depth of focus). These are \ successfully super-resolved by autofocus/super-resolution, but not by pure \ super-resolution. We conjecture that autofocus/super-resolution might \ usefully be applied to the interpretation of airborne synthetic aperture \ radar images that are subject to defocusing effects." }], "Abstract"], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], " Introduction" }], "Section 1"], Cell["\<\ Super-resolution is the name that we give to the process of increasing the \ effective bandwidth of an image (or time series) by introducing collateral \ information to augment the dataset: the classical Rayleigh resolution limit \ for distinguishing two point targets can thereby be overcome. \ Super-resolution belongs to the general class of 'inverse problems' because \ it attempts to recover an object from its image by deconvolving the imaging \ operator.\ \>", "Text"], Cell[TextData[{ "A limitation of our existing super-resolution technique [", ButtonBox["1", ButtonData:>"Ref:DelvesPrydeLuttrell1988", ButtonStyle->"Hyperlink"], "] is its assumption that the parameters of the imaging system are known \ precisely, which causes problems for some applications. Although this problem \ of imaging system calibration is quite general, this work was originally \ prompted by the need to super-resolve images obtained from synthetic aperture \ radar (SAR) systems which have time varying imaging system parameters (due to \ phase shifts caused by anomalous motion of the transmitter/receiver). An \ ideal SAR can be modelled as if it were a simple linear imaging system, and \ the anomalous motion can (in first order) be modelled as a simple defocusing \ of this linear imaging system. It is therefore convenient to visualise a SAR \ as being a microwave version of an optical bench experiment using coherent \ illumination and with the lens misplaced from its correct focus. We shall \ consider \[ScriptCapitalO](depth of focus) errors in the placement of the \ lens, which can cause severe degradation in super-resolved image quality [", ButtonBox["2", ButtonData:>"Ref:Luttrell1985a", ButtonStyle->"Hyperlink"], "]. This \[ScriptCapitalO](depth of focus) criterion applies independently \ of what the actual physical dimensions of a depth of focus happen to be for a \ particular imaging system, because it can be expressed alternatively as an \ upper bound on the quadratic phase error (which is dimensionless) that is \ acceptable at the edge of the aperture." }], "Text"], Cell["\<\ We need to use a precise autofocus method in order to super-resolve \ successfully. We therefore develop a hybrid 'autofocus/super-resolution' \ technique, in which autofocusing uses the super-resolved image (rather than \ the original image) to adjust the focusing parameter(s). The philosophy of \ this technique is to find the set of imaging system parameters and \ super-resolved image that simultaneously fit the image data and collateral \ information. In this paper we develop a theoretical framework that is \ applicable to any coherent imaging system, so it is not important which \ specific application (namely SAR image analysis) originally prompted this \ work, and our results are therefore of interest to a wide audience.\ \>", "Text"], Cell[TextData[{ "Throughout this paper we use Bayesian calculus, because it is the only \ fully consistent means of perfoming inferences from limited information [", ButtonBox["3", ButtonData:>"Ref:Jeffreys1939", ButtonStyle->"Hyperlink"], ", ", ButtonBox["4", ButtonData:>"Ref:Cox1946", ButtonStyle->"Hyperlink"], "]. Bayesian calculus uses probabilities to encode information, and makes \ inferences by manipulating these probabilities. For clarity, we use physical \ arguments to justify the form of each probability that we introduce. One \ could also note the equivalence between ", StyleBox["inverse problems", FontSlant->"Italic"], " and ", StyleBox["inference problems", FontSlant->"Italic"], " as a justification for making Bayesian calculus the appropriate language \ for formulating and solving inverse problems." }], "Text"], Cell[TextData[{ "In ", ButtonBox["section", ButtonData:>"Sect:Model", ButtonStyle->"Hyperlink"], " ", CounterBox["Section", "Sect:Model"], " we summarise our Gaussian scattering model and our linear imaging model. \ We construct our models in terms of probabilities to facilitate the use of \ Bayes' theorem to solve the inverse problem of determining what caused a \ particular dataset." }], "Text"], Cell[TextData[{ "In ", ButtonBox["section", ButtonData:>"Sect:SuperResolution", ButtonStyle->"Hyperlink"], " ", CounterBox["Section", "Sect:SuperResolution"], " we present a complete and rigorous derivation of an iterative Bayesian \ super-resolution algorithm. This is an improved version of an algorithm that \ we described in [", ButtonBox["5", ButtonData:>"Ref:Luttrell1985b", ButtonStyle->"Hyperlink"], ", ", ButtonBox["6", ButtonData:>"Ref:LuttrellOliver1986", ButtonStyle->"Hyperlink"], ", ", ButtonBox["7", ButtonData:>"Ref:Luttrell19xx", ButtonStyle->"Hyperlink"], "], and should be regarded our current definitive treatment of \ super-resolution. Our method is an application of the estimate-maximise (EM) \ method of solving maximum-likelihood problems [", ButtonBox["8", ButtonData:>"Ref:Baum1972", ButtonStyle->"Hyperlink"], ", ", ButtonBox["9", ButtonData:>"Ref:DempsterLairdRubin1977", ButtonStyle->"Hyperlink"], "]." }], "Text"], Cell[TextData[{ "In ", ButtonBox["section", ButtonData:>"Sect:SimultaneousSuperResolutionAndAutofocussing", ButtonStyle->"Hyperlink"], " ", CounterBox["Section", "Sect:SimultaneousSuperResolutionAndAutofocussing"], " we extend our iterative algorithm to account for uncertainties in the \ imaging system. The technique that we derive is a precise autofocus method, \ which effectively focuses on structure in the super-resolved image (rather \ than the original image). We also present a linearised version of the \ algorithm for use in simple cases." }], "Text"], Cell[TextData[{ "In ", ButtonBox["section", ButtonData:>"Sect:ExplanatoryNumericalExample", ButtonStyle->"Hyperlink"], " ", CounterBox["Section", "Sect:ExplanatoryNumericalExample"], " we present an explanatory numerical example to demonstrate how our \ autofocus/super-resolution method might be applied in practice. Note that we \ do not attempt to construct a robust general-purpose algorithm in this paper. \ Thus we use synthetic data that is generated by a defocused 'sinc' function, \ in which case the linearised version of autofocus/super-resolution is \ sufficient, provided that the lens is within \[ScriptCapitalO](depth of \ focus) of the correct focus. We demonstrate autofocus/super-resolution for \ both the single and double point target cases." }], "Text"], Cell[TextData[{ "In ", ButtonBox["appendix", ButtonData:>"Sect:Appendix", ButtonStyle->"Hyperlink"], " A we gather together various definitions and derivations that would \ otherwise distract the flow of the argument in the main body of the paper." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], " The Model" }], "Section", CellTags->"Sect:Model"], Cell["\<\ In this section we summarise our coherent scattering and imaging model. We \ attempt to formulate our Bayesian model by appealing to physical reasoning, \ wherever possible.\ \>", "Text"], Cell[TextData[{ "In ", ButtonBox["figure", ButtonData:>"Fig:Imaging", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:Imaging"], " we represent as a network the various stages of image formation." }], "Text"], Cell[TextData[Cell[TextData[{ " ", ButtonBox["OPEN", ButtonData:>{ URL[ "http://www.luttrell.org.uk/papers/af_sr/fig1.gif"], None}, Active->True, ButtonStyle->"Hyperlink"], " " }]]], "NumberedFigure", TextAlignment->Center, CellTags->"Fig:Imaging"], Cell["Network decomposition of imaging.", "Caption"], Cell[TextData[{ "The notation in ", ButtonBox["figure", ButtonData:>"Fig:Imaging", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:Imaging"], " is defined as" }], "Text"], Cell[BoxData[ FormBox[GridBox[{ {\(\[Sigma] \[Congruent] scattering\ cross\ section\), " ", \(f \[Congruent] scattered\ field\)}, {\(\[Theta] \[Congruent] imaging\ parameters\), " ", \(g \[Congruent] image\ data\)}, {\(P(\[Sigma]) \[Congruent] prior\ probability\ over\ cross\ sections\), " ", " "}, {\(P(\[Theta]) \[Congruent] prior\ probability\ over\ imaging\ parameters\), " ", " "}, {\(P(f | \[Sigma]) \[Congruent] scattering\ model\), " ", \(P(g | f, \[Theta]) \[Congruent] imaging\ \(\(model\)\(.\)\)\)} }], TraditionalForm]], "NumberedEquation", TextAlignment->AlignmentMarker, GridBoxOptions->{ColumnAlignments->{Left}}, CellTags->"Eq:Notation"], Cell[TextData[{ "Note that we are somewhat cavalier in our choice of notation, because for \ instance ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma])\)]], " and ", Cell[BoxData[ \(TraditionalForm\`P(\[Theta])\)]], " are different functions of their respective arguments, yet we use the \ notation ", Cell[BoxData[ \(TraditionalForm\`P( . )\)]], " for both. Thus the meaning of ", Cell[BoxData[ \(TraditionalForm\`P( . )\)]], " should be deduced from context." }], "Text"], Cell[TextData[{ ButtonBox["Figure", ButtonData:>"Fig:Imaging", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:Imaging"], " may be used to decompose the joint probability ", Cell[BoxData[ \(TraditionalForm\`P(g, f, \[Theta], \[Sigma])\)]], " into a product of factors" }], "Text"], Cell[BoxData[ \(TraditionalForm\`P(g, f, \[Theta], \[Sigma]) = \(P(g | f, \[Theta])\) \(P( f | \[Sigma])\) \(P(\[Sigma])\) \(\(P(\[Theta])\)\(.\)\)\)], \ "NumberedEquation"], Cell[TextData[{ "This has the form of a Markov tree, where each argument depends directly \ on only a limited number of other arguments, and there are no circular \ dependencies. We frequently use analogous Markovian decompositions of parts \ of ", ButtonBox["figure", ButtonData:>"Fig:Imaging", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:Imaging"], " as intermediate steps in our derivations, so it is helpful to keep ", ButtonBox["figure", ButtonData:>"Fig:Imaging", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:Imaging"], " in mind when reading this paper." }], "Text"], Cell[TextData[{ "We have deliberately omitted the annotation from the left hand part of ", ButtonBox["figure", ButtonData:>"Fig:Imaging", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:Imaging"], ". This indicates that, in general, we are ignorant of the physical origin \ of ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma])\)]], " and ", Cell[BoxData[ \(TraditionalForm\`P(\[Theta])\)]], ": they serve only as prior probabilities to encode our state of ignorance \ about the values that ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], " might have. This is not a failing of the Bayesian approach, rather it is \ an honest expression of our ignorance about the finer details of the imaging \ model." }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ".", CounterBox["Subsection"], " Prior Probability Over Cross Sections" }], "Subsection"], Cell[TextData[{ "The upper left part of ", ButtonBox["figure", ButtonData:>"Fig:Imaging", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:Imaging"], " generates the prior probability over cross sections ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma])\)]], ". The cross section ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " is an idealised model of those properties of the illuminated object that \ affect the scattered field ", Cell[BoxData[ \(TraditionalForm\`f\)]], ". This use of a cross section is phenomenological, because it is incapable \ of capturing the full range of properties of the process that generates the \ scattered field." }], "Text"], Cell[TextData[{ "For completeness, we include the ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma])\)]], " term in our Bayesian derivations. However, the main goal of this paper is \ to demonstrate autofocus/super-resolution, which we manage to simulate in \ simple cases without introducing ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma])\)]], ". In more complicated cases we might need to use explicit prior knowledge, \ such as ", Cell[BoxData[ \(TraditionalForm\`\[CapitalGamma]\)]], "-distributed cross section models [", ButtonBox["10", ButtonData:>"Ref:JakemanPusey1976", ButtonStyle->"Hyperlink"], ", ", ButtonBox["11", ButtonData:>"Ref:Ward1981", ButtonStyle->"Hyperlink"], "], or Markov random field cross section models [", ButtonBox["12", ButtonData:>"Ref:GemanGeman1984", ButtonStyle->"Hyperlink"], "]." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ".", CounterBox["Subsection"], " Stochastic Scattering Model" }], "Subsection"], Cell[TextData[{ "The upper centre part of ", ButtonBox["figure", ButtonData:>"Fig:Imaging", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:Imaging"], " generates a scattered field ", Cell[BoxData[ \(TraditionalForm\`f\)]], " according to a scattering model ", Cell[BoxData[ \(TraditionalForm\`P(f | \[Sigma])\)]], ". We assume that each cross section element ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_i\)]], " acts as if it produces a large number of scattered wavelets that combine \ coherently to produce an element of scattered field ", Cell[BoxData[ \(TraditionalForm\`f\_i\)]], ", which leads to Gaussian statistics via the central limit theorem. Note \ that ", Cell[BoxData[ \(TraditionalForm\`f\)]], " is a near field which retains essentially the same spatial structure as \ the scattering cross section itself, so it could be directly imaged without \ using a lens (in principle). For ", Cell[BoxData[ \(TraditionalForm\`m\)]], " cross section elements we obtain" }], "Text"], Cell[BoxData[ \(TraditionalForm\`P( f | \[Sigma])\[AlignmentMarker] = \(\[Product]\+\(i = 1\)\%m\( \ exp(\(-\(\(|\)\(f\_i\)\( | \^2\)\(\(/\)\(\[Sigma]\_i\)\)\)\))\)\/\(\[Pi]\ \ \[Sigma]\_i\)\)\n\(\(=\)\(\(1\/\(det(\[Pi]\ \[Sigma])\)\) \(exp(\(-f\^\ \[Dagger]\) \(\[Sigma]\^\(-1\)\) f)\)\)\)\)], "NumberedEquation", TextAlignment->AlignmentMarker, CellTags->"Eq:ScatteringModel"], Cell["where we define", "Text"], Cell[BoxData[ FormBox[ RowBox[{"\[Sigma]", "\[Congruent]", RowBox[{"(", GridBox[{ {\(\[Sigma]\_1\), "0", "0", "\[CenterEllipsis]", "0"}, {"0", \(\[Sigma]\_2\), "0", "\[CenterEllipsis]", "0"}, {"0", "0", \(\[Sigma]\_3\), "\[CenterEllipsis]", "0"}, {"\[VerticalEllipsis]", "\[VerticalEllipsis]", "\[VerticalEllipsis]", "\[DescendingEllipsis]", "\[VerticalEllipsis]"}, {"0", "0", "0", "\[CenterEllipsis]", \(\[Sigma]\_m\)} }], ")"}]}], TraditionalForm]], "NumberedEquation", CellTags->"Eq:DiagonalSigma"], Cell[BoxData[ FormBox[ RowBox[{\(\(f\^\[Dagger]\) \(\[Sigma]\^\(-1\)\) f\), "\[Congruent]", RowBox[{\(\[Sum]\+\(i, j = 1\)\%m\), RowBox[{\(f\_i\%*\), " ", SubscriptBox[\((\[Sigma]\^\(-1\))\), StyleBox["ij", FontSlant->"Italic"]], \(\(f\_j\)\(.\)\)}]}]}], TraditionalForm]], "NumberedEquation"], Cell[TextData[{ "In the first line of ", ButtonBox["equation", ButtonData:>"Eq:ScatteringModel", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:ScatteringModel"], ")", " we ignore any correlations that might exist between different components \ of ", Cell[BoxData[ \(TraditionalForm\`f\)]], ", whereas in the second line of ", ButtonBox["equation", ButtonData:>"Eq:ScatteringModel", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:ScatteringModel"], ")", " we use a notation that allows us to include off-diagonal elements in ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " to model such correlations, should we wish to do so. For generality, we \ assume that ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " is matrix-valued throughout our derivations, unless we state otherwise." }], "Text"], Cell[TextData[{ "We use operator/state notation (as in ", Cell[BoxData[ \(TraditionalForm\`\(f\^\[Dagger]\) \(\[Sigma]\^\(-1\)\) f\)]], ") throughout this paper, because it is both economical and it facilitates \ calculations that would be tedious if performed using explicit summations \ over indices. Note also that each component of the state vector ", Cell[BoxData[ \(TraditionalForm\`f\)]], " is a complex number, which explains the unusual normalisation of the \ Gaussian probability in ", ButtonBox["equation", ButtonData:>"Eq:ScatteringModel", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:ScatteringModel"], ")", "." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ".", CounterBox["Subsection"], " Stochastic Imaging Model and its Parameters" }], "Subsection"], Cell[TextData[{ "The right hand part of ", ButtonBox["figure", ButtonData:>"Fig:Imaging", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:Imaging"], " depends on both the scattered field ", Cell[BoxData[ \(TraditionalForm\`f\)]], " and on the imaging parameters ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], ". Define an imaging model ", Cell[BoxData[ \(TraditionalForm\`P(g | f, \[Theta])\)]], " as" }], "Text"], Cell[BoxData[ \(TraditionalForm\`P(g | f, \[Theta]) = \(1\/\(det(\[Pi]\ N)\)\) exp[\(-\((g - \(T(\[Theta])\) f)\)\^\[Dagger]\) \(\(N\^\(-1\)\)( g - \(T(\[Theta])\) f)\)]\)], "NumberedEquation", CellTags->"Eq:Likelihood"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`N\)]], " is a positive semi-definite covariance matrix that is used to model \ additive Gaussian image data noise. In the limit where all the eigenvalues of \ ", Cell[BoxData[ \(TraditionalForm\`N\)]], " tend to zero this reduces to ", Cell[BoxData[ \(TraditionalForm\`P(g | f, \[Theta]) = \[Delta]( g - \(T(\[Theta])\) f)\)]], " (where ", Cell[BoxData[ \(TraditionalForm\`\[Delta](g - \(T(\[Theta])\) f)\)]], " is the Dirac delta function), which describes the noiseless imaging \ equation ", Cell[BoxData[ \(TraditionalForm\`g = \(T(\[Theta])\) f\)]], "." }], "Text"], Cell[TextData[{ "We use the parameter vector ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], " to parameterise variability in the imaging system, and we use ", Cell[BoxData[ \(TraditionalForm\`P(\[Theta])\)]], " to model our prior knowledge of these parameters. In general, ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], " will be a low dimensional vector that describes those components of the \ imaging system that cannot be calibrated once and for all. We find that we do \ not need to introduce ", Cell[BoxData[ \(TraditionalForm\`P(\[Theta])\)]], " in order to demonstrate autofocus/super-resolution in simple cases, but \ we include it in our derivations, for completeness." }], "Text"], Cell[TextData[{ "In ", ButtonBox["section", ButtonData:>"Sect:GreatestLowerBound", ButtonStyle->"Hyperlink"], " ", CounterBox["Section", "Sect:GreatestLowerBound"], ".", CounterBox["Subsection", "Sect:GreatestLowerBound"], " we derive a linearised autofocus scheme which uses a Gaussian model for \ ", Cell[BoxData[ \(TraditionalForm\`P(\[Theta])\)]], ". Thus" }], "Text"], Cell[BoxData[ \(TraditionalForm\`P(\[Theta]) = \(1\/\(det(\[Pi]\ \[CapitalLambda])\)\) \ \(exp(\(-\[Theta]\^\[Dagger]\) \(\[CapitalLambda]\^\(-1\)\) \[Theta])\)\)], \ "NumberedEquation", CellTags->"Eq:GaussianTheta"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`\[CapitalLambda]\)]], " is a positive semi-definite covariance matrix. We choose ", Cell[BoxData[ \(TraditionalForm\`P(\[Theta])\)]], " to be zero mean, because all Gaussian prior probabilities can easily be \ transformed into this form." }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], " Super-Resolution" }], "Section", CellTags->"Sect:SuperResolution"], Cell[TextData[{ "In this section we assume that the imaging parameters ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], " are exactly known, and we derive an iterative 're-estimation' scheme for \ computing the cross section ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_0\)]], " that maximises the posterior probability ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma] | g)\)]], ". We call this 'super-resolution", "'", " because ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_0\)]], " can have a higher spatial resolution than the image data, although this \ effect is significant only where there are small bright regions embedded in \ the image data [", ButtonBox["13", ButtonData:>"Ref:Luttrell1987", ButtonStyle->"Hyperlink"], "]." }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ".", CounterBox["Subsection"], " Posterior Probability Over Cross Sections" }], "Subsection"], Cell[TextData[{ "Suppose one asks the question: 'What can I deduce about the cross section \ ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], ", given that I know the imaging system model (including all prior \ probabilities) and that I have available a dataset ", Cell[BoxData[ \(TraditionalForm\`g\)]], "?'. Bayesian calculus says categorically: 'The answer to your question is \ the posterior probability ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma] | g)\)]], "'", " [", ButtonBox["3", ButtonData:>"Ref:Jeffreys1939", ButtonStyle->"Hyperlink"], ", ", ButtonBox["4", ButtonData:>"Ref:Cox1946", ButtonStyle->"Hyperlink"], "]. Any reply that does not include enough information to construct ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma] | g)\)]], " has not answered the stated question. ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma] | g)\)]], " can therefore be used to deduce the answer to any other question that \ might have been asked. For instance, the ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " that maximises ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma] | g)\)]], " (let us call it ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_0\)]], ") is usually requested. If ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma] | g)\)]], " has a single well-defined peak in ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], ", then ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_0\)]], " can be used as a representative reconstruction of the cross section ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " given the data ", Cell[BoxData[ \(TraditionalForm\`g\)]], "." }], "Text"], Cell[TextData[{ "In order to calculate ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma] | g)\)]], ", we must first of all use Bayes' theorem to express ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma] | g)\)]], " in terms of quantities that are defined in the imaging system model. \ Thus" }], "Text"], Cell[BoxData[ \(TraditionalForm\`P(\[Sigma] | g) = \(\(\(P(g | \[Sigma])\) \(P(\[Sigma])\)\)\/\(\[Integral]\(d\ \[Sigma]\^\[Prime]\) \(P(g | \[Sigma]\^\[Prime])\) \ \(P(\[Sigma]\^\[Prime])\)\) = \(\(P(g | \[Sigma])\) \(P(\[Sigma])\)\)\/\(P(g)\ \)\)\)], "NumberedEquation", CellTags->"Eq:BayesTheorem"], Cell["which leads to", "Text"], Cell[BoxData[ FormBox[ RowBox[{\(\[Sigma]\_0\), "=", RowBox[{GridBox[{ {\(arg\ max\)}, {"\[Sigma]"} }], \(\({log[P(g | \[Sigma])] + log[P(\[Sigma])]}\)\(.\)\)}]}], TraditionalForm]], "NumberedEquation", CellTags->"Eq:PosteriorProbabilityMaximum"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_0\)]], " is a compromise between maximising ", Cell[BoxData[ \(TraditionalForm\`P(g | \[Sigma])\)]], " and maximising ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma])\)]], ". The ", Cell[BoxData[ \(TraditionalForm\`P(g | \[Sigma])\)]], " term attempts to maximise the probability that the data ", Cell[BoxData[ \(TraditionalForm\`g\)]], " could derive from ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], ", whereas the ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma])\)]], " term ignores the data entirely and attempts to maximise the prior \ probability that ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " could have occurred irrespective of the data. ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_0\)]], " is the cross section that best satisfies these conditions simultaneously. \ The information contained in ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_0\)]], " is less than the information contained in ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma] | g)\)]], " (except for the special case ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma] | g) = \[Delta](\[Sigma] - \[Sigma]\_0)\)]], "). ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma] | g)\)]], " contains everything that can be inferred from the data and the stated \ prior knowledge, whereas ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_0\)]], " is merely the mode of ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma] | g)\)]], "." }], "Text"], Cell[TextData[{ "In [", ButtonBox["14", ButtonData:>"Ref:Luttrell1989", ButtonStyle->"Hyperlink"], "] we presented a calculation of the derivatives of ", Cell[BoxData[ \(TraditionalForm\`P(g | \[Sigma])\)]], ", which provided a mechanism for iteratively computing ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_0\)]], " by a 'gradient ascent' (i.e. infinitesimal update step sizes) scheme. We \ now improve upon these results by deriving a 're-estimation' (", "i.e. ", "finite update step sizes) scheme. This turns out to be very similar to the \ empirical scheme that we suggested in [", ButtonBox["1", ButtonData:>"Ref:DelvesPrydeLuttrell1988", ButtonStyle->"Hyperlink"], "], and, furthermore, it is very simple to relate this to the theory that \ we presented in [", ButtonBox["14", ButtonData:>"Ref:Luttrell1989", ButtonStyle->"Hyperlink"], "]." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ".", CounterBox["Subsection"], " Lower Bound on the Posterior Probability" }], "Subsection"], Cell[TextData[{ "We now maximise a quantity that is related to ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma] | g)\)]], ", but which is constructed in such a way that it is much easier to \ maximise yet has the same local maxima as ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma] | g)\)]], ". The method that we use is based on the estimate-maximise (EM) method of \ maximising likelihood functions [", ButtonBox["9", ButtonData:>"Ref:DempsterLairdRubin1977", ButtonStyle->"Hyperlink"], "]." }], "Text"], Cell[TextData[{ "As a preliminary step we shall transform our probabilities into \ log-probabilities, because this will make our subsequent derivations much \ easier to follow. Thus define ", Cell[BoxData[ \(TraditionalForm\`\(L\_1\)(\[Sigma] | g)\)]], " as" }], "Text"], Cell[BoxData[ \(TraditionalForm\`\(L\_1\)(\[Sigma] | g) \[Congruent] \(\(log[ P(\[Sigma] | g)]\)\(.\)\)\)], "NumberedEquation", CellTags->"Eq:LogProbability"], Cell[TextData[{ "Now we shall derive an important inequality that provides a lower bound \ for ", Cell[BoxData[ \(TraditionalForm\`\(L\_1\)(\[Sigma] | g)\)]], ":" }], "Text"], Cell[BoxData[ FormBox[GridBox[{ {\(Step\ 1\), " ", \(\(L\_1\)(\[Sigma]\^\[Prime] | g)\), \(\(=\)\(log[\(\(P(g | \[Sigma]\^\[Prime])\) \(P(\ \[Sigma]\^\[Prime])\)\)\/\(P(g)\)]\)\)}, {\(Step\ 2\), " ", " ", RowBox[{"=", RowBox[{"log", "[", FractionBox[ RowBox[{"\[Integral]", RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["f", FontSlant->"Italic"]}]], " ", \(P(g | f)\), \(P( f | \[Sigma]\^\[Prime])\), \ \(P(\[Sigma]\^\[Prime])\)}]}], \(P(g)\)], "]"}]}]}, {\(Step\ 3\), " ", " ", RowBox[{"=", RowBox[{"log", "[", RowBox[{"\[Integral]", RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["f", FontSlant->"Italic"]}]], " ", \(P( f | g, \[Sigma])\), \(\(\(P(g | f)\) \(P( f | \[Sigma]\^\[Prime])\) \(P(\[Sigma]\^\[Prime])\ \)\)\/\(\(P(f | g, \[Sigma])\) \(P(g)\)\)\)}]}], "]"}]}]}, {\(Step\ 4\), " ", " ", RowBox[{"=", RowBox[{"log", "[", RowBox[{"\[Integral]", RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["f", FontSlant->"Italic"]}]], " ", \(P( f | g, \[Sigma])\), \(\(P( f | \[Sigma]\^\[Prime])\)\/\(P( f | \[Sigma])\)\), \(\(\(P( g | \[Sigma])\) \(P(\[Sigma]\^\[Prime])\)\)\/\(P( g)\)\)}]}], "]"}]}]}, {\(Step\ 5\), " ", " ", RowBox[{"\[GreaterEqual]", RowBox[{"\[Integral]", RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["f", FontSlant->"Italic"]}]], " ", \(P( f | g, \[Sigma])\), \(\(log[\(\(P( f | \[Sigma]\^\[Prime])\)\/\(P( f | \[Sigma])\)\) \(\(P(g | \[Sigma])\) \(P(\ \[Sigma]\^\[Prime])\)\)\/\(P(g)\)]\)\(.\)\)}]}]}]} }], TraditionalForm]], "NumberedEquation", GridBoxOptions->{ColumnAlignments->{Left}}, CellTags->{"Eq:Inequality", "Ed:LinkToBug1"}], Cell["\<\ We have used the following manipulations in the various steps of this \ derivation\ \>", "Text"], Cell[TextData[{ "Step 1. Use Bayes' theorem as formulated in ", ButtonBox["equation", ButtonData:>"Eq:BayesTheorem", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:BayesTheorem"], ")", " to express ", Cell[BoxData[ \(TraditionalForm\`\(L\_1\)(\[Sigma]\^\[Prime] | g)\)]], " in terms of quantities that are specified in the imaging system model." }], "Text"], Cell[TextData[{ "Step 2. Introduce the scattered field ", Cell[BoxData[ \(TraditionalForm\`f\)]], " as intermediate variables, as in ", ButtonBox["equation", ButtonData:>"Eq:IntermediateVariable", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:IntermediateVariable"], ")", " of ", ButtonBox["appendix", ButtonData:>"Sect:Appendix", ButtonStyle->"Hyperlink"], " A." }], "Text"], Cell[TextData[{ "Step 3. Introduce a factor of unity in the form ", Cell[BoxData[ \(TraditionalForm\`\(P(f | g, \[Sigma])\)\/\(P(f | g, \[Sigma])\)\)]], ". This tautology prepares the integrand for stage 5 of the manipulation." }], "Text", CellTags->"Ed:Change3"], Cell["Step 4. Use the following", "Text"], Cell[BoxData[ \(TraditionalForm\`P( f | g, \[Sigma]) = \(\(P(g, f, \[Sigma])\)\/\(P(g, \[Sigma])\) = \ \(\(P(g | f)\) \(P(f | \[Sigma])\)\)\/\(P(g | \[Sigma])\)\)\)], \ "NumberedEquation"], Cell[TextData[{ "to rearrange the ", Cell[BoxData[ \(TraditionalForm\`1\/\(P(f | g, \[Sigma])\)\)]], " term." }], "Text", CellTags->"Ed:Change4"], Cell["Step 5. Use Jensen's inequality for convex functions", "Text"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"log", "[", RowBox[{"\[Integral]", RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["x", FontSlant->"Italic"]}]], " ", \(u(x)\), \(v(x)\)}]}], "]"}], "\[GreaterEqual]", RowBox[{"\[Integral]", RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["x", FontSlant->"Italic"]}]], " ", \(u(x)\), \(log[v(x)]\)}]}]}], TraditionalForm]], "NumberedEquation"], Cell[TextData[{ "(where ", Cell[BoxData[ \(TraditionalForm\`u(x)\)]], " must satisfy ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[Integral]", RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["x", FontSlant->"Italic"]}]], " ", \(u(x)\)}]}], "=", "1"}], TraditionalForm]]], ") to move the integral outside the logarithm. Equality holds if, and only \ if, ", Cell[BoxData[ \(TraditionalForm\`P(f | \[Sigma]\^\[Prime]) = P(f | \[Sigma])\)]], " for all ", Cell[BoxData[ \(TraditionalForm\`f\)]], ". In our model, this requires ", Cell[BoxData[ \(TraditionalForm\`\[Sigma] = \[Sigma]\^\[Prime]\)]], "." }], "Text"], Cell[TextData[{ "It is convenient to rewrite the final inequality in ", ButtonBox["equation", ButtonData:>"Eq:Inequality", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:Inequality"], ")", " in the form" }], "Text"], Cell[BoxData[{ FormBox[\(\(L\_1\)(\[Sigma]\^\[Prime] | g)\[AlignmentMarker] \[GreaterEqual] \(L\_2\)(\[Sigma]\^\[Prime], \ \[Sigma] | g) + \(L\_1\)(\[Sigma] | g)\), TraditionalForm], "\n", FormBox[ RowBox[{\(\(L\_2\) \((\[Sigma]\^\[Prime], \[Sigma] | g)\)\), "\[AlignmentMarker]", "\[Congruent]", RowBox[{ RowBox[{"\[Integral]", RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["f", FontSlant->"Italic"]}]], " ", \(P( f | g, \[Sigma])\), \(log[\(P(f | \ \[Sigma]\^\[Prime])\)\/\(P(f | \[Sigma])\)]\)}]}], "+", \(log[\(P(\[Sigma]\^\[Prime])\)\/\(P(\[Sigma])\)]\)}]}], TraditionalForm]}], "NumberedEquation", TextAlignment->AlignmentMarker, CellTags->"Eq:Inequality2"], Cell[TextData[{ "where the function ", Cell[BoxData[ \(TraditionalForm\`\(L\_2\)(\[Sigma]\^\[Prime], \[Sigma] | g)\)]], " contains all the ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^\[Prime]\)]], " dependence of the lower bound of ", Cell[BoxData[ \(TraditionalForm\`\(L\_1\)(\[Sigma]\^\[Prime] | g)\)]], ", and is constructed so that ", Cell[BoxData[ \(TraditionalForm\`\(L\_2\)(\[Sigma], \[Sigma] | g) = 0\)]], ". We summarise ", ButtonBox["equation", ButtonData:>"Eq:Inequality2", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:Inequality2"], ")", " in ", ButtonBox["figure", ButtonData:>"Fig:JensensInequality", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:JensensInequality"], "." }], "Text"], Cell[TextData[Cell[TextData[{ " ", ButtonBox["OPEN", ButtonData:>{ URL[ "http://www.luttrell.org.uk/papers/af_sr/fig2.gif"], None}, Active->True, ButtonStyle->"Hyperlink"], " " }]]], "NumberedFigure", TextAlignment->Center, CellTags->"Fig:JensensInequality"], Cell["Jensen's inequality applied to the posterior probability.", "Caption"], Cell[TextData[{ "Instead of maximising ", Cell[BoxData[ \(TraditionalForm\`\(L\_1\)(\[Sigma]\^\[Prime] | g)\)]], " with respect to ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^\[Prime]\)]], ", we now maximise ", Cell[BoxData[ \(TraditionalForm\`\(L\_2\)(\[Sigma]\^\[Prime], \[Sigma] | g)\)]], " with respect to ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^\[Prime]\)]], " (for some fixed ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], "): this will recover a greatest lower bound for the true maximum of ", Cell[BoxData[ \(TraditionalForm\`\(L\_1\)(\[Sigma]\^\[Prime] | g)\)]], "." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ".", CounterBox["Subsection"], " Iteratively Maximising the Lower Bound" }], "Subsection", CellTags->"Sect:IterativelyMaximisingLowerBound"], Cell[TextData[{ "Before proceeding any further, we present an outline of our proposed \ algorithm for maximising the posterior probability ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma] | g)\)]], ". We do this to provide a concrete framework and motivation for the rather \ involved calculations that we perform later on." }], "Text"], Cell[TextData[{ "Maximising ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma] | g)\)]], " is equivalent to maximising ", Cell[BoxData[ \(TraditionalForm\`\(L\_1\)(\[Sigma] | g)\)]], ". In turn, maximising ", Cell[BoxData[ \(TraditionalForm\`\(L\_1\)(\[Sigma]\^\[Prime] | g)\)]], " can be replaced by maximising ", Cell[BoxData[ \(TraditionalForm\`\(L\_2\)(\[Sigma]\^\[Prime], \[Sigma] | g)\)]], ", although this leads only to a greatest lower bound for ", Cell[BoxData[ \(TraditionalForm\`\(L\_1\)(\[Sigma]\^\[Prime] | g)\)]], ", as given in ", ButtonBox["equation", ButtonData:>"Eq:Inequality2", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:Inequality2"], ")", ". If this greatest lower bound process is iterated by replacing ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " with the optimum value of ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^\[Prime]\)]], " that was found during the previous iteration, then the greatest lower \ bound converges towards a local maximum of ", Cell[BoxData[ \(TraditionalForm\`\(L\_1\)(\[Sigma]\^\[Prime] | g)\)]], "." }], "Text"], Cell[TextData[{ "In broad outline, our proposed algorithm for maximising ", Cell[BoxData[ \(TraditionalForm\`\(L\_1\)(\[Sigma]\^\[Prime] | g)\)]], " is:" }], "Text"], Cell[TextData[{ "1. Initialisation step: Make an initial choice of ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], "." }], "Text"], Cell[TextData[{ "2. Re-estimation step: Maximise ", Cell[BoxData[ \(TraditionalForm\`\(L\_2\)(\[Sigma]\^\[Prime], \[Sigma] | g)\)]], " with respect to ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^\[Prime]\)]], "." }], "Text"], Cell[TextData[{ "3. Update step: ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\[LongRightArrow]\[Sigma]\^\[Prime]\)]], "." }], "Text"], Cell[TextData[{ "4. Iteration step: If the update in ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " does not satisfy some convergence criterion, then go to step 2." }], "Text"], Cell[TextData[{ "5. ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " is now close to a local maximum of ", Cell[BoxData[ \(TraditionalForm\`\(L\_1\)(\[Sigma] | g)\)]], ". This does not guarantee that ", Cell[BoxData[ \(TraditionalForm\`\[Sigma] \[TildeTilde] \[Sigma]\_0\)]], ", which is the required global maximum. Jensen's inequality, together with \ our model, guarantees only that fixed points of the re-estimation step are \ local maxima of ", Cell[BoxData[ \(TraditionalForm\`\(L\_1\)(\[Sigma] | g)\)]], "." }], "Text"], Cell[TextData[{ "In ", ButtonBox["figure", ButtonData:>"Fig:MaximisationAlgorithm", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:MaximisationAlgorithm"], " we show three iterations of this algorithm." }], "Text"], Cell[TextData[Cell[TextData[{ " ", ButtonBox["OPEN", ButtonData:>{ URL[ "http://www.luttrell.org.uk/papers/af_sr/fig3.gif"], None}, Active->True, ButtonStyle->"Hyperlink"], " " }]]], "NumberedFigure", TextAlignment->Center, CellTags->"Fig:MaximisationAlgorithm"], Cell["Three iterations of the maximisation algorithm.", "Caption"], Cell[TextData[{ "We represent ", Cell[BoxData[ \(TraditionalForm\`\(L\_1\)(\[Sigma] | g)\)]], " as a large inverted cup, and ", Cell[BoxData[ \(TraditionalForm\`\(L\_2\)(\[Sigma]\^\[Prime], \[Sigma] | g) + \(L\_1\)(\[Sigma] | g)\)]], " as a small inverted cup. The initial choice is ", Cell[BoxData[ \(TraditionalForm\`\[Sigma] = \[Sigma]\_1\)]], ", so ", Cell[BoxData[ \(TraditionalForm\`\(\(\(L\_2\)(\[Sigma]\^\[Prime], \[Sigma]\_1 | g) + \(L\_1\)(\[Sigma]\_1 | g)\)\(\[LessEqual]\)\(\(L\_1\)(\[Sigma]\^\[Prime] | g)\)\(\[AlignmentMarker]\)\)\)]], " with equality at ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^\[Prime] = \[Sigma]\_1\)]], ", as shown. ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\[LongRightArrow]\[Sigma]\_2\)]], " is then the outcome of the re-estimation step. Accordingly, we show ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_2\)]], " as the position of the maximum of ", Cell[BoxData[ \(TraditionalForm\`\(L\_2\)(\[Sigma]\^\[Prime], \[Sigma]\_1 | g)\)]], " in ", ButtonBox["figure", ButtonData:>"Fig:MaximisationAlgorithm", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:MaximisationAlgorithm"], ". The remainder of ", ButtonBox["figure", ButtonData:>"Fig:MaximisationAlgorithm", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:MaximisationAlgorithm"], " depicts two further iterations, producing ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_3\)]], " then ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_4\)]], ", and making steady progress towards a maximum of ", Cell[BoxData[ \(TraditionalForm\`\(L\_1\)(\[Sigma] | g)\)]], ". It is geometrically obvious how this algorithm is guaranteed to find a \ local maximum of ", Cell[BoxData[ \(TraditionalForm\`\(L\_1\)(\[Sigma] | g)\)]], "." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ".", CounterBox["Subsection"], " Greatest Lower Bound on the Posterior Probability" }], "Subsection"], Cell[TextData[{ "The algorithm in ", ButtonBox["subsection", ButtonData:>"Sect:IterativelyMaximisingLowerBound", ButtonStyle->"Hyperlink"], " ", CounterBox["Section", "Sect:SuperResolution"], ".", CounterBox["Subsection", "Sect:IterativelyMaximisingLowerBound"], " relies critically on the re-estimation step. We therefore derive the \ stationary point(s) of ", Cell[BoxData[ \(TraditionalForm\`\(L\_2\)(\[Sigma]\^\[Prime], \[Sigma] | g)\)]], ", as defined in ", ButtonBox["equation", ButtonData:>"Eq:Inequality2", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:Inequality2"], ")", "." }], "Text"], Cell[TextData[{ "From ", ButtonBox["equation", ButtonData:>"Eq:Inequality2", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:Inequality2"], ")", " and ", ButtonBox["equation", ButtonData:>"Eq:ScatteringModel", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:ScatteringModel"], ")", " we obtain" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{\(\(L\_2\)(\[Sigma]\^\[Prime], \[Sigma] | g)\), "=", RowBox[{"\[Integral]", RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["f", FontSlant->"Italic"]}]], " ", \(P( f | g, \[Sigma])\), \(\({\(\([\)\(\(-log[ det\ \((\[Pi]\ \[Sigma]\^\[Prime])\)]\) - \(f\^\ \[Dagger]\) \(\[Sigma]\^\(-1\)\) f + log[P(\[Sigma]\^\[Prime])]\)\(]\)\) - \(\([\)\(\[Sigma]\^\ \[Prime]\[LongRightArrow]\[Sigma]\)\(]\)\)}\)\(.\)\)}]}]}], TraditionalForm]], "NumberedEquation", CellTags->"Eq:LowerBound"], Cell[TextData[{ "Neither the ", Cell[BoxData[ \(TraditionalForm\`log[det(\[Pi]\ \[Sigma]\^\[Prime])]\)]], " term nor the ", Cell[BoxData[ \(TraditionalForm\`log[P(\[Sigma]\^\[Prime])]\)]], " term depend on ", Cell[BoxData[ \(TraditionalForm\`f\)]], ", so their ", Cell[BoxData[ \(TraditionalForm\`f\)]], " integrals can be discarded (using ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[Integral]", RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["f", FontSlant->"Italic"]}]], " ", "P", \((f | g, \[Sigma])\)}]}], "=", "1"}], TraditionalForm]]], ")." }], "Text"], Cell[TextData[{ "However, the ", Cell[BoxData[ \(TraditionalForm\`\(f\^\[Dagger]\) \(\[Sigma]\^\(-1\)\) f\)]], " term requires more attention. ", Cell[BoxData[ \(TraditionalForm\`P(f | g, \[Sigma])\)]], " is given in ", ButtonBox["equation", ButtonData:>"Eq:PosteriorProbability", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:PosteriorProbability"], ")", " of ", ButtonBox["appendix", ButtonData:>"Sect:Appendix", ButtonStyle->"Hyperlink"], " A, where we see that it is a Gaussian with mean ", Cell[BoxData[ \(TraditionalForm\`f\&_\)]], ". This makes the ", Cell[BoxData[ \(TraditionalForm\`\(f\^\[Dagger]\) \(\[Sigma]\^\(-1\)\) f\)]], " term of ", ButtonBox["equation", ButtonData:>"Eq:LowerBound", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:LowerBound"], ")", " relatively easy to evaluate. To facilitate this integration we rearrange \ ", Cell[BoxData[ \(TraditionalForm\`\(f\^\[Dagger]\) \(\[Sigma]\^\(-1\)\) f\)]], " so that ", Cell[BoxData[ \(TraditionalForm\`f\)]], " appears only in the combination ", Cell[BoxData[ \(TraditionalForm\`f - f\&_\)]], ":" }], "Text"], Cell[BoxData[ \(TraditionalForm\`\(f\^\[Dagger]\) \(\[Sigma]\^\(-1\)\) f = \(\((f - f\&_)\)\^\[Dagger]\) \(\(\(\[Sigma]\^\[Prime]\)\^\(-1\)\)( f - f\&_)\) + \(\([\)\(\(f\&_\^\[Dagger]\) \(\(\(\[Sigma]\^\ \[Prime]\)\^\(-1\)\)(f - f\&_)\) + CC\)\(]\)\) + \(f\&_\^\[Dagger]\) \ \(\(\[Sigma]\^\[Prime]\)\^\(-1\)\) f\&_\)], "NumberedEquation", CellTags->"Eq:HermitianForm"], Cell["where 'CC' denotes 'complex conjugate'. This yields", "Text"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[Integral]", RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["f", FontSlant->"Italic"]}]], " ", \(P( f | g, \[Sigma])\), \(f\^\[Dagger]\), \ \(\(\[Sigma]\^\[Prime]\)\^\(-1\)\), "f"}]}], "=", \(tr[\(\(\[Sigma]\^\[Prime]\)\^\(-1\)\) C] + \(f\&_\^\[Dagger]\) \(\(\[Sigma]\^\[Prime]\)\^\(-1\)\) \ \(\(f\&_\)\(.\)\)\)}], TraditionalForm]], "NumberedEquation", CellTags->"Eq:PosteriorProbabilityHermitianForm"], Cell[TextData[{ "To obtain this result we rewrite the ", Cell[BoxData[ \(TraditionalForm\`\(\((f - f\&_)\)\^\[Dagger]\) \(\(\(\[Sigma]\^\[Prime]\)\^\(-1\)\)( f - f\&_)\)\)]], " term by using ", Cell[BoxData[ \(TraditionalForm\`\(\((f - f\&_)\)\^\[Dagger]\) \(\(\(\[Sigma]\^\[Prime]\)\^\(-1\)\)( f - f\&_)\) = tr[\(\(\(\[Sigma]\^\[Prime]\)\^\(-1\)\)( f - f\&_)\) \((f - f\&_)\)\^\[Dagger]]\)]], ", and then using the covariance ", Cell[BoxData[ \(TraditionalForm\`C\)]], " of ", Cell[BoxData[ \(TraditionalForm\`f - f\&_\)]], " (see ", ButtonBox["equation", ButtonData:>"Eq:PosteriorProbability", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:PosteriorProbability"], ")", " of ", ButtonBox["appendix", ButtonData:>"Sect:Appendix", ButtonStyle->"Hyperlink"], " A) to average under the trace. The term linear in ", Cell[BoxData[ \(TraditionalForm\`f - f\&_\)]], " averages to zero, by symmetry." }], "Text"], Cell[TextData[{ "Finally, insert ", ButtonBox["equation", ButtonData:>"Eq:PosteriorProbabilityHermitianForm", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:PosteriorProbabilityHermitianForm"], ")", " into ", ButtonBox["equation", ButtonData:>"Eq:LowerBound", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:LowerBound"], ")", " to obtain" }], "Text"], Cell[BoxData[ \(TraditionalForm\`\(L\_2\)(\[Sigma]\^\[Prime], \[Sigma] | g) = \(\([\)\(\(-log[det(\[Pi]\ \[Sigma]\^\[Prime])]\) - tr[\(\(\[Sigma]\^\[Prime]\)\^\(-1\)\) C] - \(f\&_\^\[Dagger]\) \(\(\[Sigma]\^\[Prime]\)\^\(-1\)\) f\&_ + log[ P(\[Sigma]\^\[Prime])]\)\(]\)\) - \(\([\)\(\[Sigma]\^\[Prime]\ \[LongRightArrow]\[Sigma]\)\(]\)\)\)], "NumberedEquation", CellTags->"Eq:LowerBoundIntegrated"], Cell["which may also be written in the form", "Text"], Cell[BoxData[{ \(TraditionalForm\`\(L\_2\)(\[Sigma]\^\[Prime], \[Sigma] | g)\[AlignmentMarker] = log[\(\(P\_eff\)(g | \[Sigma]\^\[Prime])\)\/\(\(P\_eff\)(g | \ \[Sigma])\)] + log[\(\(P\_eff\)(\[Sigma]\^\[Prime])\)\/\(\(P\_eff\)(\[Sigma])\)]\), \ "\n", \(TraditionalForm\`\(P\_eff\)( g | \[Sigma]\^\[Prime])\[AlignmentMarker] \[Proportional] exp(\(-f\&_\^\[Dagger]\) \(\(\[Sigma]\^\[Prime]\)\^\(-1\)\) f\&_)\), "\n", \(TraditionalForm\`\(P\_eff\)(\[Sigma]\^\[Prime])\[AlignmentMarker] \ \[Proportional] \(\(P(\[Sigma]\^\[Prime])\)\/\(det(\[Pi]\ \[Sigma]\^\[Prime])\ \)\) \(exp(\(-tr[\(\(\[Sigma]\^\[Prime]\)\^\(-1\)\) C]\))\)\)}], "NumberedEquation", TextAlignment->AlignmentMarker], Cell[TextData[{ "where we include only the data dependent part in ", Cell[BoxData[ \(TraditionalForm\`\(P\_eff\)(g | \[Sigma]\^\[Prime])\)]], ". For a diagonal ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], "-matrix (as in ", ButtonBox["equation", ButtonData:>"Eq:DiagonalSigma", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:DiagonalSigma"], ")", ") this reduces to" }], "Text"], Cell[BoxData[{ FormBox[ RowBox[{\(\(P\_eff\) \((g | \[Sigma]\^\[Prime])\)\), "\[AlignmentMarker]", "\[Proportional]", RowBox[{\(\[Product]\+\(i = 1\)\%m\), RowBox[{"exp", "(", RowBox[{"-", FractionBox[ FormBox[\(\[LeftBracketingBar]f\&_\_i\[RightBracketingBar]\^2\ \), "TraditionalForm"], \(\[Sigma]\_i\^\[Prime]\)]}], ")"}]}]}], TraditionalForm], "\n", FormBox[ RowBox[{\(\(P\_eff\) \((\[Sigma]\^\[Prime])\)\), "\[AlignmentMarker]", "\[Proportional]", RowBox[{\(P(\[Sigma]\^\[Prime])\), RowBox[{\(\[Product]\+\(i = 1\)\%m\), RowBox[{\(1\/\(\[Pi]\ \[Sigma]\_i\^\[Prime]\)\), RowBox[{ RowBox[{"exp", "(", RowBox[{"-", FractionBox[ SubscriptBox["C", StyleBox["ii", FontSlant->"Italic"]], \(\[Sigma]\_i\^\[Prime]\)]}], ")"}], "."}]}]}]}]}], TraditionalForm]}], "NumberedEquation", TextAlignment->AlignmentMarker, CellTags->"Ed:Change16"], Cell[TextData[{ "We have now greatly simplified ", Cell[BoxData[ \(TraditionalForm\`\(L\_2\)(\[Sigma]\^\[Prime], \[Sigma] | g)\)]], " (", ButtonBox["equation", ButtonData:>"Eq:LowerBound", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:LowerBound"], ")", ") because we have succeeded in integrating out the intermediate variable \ ", Cell[BoxData[ \(TraditionalForm\`f\)]], "." }], "Text"], Cell[TextData[Cell[TextData[{ " ", ButtonBox["OPEN", ButtonData:>{ URL[ "http://www.luttrell.org.uk/papers/af_sr/fig4.gif"], None}, Active->True, ButtonStyle->"Hyperlink"], " " }]]], "NumberedFigure", TextAlignment->Center, CellTags->"Fig:NetworkDecompositionReconstruction"], Cell["Network decomposition of reconstruction.", "Caption"], Cell[TextData[{ "In ", ButtonBox["figure", ButtonData:>"Fig:NetworkDecompositionReconstruction", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:NetworkDecompositionReconstruction"], " we show the relationship between the various probabilities that we have \ introduced. ", ButtonBox["Figure", ButtonData:>"Fig:NetworkDecompositionReconstruction", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:NetworkDecompositionReconstruction"], " contains two coupled inference processes. Firstly, the top half of ", ButtonBox["figure", ButtonData:>"Fig:NetworkDecompositionReconstruction", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:NetworkDecompositionReconstruction"], " constructs the posterior probability ", Cell[BoxData[ \(TraditionalForm\`P(f | g, \[Sigma])\)]], ". This is a Gaussian probability with mean ", Cell[BoxData[ \(TraditionalForm\`f\&_\)]], " that is normally used as the 'maximum posterior probability' \ reconstruction. Secondly, we input ", Cell[BoxData[ \(TraditionalForm\`f\&_\)]], " to the bottom half of ", ButtonBox["figure", ButtonData:>"Fig:NetworkDecompositionReconstruction", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:NetworkDecompositionReconstruction"], ", to construct an effective posterior probability (proportional to ", Cell[BoxData[ \(TraditionalForm\`\(\(P\_eff\)(\[Sigma]\^\[Prime])\) \(\(P\_eff\)( g | \[Sigma]\^\[Prime])\)\)]], ") whose logarithm is (up to an additive constant) ", Cell[BoxData[ \(TraditionalForm\`\(L\_2\)(\[Sigma]\^\[Prime], \[Sigma] | g)\)]], " in ", ButtonBox["equation", ButtonData:>"Eq:LowerBoundIntegrated", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:LowerBoundIntegrated"], ")", "." }], "Text"], Cell[TextData[{ "It is important to note that the second stage of the above inference \ process does not construct a true Bayesian posterior probability, because ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^\[Prime]\)]], " is merely a mathematical convenience, not a physical reality. We have \ written the second stage in the style of a posterior probability merely to \ aid its interpretation. The true Bayesian posterior probability is ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma] | g)\)]], ", which is maximised by an algorithm that consists of several iterations \ as described above." }], "Text"], Cell[TextData[{ "We now differentiate each term in ", ButtonBox["equation", ButtonData:>"Eq:LowerBoundIntegrated", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:LowerBoundIntegrated"], ")", " with respect to ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^\[Prime]\)]], ". It turns out to be convenient to differentiate with respect to the \ transpose of the inverse matrix ", Cell[BoxData[ \(TraditionalForm\`\((\(\[Sigma]\&~\)\^\[Prime])\)\^\(-1\)\)]], ". Differentiating the ", Cell[BoxData[ \(TraditionalForm\`log\ [det\ \((\[Pi]\ \[Sigma]\^\[Prime])\)]\)]], " term requires considerable care, because matrix-valued quantities do not \ necessarily commute, so we give a compact derivation in ", ButtonBox["appendix", ButtonData:>"Sect:Appendix", ButtonStyle->"Hyperlink"], " A. The other terms in ", ButtonBox["equation", ButtonData:>"Eq:LowerBoundIntegrated", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:LowerBoundIntegrated"], ")", " pose no problem, and we obtain finally" }], "Text"], Cell[BoxData[{ \(TraditionalForm\`\(\[PartialD]\/\[PartialD]\((\(\[Sigma]\&~\)\^\[Prime])\ \)\^\(-1\)\) log[det(\[Pi]\ \[Sigma]\^\[Prime])]\[AlignmentMarker] = \(-\[Sigma]\^\ \[Prime]\)\n\(\(\[PartialD]\/\[PartialD]\((\(\[Sigma]\&~\)\^\[Prime])\)\^\(-1\ \)\) tr[\(\(\[Sigma]\^\[Prime]\)\^\(-1\)\) C]\[AlignmentMarker] = C\)\), "\n", \(TraditionalForm\`\(\[PartialD]\/\[PartialD]\((\(\[Sigma]\&~\)\^\[Prime])\ \)\^\(-1\)\) \(f\&_\^\[Dagger]\) \(\(\[Sigma]\^\[Prime]\)\^\(-1\)\) f\&_\[AlignmentMarker] = \(f\&_\) \(\(f\&_\^\[Dagger]\)\(.\)\)\)}], \ "NumberedEquation", TextAlignment->AlignmentMarker, CellTags->"Eq:Derivatives"], Cell[TextData[{ "Combine ", ButtonBox["equation", ButtonData:>"Eq:Derivatives", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:Derivatives"], ")", " with ", ButtonBox["equation", ButtonData:>"Eq:LowerBoundIntegrated", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:LowerBoundIntegrated"], ")", " to obtain" }], "Text"], Cell[BoxData[ \(TraditionalForm\`\(\[PartialD]\/\[PartialD]\((\(\[Sigma]\&~\)\^\[Prime])\ \)\^\(-1\)\) \(\(L\_2\)(\[Sigma]\^\[Prime], \[Sigma] | g)\) = \[Sigma]\^\[Prime] - \(f\&_\) f\&_\^\[Dagger] - C + \(\[PartialD]\/\[PartialD]\((\(\[Sigma]\&~\)\^\[Prime])\)\^\(-1\)\ \) \(\(log[P(\[Sigma]\^\[Prime])]\)\(.\)\)\)], "NumberedEquation", CellTags->"Eq:LowerBoundIntegratedDerivative"], Cell["\<\ This is the central result from which our super-resolution 're-estimation' \ algorithm can be deduced.\ \>", "Text"], Cell[TextData[{ "We may express ", ButtonBox["equation", ButtonData:>"Eq:LowerBoundIntegratedDerivative", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:LowerBoundIntegratedDerivative"], ")", " in a form that is appropriate for a diagonal ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], "-matrix. Thus" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{\(\(\[PartialD]\/\[PartialD]\ log[\[Sigma]\_i\^\[Prime]]\) \(\(L\_2\)(\[Sigma]\^\[Prime], \ \[Sigma] | g)\)\), "\[AlignmentMarker]", "=", RowBox[{ FractionBox[ RowBox[{\(\[LeftBracketingBar]f\&_\_i\[RightBracketingBar]\^2\), "+", SubscriptBox["C", StyleBox["ii", FontSlant->"Italic"]], "-", \(\[Sigma]\_i\^\[Prime]\)}], \(\[Sigma]\_i\^\[Prime]\)], "+", \(\(\[PartialD]\/\[PartialD]\ log[\[Sigma]\_i\^\[Prime]]\) log[P(\[Sigma]\^\[Prime])]\)}]}], TraditionalForm]], "NumberedEquation", CellTags->"Eq:LowerBoundIntegratedDerivativeDiagonal1"], Cell[BoxData[ \(TraditionalForm\`\(\[PartialD]\/\[PartialD]\ log[\[Sigma]\_i\^\[Prime]]\) \(\(L\_2\)(\[Sigma]\^\[Prime], \ \[Sigma] | g)\)\[AlignmentMarker] = \ \(\(\([\)\(\(\(\[LeftBracketingBar]f\&_\_i\[RightBracketingBar]\^2\)\(-\)\) < \ \[LeftBracketingBar]f\&_\_i\[RightBracketingBar]\^2 > \)\(]\)\) + \(\([\)\(\ \[Sigma]\_i - \[Sigma]\_i\^\[Prime]\)\(]\)\)\)\/\[Sigma]\_i\^\[Prime] + \(\ \[PartialD]\/\[PartialD]\ log[\[Sigma]\_i\^\[Prime]]\) log[P(\[Sigma]\^\[Prime])]\)], "NumberedEquation", CellTags->"Eq:LowerBoundIntegratedDerivativeDiagonal2"], Cell[TextData[{ "where we have used the results in ", ButtonBox["equation", ButtonData:>"Eq:Definitions", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:Definitions"], ")", " and ", ButtonBox["equation", ButtonData:>"Eq:DefinitionsInverse", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:DefinitionsInverse"], ")", ". The appropriate update ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\[LongRightArrow]\[Sigma]\^\[Prime]\)]], " (to obtain the required greatest lower bound of ", Cell[BoxData[ \(TraditionalForm\`\(L\_1\)(\[Sigma]\^\[Prime] | g)\)]], ") is to replace ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " by the value of ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^\[Prime]\)]], " that makes the right hand side of ", ButtonBox["equation", ButtonData:>"Eq:LowerBoundIntegratedDerivativeDiagonal1", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:LowerBoundIntegratedDerivativeDiagonal1"], ")", " (and ", ButtonBox["equation", ButtonData:>"Eq:LowerBoundIntegratedDerivativeDiagonal2", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:LowerBoundIntegratedDerivativeDiagonal2"], ")", ") equal to zero." }], "Text"], Cell[TextData[{ "On the other hand, we may relate ", ButtonBox["equation", ButtonData:>"Eq:LowerBoundIntegratedDerivative", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:LowerBoundIntegratedDerivative"], ") to the results of [", ButtonBox["14", ButtonData:>"Ref:Luttrell1989", ButtonStyle->"Hyperlink"], "]. Thus we paraphrase ", "equation (5.14)", " of [", ButtonBox["14", ButtonData:>"Ref:Luttrell1989", ButtonStyle->"Hyperlink"], "] in the notation of this paper as follows" }], "Text"], Cell[BoxData[ \(TraditionalForm\`\(\[PartialD]\/\[PartialD]\ log[\[Sigma]\_i]\) \(\(L\_1\)(\[Sigma] | g)\)\[AlignmentMarker] = \(\(\(\[LeftBracketingBar]f\&_\_i\ \[RightBracketingBar]\^2\)\(-\)\) < \[LeftBracketingBar]f\&_\_i\ \[RightBracketingBar]\^2 > \)\/\[Sigma]\_i + \(\[PartialD]\/\[PartialD]\ log[\[Sigma]\_i]\) \(\(log[ P(\[Sigma])]\)\(.\)\)\)], "NumberedEquation", CellTags->"Eq:DerivativeDiagonalOther"], Cell[TextData[{ "When we set ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\_i\^\[Prime] = \[Sigma]\_i\)]], " in ", ButtonBox["equation", ButtonData:>"Eq:LowerBoundIntegratedDerivativeDiagonal2", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:LowerBoundIntegratedDerivativeDiagonal2"], ")", " we recover ", ButtonBox["equation", ButtonData:>"Eq:DerivativeDiagonalOther", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:DerivativeDiagonalOther"], ")", ", as required. This demonstrates the consistency of the expressions for \ the gradient of ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma] | g)\)]], " that are contained in this paper and in [", ButtonBox["14", ButtonData:>"Ref:Luttrell1989", ButtonStyle->"Hyperlink"], "]. In a sense, [", ButtonBox["14", ButtonData:>"Ref:Luttrell1989", ButtonStyle->"Hyperlink"], "] implements the bottom half of ", ButtonBox["figure", ButtonData:>"Fig:NetworkDecompositionReconstruction", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:NetworkDecompositionReconstruction"], " in a half-hearted fashion, by suggesting small 'gradient ascent' style \ updates using ", ButtonBox["equation", ButtonData:>"Eq:DerivativeDiagonalOther", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:DerivativeDiagonalOther"], ")", "), rather than by making the better 're-estimation' style updates using \ the zero(s) of ", ButtonBox["equation", ButtonData:>"Eq:LowerBoundIntegratedDerivativeDiagonal1", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:LowerBoundIntegratedDerivativeDiagonal1"], ")", " or ", ButtonBox["equation", ButtonData:>"Eq:LowerBoundIntegratedDerivativeDiagonal2", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:LowerBoundIntegratedDerivativeDiagonal2"], ")", "." }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], " Simultaneous Super-Resolution and Autofocusing" }], "Section", CellTags->"Sect:SimultaneousSuperResolutionAndAutofocussing"], Cell[TextData[{ "In this section we extend the results of ", ButtonBox["section", ButtonData:>"Sect:SuperResolution", ButtonStyle->"Hyperlink"], " ", CounterBox["Section", "Sect:SuperResolution"], " to account for uncertain imaging parameters ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], ". We derive an iterative scheme for computing ", Cell[BoxData[ \(TraditionalForm\`\((\[Sigma]\_0, \[Theta]\_0)\)\)]], " that maximises the posterior probability ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma], \[Theta] | g)\)]], ". We call this 'autofocus/super-resolution' because we recover the imaging \ parameters ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], " (", "i.e. ", "we 'focus' the imaging system) at the same time as we recover the cross \ section ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " (i.e. 'super-resolve' the image)." }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ".", CounterBox["Subsection"], " Derivation of the Re-estimation Equation" }], "Subsection"], Cell[TextData[{ ButtonBox["Equation", ButtonData:>"Eq:PosteriorProbabilityMaximum", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:PosteriorProbabilityMaximum"], ")", " becomes" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{\((\(\[Sigma]\_0\) \[Theta]\_0)\), "=", RowBox[{GridBox[{ {\(arg\ max\)}, {\(\[Sigma], \[Theta]\)} }], \({log[P(g | \[Sigma], \[Theta])] + log[P(\[Sigma])] + log[P(\[Theta])]}\)}]}], TraditionalForm]], "NumberedEquation", CellTags->"Eq:PosteriorProbabilityMaximumSimultaneous"], Cell[TextData[{ "where we assume that the prior probabilities of ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], " are independent. ", ButtonBox["Equation", ButtonData:>"Eq:LogProbability", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:LogProbability"], ")", " becomes" }], "Text"], Cell[BoxData[ \(TraditionalForm\`\(L\_1\)(\[Sigma], \[Theta] | g) \[Congruent] log[P(\[Sigma], \[Theta] | g)]\)], "NumberedEquation"], Cell[TextData[{ "and the derivation in ", ButtonBox["equation", ButtonData:>"Eq:Inequality", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:Inequality"], ")", " becomes" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{\(\(L\_1\)(\[Sigma]\^\[Prime], \[Theta]\^\[Prime] | g)\), "\[AlignmentMarker]", "=", RowBox[{\(log[\(\(P(g | \[Sigma]\^\[Prime], \[Theta]\^\[Prime])\) \ \(P(\[Sigma]\^\[Prime])\) \(P(\[Theta]\^\[Prime])\)\)\/\(P(g)\)]\), "\n", "=", RowBox[{ RowBox[{"log", "[", FractionBox[ RowBox[{"\[Integral]", RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["f", FontSlant->"Italic"]}]], " ", \(P(g | f, \[Theta]\^\[Prime])\), \(P( f | \[Sigma]\^\[Prime])\), \(P(\[Sigma]\^\[Prime])\), \ \(P(\[Theta]\^\[Prime])\)}]}], \(P(g)\)], "]"}], "\n", "=", RowBox[{ RowBox[{"log", "[", RowBox[{"\[Integral]", RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["f", FontSlant->"Italic"]}]], " ", \(P( f | g, \[Sigma], \[Theta])\), \(\(\(P( g | f, \[Theta]\^\[Prime])\) \(P( f | \[Sigma]\^\[Prime])\) \(P(\[Sigma]\^\[Prime])\ \) \(P(\[Theta]\^\[Prime])\)\)\/\(\(P(f | g, \[Sigma], \[Theta])\) \(P( g)\)\)\)}]}], "]"}], "\n", "=", RowBox[{ RowBox[{"log", "[", RowBox[{"\[Integral]", RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["f", FontSlant->"Italic"]}]], " ", \(P( f | g, \[Sigma], \[Theta])\), \(\(P( f | \[Sigma]\^\[Prime])\)\/\(P( f | \[Sigma])\)\), \(\(P( g | f, \[Theta]\^\[Prime])\)\/\(P( g | f, \[Theta])\)\), \(\(\(P( g | \[Sigma], \[Theta])\) \ \(P(\[Sigma]\^\[Prime])\) \(P(\[Theta]\^\[Prime])\)\)\/\(P(g)\)\)}]}], "]"}], "\n", "\[GreaterEqual]", RowBox[{"\[Integral]", RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["f", FontSlant->"Italic"]}]], " ", \(P( f | g, \[Sigma], \[Theta])\), \(log[\(\(P( f | \[Sigma]\^\[Prime])\)\/\(P( f | \[Sigma])\)\) \(\(P( g | f, \[Theta]\^\[Prime])\)\/\(P( g | f, \[Theta])\)\) \(\(P(g | \[Sigma], \ \[Theta])\) \(P(\[Sigma]\^\[Prime])\) \ \(P(\[Theta]\^\[Prime])\)\)\/\(P(g)\)]\)}]}]}]}]}]}]}], TraditionalForm]], "NumberedEquation", TextAlignment->AlignmentMarker, CellTags->"Ed:LinkToBug2"], Cell[TextData[{ "where we have manipulated the expressions in the same way as in ", ButtonBox["equation", ButtonData:>"Eq:Inequality", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:Inequality"], ")", ". ", ButtonBox["Equation", ButtonData:>"Eq:Inequality2", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:Inequality2"], ")", " becomes" }], "Text"], Cell[BoxData[{ FormBox[\(\(L\_1\)(\[Sigma]\^\[Prime], \[Theta]\^\[Prime] | g)\[AlignmentMarker] \[GreaterEqual] \(L\_\(2, \[Sigma]\)\)(\ \[Sigma]\^\[Prime], \[Sigma], \[Theta] | g) + \(L\_\(2, \[Theta]\)\)(\[Theta]\^\[Prime], \[Sigma], \ \[Theta] | g) + \(L\_1\)(\[Sigma], \[Theta] | g)\), TraditionalForm], "\n", FormBox[ RowBox[{\(\(L\_\(2, \[Sigma]\)\) \((\[Sigma]\^\[Prime], \[Sigma], \ \[Theta] | g)\)\), "\[AlignmentMarker]", "\[Congruent]", RowBox[{"\[Integral]", RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["f", FontSlant->"Italic"]}]], " ", \(P( f | g, \[Sigma], \[Theta])\), \({\(\([\)\(\(-log[ det(\[Pi]\ \[Sigma]\^\[Prime])]\) - \(f\^\[Dagger]\) \(\ \(\[Sigma]\^\[Prime]\)\^\(-1\)\) f + log[P(\[Sigma]\^\[Prime])]\)\(]\)\) - \(\([\)\(\[Sigma]\^\ \[Prime]\[LongRightArrow]\[Sigma]\)\(]\)\)}\)}]}]}], TraditionalForm], "\n", FormBox[ RowBox[{\(\(L\_\(2, \[Theta]\)\) \((\[Theta]\^\[Prime], \[Sigma], \ \[Theta] | g)\)\), "\[AlignmentMarker]", "\[Congruent]", RowBox[{"\[Integral]", RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["f", FontSlant->"Italic"]}]], " ", \(P( f | g, \[Sigma], \[Theta])\), \({\(\([\)\(\(\([\)\(\(f\^\ \[Dagger]\) \(\(T\^\[Prime]\)\^\[Dagger]\) \(N\^\(-1\)\) g + CC\)\(]\)\) - \(f\^\[Dagger]\) \(\(T\^\[Prime]\)\^\ \[Dagger]\) \(N\^\(-1\)\) \(T\^\[Prime]\) f + log[P(\[Theta]\^\[Prime])]\)\(]\)\) - \(\([\)\(\[Theta]\^\ \[Prime]\[LongRightArrow]\[Theta]\)\(]\)\)}\)}]}]}], TraditionalForm]}], "NumberedEquation", TextAlignment->AlignmentMarker, CellTags->{"Eq:Inequality2Simultaneous", "Ed:Change7"}], Cell[TextData[{ "which should be compared with the result in ", ButtonBox["equation", ButtonData:>"Eq:LowerBound", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:LowerBound"], ")", "." }], "Text"], Cell[TextData[{ "The most important property of ", ButtonBox["equation", ButtonData:>"Eq:Inequality2Simultaneous", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:Inequality2Simultaneous"], ")", " is the separation of the ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^\[Prime]\)]], " and the ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\^\[Prime]\)]], " dependences. This implies that we can optimise ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^\[Prime]\)]], " and the ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\^\[Prime]\)]], " independently. We take advantage of this by interleaving separate ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^\[Prime]\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\^\[Prime]\)]], " update iterations in our implementation of autofocus/super-resolution in \ ", ButtonBox["section", ButtonData:>"Sect:ExplanatoryNumericalExample", ButtonStyle->"Hyperlink"], " ", CounterBox["Section", "Sect:ExplanatoryNumericalExample"], ". The ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^\[Prime]\)]], " re-estimation process is the same as in ", ButtonBox["section", ButtonData:>"Sect:SuperResolution", ButtonStyle->"Hyperlink"], " ", CounterBox["Section", "Sect:SuperResolution"], ", except that the imaging operator ", Cell[BoxData[ \(TraditionalForm\`T(\[Theta])\)]], " is used with the current value of ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], " inserted. On the other hand, the ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\^\[Prime]\)]], " re-estimation process requires some further calculation in order to \ obtain the corresponding re-estimation equation." }], "Text"], Cell[TextData[{ "There are two terms in ", Cell[BoxData[ \(TraditionalForm\`\(L\_\(2, \[Theta]\)\)(\[Theta]\^\[Prime], \[Sigma], \ \[Theta] | g)\)]], " that require attention. The ", Cell[BoxData[ \(TraditionalForm\`\(f\^\[Dagger]\) \(\(T\^\[Prime]\)\^\[Dagger]\) \ \(N\^\(-1\)\) \(T\^\[Prime]\) f\)]], " term can be obtained by a derivation analogous to ", ButtonBox["equation", ButtonData:>"Eq:HermitianForm", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:HermitianForm"], ")", " and ", ButtonBox["equation", ButtonData:>"Eq:PosteriorProbabilityHermitianForm", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:PosteriorProbabilityHermitianForm"], ")", " to yield" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[Integral]", RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["f", FontSlant->"Italic"]}]], " ", \(P( f | g, \[Sigma], \[Theta])\), \(f\^\[Dagger]\), \(\(T\^\[Prime]\ \)\^\[Dagger]\), \(N\^\(-1\)\), \(T\^\[Prime]\), "f"}]}], "=", \(tr[\(\(T\^\[Prime]\)\^\[Dagger]\) \(N\^\(-1\)\) \ \(T\^\[Prime]\) C] + \(f\&_\^\[Dagger]\) \(\(T\^\[Prime]\)\^\[Dagger]\) \ \(N\^\(-1\)\) \(T\^\[Prime]\) \(\(f\&_\)\(.\)\)\)}], TraditionalForm]], "NumberedEquation"], Cell[TextData[{ "The ", Cell[BoxData[ \(TraditionalForm\`\(f\^\[Dagger]\) \(\(T\^\[Prime]\)\^\[Dagger]\) \ \(N\^\(-1\)\) g\)]], " term simply yields" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[Integral]", RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["f", FontSlant->"Italic"]}]], " ", \(P( f | g, \[Sigma], \[Theta])\), \(f\^\[Dagger]\), \(\(T\^\[Prime]\ \)\^\[Dagger]\), \(N\^\(-1\)\), "g"}]}], "=", \(\(f\&_\^\[Dagger]\) \(\(T\^\[Prime]\)\^\[Dagger]\) \(N\^\(-1\)\ \) \(\(g\)\(.\)\)\)}], TraditionalForm]], "NumberedEquation"], Cell[TextData[{ "Finally, ", Cell[BoxData[ \(TraditionalForm\`\(L\_\(2, \[Theta]\)\)(\[Theta]\^\[Prime], \[Sigma], \ \[Theta] | g)\)]], " may be expressed (together with ", Cell[BoxData[ \(TraditionalForm\`\(L\_\(2, \[Sigma]\)\)(\[Sigma]\^\[Prime], \[Sigma], \ \[Theta] | g)\)]], ", for completeness) as" }], "Text"], Cell[BoxData[{ \(TraditionalForm\`\(L\_\(2, \[Sigma]\)\)(\[Sigma]\^\[Prime], \[Sigma], \ \[Theta] | g)\[AlignmentMarker] \[Congruent] \(\([\)\(\(-log[ det(\[Pi]\ \[Sigma]\^\[Prime])]\) - tr[\(\(\[Sigma]\^\[Prime]\)\^\(-1\)\) C] - \(f\^\[Dagger]\) \(\(\[Sigma]\^\[Prime]\)\^\(-1\)\) f + log[P(\[Sigma]\^\[Prime])]\)\(]\)\) - \(\([\)\(\[Sigma]\^\[Prime]\ \[LongRightArrow]\[Sigma]\)\(]\)\)\), "\n", \(TraditionalForm\`\(L\_\(2, \[Theta]\)\)(\[Theta]\^\[Prime], \[Sigma], \ \[Theta] | g)\[AlignmentMarker] \[Congruent] \ \(\([\)\(\(-tr[\(\(T\^\[Prime]\)\^\[Dagger]\) \(N\^\(-1\)\) \(T\^\[Prime]\) C]\) - \(f\&_\^\[Dagger]\) \(\(T\^\[Prime]\)\^\[Dagger]\) \(N\ \^\(-1\)\) \(T\^\[Prime]\) f\&_ + \(\([\)\(\(f\&_\^\[Dagger]\) \ \(\(T\^\[Prime]\)\^\[Dagger]\) \(N\^\(-1\)\) g + CC\)\(]\)\) + log[P(\[Theta]\^\[Prime])]\)\(]\)\) - \(\(\([\)\(\[Theta]\^\[Prime]\ \[LongRightArrow]\[Theta]\)\(]\)\)\(.\)\)\)}], "NumberedEquation", TextAlignment->AlignmentMarker, CellTags->{"Eq:LowerBoundIntegratedSimultaneous", "Ed:Change8"}], Cell[TextData[{ "We use the result for ", Cell[BoxData[ \(TraditionalForm\`\(L\_\(2, \[Theta]\)\)(\[Theta]\^\[Prime], \[Sigma], \ \[Theta] | g)\)]], " to derive a re-estimation equation for ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], "." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ".", CounterBox["Subsection"], " Greatest Lower Bound on the Posterior Probability" }], "Subsection", CellTags->"Sect:GreatestLowerBound"], Cell[TextData[{ " We now introduce a linearised model of ", Cell[BoxData[ \(TraditionalForm\`T(\[Theta])\)]] }], "Text"], Cell[BoxData[ \(TraditionalForm\`T(\[Theta]) = T\_0 + \[Sum]\+\(i = 1\)\%r\( \[Theta]\_i\) T\_i\)], "NumberedEquation"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`T\_0\)]], " is the ideal imaging operator, and ", Cell[BoxData[ \(TraditionalForm\`T\_i, i = 1, 2, \[CenterEllipsis], r\)]], " is a set of operators that we use to model the variation in ", Cell[BoxData[ \(TraditionalForm\`T(\[Theta])\)]], ". 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With these substitutions, ", ButtonBox["equation", ButtonData:>"Eq:LowerBoundIntegratedSimultaneous", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:LowerBoundIntegratedSimultaneous"], ")", " for ", Cell[BoxData[ \(TraditionalForm\`\(L\_\(2, \[Theta]\)\)(\[Theta]\^\[Prime], \[Sigma], \ \[Theta] | g)\)]], " becomes" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{\(\(L\_\(2, \[Theta]\)\)(\[Theta]\^\[Prime], \[Sigma], \[Theta] \ | g)\), "=", RowBox[{ RowBox[{"[", RowBox[{\(-tr[\((T\_0\^\[Dagger] + \[Sum]\+\(i = 1\)\%r\(\( \ \[Theta]\^\[Prime]\)\_i\%*\) T\_i\^\[Dagger])\) \(\(N\^\(-1\)\)( T\_0 + \[Sum]\+\(j = 1\)\%r\(\( \[Theta]\^\[Prime]\)\_j\ \) T\_j)\) C]\), "-", \(f\&_\^\[Dagger]\ \((T\_0\^\[Dagger] + \[Sum]\+\(i = \ 1\)\%r\(\( \[Theta]\^\[Prime]\)\_i\%*\) T\_i\^\[Dagger])\) \(\(N\^\(-1\)\)( T\_0 + \[Sum]\+\(j = 1\)\%r\(\( \[Theta]\^\[Prime]\)\_j\) T\_j)\) f\&_\), "+", \(\([\)\(f\&_\^\[Dagger]\ \((T\_0\^\[Dagger] + \[Sum]\+\(i \ = 1\)\%r\(\( \[Theta]\^\[Prime]\)\_i\%*\) T\_i\^\[Dagger])\) \(N\^\(-1\)\) g + CC\)\(]\)\), "-", \(log[det(\[Pi]\ \[CapitalLambda])]\), "-", RowBox[{\(\[Sum]\+\(i, j = 1\)\%r\), RowBox[{\(\(\[Theta]\^\[Prime]\)\_i\%*\), \(\(\[Theta]\^\ \[Prime]\)\_j\), " ", SubscriptBox[\((\[CapitalLambda]\^\(-1\))\), StyleBox["ij", FontSlant->"Italic"]]}]}]}], "]"}], "-", \(\(\([\)\(\[Theta]\^\[Prime]\[LongRightArrow]\[Theta]\)\(]\)\)\ \(.\)\)}]}], TraditionalForm]], "NumberedEquation", CellTags->"Ed:Change9"], Cell[TextData[{ "Differentiating this result to obtain ", Cell[BoxData[ \(TraditionalForm\`\[PartialD]\(\(L\_\(2, \[Theta]\)\)(\[Theta]\^\ \[Prime], \[Sigma], \[Theta] | \ g)\)\/\[PartialD]\(\(\[Theta]\^\[Prime]\)\_i\%*\)\)]], " yields" }], "Text", CellTags->"Ed:Change10"], Cell[BoxData[ FormBox[ RowBox[{\(\(\[PartialD]\/\[PartialD]\(\(\[Theta]\^\[Prime]\)\_i\%*\)\) \ \(\(L\_\(2, \[Theta]\)\)(\[Theta]\^\[Prime], \[Sigma], \[Theta] | g)\)\), "=", RowBox[{\(-tr[\(T\_i\^\[Dagger]\) \(\(N\^\(-1\)\)( T\_0 + \[Sum]\+\(j = 1\)\%r\(\( \[Theta]\^\[Prime]\)\_j\) T\_j)\) C]\), "-", \(\(f\&_\^\[Dagger]\) \(T\_i\^\[Dagger]\) \(\(N\^\(-1\)\)( T\_0 + \[Sum]\+\(j = 1\)\%r\(\( \[Theta]\^\[Prime]\)\_j\) T\_j)\) f\&_\), "+", \(f\&_\^\[Dagger]\ \(T\_i\^\[Dagger]\) \(N\^\(-1\)\) g\), "-", RowBox[{\(\[Sum]\+\(j = 1\)\%r\), RowBox[{\(\(\[Theta]\^\[Prime]\)\_j\), " ", SubscriptBox[\((\[CapitalLambda]\^\(-1\))\), StyleBox["ij", FontSlant->"Italic"]]}]}]}]}], TraditionalForm]], "NumberedEquation", CellTags->{"Eq:LowerBoundIntegratedSimultaneousDerivative", "Ed:Change5"}], Cell[TextData[{ "which depends linearly on the quantity of interest ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\^\[Prime]\)]], ". We may simplify ", ButtonBox["equation", ButtonData:>"Eq:LowerBoundIntegratedSimultaneousDerivative", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:LowerBoundIntegratedSimultaneousDerivative"], ")", " by making the definitions" }], "Text"], Cell[BoxData[{ FormBox[ RowBox[{ SubscriptBox["A", StyleBox["ij", FontSlant->"Italic"]], "\[AlignmentMarker]", "\[Congruent]", RowBox[{\(-tr[\(T\_i\^\[Dagger]\) \(N\^\(-1\)\) \(T\_j\) C]\), "-", \(\(f\&_\^\[Dagger]\) \(T\_i\^\[Dagger]\) \(N\^\(-1\)\) \(T\_j\ \) f\&_\), "-", SubscriptBox[\((\[CapitalLambda]\^\(-1\))\), StyleBox["ij", FontSlant->"Italic"]]}]}], TraditionalForm], "\n", FormBox[\(b\_i\[AlignmentMarker] \[Congruent] tr[\(T\_i\^\[Dagger]\) \(N\^\(-1\)\) \(T\_0\) C] + \(f\&_\^\[Dagger]\) \(T\_i\^\[Dagger]\) \(N\^\(-1\)\) \ \(T\_0\) f\&_ - f\&_\^\[Dagger]\ \(T\_i\^\[Dagger]\) \(N\^\(-1\)\) \(\(g\)\(.\)\)\), TraditionalForm]}], "NumberedEquation", TextAlignment->AlignmentMarker, CellTags->"Eq:DefineAb"], Cell[TextData[{ "Note that the matrix ", Cell[BoxData[ \(TraditionalForm\`A\)]], " and the vector ", Cell[BoxData[ \(TraditionalForm\`b\)]], " depend only on the old parameter values ", Cell[BoxData[ \(TraditionalForm\`\((\[Sigma], \[Theta])\)\)]], "." }], "Text"], Cell[TextData[{ "Finally, we obtain the re-estimation equation for ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], " by locating the stationary point ", Cell[BoxData[ \(TraditionalForm\`\[PartialD]\(\(L\_\(2, \[Theta]\)\)(\[Theta]\^\ \[Prime], \[Sigma], \[Theta] | \ g)\)\/\[PartialD]\(\(\[Theta]\^\[Prime]\)\_i\%*\) = 0\)]], ", which is the solution of the linear matrix equation" }], "Text", CellTags->"Ed:Change11"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\[Sum]\+\(j = 1\)\%r\), RowBox[{ SubscriptBox["A", StyleBox["ij", FontSlant->"Italic"]], " \[AlignmentMarker]", \(\(\[Theta]\^\[Prime]\)\_j\)}]}], "=", \(\(b\_i\)\(\[AlignmentMarker]\)\(.\)\)}], TraditionalForm]], "NumberedEquation", CellTags->"Eq:MatrixTheta"], Cell[TextData[{ "This is a remarkably simple re-estimation formula for the imaging system \ parameters ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], ", which depends on quantities that are easily computed. For completeness, \ we quote the analogous result for real-valued imaging parameters ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], " (which have a real symmetric covariance matrix ", Cell[BoxData[ \(TraditionalForm\`\[CapitalLambda]\)]], ")" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["A", StyleBox["ij", FontSlant->"Italic"]], "\[AlignmentMarker]", "\[Congruent]", RowBox[{\(Re {\(-tr[\(T\_i\^\[Dagger]\) \(N\^\(-1\)\) \(T\_j\) C]\) - \(f\&_\^\[Dagger]\) \(T\_i\^\[Dagger]\) \ \(N\^\(-1\)\) \(T\_j\) f\&_}\), "-", RowBox[{\(1\/2\), SubscriptBox[\((\[CapitalLambda]\^\(-1\))\), StyleBox["ij", FontSlant->"Italic"]], "\n", \(b\_i\)}]}], "\[AlignmentMarker]", "\[Congruent]", \(Re \(\({tr[\(T\_i\^\[Dagger]\) \(N\^\(-1\)\) \(T\_0\ \) C] + \(f\&_\^\[Dagger]\) \(T\_i\^\[Dagger]\) \(N\^\(-1\)\) \(T\_0\) f\&_ - f\&_\^\[Dagger]\ \(T\_i\^\[Dagger]\) \(N\^\(-1\)\) g}\)\(.\)\)\)}], TraditionalForm]], "NumberedEquation", TextAlignment->AlignmentMarker, CellTags->"Eq:DefineAb2"], Cell[TextData[{ "In our numerical simulations in ", ButtonBox["section", ButtonData:>"Sect:ExplanatoryNumericalExample", ButtonStyle->"Hyperlink"], " ", CounterBox["Section", "Sect:ExplanatoryNumericalExample"], " we use a single real imaging parameter, so ", ButtonBox["equation", ButtonData:>"Eq:DefineAb2", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:DefineAb2"], ")", " will be the appropriate re-estimation equation to use. In ", ButtonBox["section", ButtonData:>"Sect:ExplanatoryNumericalExample", ButtonStyle->"Hyperlink"], " ", CounterBox["Section", "Sect:ExplanatoryNumericalExample"], " we also explain an important modification to the true noise covariance ", Cell[BoxData[ \(TraditionalForm\`N\)]], " that we found to be necessary in order to obtain convergence in a \ reasonable number of ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], " re-estimation iterations." }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], " An Explanatory Numerical Example" }], "Section", CellTags->"Sect:ExplanatoryNumericalExample"], Cell[TextData[{ "In this section we apply our super-resolution/autofocus re-estimation \ method to a variety of one dimensional images (e.g. time series). We wish \ only to demonstrate the principle of our method, so we provide an explanatory \ numerical example, rather than extensive numerical simulations and refinement \ of algorithms. A detailed analysis of a low complexity algorithm for the \ super-resolution part of our theory can be found in [", ButtonBox["1", ButtonData:>"Ref:DelvesPrydeLuttrell1988", ButtonStyle->"Hyperlink"], "]." }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ".", CounterBox["Subsection"], " Imaging system - a defocused lens" }], "Subsection"], Cell[TextData[{ "For simplicity we consider the case of a scalar imaging parameter ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], ". In order to ensure that our simulations are relevant to a commonly \ encountered practical situation, we consider the problem of super-resolution \ in a defocused linear imaging system. Specifically, we consider the case \ where a lens is defocused by \[ScriptCapitalO](depth of focus)." }], "Text"], Cell["We model a defocused lens as follows", "Text"], Cell[BoxData[ FormBox[ RowBox[{\(T(\[Theta])\), "\[AlignmentMarker]", "=", RowBox[{ RowBox[{ RowBox[{\(1\/\(2 c\)\), RowBox[{\(\[Integral]\_\(-c\)\%\(+c\)\), RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["k", FontSlant->"Italic"]}]], " ", \(exp(i\ k\ x + i\ \[Theta]\ \(k\^2\) x\^2)\)}]}]}], "\n", "\[TildeTilde]", RowBox[{\(1\/\(2 c\)\), RowBox[{\(\[Integral]\_\(-c\)\%\(+c\)\), RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["k", FontSlant->"Italic"]}]], " ", \(\(exp(i\ k\ x)\)\ [ 1 + i\ \[Theta]\ \(k\^2\) x\^2]\)}]}]}]}], "\n", "=", \(T\_0 + \[Theta]\ T\_1\)}]}], TraditionalForm]], "NumberedEquation", TextAlignment->AlignmentMarker, CellTags->"Eq:Perturb"], Cell["where we have defined", "Text"], Cell[BoxData[{ \(TraditionalForm\`T\_0\[AlignmentMarker] \[Congruent] \(sin(c\ x)\)\/\(c\ \ x\)\), "\n", \(TraditionalForm\`\(-i\)\ T\_1\[AlignmentMarker] \[Congruent] c\ x\ \(sin(c\ x)\) + 2 \( cos( c\ x)\) - \(2 \( sin(c\ x)\)\)\/\(c\ \(\(x\)\(.\)\)\)\)}], \ "NumberedEquation", TextAlignment->AlignmentMarker, CellTags->"Eq:DefineT0T1"], Cell[TextData[{ "The perturbation expansion in ", ButtonBox["equation", ButtonData:>"Eq:Perturb", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:Perturb"], ")", " is appropriate when ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]\[Theta]\ \(c\^2\) x\^2\[RightBracketingBar] \[LessTilde] 1\)]], ", which ensures that the quadratic phase term ", Cell[BoxData[ \(TraditionalForm\`exp\ \((i\ \[Theta]\ \(k\^2\) x\^2)\)\)]], " is small at the edges (", Cell[BoxData[ \(TraditionalForm\`k = \(\[PlusMinus]c\)\)]], ") of the aperture. Physically, this condition requires that the lens be \ within \[ScriptCapitalO](depth of focus) of perfect focus. For ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]x\[RightBracketingBar] < \ \[Pi]\/c\)]], " (i.e. within the main lobe) we therefore require ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]\[Theta]\[RightBracketingBar] < 1\/\[Pi]\^2 \[TildeEqual] 0.1\)]], ". In our numerical simulations we use ", Cell[BoxData[ \(TraditionalForm\`\[Theta] = 0.1\)]], " in order to defocus the lens by \[ScriptCapitalO](depth of focus)." }], "Text", CellTags->"Ed:Change6"], Cell[TextData[{ "We sample the output space of the imaging system defined in ", ButtonBox["equation", ButtonData:>"Eq:Perturb", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:Perturb"], ")", " and ", ButtonBox["equation", ButtonData:>"Eq:DefineT0T1", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:DefineT0T1"], ")", " at intervals ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ RowBox[{ StyleBox["\[CapitalDelta]", FontSlant->"Plain"], StyleBox["x", FontSlant->"Italic"]}]], "=", \(0.8 \[Pi]\/c\)}], TraditionalForm]]], " (i.e. 0.8 of the Nyquist length), and we restrict our attention to an \ output space containing just five such samples. For simplicity, we do not use \ a continuous variable for position in input space, instead we merely sample \ it more finely than the output space. This strategy simplifies the software, \ without losing the essential properties of super-resolution and autofocusing. \ For our purposes we choose to sample the input space at intervals ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ RowBox[{ StyleBox["\[CapitalDelta]", FontSlant->"Plain"], StyleBox["x", FontSlant->"Italic"]}]], "=", \(0.4 \[Pi]\/c\)}], TraditionalForm]]], " (i.e. two samples per output sample), which we superimpose on the output \ sample positions (and the midpoints of the intersample intervals). Thus we \ have 9 input samples. In summary, our sampling lattices are (in Nyquist \ units)" }], "Text", CellTags->"Ed:Change12"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(input\ sample\ positions\[AlignmentMarker] = {\ \(-1.6\), \ \(-1.2\), \(-0.8\), \(-0.4\), 0.0, 0.4, 0.8, 1.2, 1.6}\), "\n", FormBox[\(output\ sample\ positions\[AlignmentMarker] = {\ \ \(-1.6\), \(-0.8\), 0.0, 0.8, 1.6}\), "TraditionalForm"]}], " "}], TraditionalForm]], "NumberedEquation",\ TextAlignment->AlignmentMarker], Cell["\<\ These sampling lattices are not very long, but they are sufficient for us to \ demonstrate the properties of our re-estimation method.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ".", CounterBox["Subsection"], " One-target case" }], "Subsection"], Cell[TextData[{ "We now perform the simplest possible numerical simulation to demonstrate \ autofocus/super-resolution, so our numerical simulation should be regarded as \ an explanatory example, rather than a detailed practical implementation of \ our method. Furthermore, we omit the prior probability terms ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma])\)]], " and ", Cell[BoxData[ \(TraditionalForm\`P(\[Theta])\)]], ", to perform a maximum likelihood (rather than maximum posterior \ probability) fit to the data. We thus rob our method of some of its power in \ order to produce a simple and uncluttered demonstration of its effectiveness. \ It is easy to reintroduce the effect of ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma])\)]], " and ", Cell[BoxData[ \(TraditionalForm\`P(\[Theta])\)]], " into the re-estimation equations when more complicated problems (which ", StyleBox["do", FontSlant->"Italic"], " require prior knowledge to resolve ambiguous interpretations of the data) \ need to be solved." }], "Text"], Cell["\<\ We use the following point target embedded in a weak surrounding\ \>", "Text"], Cell[BoxData[ \(TraditionalForm\`\[Sigma] = \((1, 1, 1, 1, 1000, 1, 1, 1, 1)\)\)], "NumberedEquation"], Cell[TextData[{ "which can produce a variety of scattered fields ", Cell[BoxData[ \(TraditionalForm\`f\)]], " distributed according to ", Cell[BoxData[ \(TraditionalForm\`P(f | \[Sigma])\)]], ". The particular realisation that occurred in our simulation happened to \ be" }], "Text"], Cell[BoxData[{ \(TraditionalForm\`Re(f)\[AlignmentMarker] = \((\(-0.4812\), 0.0352, \(-0.5281\), \(-0.6611\), 23.8763, \(-0.0822\), 0.1760, 0.4499, 0.6768)\)\), "\n", \(TraditionalForm\`Im(f)\[AlignmentMarker] = \((\(-0.1526\), \(-0.3091\), 0.0117, 0.5673, \(-15.4640\), 0.0822, 0.3951, 0.0117, 1.1462)\)\), "\n", \(TraditionalForm\`\[LeftBracketingBar]f\[RightBracketingBar]\^2\ \[AlignmentMarker] = \((0.2548, 0.0968, 0.2791, 0.7589, 809.2133, 0.0135, 0.1871, 0.2025, 1.7719)\)\)}], "NumberedEquation", TextAlignment->AlignmentMarker], Cell[TextData[{ "where we use the notation ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]f\[RightBracketingBar]\^2\)]], " informally to denote the vector of modulus squared components of ", Cell[BoxData[ \(TraditionalForm\`f\)]], ". We initialise the defocus parameter to ", Cell[BoxData[ \(TraditionalForm\`\[Theta] = 0.1\)]], ", which is \[ScriptCapitalO](depth of focus) away from perfect focus, \ which yields an image" }], "Text"], Cell[BoxData[{ \(TraditionalForm\`Re(g)\[AlignmentMarker] = \((\(-10.0906\), 2.8887, 23.3954, 5.8459, \(-9.4112\))\)\), "\n", \(TraditionalForm\`Im( g)\[AlignmentMarker] = \(\((\(-6.5631\), \(-4.2681\), \(-15.2647\), \ \(-4.3982\), \(-5.0146\))\)\(.\)\)\)}], "NumberedEquation", TextAlignment->AlignmentMarker], Cell["We have not included the effects of noise yet.", "Text"], Cell[TextData[{ "The norm of this image is ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]g\[RightBracketingBar]\^2 = 1119.0\)]], ", and we add complex Gaussian noise to each image sample so that the \ expected norm of the total noise is ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]g\[RightBracketingBar]\^2\/100\)]\ ], " (i.e. the image signal to noise ratio (SNR) is 100, or equivalently \ 20dB). Note that we define SNR as a ratio of total energies, because it is \ this quantity that determines the information content of the data (the \ conventional way of measuring SNR by using the peak amplitude is not \ information theoretically meaningful). A SNR of 20dB requires that the \ variance of each Gaussian noise variate (one for each real and each imaginary \ part) is 1.1190 (", Cell[BoxData[ \(TraditionalForm\`\(\(=\)\(1119.0\/\(2\[Times]5\[Times]100\)\)\)\)]], "). The noisy image so generated turned out to be" }], "Text", CellTags->"Ed:Change13"], Cell[BoxData[{ \(TraditionalForm\`Re(g)\[AlignmentMarker] = \((\(-10.6700\), 3.2223, 22.3595, 5.7816, \(-7.9188\))\)\), "\n", \(TraditionalForm\`Im( g)\[AlignmentMarker] = \((\(-7.0723\), \(-6.2404\), \(-15.4110\), \ \(-4.0880\), \(-5.0673\))\)\), "\n", \(TraditionalForm\`\[LeftBracketingBar]g\[RightBracketingBar]\^2\ \[AlignmentMarker] = \((163.8664, 49.3264, 737.4455, 50.1384, 88.3842)\)\)}], "NumberedEquation", TextAlignment->AlignmentMarker], Cell[TextData[{ "where we informally use the notation ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]g\[RightBracketingBar]\^2\)]], " to denote the vector of modulus squared components of ", Cell[BoxData[ \(TraditionalForm\`g\)]], ". This is the image that we use to explore the basic properties of our \ autofocus/super-resolution method." }], "Text"], Cell[TextData[{ "In each of our experiments we initialise the cross section to be ", Cell[BoxData[ \(TraditionalForm\`\[Sigma] = \((1, 1, 1, 1, 1, 1, 1, 1, 1)\)\)]], ". This is a featureless first guess in which we do not even bother to \ initialise the overall normalisation correctly. We perform experiments under \ three different focusing conditions: autofocus, defocus, and focus." }], "Text"], Cell[TextData[{ "(i) 'Autofocus' initialises ", Cell[BoxData[ \(TraditionalForm\`\[Theta] = 0\)]], " and then recovers both ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], " from the data. This demonstrates the full power of our \ autofocus/super-resolution method." }], "Text"], Cell[TextData[{ "(ii) 'Defocus' constrains ", Cell[BoxData[ \(TraditionalForm\`\[Theta] = 0\)]], " (out of focus) and then recovers ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " from the data. This demonstrates what super-resolution (without \ autofocus) would do if an inappropriate imaging operator were used." }], "Text"], Cell[TextData[{ "(iii) 'Focus' constrains ", Cell[BoxData[ \(TraditionalForm\`\[Theta] = 0.1\)]], " (in focus) and then recovers ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " from the data. This demonstrates what super-resolution (without \ autofocus) would ideally do." }], "Text"], Cell["\<\ By comparing the results of each of these types of super-resolution we may \ determine the effectiveness of autofocus/super-resolution.\ \>", "Text"], Cell[TextData[{ "The re-estimation algorithm that we use in all of our experiments consists \ firstly of two iterations in which we re-estimate only ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " by setting the gradient in ", ButtonBox["equation", ButtonData:>"Eq:LowerBoundIntegratedDerivativeDiagonal1", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:LowerBoundIntegratedDerivativeDiagonal1"], ")", " to zero (and dropping the ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma])\)]], " term), followed by one iteration in which we re-estimate only the ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], " by using ", ButtonBox["equation", ButtonData:>"Eq:MatrixTheta", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:MatrixTheta"], ")", " with the results in ", ButtonBox["equation", ButtonData:>"Eq:DefineAb2", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:DefineAb2"], ")", " inserted (and dropping the ", Cell[BoxData[ \(TraditionalForm\`\[CapitalLambda]\)]], " term). Note that because we have only one defocus parameter, ", ButtonBox["equation", ButtonData:>"Eq:MatrixTheta", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:MatrixTheta"], ")", " reduces to a scalar equation. We repeat this prescription of two \ iterations of ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " re-estimation followed by one iteration of \[Theta] re-estimation until \ the algorithm converges. Note that any sequence of re-estimations is \ permitted within our theoretical model, and they are all guaranteed to \ converge towards a local maximum of ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma], \[Theta] | g)\)]], "." }], "Text"], Cell[TextData[{ "For compactness we denote a ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " re-estimation iteration as ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\)]], ", and a ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], " re-estimation iteration as \[CapitalTheta], so our algorithm consists of \ ", Cell[BoxData[ \(TraditionalForm\`\[CapitalTheta]\ \[CapitalSigma]\^2\)]], " (reading from right to left) repeatedly applied to the data. We \ discovered this re-estimation algorithm by experimentation: it is one of the \ few schemes that we discovered that seems to avoid the problems of local \ maxima (and large flat regions as well) of ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma], \[Theta] | g)\)]], ". We do not fully understand the reason why this particular algorithm \ works, and it is likely that it will need to be extended in order to handle \ more complicated situations successfully. Note that any re-estimation \ algorithm based on ", ButtonBox["equation", ButtonData:>"Eq:Inequality2Simultaneous", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:Inequality2Simultaneous"], ")", " is guaranteed to make steady progress towards a local maximum of ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma], \[Theta] | g)\)]], ", because at each stage it locates a greatest lower bound for ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma], \[Theta] | g)\)]], ". The problem of designing an algorithm that makes rapid progress towards \ the global maximum of ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma], \[Theta] | g)\)]], " is an extensive research topic in it own right, and it would require that \ the topography of ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma], \[Theta] | g)\)]], " be investigated in detail." }], "Text"], Cell[TextData[{ "To improve the rate of convergence we artificially increase the noise ", Cell[BoxData[ \(TraditionalForm\`N\[LongRightArrow]100 N\)]], " in ", Cell[BoxData[ \(TraditionalForm\`\[CapitalTheta]\)]], " (i.e. we evaluate ", ButtonBox["equation", ButtonData:>"Eq:DefineAb2", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:DefineAb2"], ")", " using an artificially boosted noise level), because this permits much \ larger ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], " updates to occur. If we left the noise at its correct value, then both \ the old and the new values of ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], " would have to be consistent with the observed image to a high tolerance, \ which forces the ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], " updates to be minuscule. On the other hand, if we boost the noise level, \ then consistency with the image need only be true to a low tolerance, so \ large ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], " updates become possible. We do not bother to change this artificial noise \ level as the algorithm converges. No doubt some further improvements in the \ rate of convergence could be won at the cost of making the iteration schedule \ more complicated. Note that this prescription is not part of our basic \ theoretical machinery, so it must be used with extreme care. For instance, it \ is important ", StyleBox["not", FontSlant->"Italic"], " to use an increased noise level in ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\)]], ", because this would destroy our ability to super-resolve." }], "Text"], Cell[TextData[{ "We show the 'autofocus' result in ", ButtonBox["figure", ButtonData:>"Fig:1TargetAutofocus", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:1TargetAutofocus"], ", the 'defocus' result in ", ButtonBox["figure", ButtonData:>"Fig:1TargetDefocus", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:1TargetDefocus"], ", and the 'focus' result in ", ButtonBox["figure", ButtonData:>"Fig:1TargetFocus", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:1TargetFocus"], ". In each of these figures we present the final result as a bold line, and \ some of the intermediate results (before convergence) as dotted lines. \ Reading from right to left, we obtain the 'autofocus' result by applying ", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalSigma]\^2\)[\[CapitalTheta]\ \ \[CapitalSigma]\^2]\^4\)]], ", the 'defocus' result by applying ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\^10\)]], ", and the 'focus' result by applying ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\^5\)]], "." }], "Text"], Cell[TextData[Cell[TextData[{ " ", ButtonBox["OPEN", ButtonData:>{ URL[ "http://www.luttrell.org.uk/papers/af_sr/fig5.gif"], None}, Active->True, ButtonStyle->"Hyperlink"], " " }]]], "NumberedFigure", TextAlignment->Center, CellTags->"Fig:1TargetAutofocus"], Cell["Super-resolve a target (autofocus).", "Caption"], Cell[TextData[Cell[TextData[{ " ", ButtonBox["OPEN", ButtonData:>{ URL[ "http://www.luttrell.org.uk/papers/af_sr/fig6.gif"], None}, Active->True, ButtonStyle->"Hyperlink"], " " }]]], "NumberedFigure", TextAlignment->Center, CellTags->"Fig:1TargetDefocus"], Cell["Super-resolve a target (defocus).", "Caption"], Cell[TextData[Cell[TextData[{ " ", ButtonBox["OPEN", ButtonData:>{ URL[ "http://www.luttrell.org.uk/papers/af_sr/fig7.gif"], None}, Active->True, ButtonStyle->"Hyperlink"], " " }]]], "NumberedFigure", TextAlignment->Center, CellTags->"Fig:1TargetFocus"], Cell["Super-resolve a target (focus).", "Caption"], Cell[TextData[{ "In ", ButtonBox["figure", ButtonData:>"Fig:1TargetAutofocus", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:1TargetAutofocus"], " the algorithm responds initially by fitting a W-shaped ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " to the data, because at this stage ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], " is incorrect. The algorithm then gradually corrects ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], " and simultaneously recovers the correct cross section. In ", ButtonBox["figure", ButtonData:>"Fig:1TargetDefocus", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:1TargetDefocus"], " we show what would have happened had we not allowed ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], " to be adjusted as in ", ButtonBox["figure", ButtonData:>"Fig:1TargetAutofocus", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:1TargetAutofocus"], ": the W-shaped ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " is retained. It is one of the main aims of this paper to remove such \ artefacts. In ", ButtonBox["figure", ButtonData:>"Fig:1TargetFocus", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:1TargetFocus"], " we present the ideal result that would be obtained if somehow we knew in \ advance what ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], " was, and used it throughout the super-resolution iterations without \ change: the algorithm converges directly towards the correct solution." }], "Text"], Cell[TextData[{ "Differences between ", ButtonBox["figure", ButtonData:>"Fig:1TargetAutofocus", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:1TargetAutofocus"], " and ", ButtonBox["figure", ButtonData:>"Fig:1TargetFocus", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:1TargetFocus"], " are largely because we have not run our algorithm all the way to \ convergence. Figures ", CounterBox["NumberedFigure", "Fig:1TargetAutofocus"], ", ", CounterBox["NumberedFigure", "Fig:1TargetDefocus"], " and ", CounterBox["NumberedFigure", "Fig:1TargetFocus"], " provide the simplest demonstration of the utility of our \ autofocus/super-resolution method. We commented in [", ButtonBox["2", ButtonData:>"Ref:Luttrell1985a", ButtonStyle->"Hyperlink"], "] that a scattered field that consisted of a point target placed between a \ pair of appropriately chosen point targets could produce a correctly focused \ image that mimicked the image that we would obtain from a single point target \ observed through a defocused imaging system. The artefacts that we observe in \ ", ButtonBox["figure", ButtonData:>"Fig:1TargetDefocus", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:1TargetDefocus"], " are consistent with this. If we have available any prior knowledge, we \ can introduce ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma])\)]], " and/or ", Cell[BoxData[ \(TraditionalForm\`P(\[Theta])\)]], " terms into the re-estimation equations in order to reduce the effect of \ possibly ambiguous interpretations of the data." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ".", CounterBox["Subsection"], " Two-target case" }], "Subsection"], Cell["\<\ We now repeat the above experiment using two point targets embedded in a weak \ surrounding. The various vectors become\ \>", "Text"], Cell[BoxData[{ \(TraditionalForm\`\[Sigma]\[AlignmentMarker] = \((1, 1, 1, 1000, 1, 1000, 1, 1, 1)\)\), "\n", \(TraditionalForm\`Re( f)\[AlignmentMarker] = \((\(-0.5751\), \(-0.3091\), \(-0.6768\), \ \(-29.3196\), 0.3482, \(-17.6908\), \(-0.0430\), 0.5125, 0.1682)\)\), "\n", \(TraditionalForm\`Im(f)\[AlignmentMarker] = \((0.0743, 0.5673, 0.4264, \(-1.1134\), 0.1917, \(-7.2990\), 0.3795, \(-0.3482\), 0.3560)\)\), "\n", \(TraditionalForm\`\[LeftBracketingBar]f\[RightBracketingBar]\^2\ \[AlignmentMarker] = \((0.3362, 0.4173, 0.6399, 860.8815, 0.1580, 366.2381, 0.1459, 0.3839, 0.1550)\)\), "\n", \(TraditionalForm\`\t Re(g)\[AlignmentMarker] = \((4.0108, \(-21.2726\), \(-35.2127\), \ \(-9.4149\), 0.5343)\)\), "\n", \(TraditionalForm\`Im(g)\[AlignmentMarker] = \((6.8472, 5.4006, \(-6.9683\), 4.4031, 2.1164)\)\), "\n", \(TraditionalForm\`\[LeftBracketingBar]g\[RightBracketingBar]\^2\ \[AlignmentMarker] = \((62.9714, 481.6923, 1288.4935, 108.0277, 4.7648)\)\)}], "NumberedEquation", TextAlignment->AlignmentMarker], Cell["\<\ where we have quoted only the noisy version of the image (which has a SNR of \ 20dB as before). Note that the fields scattered by the two point targets are \ approximately in phase, and the targets are 0.8 Rayleigh resolution cells \ apart.\ \>", "Text"], Cell[TextData[{ "As before we present the results that we obtain under three different \ focusing conditions. We show the 'autofocus' result in ", ButtonBox["figure", ButtonData:>"Fig:2TargetAutofocus", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:2TargetAutofocus"], ", the 'defocus' result in ", ButtonBox["figure", ButtonData:>"Fig:2TargetDefocus", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:2TargetDefocus"], ", and the 'focus' result in ", ButtonBox["figure", ButtonData:>"Fig:2TargetFocus", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:2TargetFocus"], ". Reading from right to left, we obtain the 'autofocus' result by applying \ ", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalSigma]\^2\)[\[CapitalTheta]\ \ \[CapitalSigma]\^2]\^4\)]], ", the 'defocus' result by applying ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\^10\)]], ", and the 'focus' result by applying ", Cell[BoxData[ \(TraditionalForm\`\[CapitalSigma]\^10\)]], "." }], "Text"], Cell[TextData[Cell[TextData[{ " ", ButtonBox["OPEN", ButtonData:>{ URL[ "http://www.luttrell.org.uk/papers/af_sr/fig8.gif"], None}, Active->True, ButtonStyle->"Hyperlink"], " " }]]], "NumberedFigure", TextAlignment->Center, CellTags->"Fig:2TargetAutofocus"], Cell["Super-resolve two targets (autofocus, in-phase).", "Caption"], Cell[TextData[Cell[TextData[{ " ", ButtonBox["OPEN", ButtonData:>{ URL[ "http://www.luttrell.org.uk/papers/af_sr/fig9.gif"], None}, Active->True, ButtonStyle->"Hyperlink"], " " }]]], "NumberedFigure", TextAlignment->Center, CellTags->"Fig:2TargetDefocus"], Cell["Super-resolve two targets (defocus, in-phase).", "Caption"], Cell[TextData[Cell[TextData[{ " ", ButtonBox["OPEN", ButtonData:>{ URL[ "http://www.luttrell.org.uk/papers/af_sr/fig10.gif"], None}, Active->True, ButtonStyle->"Hyperlink"], " " }]]], "NumberedFigure", TextAlignment->Center, CellTags->"Fig:2TargetFocus"], Cell["Super-resolve two targets (focus, in-phase).", "Caption"], Cell[TextData[{ "In ", ButtonBox["figure", ButtonData:>"Fig:2TargetAutofocus", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:2TargetAutofocus"], " the algorithm responds initially by constructing a ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " that has a featureless bump in the centre. This gradually evolves to a \ bimodal form, which then converges towards the correct ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], ". In ", ButtonBox["figure", ButtonData:>"Fig:2TargetDefocus", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:2TargetDefocus"], " we do not update ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], ", and the result is a featureless bump. In ", ButtonBox["figure", ButtonData:>"Fig:2TargetFocus", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:2TargetFocus"], ", we apply the correct value of ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], " throughout the super-resolution iterations, and the algorithm converges \ directly towards the correct solution. Differences between ", ButtonBox["figure", ButtonData:>"Fig:2TargetAutofocus", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:2TargetAutofocus"], " and ", ButtonBox["figure", ButtonData:>"Fig:2TargetFocus", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:2TargetFocus"], " are largely because we have not run the algorithm all the way to \ convergence." }], "Text"], Cell[TextData[{ "These results demonstrate very clearly how autofocus/super-resolution (", ButtonBox["figure", ButtonData:>"Fig:2TargetAutofocus", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:2TargetAutofocus"], ") can produce superior results to pure super-resolution (", ButtonBox["figure", ButtonData:>"Fig:2TargetDefocus", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:2TargetDefocus"], ")." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], ".", CounterBox["Subsection"], " Discussion" }], "Subsection"], Cell["\<\ The results of our numerical experiments show that our \ autofocus/super-resolution method performs as expected. In simple cases, we \ have demonstrated successful super-resolution when the lens is defocused by \ \[ScriptCapitalO](depth of focus).\ \>", "Text", CellTags->"Ed:Change1"], Cell[TextData[{ "Our numerical experiments were successful because point targets have \ images with a lot of contrast, which are easy to autofocus. However, we \ anticipate that our method will not work in all circumstances. For instance, \ images containing clutter usually have a lower contrast than a point target \ image, and might therefore have several alternative feasible interpretations \ at different setting of the defocus parameter. The same problem would be \ encountered in any other method that had available the same information, \ because ambiguities can be resolved only by supplying appropriate additional \ information. In [", ButtonBox["2", ButtonData:>"Ref:Luttrell1985a", ButtonStyle->"Hyperlink"], "] we discussed some examples of target configurations that conspire to \ fool an autofocus algorithm in this way." }], "Text"], Cell[TextData[{ "The imaging parameter re-estimation method in ", ButtonBox["equation", ButtonData:>"Eq:MatrixTheta", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:MatrixTheta"], ")", " caters for a vector of parameters, so we could extend the model in ", ButtonBox["equation", ButtonData:>"Eq:Perturb", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:Perturb"], ")", " to include the effect of other terms if we wished." }], "Text"], Cell[TextData[{ "We did not introduce prior knowledge terms ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma])\)]], " or ", Cell[BoxData[ \(TraditionalForm\`P(\[Theta])\)]], " into our numerical simulations. Such terms can only improve the \ performance of our method, provided that the prior knowledge is correct." }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ CounterBox["Section"], " ", "Conclusions" }], "Section"], Cell[TextData[{ "Firstly, we have shown how to use Jensen's inequality (in the form of an \ estimate-maximise (EM) algorithm) to maximise the posterior probability ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma] | g)\)]], " for recovering a scattering cross section ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " from a coherent image $g$: we call this 'super-resolution'. Secondly, we \ have extended this result to maximise the posterior probability ", Cell[BoxData[ \(TraditionalForm\`P(\[Sigma], \[Theta] | g)\)]], " for simultaneously recovering both the imaging system parameters ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], " and the cross section ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], ": we call this 'autofocus/super-resolution'. The first result proves the \ validity of our heuristic iterative super-resolution algorithm [", ButtonBox["1", ButtonData:>"Ref:DelvesPrydeLuttrell1988", ButtonStyle->"Hyperlink"], "]. The second result extends this algorithm to compensate for the effects \ of an uncertain imaging system." }], "Text"], Cell[TextData[{ "Although our results apply to coherent imaging systems in general, they \ have the potential to super-resolve synthetic aperture radar (SAR) images. \ SAR can be modelled as a linear imaging system, and anomalous motion of the \ transmitter/receiver can be modelled (in simple cases) as a simple defocusing \ of the type exemplified in ", ButtonBox["section", ButtonData:>"Sect:ExplanatoryNumericalExample", ButtonStyle->"Hyperlink"], " ", CounterBox["Section", "Sect:ExplanatoryNumericalExample"], ". There is a contrast maximisation autofocus method [", ButtonBox["15", ButtonData:>"Ref:FinleyWood1985", ButtonStyle->"Hyperlink"], ", ", ButtonBox["16", ButtonData:>"Ref:Oliver1989", ButtonStyle->"Hyperlink"], "] that corrects the focus of a SAR to within \[ScriptCapitalO](depth of \ focus) of the correct value, but this is inadequate for successful \ super-resolution to be attempted [", ButtonBox["2", ButtonData:>"Ref:Luttrell1985a", ButtonStyle->"Hyperlink"], "]. Our current autofocus/super-resolution method shows potential for \ correcting these residual focusing uncertainties, and thus offers the \ possibility of a robust super-resolution algorithm for SAR image analysis." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Acknowledgements", "Section"], Cell["\<\ I thank J S Bridle for explaining to me the advantages that 're-estimation' \ methods have over 'gradient ascent' methods.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Appendix A: Miscellaneous Derivations", "Section", CellTags->"Sect:Appendix"], Cell["\<\ First of all let us make some definitions, whose purpose will become clear \ later on\ \>", "Text"], Cell[BoxData[{ \(TraditionalForm\`M\[AlignmentMarker] \[Congruent] T\ \[Sigma]\ T\^\[Dagger] + N\), "\n", \(TraditionalForm\`C\^\(-1\)\[AlignmentMarker] \[Congruent] \ \[Sigma]\^\(-1\) + \(T\^\[Dagger]\) \(N\^\(-1\)\) T\), "\n", \(TraditionalForm\`f\&_\[AlignmentMarker] \[Congruent] \[Sigma]\ \(T\^\ \[Dagger]\) \(M\^\(-1\)\) g = C\ \(T\^\[Dagger]\) \(N\^\(-1\)\) g\)}], "NumberedEquation", TextAlignment->AlignmentMarker, CellTags->"Eq:Definitions"], Cell["whence", "Text"], Cell[BoxData[{ \(TraditionalForm\`M\^\(-1\)\[AlignmentMarker] = N\^\(-1\) - \(N\^\(-1\)\) T\ C\ \(T\^\[Dagger]\) N\^\(-1\)\), "\n", \(TraditionalForm\`C\[AlignmentMarker] = \(\[Sigma] - \[Sigma]\ \(T\^\ \[Dagger]\) \(M\^\(-1\)\) T\ \[Sigma]\)\n\(\(\(<\)\(\(f\&_\) f\&_\^\[Dagger]\)\(>\)\)\[AlignmentMarker] = \[Sigma]\ \(T\^\ \[Dagger]\) \(M\^\(-1\)\) T\ \[Sigma]\)\)}], "NumberedEquation", TextAlignment->AlignmentMarker, CellTags->"Eq:DefinitionsInverse"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`\(\(<\)\(\[CenterEllipsis]\)\(>\)\)\)]], " denotes an average over the ", Cell[BoxData[ \(TraditionalForm\`g\)]], " as described by the model probabilities (which is not necessarily the \ same as the average over observed ", Cell[BoxData[ \(TraditionalForm\`g\)]], ", because one's model can be, and usually is, wrong)." }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["A", FontSlant->"Italic"], StyleBox[ CounterBox["Subsection"], FontSlant->"Italic"], StyleBox[". Deriving", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`P(g | \[Sigma])\)]] }], "Subsection"], Cell[TextData[{ "We derive the probability ", Cell[BoxData[ \(TraditionalForm\`P(g | \[Sigma])\)]], " that a cross section ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " can give rise to dataset ", Cell[BoxData[ \(TraditionalForm\`g\)]], ". Potentially, there are two types of hidden variable to consider: the \ scattered field ", Cell[BoxData[ \(TraditionalForm\`f\)]], ", and the imaging parameters ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], ", which we showed in ", ButtonBox["figure", ButtonData:>"Fig:Imaging", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:Imaging"], ". Let us assume that ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], " is known, so that it can be removed from the problem, to yield" }], "Text"], Cell[BoxData[ FormBox[ RowBox[{\(P(g | \[Sigma])\), "=", RowBox[{"\[Integral]", RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["f", FontSlant->"Italic"]}]], " ", \(P(g | f)\), \(\(P(f | \[Sigma])\)\(.\)\)}]}]}], TraditionalForm]], "NumberedEquation", CellTags->"Eq:IntermediateVariable"], Cell[TextData[{ ButtonBox["Equation", ButtonData:>"Eq:IntermediateVariable", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:IntermediateVariable"], ")", " represents the top half of ", ButtonBox["figure", ButtonData:>"Fig:Imaging", ButtonStyle->"Hyperlink"], " ", CounterBox["NumberedFigure", "Fig:Imaging"], " with ", Cell[BoxData[ \(TraditionalForm\`f\)]], " integrated out. We obtain ", Cell[BoxData[ \(TraditionalForm\`P(f | \[Sigma])\)]], " from ", ButtonBox["equation", ButtonData:>"Eq:ScatteringModel", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:ScatteringModel"], ")", " and ", Cell[BoxData[ \(TraditionalForm\`P(g | f)\)]], " from ", ButtonBox["equation", ButtonData:>"Eq:Likelihood", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:Likelihood"], ")", " (with ", Cell[BoxData[ \(TraditionalForm\`\[Theta]\)]], " removed). The integrand of ", ButtonBox["equation", ButtonData:>"Eq:IntermediateVariable", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:IntermediateVariable"], ")", " is Gaussian, so it is easy to perform the integral. The steps are as \ follows:" }], "Text"], Cell[BoxData[{ FormBox[ RowBox[{\(P \((g | \[Sigma])\)\), "\[AlignmentMarker]", "=", RowBox[{ RowBox[{\(1\/\(\(det(\[Pi]\ N)\) \(det(\[Pi]\ \[Sigma])\)\)\), RowBox[{"\[Integral]", RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["f", FontSlant->"Italic"]}]], " ", \(exp[\(-f\)\ \(\[Sigma]\^\(-1\)\) f - \(\((g - T\ f)\)\^\[Dagger]\) \(\(N\^\(-1\)\)( g - T\ f)\)]\)}]}]}], "\n", "=", RowBox[{\(1\/\(\(det(\[Pi]\ N)\) \(det(\[Pi]\ \[Sigma])\)\)\), RowBox[{"\[Integral]", RowBox[{ StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["f", FontSlant->"Italic"]}]], " ", \(exp[\(-\((f - f\&_)\)\^\[Dagger]\) \(\(C\^\(-1\)\)( f - f\&_)\) + \(\(g\^\[Dagger]\)(\(-N\^\(-1\)\) + \(N\ \^\(-1\)\) T\ C\ \(T\^\[Dagger]\) N\^\(-1\))\) g]\)}]}]}]}]}], TraditionalForm], "\n", FormBox[\(\(=\)\(\(\(\(det(\[Pi]\ C)\)\/\(\(det(\[Pi]\ N)\) \(det(\[Pi]\ \ \[Sigma])\)\)\) exp[\(\(g\^\[Dagger]\)(\(-N\^\(-1\)\) + \((N\^\(-1\) - M\^\(-1\))\))\) g]\)\n\(\(=\)\(\(1\/\(det(\[Pi]\ M)\)\) \(\(exp(\(-g\^\ \[Dagger]\) \(M\^\(-1\)\) g)\)\(.\)\)\)\)\)\), TraditionalForm]}], "NumberedEquation", TextAlignment->AlignmentMarker, CellTags->{"Eq:IntermediateVariable2", "Ed:Change14"}], Cell[TextData[{ "In the last step we simplify the normalisation factor by noting (", StyleBox["a posteriori", FontSlant->"Italic"], ") that the result must be a correctly normalised probability. ", Cell[BoxData[ \(TraditionalForm\`P(g | \[Sigma])\)]], " is a zero mean complex Gaussian probability with covariance ", Cell[BoxData[ \(TraditionalForm\`M\)]], ". The form of ", Cell[BoxData[ \(TraditionalForm\`M\)]], " given in ", ButtonBox["equation", ButtonData:>"Eq:Definitions", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:Definitions"], ")", " is very simple: the ", Cell[BoxData[ \(TraditionalForm\`T\ \[Sigma]\ T\^\[Dagger]\)]], " piece is the linearly filtered scattered field covariance matrix ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], ", and the ", Cell[BoxData[ \(TraditionalForm\`N\)]], " piece is the data noise covariance. These terms sum together because the \ data noise is statistically independent of the scattered field." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["A", FontSlant->"Italic"], StyleBox[ CounterBox["Subsection"], FontSlant->"Italic"], StyleBox[". Deriving", FontSlant->"Italic"], " ", Cell[BoxData[ \(TraditionalForm\`P(f | g, \[Sigma])\)]] }], "Subsection"], Cell[TextData[{ "We derive the posterior probability ", Cell[BoxData[ \(TraditionalForm\`P(f | g, \[Sigma])\)]], " that a scattered field ", Cell[BoxData[ \(TraditionalForm\`f\)]], " can occur, given that both the scattering cross section ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " and the data ", Cell[BoxData[ \(TraditionalForm\`g\)]], " are known. The steps in the derivation are" }], "Text"], Cell[BoxData[ \(TraditionalForm\`P( f | g, \[Sigma])\[AlignmentMarker] = \(\(P(g | f)\) \(P(f | \[Sigma])\ \)\)\/\(P(g | \[Sigma])\)\n\(\(=\)\(\(\(\(det(\[Pi]\ M)\)\/\(\(det(\[Pi]\ \ N)\) \(det(\[Pi]\ \[Sigma])\)\)\) exp[\(-f\)\ \(\[Sigma]\^\(-1\)\) f - \(\((g - T\ f)\)\^\[Dagger]\) \(\(N\^\(-1\)\)( g - T\ f)\) + \(g\^\[Dagger]\) \(M\^\(-1\)\) g]\)\n\(\(=\)\(\(\(det(\[Pi]\ M)\)\/\(\(det(\[Pi]\ N)\) \ \(det(\[Pi]\ \[Sigma])\)\)\) exp[\(-\((f - f\&_)\)\^\[Dagger]\) \(\(C\^\(-1\)\)( f - f\&_)\) + \(\(g\^\[Dagger]\)( M\^\(-1\) - N\^\(-1\) + \(N\^\(-1\)\) T\ C\ \(T\^\[Dagger]\) N\^\(-1\))\) g]\)\)\n\(\(=\)\(\(1\/\(det(\[Pi]\ C)\)\) \ \(\(exp[\(-\((f - f\&_)\)\^\[Dagger]\) \(\(C\^\(-1\)\)( f - f\&_)\)]\)\(.\)\)\)\)\)\)\)], "NumberedEquation", TextAlignment->AlignmentMarker, CellTags->{"Eq:PosteriorProbability", "Ed:Change15"}], Cell[TextData[{ "In the last step we simplify the normalisation factor by noting (", StyleBox["a posteriori", FontSlant->"Italic"], ") that the result must be a correctly normalised probability. ", Cell[BoxData[ \(TraditionalForm\`P(f | g, \[Sigma])\)]], " is a Gaussian with mean ", Cell[BoxData[ \(TraditionalForm\`f\&_\)]], " and covariance ", Cell[BoxData[ \(TraditionalForm\`C\)]], "." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["A", FontSlant->"Italic"], StyleBox[ CounterBox["Subsection"], FontSlant->"Italic"], StyleBox[". Differentiating", FontSlant->"Italic"], " log det" }], "Subsection"], Cell[TextData[{ "We differentiate the logarithm of the determinant of a matrix-valued \ quantity. We use this in order to differentiate ", ButtonBox["equation", ButtonData:>"Eq:LowerBoundIntegrated", ButtonStyle->"Hyperlink"], " (", CounterBox["NumberedEquation", "Eq:LowerBoundIntegrated"], ")", ", so we present the derivation using an appropriate notation." }], "Text"], Cell[BoxData[ FormBox[GridBox[{ {\(Step\ 1\), \(\[Delta]\ log[ det(\[Pi]\ \[Sigma]\^\[Prime])]\), \(\(=\)\(\(-log[ det(\[Pi](\(\[Sigma]\^\[Prime]\)\^\(-1\) + \(\[Delta]\ \[Sigma]\^\[Prime]\)\^\(-1\)))]\) + log[det(\[Pi]\ \(\[Sigma]\^\[Prime]\)\^\(-1\))]\)\)}, {\(Step\ 2\), " ", \(\(=\)\(\(-tr[ log(\(\(\[Sigma]\^\[Prime]\)\^\(-1\)\)( 1 + \(\[Sigma]\^\[Prime]\) \(\[Delta]\[Sigma]\^\ \[Prime]\)\^\(-1\)))]\) + tr[log(\(\[Sigma]\^\[Prime]\)\^\(-1\))]\)\)}, {\(Step\ 3\), " ", \(\(=\)\(-tr[ log(1 + \(\[Sigma]\^\[Prime]\) \(\[Delta]\[Sigma]\^\[Prime]\ \)\^\(-1\))]\)\)}, {\(Step\ 4\), " ", \(\(\[TildeEqual]\)\(-\(\(tr[\(\[Sigma]\^\[Prime]\) \(\ \[Delta]\[Sigma]\^\[Prime]\)\^\(-1\)]\)\(.\)\)\)\)} }], TraditionalForm]], "NumberedEquation", TextAlignment->AlignmentMarker, GridBoxOptions->{ColumnAlignments->{Left}}], Cell["\<\ We justify the various stages of this manipulation as follows:\ \>", "Text"], Cell[TextData[{ "Step 1. Matrix invert ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\^\[Prime]\)]], ", which introduces a minus sign outside the logarithm function. In order \ to calculate the derivative, write the difference that results from changing \ ", Cell[BoxData[ \(TraditionalForm\`\(\[Sigma]\^\[Prime]\)\^\(-1\)\)]], " infinitesimally." }], "Text"], Cell[TextData[{ "Step 2. Use the identity ", Cell[BoxData[ \(TraditionalForm\`log[det(X)] = tr[log(X)]\)]], "." }], "Text"], Cell[TextData[{ "Step 3. Use the identity ", Cell[BoxData[ \(TraditionalForm\`log(X\ Y) = log(X) + log(Y) + \((commutator\ terms\ from\ the\ Baker - Hausdorff\ identity)\)\)]], " to obtain ", Cell[BoxData[ \(TraditionalForm\`tr[log(X\ Y)] = tr[log(X)] + tr[log(Y)]\)]], " which causes a pair of terms to cancel, leaving only the infinitesimal \ part. Note that the trace of any commutator is zero." }], "Text"], Cell[TextData[{ "Step 4. Expand the logarithm using ", Cell[BoxData[ \(TraditionalForm\`log(1 + X) = X + \[ScriptCapitalO](X\^2)\)]], "." }], "Text", CellTags->"Ed:Change2"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["References", "Section"], Cell[TextData[{ Cell[TextData[{ " ", ButtonBox["OPEN", ButtonData:>{ URL[ "http://www.luttrell.org.uk/papers/sr_alg/sr_alg.nb"], None}, Active->True, ButtonStyle->"Hyperlink"], " " }]], "[1] Delves L M, Pryde G and Luttrell S P, A super-resolution algorithm for \ SAR images, ", StyleBox["Inverse Problems", FontSlant->"Italic"], ", ", StyleBox["4", FontWeight->"Bold"], ", 681-703, 1988." }], "Reference", CellTags->"Ref:DelvesPrydeLuttrell1988"], Cell[TextData[{ Cell[TextData[{ " ", ButtonBox["OPEN", ButtonData:>{ URL[ "http://www.luttrell.org.uk/papers/3785/3785.nb"], None}, Active->True, ButtonStyle->"Hyperlink"], " " }]], "[2] Luttrell S P, A super-resolution model for synthetic aperture radar, \ Technical Report ", StyleBox["3785", FontWeight->"Bold"], ", RSRE, 1985." }], "Reference", CellTags->"Ref:Luttrell1985a"], Cell[TextData[{ "[3] Jeffreys H, ", StyleBox["Theory of Probability", FontSlant->"Italic"], ", Clarendon Press, 1939." }], "Reference", CellTags->"Ref:Jeffreys1939"], Cell[TextData[{ "[4] Cox R T, Probability, frequency and reasonable expectation, ", StyleBox["Am. J. Phys.", FontSlant->"Italic"], ", ", StyleBox["14", FontWeight->"Bold"], ", 1-13, 1946." }], "Reference", CellTags->"Ref:Cox1946"], Cell[TextData[{ Cell[TextData[{ " ", ButtonBox["OPEN", ButtonData:>{ URL[ "http://www.luttrell.org.uk/papers/ble/ble.nb"], None}, Active->True, ButtonStyle->"Hyperlink"], " " }]], "[5] Luttrell S P, Prior knowledge and resolution enhancement using the \ best linear estimate technique, ", StyleBox["Opt. Acta", FontSlant->"Italic"], ", ", StyleBox["32", FontWeight->"Bold"], ", 703-716, 1985." }], "Reference", CellTags->"Ref:Luttrell1985b"], Cell[TextData[{ Cell[TextData[{ " ", ButtonBox["OPEN", ButtonData:>{ URL[ "http://www.luttrell.org.uk/papers/priorsar/priorsar.nb"], None}, Active->True, ButtonStyle->"Hyperlink"], " " }]], "[6] Luttrell S P and Oliver C J, Prior knowledge in synthetic aperture \ radar processing, ", StyleBox["J. Phys. D: Appl. Phys.", FontSlant->"Italic"], ", ", StyleBox["19", FontWeight->"Bold"], ", 333-356, 1986." }], "Reference", CellTags->"Ref:LuttrellOliver1986"], Cell[TextData[{ Cell[TextData[{ " ", ButtonBox["OPEN", ButtonData:>{ URL[ "http://www.luttrell.org.uk/papers/bayessup/bayessup.nb"], None}, Active->True, ButtonStyle->"Hyperlink"], " " }]], "[7] Luttrell S P, Bayesian super-resolution theory for SAR, ", StyleBox["Int. J. Remote Sensing", FontSlant->"Italic"], ", in press." }], "Reference", CellTags->"Ref:Luttrell19xx"], Cell[TextData[{ "[8] Baum L E, An inequality and associated maximisation technique in \ statistical estimation for probabilistic functions of Markov processes, ", StyleBox["Inequalities", FontSlant->"Italic"], ", ", StyleBox["3", FontWeight->"Bold"], ", 1-8, 1972." }], "Reference", CellTags->"Ref:Baum1972"], Cell[TextData[{ "[9] Dempster A P, Laird N M and Rubin D B, Maximum likelihood from \ incomplete data via the EM algorithm, ", StyleBox["J. Roy. Statist. Soc. Ser. B", FontSlant->"Italic"], ", ", StyleBox["39", FontWeight->"Bold"], ", 1-37, 1977." }], "Reference", CellTags->"Ref:DempsterLairdRubin1977"], Cell[TextData[{ "[10] Jakeman E and Pusey P N, A model for non-Rayleigh sea echo, ", StyleBox["IEEE Trans. AP", FontSlant->"Italic"], ", ", StyleBox["24", FontWeight->"Bold"], ", 806-814, 1976." }], "Reference", CellTags->"Ref:JakemanPusey1976"], Cell[TextData[{ "[11] Ward K D, Compound representation of high resolution sea clutter, ", StyleBox["Electr. Lett.", FontSlant->"Italic"], ", ", StyleBox["17", FontWeight->"Bold"], ", 561-563, 1981." }], "Reference", CellTags->"Ref:Ward1981"], Cell[TextData[{ "[12] Geman S and Geman D, Stochastic relaxation, Gibbs distributions, and \ the Bayesian restoration of images, ", StyleBox["IEEE Trans. PAMI", FontSlant->"Italic"], ", ", StyleBox["6", FontWeight->"Bold"], ", 721-741, 1984." }], "Reference", CellTags->"Ref:GemanGeman1984"], Cell[TextData[{ Cell[TextData[{ " ", ButtonBox["OPEN", ButtonData:>{ URL[ "http://www.luttrell.org.uk/papers/bs1_41/bs1_41.nb"], None}, Active->True, ButtonStyle->"Hyperlink"], " " }]], "[13] Luttrell S P, The relationship between super-resolution and phase \ imaging of SAR data, Technical Report ", StyleBox["BS1/41", FontWeight->"Bold"], ", RSRE, 1987." }], "Reference", CellTags->"Ref:Luttrell1987"], Cell[TextData[{ Cell[TextData[{ " ", ButtonBox["OPEN", ButtonData:>{ URL[ "http://www.luttrell.org.uk/papers/invcross/invcross.nb"], None}, Active->True, ButtonStyle->"Hyperlink"], " " }]], "[14] Luttrell S P, The inverse cross section problem for complex data, ", StyleBox["Inverse Problems", FontSlant->"Italic"], ", ", StyleBox["5", FontWeight->"Bold"], ", 35-50, 1989." }], "Reference", CellTags->"Ref:Luttrell1989"], Cell[TextData[{ "[15] Finley I P and Wood J W, An investigation of synthetic aperture radar \ autofocus, Technical Report ", StyleBox["3790", FontWeight->"Bold"], ", RSRE, 1985." }], "Reference", CellTags->"Ref:FinleyWood1985"], Cell[TextData[{ "[16] Oliver C J, Synthetic aperture radar imaging, J. Phys. D: Appl. \ Phys., ", StyleBox["22", FontWeight->"Bold"], ", 871-890, 1989." }], "Reference", CellTags->"Ref:Oliver1989"] }, Closed]] }, Open ]] }, FrontEndVersion->"5.0 for Microsoft Windows", ScreenRectangle->{{0, 1280}, {0, 941}}, WindowSize->{665.375, 641}, WindowMargins->{{307.25, Automatic}, {Automatic, 50}}, Magnification->1, StyleDefinitions -> "Report.nb" ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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