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Cell[TextData[{ "Notation ", Cell[BoxData[ \(TraditionalForm\`\[SelectionPlaceholder]\_\(\[SelectionPlaceholder]\ \ \[SelectionPlaceholder]\)\)]], " changed to ", Cell[BoxData[ \(TraditionalForm\`\[SelectionPlaceholder]\_\(\[SelectionPlaceholder], \ \[SelectionPlaceholder]\)\)]], " throughout the paper." }], "Text"], Cell["\"ie\" changed to \"i.e.\" throughout the paper.", "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change1", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"Blackledge J M, Burge R E and Barrett N R, \"Inversion method to extract \ the phase information generated by synthetic aperature radar systems\", 1987, \ ", StyleBox["Inverse Problem in Optics", FontSlant->"Italic"], ", The Hague\" changed to \"Blackledge J M, Burge R E and Barrett N R, \ 1987, \"Inversion method to extract the phase information generated by \ synthetic aperature radar systems\", ", StyleBox["Inverse Problems in Optics", FontSlant->"Italic"], ", The Hague\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change2", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"Luttrell S P and Oliver C J, \"Prior knowledge in synthetic aperture \ radar processing\", 1986, ", StyleBox["J Phys D", FontSlant->"Italic"], ", ", StyleBox["19", FontWeight->"Bold"], ", 333-356\" changed to \"Luttrell S P and Oliver C J, 1986, \"Prior \ knowledge in synthetic aperture radar processing\", ", StyleBox["J Phys D", FontSlant->"Italic"], ", ", StyleBox["19", FontWeight->"Bold"], ", 333-356\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change3", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ \(TraditionalForm\`a/\(A(t)\)\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`a\/\(A(t)\)\)]], "\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change4", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ \(TraditionalForm\`P[g | \[Sigma]]\)]], " contains all the information about ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`P[\[Sigma] | g]\)]], " contains all the information about ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], "\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change5", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]f\_j\[RightBracketingBar]\^2/\ \[Sigma]\_j\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]f\_j\[RightBracketingBar]\^2\/\ \[Sigma]\_j\)]], "\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change6", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]g\_k\[RightBracketingBar]\^2/\((\ \[Sigma]\_0 + \[Nu])\)\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]g\_k\[RightBracketingBar]\^2\/\(\ \[Sigma]\_0 + \[Nu]\)\)]], "\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change7", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ \(TraditionalForm\`diag(1/\[Lambda]\_1, 1/\[Lambda]\_2, \[CenterEllipsis])\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`diag(1\/\[Lambda]\_1, 1\/\[Lambda]\_2, \[CenterEllipsis])\)]], "\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change8", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]g\_j\^\[Prime]\ \[RightBracketingBar]\^2/\[Lambda]\_j\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]g\_j\^\[Prime]\ \[RightBracketingBar]\^2\/\[Lambda]\_j\)]], "\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change9", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "Equation reformatted to make it more readable." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change10", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ \(TraditionalForm\`g\_j\^\[Prime]/\[Lambda]\_j\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`g\_j\^\[Prime]\/\[Lambda]\_j\)]], "\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change11", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"", Cell[BoxData[ \(TraditionalForm\`1/\[Lambda]\_j\)]], "\" changed to \"", Cell[BoxData[ \(TraditionalForm\`1\/\[Lambda]\_j\)]], "\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["TYPO", ButtonData:>"Ed:Change12", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "\"requirementy\" changed to \"requirements\"." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ ButtonBox["PROBLEM", ButtonData:>"Ed:Problem1", Active->True, ButtonStyle->"Hyperlink"], TextForm]]], "This is not cited anywhere in this paper." }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ The relationship between super-resolution and phase imaging of SAR data \ \>", "Title"], Cell["\<\ Dr S P Luttrell (approved by Dr J M Hinsley)\ \>", "Author"], Cell["\<\ This paper appeared as BS1 Divisional Memo, No. 41, April 1987.\ \>", "Text"], Cell["\<\ Summary I review the real zero conversion (RZC) method of producing a phase image \ from complex SAR data, and I find that the phase image so produced is \ trivially related to the output voltage of the SAR receiver. I argue that the \ only useful type of image processing must be related to specific questions \ which one asks of the original SAR data, and so phase imaging must be related \ to the specific demands of the user. Super-resolution imaging is a technique \ for processing an image to enhance its bandwidth by the introduction of prior \ knowledge. I shall present super-resolution in such a way as to make clear \ its relationship to phase imaging, and also make clear that complex images of \ targets are likely to contain useful phase information. The principal \ practical consequence of this work is that phase information in SAR images \ must be retained in the vicinity of targets, but it may be discarded \ elsewhere.\ \>", "Abstract"], Cell[CellGroupData[{ Cell["Notation", "Section 1"], Cell[TextData[{ "\t", Cell[BoxData[ FormBox[GridBox[{ {\(a\ e\^\(i\ \[Omega]\ t\)\), \(complex\ exponential\ used\ in\ \ RZC\ phase\ imaging\)}, {\(A( t)\), \(envelope\ of\ complex\ SAR\ image\ \((range\ trace)\)\ \)}, {\(\[Phi]( t)\), \(phase\ shift\ of\ complex\ SAR\ image\ \((range\ \ trace)\)\)}, {\(V( t)\), \(voltage\ at\ output\ of\ SAR\ rerceiver\ \ \((focused)\)\)}, {\(\[CapitalPhi](t)\), \(phase\ function\ produced\ by\ RZC\)}, { "\[Sigma]", \(scatterer\ cross\ section\ \((per\ unit\ \ area)\)\)}, {\(\[Sigma]\_0\), \(constant\ scatterer\ cross\ section\)}, {"f", \(scattered\ field\)}, {"g", \(complex\ image\ data\)}, {\(g\_k\), \(pixel\ k\ of\ g\)}, {"T", \(linear\ imaging\ operator\ \((band - limited)\)\)}, {"B", \(band - limiting\ operator\)}, {"\[Nu]", \(covariance\ matrix\ of\ additive\ image\ noise\)}, {"M", \(covariance\ matrix\ of\ complex\ image\ data\)}, {"U", \(unitary\ matrix\ which\ diagonalises\ M\)}, {"\[CapitalLambda]", \(diagonal\ form\ of\ M\)}, {\(\[Lambda]\_j\), \(eigenvalue\ j\ of\ M\)}, {\(g\_j\^\[Prime]\), \(component\ j\ of\ g\ in\ eigenbasis\ of\ M\ \)}, {\(P[\[Sigma]]\), \(prior\ PDF\ over\ \[Sigma]\)}, {\(P[ f | \[Sigma]]\), \(conditional\ probability\ of\ f\ given\ \ \[Sigma]\)}, {\(P[ g | \[Sigma]]\), \(conditional\ probability\ of\ g\ given\ \ \[Sigma]\)}, {\(P[g | f]\), \(conditional\ probability\ of\ g\ given\ f\)}, {\(P[g]\), \(total\ probability\ of\ g\)}, {\(P[\[Sigma] | g]\), RowBox[{"a", " ", StyleBox["posteriori", FontSlant->"Italic"], " ", "probability", " ", "of", " ", "\[Sigma]", " ", "given", " ", "g"}]}, {\(P[f | g, \[Sigma]]\), RowBox[{"a", " ", StyleBox["posteriori", FontSlant->"Italic"], " ", "probability", " ", "of", " ", "f", " ", "given", " ", "g", " ", "and", " ", "\[Sigma]"}]} }], TraditionalForm]]] }], "Text", GridBoxOptions->{ColumnAlignments->{Left}}] }, Closed]], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell["\<\ We shall be concerned with the interpretation of synthetic aperture radar \ (SAR) imagery. The RSRE SAR has great capabilities, and there is an active \ and successful programme of research (in the sensor information processing \ section of BS1) into ways of processing and interpreting its data. SAR data \ is complex valued there are two degrees of freedom per pixel - and image \ processing techniques must decide how to deal with such data.\ \>", "Text"], Cell["\<\ Most techniques (such as image display, segmentation, etc) discard the phase \ of the complex data and use only its modulus. Visually a modulus (i.e. \ conventional) image conveys a lot of information, so it is natural to suspect \ that adequate data interpretation can be achieved using the modulus alone. In \ fact this supposition turns out to be correct for most image interpretation \ purposes which are concerned with large scale objects such as fields, woods, \ etc. Therefore conventional segmentation algorithms are not handicapped by \ lack of phase information. Discarding the phase also has the side effect of \ reducing the bandwidth required to transmit a SAR image, which is \ desirable.\ \>", "Text"], Cell[TextData[{ "However I shall show how the phase of a SAR image contains useful \ information in the vicinity of targets, which are generically the small scale \ features of SAR images. This phase information is extracted very naturally by \ using the known super-resolution technique of ", ButtonBox["Luttrell", ButtonData:>"Ref:Luttrell1985a", ButtonStyle->"Hyperlink"], ". This useful phase information should not be discarded in any bandwidth \ reduction technique: full complex data should be retained in the vicinity of \ targets, whilst only modulus data needs be retained elsewhere." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Phase Imaging", "Section"], Cell[TextData[{ "The display of complex valued images presents a unique problem. There are \ two degrees of freedom per pixel (real and imaginary parts, or ", Cell[BoxData[ \(TraditionalForm\`I\)]], " and ", Cell[BoxData[ \(TraditionalForm\`Q\)]], "), and a compromise must be found for the best way of displaying them. The \ most common solution is to display the modulus of the complex data and to \ discard the phase altogether, which is fine as long as there is an \ insignificant amount of information in the phase. In situations where the \ phase of the image might contain a significant amount of information an \ alternative image display scheme is needed. A means of transforming the \ original complex image into one in which each pixel's information content is \ recorded in a single real degree of freedom would serve as an ideal \ preprocessing stage before displaying the image." }], "Text"], Cell["\<\ I shall assume throughout that the complex image is produced by a \ band-limiting process and that Nyquist sampling (or better) is used. In \ addition to these constraints I shall assume that each complex sample \ contains two genuinely independent observations. Synthetic aperture radar \ (SAR) is an example of such imagery, where the two degrees of freedom per \ sample are:\ \>", "Text"], Cell["\<\ (i) The envelope of the returned signal (modulus) (ii) The phase shift of the returned signal (phase)\ \>", "Text"], Cell["\<\ For SAR these are physically independent degrees of freedom both of which \ vary on a length scale of order of the resolution cell size (\[TildeEqual]2m) \ which is much greater than the wavelength (\[Lambda]\[TildeEqual]3cm). The \ wavelength serves only to define the carrier frequency, and the frequency at \ which scattering properties are to be measured. The independence of the \ modulus and the phase of each sample mean than although the complex image is \ analytic (because it is bandlimited) its real and imaginary parts can not be \ related to each other in any simple way: a model of the scatterers and of the \ point spread function is required for this. In particular, it is incorrect to \ use the Hilbert transform to interrelate the real and imaginary components \ (and hence envelope and phase shift). However it would be permissable to use \ the Hilbert transform if the signal were first of all translated to an offset \ carrier to ensure that only positive (or alternatively negative) frequencies \ are present, and then sampled at the new Nyquist rate. Note that the Hilbert \ transform does not relate the original envelope and phase shift in this new \ representation because they are not simply related to the new complex signal, \ and so the physical independence of these two degrees of freedom is \ preserved.\ \>", "Text"], Cell[TextData[{ "There has been an attempt [", ButtonBox["Blackledge ", ButtonData:>"Ref:BlackledgeBurgeBarrett1987", ButtonStyle->"Hyperlink"], StyleBox[ButtonBox["et al", ButtonData:>"Ref:BlackledgeBurgeBarrett1987", ButtonStyle->"Hyperlink"], FontSlant->"Italic"], "] to produce phase imaging by real zero conversion (RZC) of complex SAR \ images. This is a one-dimensional technique, where they treat each range \ trace of the focused complex SAR image independently, and then obtain RZC by \ adding a complex exponential ", Cell[BoxData[ \(TraditionalForm\`a\ e\^\(i\ \[Omega]\ t\)\)]], " to each range trace. The amplitude ", Cell[BoxData[ \(TraditionalForm\`a\)]], " is chosen to move the complex zeros of the real and imaginary parts of \ the SAR data onto the real line, and the frequency ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\)]], " is chosen to be outside the band of the SAR data. They argue that the \ behaviour of the phase is determined predominantly by the presence of real \ zeros, and so RZC must move information in the SAR image into its phase. They \ then display an image which records the modulus of the rate of change of this \ phase (for each range trace)." }], "Text"], Cell[TextData[{ "There is a simple physical interpretation of the RZC method applied to SAR \ imagery. The effect of adding ", Cell[BoxData[ \(TraditionalForm\`a\ e\^\(i\ \[Omega]\ t\)\)]], " to the trace of a complex function in the Argand plane is to move its \ trace away from the origin, provided that the amplitude ", Cell[BoxData[ \(TraditionalForm\`a\)]], " is chosen to be large enough. Loosely speaking, the trace of the data is \ then contained within an annulus which is centred on the origin of the Argand \ plane. The effect of choosing ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\)]], " to be outside the band of the original data is to ensure that the phase \ of the trace monontonically increases (or decreases): this is the phase which \ determines the form of the phase image presented by ", ButtonBox["Blackledge ", ButtonData:>"Ref:BlackledgeBurgeBarrett1987", ButtonStyle->"Hyperlink"], StyleBox[ButtonBox["et al", ButtonData:>"Ref:BlackledgeBurgeBarrett1987", ButtonStyle->"Hyperlink"], FontSlant->"Italic"], ". Assuming that azimuth focusing has been applied, the output voltage of \ the SAR receiver is given by" }], "Text"], Cell[TextData[{ "\t", Cell[BoxData[ \(TraditionalForm\`V(t) = \(A(t)\)\ cos[\[Omega]\ t - \[Phi](t)]\)]] }], "Text"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`A(t)\)]], " is an amplitude, ", Cell[BoxData[ \(TraditionalForm\`\[Phi](t)\)]], " is a phase shift, and ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\)]], " is an intermediate frequency which lies outside the band of the complex \ image ", Cell[BoxData[ \(TraditionalForm\`\(A(t)\) e\^\(i\ \(\[Phi](t)\)\)\)]], ". On the other hand the phase of of the RZC signal ", Cell[BoxData[ \(TraditionalForm\`\(A(t)\) e\^\(i\ \(\[Phi](t)\)\) + a\ e\^\(i\ \[Omega]\ t\)\)]], " is given approximately by" }], "Text"], Cell[TextData[{ "\t", Cell[BoxData[ \(TraditionalForm\`\[CapitalPhi]( t) \[TildeEqual] \(\(A(t)\) cos[\[Omega]\ t - \[Phi](t)]\)\/a\)]] }], "Text"], Cell[TextData[{ "where the approximation becomes better the larger ", Cell[BoxData[ \(TraditionalForm\`a\/\(A(t)\)\)]], " becomes. On comparing the expressions for ", Cell[BoxData[ \(TraditionalForm\`V(t)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\[CapitalPhi](t)\)]], " it is obvious that they are essentially identical, so the phase function \ which is used by ", ButtonBox["Blackledge ", ButtonData:>"Ref:BlackledgeBurgeBarrett1987", ButtonStyle->"Hyperlink"], StyleBox[ButtonBox["et al", ButtonData:>"Ref:BlackledgeBurgeBarrett1987", ButtonStyle->"Hyperlink"], FontSlant->"Italic"], " to produce a phase image measures the output voltage of the SAR receiver \ for some suitably chosen intermediate frequency ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\)]], ". However it is not at all obvious what the interpretation of this phase \ function is in terms of physical scatterers, so it is not clear what progress \ has been made by producing such a phase image. Despite these criticisms their \ phase image has a pleasing visual appearance, not unlike the modulus image \ had in the first place." }], "Text", CellTags->"Ed:Change3"], Cell[TextData[{ "There is also a true two-dimensional phase imaging technique for SAR data \ [", ButtonBox["Scivier ", ButtonData:>"Ref:ScivierFiddyBurge1986", ButtonStyle->"Hyperlink"], StyleBox[ButtonBox["et al", ButtonData:>"Ref:ScivierFiddyBurge1986", ButtonStyle->"Hyperlink"], FontSlant->"Italic"], "]. A consistent two-dimensional phase image must take into account the \ point zeros which frequently occur, at which the phase is undefined (this \ problem is far less severe in one-dimension). This delicate operation has \ become known as \"phase unwrapping\" because it recovers the phase without \ the modulo ", Cell[BoxData[ \(TraditionalForm\`2 \[Pi]\)]], " restriction which normally accompanies it, and so the unwrapped phase is \ a well behaved function. However the effect of point zeros is to introduce \ dislocations in the unwrapped phase of the complex SAR image, which ", ButtonBox["Scivier ", ButtonData:>"Ref:ScivierFiddyBurge1986", ButtonStyle->"Hyperlink"], StyleBox[ButtonBox["et al", ButtonData:>"Ref:ScivierFiddyBurge1986", ButtonStyle->"Hyperlink"], FontSlant->"Italic"], " regularise by using a weighted least squares method when fitting the \ unwrapped phase function to the complex SAR data. The weighting function used \ is a smoothed version of the original modulus image which therefore has no \ zeros. Whilst this method aims to display an image which contains phase \ information alone, it is spoilt by the somewhat ", StyleBox["ad hoc", FontSlant->"Italic"], " inclusion of modulus information together with the phase in the fitting \ procedure. As with the method of ", ButtonBox["Blackledge ", ButtonData:>"Ref:BlackledgeBurgeBarrett1987", ButtonStyle->"Hyperlink"], StyleBox[ButtonBox["et al", ButtonData:>"Ref:BlackledgeBurgeBarrett1987", ButtonStyle->"Hyperlink"], FontSlant->"Italic"], " this method also suffers from difficulty of physical interpretation of \ the output image, although their unwrapped phase image has a pleasing visual \ appearance." }], "Text"], Cell["\<\ I shall adopt the attitude that a scheme for extracting information from a \ complex image must have an underlying physical basis, so that the output \ image can be physically interpreted. The ultimate goal of image processing is \ not to produce a modulus or a phase image, but rather to provide an \ interpretation of the complex image in terms of specific questions which \ might be asked of the data. In this picture the modulus and phase images \ would be possible (but not necessary) intermediate stages in the overall \ image processing procedure. In order to obtain such a scheme I shall \ construct a probabilistic model of the processes which lead to image \ formation, so that the information content of the complex image pixels may be \ examined in the light of the model.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Bayesian analysis", "Section"], Cell[TextData[{ "Using Bayes' theorem to express the ", StyleBox["a posteriori", FontSlant->"Italic"], " PDF for the cross section ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " of the physical scatterers conditioned on the complex image data ", Cell[BoxData[ \(TraditionalForm\`g\)]], ", I obtain" }], "Text"], Cell[TextData[{ "\t", Cell[BoxData[ \(TraditionalForm\`P[\[Sigma] | g] = \(P[\[Sigma]] P[g | \[Sigma]]\)\/P[g]\)]] }], "Text"], Cell["where", "Text"], Cell[TextData[{ "\t", Cell[BoxData[ \(TraditionalForm\`P[ g] \[Congruent] \[Integral]\(\([\)\(d\[Sigma]\)\(]\)\) P[\[Sigma]] P[g | \[Sigma]]\)]] }], "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`P[\[Sigma] | g]\)]], " contains all the information about ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " given our prior knowledge ", Cell[BoxData[ \(TraditionalForm\`P[\[Sigma]]\)]], " and knowledge of the imaging process and image data ", Cell[BoxData[ \(TraditionalForm\`P[g | \[Sigma]]\)]], "." }], "Text", CellTags->"Ed:Change4"], Cell[TextData[{ "Define a probabilistic model ", Cell[BoxData[ \(TraditionalForm\`P[f | \[Sigma]]\)]], " to describe the (scalar) field ", Cell[BoxData[ \(TraditionalForm\`f\)]], " which is scattered off ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]] }], "Text"], Cell[TextData[{ "\t", Cell[BoxData[ \(TraditionalForm\`P[ f | \[Sigma]] \[Congruent] \(1\/det[\[Pi]\ \[Sigma]]\) exp {\(-f\^\[Dagger]\) \(\[Sigma]\^\(-1\)\) f}\)]] }], "Text"], Cell[TextData[{ "Because ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " is diagonal this can be written in discrete form as" }], "Text"], Cell[TextData[{ "\t", Cell[BoxData[ \(TraditionalForm\`P[ f | \[Sigma]] = \[Product]\+j\( 1\/\(\[Pi]\ \[Sigma]\_j\)\) exp {\(-\(\[LeftBracketingBar]f\_j\[RightBracketingBar]\^2\/\ \[Sigma]\_j\)\)}\)]] }], "Text", CellTags->"Ed:Change5"], Cell[TextData[{ "This model for ", Cell[BoxData[ \(TraditionalForm\`P[f | \[Sigma]]\)]], " is widely used, and it assumes that the amplitude which is scattered off \ each small patch of ground is Gaussian distributed once its cross section is \ known. This will indeed be the case if all we know is that there are many \ scatterers within the patch each contributing to the field (by superposition) \ with a random phase. Also define a probabilistic model ", Cell[BoxData[ \(TraditionalForm\`P[g | f]\)]], " for the imaging process" }], "Text"], Cell[TextData[{ "\t", Cell[BoxData[ \(TraditionalForm\`P[g | f] \[Congruent] \(1\/det[\[Pi]\ \[Nu]]\) exp {\(-\((g - T\ f)\)\^\[Dagger]\) \(\(\[Nu]\^\(-1\)\)( g - T\ f)\)}\)]] }], "Text"], Cell[TextData[{ "where I describe the imaging process by a band-limited linear operator ", Cell[BoxData[ \(TraditionalForm\`T\)]], ", and I assume that the data is corrupted with additive zero mean Gaussian \ noise with covariance ", Cell[BoxData[ \(TraditionalForm\`\[Nu]\)]], ". The linearity of ", Cell[BoxData[ \(TraditionalForm\`T\)]], " is a good approximation to the imaging process which occurs in a SAR. The \ assumption that the data noise can be modelled as an additive Gaussian \ process is adequate because it only acts as a regularisation constraint on \ any interpretation that we place on the data. The \"chain rule\" for PDFs may \ be used to obtain" }], "Text"], Cell[TextData[{ "\t", Cell[BoxData[ FormBox[ RowBox[{\(P[g | f]\), "=", RowBox[{"\[Integral]", RowBox[{ RowBox[{"[", StyleBox[ RowBox[{ StyleBox["d", FontSlant->"Plain"], StyleBox["f", FontSlant->"Italic"]}]], "]"}], \(P[g | f]\), \(P[ f | \[Sigma]]\)}]}]}], TraditionalForm]]] }], "Text"], Cell[TextData[{ "which yields after some manipulation the so-called \"likelihood function\" \ ", Cell[BoxData[ \(TraditionalForm\`P[g | \[Sigma]]\)]] }], "Text"], Cell[TextData[{ "\t", Cell[BoxData[ \(TraditionalForm\`P[g | \[Sigma]] \[Congruent] \(1\/det[\[Pi]\ M]\) exp {\(-g\^\[Dagger]\) \(M\^\(-1\)\) g}\)]] }], "Text"], Cell["where", "Text"], Cell[TextData[{ "\t", Cell[BoxData[ \(TraditionalForm\`M \[Congruent] T\ \[Sigma]\ T\^\[Dagger] + \[Nu]\)]] }], "Text"], Cell[TextData[{ "This result together with an ", StyleBox["a priori", FontSlant->"Italic"], " model ", Cell[BoxData[ \(TraditionalForm\`P[\[Sigma]]\)]], " for a uniquely define the ", StyleBox["a posteriori", FontSlant->"Italic"], " PDF ", Cell[BoxData[ \(TraditionalForm\`P[\[Sigma] | g]\)]], ". The likelihood function ", Cell[BoxData[ \(TraditionalForm\`P[g | \[Sigma]]\)]], " contains all the dependence of ", Cell[BoxData[ \(TraditionalForm\`P[\[Sigma] | g]\)]], " on the image data ", Cell[BoxData[ \(TraditionalForm\`g\)]], ". I shall therefore consider only the form of ", Cell[BoxData[ \(TraditionalForm\`P[g | \[Sigma]]\)]], " when assessing the information content of ", Cell[BoxData[ \(TraditionalForm\`g\)]], "." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["A special case: constant cross section", "Section"], Cell["\<\ I shall consider the special case where the cross section is constant (but of \ unknown value)\ \>", "Text"], Cell[TextData[{ "\t", Cell[BoxData[ \(TraditionalForm\`\[Sigma] = \[Sigma]\_0\)]] }], "Text"], Cell[TextData[{ "For simplicity I shall assume that ", Cell[BoxData[ \(TraditionalForm\`T\)]], " is the pure band-limiting operator (", Cell[BoxData[ \(TraditionalForm\`T = B\)]], ") and that the image data is Nyquist sampled, so that" }], "Text"], Cell[TextData[{ "\t", Cell[BoxData[ \(TraditionalForm\`T\ \[Sigma]\ T\^\[Dagger] = \[Sigma]\_0\)]] }], "Text"], Cell[TextData[{ "where I have used ", Cell[BoxData[ \(TraditionalForm\`B\ B\^\[Dagger] = I\)]], ". This restriction is not a valid approximation to the RSRE SAR system, \ but nevertheless it serves to demonstrate the image processing principles \ which I wish to explain. The generalisation to arbitrary ", Cell[BoxData[ \(TraditionalForm\`T\)]], " is straightforward." }], "Text"], Cell[TextData[{ "I shall also assume that the data noise covariance matrix is diagonal \ (which is a good approximation in practice) so that the matrix ", Cell[BoxData[ \(TraditionalForm\`M\)]], " is given in component form by" }], "Text"], Cell[TextData[{ "\t", Cell[BoxData[ \(TraditionalForm\`M\_\(j, k\) = \((\[Sigma]\_0 + \[Nu])\) \ \[Delta]\_\(j, k\)\)]] }], "Text"], Cell["The likelihood function therefore reduces to", "Text"], Cell[TextData[{ "\t", Cell[BoxData[ \(TraditionalForm\`P[ g | \[Sigma]] = \[Product]\+k\( 1\/\(\[Pi]\ \((\[Sigma]\_0 + \[Nu])\)\)\) exp {\(-\(\[LeftBracketingBar]g\_k\[RightBracketingBar]\^2\/\(\ \[Sigma]\_0 + \[Nu]\)\)\)}\)]] }], "Text", CellTags->"Ed:Change6"], Cell[TextData[{ "where the product is over image samples. The most important feature of ", Cell[BoxData[ \(TraditionalForm\`P[g | \[Sigma]]\)]], " is that dependence on ", Cell[BoxData[ \(TraditionalForm\`g\_k\)]], " is reduced to dependence on ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]g\_k\[RightBracketingBar]\)]], " alone: the phase ", Cell[BoxData[ \(TraditionalForm\`arg(g\_k)\)]], " does not affect the likelihood function, and so there is no information \ about ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " in ", Cell[BoxData[ \(TraditionalForm\`arg(g)\)]], "." }], "Text"], Cell[TextData[{ "This is an intuitively obvious result, because all that I have derived is \ that the estimated the cross section of a large region of assumed constant \ cross section depends only on the average of the modulus squared of the image \ pixels; this is a form of energy conservation. The most stringent assumption \ which I made in order to obtain this result was ", Cell[BoxData[ \(TraditionalForm\`\[Sigma] = \[Sigma]\_0\)]], ", which I shall now relax." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["General case: variable cross section", "Section"], Cell[TextData[{ "Consider the same case as above but with arbitrary ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], ". The ", Cell[BoxData[ \(TraditionalForm\`T\ \[Sigma]\ T\^\[Dagger]\)]], " term now has non-zero off-diagonal elements, so ", Cell[BoxData[ \(TraditionalForm\`M\^\(-1\)\)]], " now has non-zero off-diagonal elements which causes ", Cell[BoxData[ \(TraditionalForm\`P[g | \[Sigma]]\)]], " to depend on terms like ", Cell[BoxData[ \(TraditionalForm\`\(g\_j\%*\) g\_k\)]], ". Therefore ", Cell[BoxData[ \(TraditionalForm\`arg(g)\)]], " manifestly contains information about ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], ". Of course off-diagonal elements can also arise through any mix of ", Cell[BoxData[ \(TraditionalForm\`T \[NotEqual] B\)]], ", non-Nyquist sampling, and non-diagonal \[Nu]." }], "Text"], Cell[TextData[{ "I shall now diagonalise ", Cell[BoxData[ \(TraditionalForm\`M\)]], ". Thus define" }], "Text"], Cell[TextData[{ "\t", Cell[BoxData[ \(TraditionalForm\`g \[Congruent] U\ g\^\[Prime]\)]] }], "Text"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`U\)]], " is the unitary transformation which diagonalises ", Cell[BoxData[ \(TraditionalForm\`M\)]] }], "Text"], Cell[TextData[{ "\t", Cell[BoxData[{ \(TraditionalForm\`\(U\^\[Dagger]\) M\ U = diag(\[Lambda]\_1, \[Lambda]\_2, \[CenterEllipsis]) \[Congruent] \ \[CapitalLambda]\), "\[IndentingNewLine]", \(TraditionalForm\`\(U\^\[Dagger]\) M\^\(-1\)\ U = diag(1\/\[Lambda]\_1, 1\/\[Lambda]\_2, \[CenterEllipsis]) \[Congruent] \[CapitalLambda]\ \^\(-1\)\)}]] }], "Text", CellTags->"Ed:Change7"], Cell[TextData[{ "where I have denoted the eigenvalues of ", Cell[BoxData[ \(TraditionalForm\`M\)]], " by ", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\_j\)]], ". The likelihood function then becomes" }], "Text"], Cell[TextData[{ "\t", Cell[BoxData[ \(TraditionalForm\`P[ g | \[Sigma]] = \[Product]\+j\( 1\/\(\[Pi]\ \[Lambda]\_j\)\) exp {\(-\(\[LeftBracketingBar]g\_j\^\[Prime]\[RightBracketingBar]\ \^2\/\[Lambda]\_j\)\)}\)]] }], "Text", CellTags->"Ed:Change8"], Cell[TextData[{ "Clearly there is no information about ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " in ", Cell[BoxData[ \(TraditionalForm\`arg(g\^\[Prime])\)]], ". The basic lesson which we learn from this analysis is that the original \ image data ", Cell[BoxData[ \(TraditionalForm\`g\)]], " contains information in its phase about non-uniform ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], ". Furthermore by diagonalising ", Cell[BoxData[ \(TraditionalForm\`M\)]], " we may combine the ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]g\[RightBracketingBar]\)]], " and ", Cell[BoxData[ \(TraditionalForm\`arg(g)\)]], " in such a way that the information about ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " is contained entirely in ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]g\^\[Prime]\[RightBracketingBar]\ \)]], ". I shall call this transformation \"linear phase imaging\", because the \ linear transformation ", Cell[BoxData[ \(TraditionalForm\`U\)]], " combines amplitude and phase in such a way as to produce ", Cell[BoxData[ \(TraditionalForm\`g\^\[Prime]\)]], " where modulus information alone contains all the relevant information. \ Note that ", Cell[BoxData[ \(TraditionalForm\`g\^\[Prime]\)]], " is not strictly an image, and so it does not have a simple visual \ interpretation. However it does have a physical interpretation in the light \ of the Bayesian model which we have used, and so ", Cell[BoxData[ \(TraditionalForm\`g\^\[Prime]\)]], " may be processed in a well-defined way." }], "Text"], Cell[TextData[{ "This procedure must be applied for each ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " which we wish to consider in a Bayesian analysis. Each ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " implies a different diagonalisation matrix ", Cell[BoxData[ \(TraditionalForm\`U\)]], " in general, and so the way in which the image amplitude and phase are \ combined is ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], "-dependent. This is an important point: we can not process the complex \ image data until we ask a specific question such as \"What is the likelihood \ that ", Cell[BoxData[ \(TraditionalForm\`g\)]], " arose from this particular ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], "?\". Any processing scheme which claims to transform the complex image in \ some meaningful manner without at the same time stating what question is \ being asked of the image is at best ", StyleBox["ad hoc", FontSlant->"Italic"], "." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["The relationship to super-resolution", "Section"], Cell[TextData[{ "Super-resolution is one aspect of the Bayesian analysis of the recovery of \ the scattered field ", Cell[BoxData[ \(TraditionalForm\`f\)]], " (not the cross section a) from knowledge of the image data g and the \ scattering cross section a. In practice ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " is not known in advance, so an estimate is provided in order to obtain \ the ", StyleBox["a posteriori", FontSlant->"Italic"], " PDF ", Cell[BoxData[ \(TraditionalForm\`P[f | g, \[Sigma]]\)]], ". A simple example of this approach arises when ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " is known to be exactly zero outside some domain ", Cell[BoxData[ \(TraditionalForm\`D\)]], ". A non-committal estimate of ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " is then" }], "Text"], Cell[TextData[{ "\t", Cell[BoxData[ FormBox[GridBox[{ {\(\[Sigma] = \[Sigma]\_0\), \(inside\ D\)}, {\(\(\[Sigma]\)\(=\)\(0\)\(\ \)\), \(outside\ D\)} }], TraditionalForm]]] }], "Text", GridBoxOptions->{ColumnAlignments->{Left}}], Cell[TextData[{ "A generalisation of this approach has been given by ", ButtonBox["Luttrell", ButtonData:>"Ref:Luttrell1985a", ButtonStyle->"Hyperlink"], "." }], "Text"], Cell[TextData[{ "Using Bayes' theorem the relevant ", StyleBox["a posteriori", FontSlant->"Italic"], " PDF is given by" }], "Text"], Cell[TextData[{ "\t", Cell[BoxData[ \(TraditionalForm\`P[ f | g, \[Sigma]] = \[AlignmentMarker]\(P[f, \[Sigma]] P[g | f, \ \[Sigma]]\)\/P[g, \[Sigma]]\[IndentingNewLine]\(\(=\)\(\[AlignmentMarker]\)\(\ \(P[f | \[Sigma]] P[g | f]\)\/P[g | \[Sigma]]\)\)\)], TextAlignment->AlignmentMarker] }], "Text"], Cell["where I have used", "Text"], Cell[TextData[{ "\t", Cell[BoxData[{ \(TraditionalForm\`P[ f, \[Sigma]] \[Congruent] \[AlignmentMarker]P[f | \[Sigma]] P[\[Sigma]]\), "\[IndentingNewLine]", \(TraditionalForm\`P[ g, \[Sigma]] \[Congruent] \[AlignmentMarker]P[g | \[Sigma]] P[\[Sigma]]\), "\[IndentingNewLine]", \(TraditionalForm\`P[g | f, \[Sigma]] \[Congruent] \[AlignmentMarker]P[ g | f]\)}], TextAlignment->AlignmentMarker] }], "Text", CellTags->"Ed:Change9"], Cell[TextData[{ "The last relationship holds because the sequence ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\[LongRightArrow]f\[LongRightArrow]g\)]], " is Markovian with each state depending directly only on the previous \ state. After some manipulation the ", StyleBox["a posteriori", FontSlant->"Italic"], " PDF is obtained as" }], "Text"], Cell[TextData[{ "\t", Cell[BoxData[ \(TraditionalForm\`P[f | g, \[Sigma]] \[Proportional] exp {\(-\((f - f\_0)\)\^\[Dagger]\) \(\(A\^\(-1\)\)(f - f\_0)\)}\)]] }], "Text"], Cell["where", "Text"], Cell[TextData[{ "\t", Cell[BoxData[ \(TraditionalForm\`f\_0 \[Congruent] \[Sigma]\ \(T\^\[Dagger]\) \ \(M\^\(-1\)\) g\)]] }], "Text"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`M\)]], " is the same as I used above, and ", Cell[BoxData[ \(TraditionalForm\`A\)]], " is a covariance matrix Clearly the f which maximises the ", StyleBox["a posteriori", FontSlant->"Italic"], " PDF is ", Cell[BoxData[ \(TraditionalForm\`f\_0\)]], ", and so ", Cell[BoxData[ \(TraditionalForm\`f\_0\)]], " could be used as a representative scattered field corresponding to the \ particular cross section ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " and image data ", Cell[BoxData[ \(TraditionalForm\`g\)]], " which are available." }], "Text"], Cell[TextData[{ "I shall now diagonalise ", Cell[BoxData[ \(TraditionalForm\`M\)]], " as before to yield" }], "Text"], Cell[TextData[{ "\t", Cell[BoxData[ \(TraditionalForm\`f\_0 = \[AlignmentMarker]\(\[Sigma]\ \ \(T\^\[Dagger]\) U\ \(U\^\[Dagger]\) \(M\^\(-1\)\) U\ g\^\[Prime]\)\[IndentingNewLine]\(\(=\)\(\[AlignmentMarker]\)\(\ \[Sigma]\ \(T\^\[Dagger]\) U\ \[CapitalLambda]\ g\^\[Prime]\)\)\)], TextAlignment->AlignmentMarker] }], "Text"], Cell["This may be be written out in component form as", "Text"], Cell[TextData[{ "\t", Cell[BoxData[ \(TraditionalForm\`f\_\(0, a\) = \[Sum]\+j\( C\_\(a, j\)\) g\_j\^\[Prime]\/\[Lambda]\_j\)]] }], "Text", CellTags->"Ed:Change10"], Cell["where", "Text"], Cell[TextData[{ "\t", Cell[BoxData[ \(TraditionalForm\`C\_\(a, j\) \[Congruent] \((\[Sigma]\ \(T\^\[Dagger]\ \) U)\)\_\(a, j\)\)]] }], "Text"], Cell[TextData[{ "For fixed ", Cell[BoxData[ \(TraditionalForm\`j\)]], " the ", Cell[BoxData[ \(TraditionalForm\`C\_\(a, j\)\)]], " form an \"object function\" which is derived directly from a \ corresponding \"image function\" ", Cell[BoxData[ \(TraditionalForm\`U\_\(k, j\)\)]], ". The overall structure of the expression for ", Cell[BoxData[ \(TraditionalForm\`f\_0\)]], " shows that ", Cell[BoxData[ \(TraditionalForm\`f\_0\)]], " is built up as a linear combination of object functions ", Cell[BoxData[ \(TraditionalForm\`C\_\(a, j\)\)]], " where the coefficients are proportional to the projections ", Cell[BoxData[ \(TraditionalForm\`g\_j\^\[Prime]\)]], " of the image data ", Cell[BoxData[ \(TraditionalForm\`g\)]], " onto the corresponding image functions ", Cell[BoxData[ \(TraditionalForm\`U\_\(k, j\)\)]], " (the constants of proportionality are the ", Cell[BoxData[ \(TraditionalForm\`1\/\[Lambda]\_j\)]], ")." }], "Text", CellTags->"Ed:Change11"], Cell[TextData[{ "This expression for ", Cell[BoxData[ \(TraditionalForm\`f\_0\)]], " has been used successfully to produce super-resolution from image data \ when prior knowledge ", Cell[BoxData[ \(TraditionalForm\`P[f | \[Sigma]]\)]], " is available [", ButtonBox["Luttrell", ButtonData:>"Ref:Luttrell1985a", ButtonStyle->"Hyperlink"], ", ", ButtonBox["Luttrell and Oliver", ButtonData:>"Ref:LuttrellOliver1986", ButtonStyle->"Hyperlink"], "]. The super-resolved image ", Cell[BoxData[ \(TraditionalForm\`f\_0\)]], " is derived from precisely those components ", Cell[BoxData[ \(TraditionalForm\`g\_j\)]], " of the image data ", Cell[BoxData[ \(TraditionalForm\`g\)]], " which also serve to diagonalise the likelihood function ", Cell[BoxData[ \(TraditionalForm\`P[g | \[Sigma]]\)]], " in linear phase imaging of ", Cell[BoxData[ \(TraditionalForm\`g\)]], ". Therefore super-resolution is merely a representation in terms of \ \"object functions\" of the output of the linear phase imaging procedure." }], "Text"], Cell[TextData[{ "This analysis completes the connection between Bayesian super-resolution \ and the linear phase imaging procedure which I introduced here. \ Super-resolution provides a means of processing ", Cell[BoxData[ \(TraditionalForm\`g\)]], " where firstly the linear phase imaging transformation is applied to \ produce ", Cell[BoxData[ \(TraditionalForm\`g\^\[Prime]\)]], ", and secondly ", Cell[BoxData[ \(TraditionalForm\`g\^\[Prime]\)]], " is linearly transformed to a scattered field ", Cell[BoxData[ \(TraditionalForm\`f\_0\)]], " which is more easily interpreted. ", Cell[BoxData[ \(TraditionalForm\`g\^\[Prime]\)]], " is not a simple visual image, but its information content is very \ naturally expressed as a super-resolved image ", Cell[BoxData[ \(TraditionalForm\`f\_0\)]], "." }], "Text"], Cell[TextData[{ "Using super-resolution theory [", ButtonBox["Luttrell", ButtonData:>"Ref:Luttrell1985a", ButtonStyle->"Hyperlink"], "] we may deduce immediately which parts of ", Cell[BoxData[ \(TraditionalForm\`g\)]], " contain a useful amount of phase information. Super-resolution depends \ essentially on the transformation ", Cell[BoxData[ \(TraditionalForm\`g\[LongRightArrow]g\^\[Prime]\)]], " which produces the \"linear phase image\". For the particular case ", Cell[BoxData[ \(TraditionalForm\`T = B\)]], " with Nyquist sampling and white noise this transformation causes ", Cell[BoxData[ \(TraditionalForm\`g\^\[Prime]\)]], " to depend on ", Cell[BoxData[ \(TraditionalForm\`arg(g)\)]], " only in regions where ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " fluctuates strongly on a length scale of order of the resolution cell \ size. Typically such fluctuations only occur in targets, and so background \ clutter does not cause there to be a significant amount of information in ", Cell[BoxData[ \(TraditionalForm\`arg(g)\)]], ". Some extremely spiky types of clutter might violate this, but then the \ spikes would be picked up as possible target candidates." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Conclusions", "Section"], Cell[TextData[{ "I have demonstrated that the one-dimensional SAR phase imaging technique \ of ", ButtonBox["Blackledge ", ButtonData:>"Ref:BlackledgeBurgeBarrett1987", ButtonStyle->"Hyperlink"], StyleBox[ButtonBox["et al", ButtonData:>"Ref:BlackledgeBurgeBarrett1987", ButtonStyle->"Hyperlink"], FontSlant->"Italic"], " using RZC is related directly to the output voltage of the SAR receiver. \ However, this voltage has no simple physical interpretation because it \ depends jointly on the envelope and phase shift of the returned signal. There \ is no obvious physical question which one can ask for which the answer is to \ be found directly from their phase image." }], "Text"], Cell[TextData[{ ButtonBox["Scivier ", ButtonData:>"Ref:ScivierFiddyBurge1986", ButtonStyle->"Hyperlink"], StyleBox[ButtonBox["et al", ButtonData:>"Ref:ScivierFiddyBurge1986", ButtonStyle->"Hyperlink"], FontSlant->"Italic"], " are more rigorous in their attempt to produce a true two-dimensional \ phase image. However their method also suffers from the problem that the \ output is not a pure phase image, and its physical interpretation is \ questionable." }], "Text"], Cell[TextData[{ "I have demonstrated within the Bayesian approach to complex image analysis \ how the information content (amplitude and phase) of an image ", Cell[BoxData[ \(TraditionalForm\`g\)]], " may be reorganised according to specific questions which are asked of the \ image data. In particular I have shown how the likelihood function ", Cell[BoxData[ \(TraditionalForm\`P[g | \[Sigma]]\)]], " (and hence the ", StyleBox["a posteriori", FontSlant->"Italic"], " PDF ", Cell[BoxData[ \(TraditionalForm\`P[\[Sigma] | g]\)]], ") may be expressed in a form in which depends only on ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]g\^\[Prime]\[RightBracketingBar]\ \)]], ", where ", Cell[BoxData[ \(TraditionalForm\`g\^\[Prime]\)]], " is a linearly transformed image: I have called this linear phase imaging. \ ", Cell[BoxData[ \(TraditionalForm\`g\^\[Prime]\)]], " is not a pure phase image, rather it is that image which is formed by \ combining together ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]g\[RightBracketingBar]\)]], " and ", Cell[BoxData[ \(TraditionalForm\`arg(g)\)]], " in a way which is uniquely defined by the requirements of Bayesian image \ processing. Furthermore I have shown that Bayesian super-resolution is \ obtained by a further linear transformation of ", Cell[BoxData[ \(TraditionalForm\`g\^\[Prime]\)]], " into a scattered field ", Cell[BoxData[ \(TraditionalForm\`f\_0\)]], "." }], "Text", CellTags->"Ed:Change12"], Cell[TextData[{ "This flow of information in super-resolution processing demonstrates how \ the method is an extension of the linear phase imaging technique. ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]g\^\[Prime]\[RightBracketingBar]\ \)]], " contains all the image dependent information which is relevant to \ deciding what ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " gave rise to ", Cell[BoxData[ \(TraditionalForm\`g\)]], ", but ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]g\^\[Prime]\[RightBracketingBar]\ \)]], " does not by itself form an image. If you wish to display the information \ which is contained in ", Cell[BoxData[ \(TraditionalForm\`g\^\[Prime]\)]], " in a visually pleasing form, then the super-resolved image ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]f\_0\[RightBracketingBar]\)]], " provides one means of doing so. Note that ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]f\_0\[RightBracketingBar]\)]], " depends both on ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]g\^\[Prime]\[RightBracketingBar]\ \)]], " and ", Cell[BoxData[ \(TraditionalForm\`arg(g\^\[Prime])\)]], " so it mixes together modulus and phase information in a complicated way. \ However the beauty of the method is that the super-resolved image ", Cell[BoxData[ \(TraditionalForm\`f\_0\)]], " has a simple physical interpretation: it is the scattered field which is \ most likely to have occurred given ", Cell[BoxData[ \(TraditionalForm\`g\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\[Sigma]\)]], " (assuming the Gaussian scattering model is correct). Such physical \ interpretation(s) are not to be found in the methods of ", ButtonBox["Blackledge ", ButtonData:>"Ref:BlackledgeBurgeBarrett1987", ButtonStyle->"Hyperlink"], StyleBox[ButtonBox["et al", ButtonData:>"Ref:BlackledgeBurgeBarrett1987", ButtonStyle->"Hyperlink"], FontSlant->"Italic"], " or ", ButtonBox["Scivier ", ButtonData:>"Ref:ScivierFiddyBurge1986", ButtonStyle->"Hyperlink"], StyleBox[ButtonBox["et al", ButtonData:>"Ref:ScivierFiddyBurge1986", ButtonStyle->"Hyperlink"], FontSlant->"Italic"], "." }], "Text"], Cell["\<\ The most useful practical result which emerges from this work is that the \ phase of a SAR image need only be retained in the vicinity of targets. This \ leads to a small increase in the storage space and transmission bandwidth \ which a SAR image requires, but the information which is preserved is worth \ the effort.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["References", "Section"], Cell[TextData[{ "Blackledge J M, Burge R E and Barrett N R, 1987, \"Inversion method to \ extract the phase information generated by synthetic aperature radar \ systems\", ", StyleBox["Inverse Problems in Optics", FontSlant->"Italic"], ", The Hague" }], "Reference", CellTags->{"Ed:Change1", "Ref:BlackledgeBurgeBarrett1987"}], Cell[TextData[{ "Scivier M S, Fiddy M A and Burge R E, 1986, \"Estimating SAR phase from \ complex SAR imagery\", ", StyleBox["J Phys D: Appl Phys", FontSlant->"Italic"], ", ", StyleBox["19", FontWeight->"Bold"], ", 357-362" }], "Reference", CellTags->"Ref:ScivierFiddyBurge1986"], Cell[TextData[{ Cell[TextData[{ " ", ButtonBox["OPEN", ButtonData:>{ URL[ "http://www.luttrell.org.uk/papers/ble/ble.nb"], None}, Active->True, ButtonStyle->"Hyperlink"], " " }]], "Luttrell S P, 1985, \"Prior knowledge and object reconstruction using the \ best linear estimate technique\", 1985, ", StyleBox["Opt Acta", FontSlant->"Italic"], ", ", StyleBox["32", FontWeight->"Bold"], "(6), 703-716" }], "Reference", CellTags->"Ref:Luttrell1985a"], Cell[TextData[{ Cell[TextData[{ " ", ButtonBox["OPEN", ButtonData:>{ URL[ "http://www.luttrell.org.uk/papers/transinf/transinf.nb"], None}, Active->True, ButtonStyle->"Hyperlink"], " " }]], "Luttrell S P, 1985, \"The use of transinformation in the design of data \ sampling schemes for inverse problems\", 1985, ", StyleBox["Inv Prob", FontSlant->"Italic"], ", ", StyleBox["1", FontWeight->"Bold"], ", 199-218" }], "Reference", CellTags->{"Ref:Luttrell1985b", "Ed:Problem1"}], Cell[TextData[{ Cell[TextData[{ " ", ButtonBox["OPEN", ButtonData:>{ URL[ "http://www.luttrell.org.uk/papers/priorsar/priorsar.nb"], None}, Active->True, ButtonStyle->"Hyperlink"], " " }]], "Luttrell S P and Oliver C J, 1986, \"Prior knowledge in synthetic aperture \ radar processing\", ", StyleBox["J Phys D", FontSlant->"Italic"], ", ", StyleBox["19", FontWeight->"Bold"], ", 333-356" }], "Reference", CellTags->{"Ref:LuttrellOliver1986", "Ed:Change2"}] }, Closed]] }, Open ]] }, FrontEndVersion->"5.0 for Microsoft Windows", ScreenRectangle->{{0, 1280}, {0, 941}}, WindowToolbars->{}, WindowSize->{665.375, 641}, WindowMargins->{{307.25, Automatic}, {Automatic, 50}}, Magnification->1, StyleDefinitions -> "Report.nb" ] (******************************************************************* Cached data follows. 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