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Cell[CellGroupData[{
Cell["Notes", "Section 1"],

Cell[CellGroupData[{

Cell["Editorial Changes", "Subsection"],

Cell["\"ie\" changed to \"i.e.\" throughout the paper.", "Text"],

Cell["\<\
\"estimate-maximise\" changed to \"expectation-maximisation\" thoughout the \
paper.\
\>", "Text"],

Cell[TextData[{
  "Notation \"",
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\[SelectionPlaceholder]\)\)]],
  "\" changed to \"",
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\[SelectionPlaceholder]\)\)]],
  "\" throughout the paper."
}], "Text"],

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  "\"ariose form\" changed to \"arise from\"."
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Cell[CellGroupData[{

Cell[TextData[{
  "An EM approach to iterative Bayesian super-resolution with applications to \
",
  Cell[BoxData[
      FormBox[
        StyleBox["K",
          FontSlant->"Plain"], TraditionalForm]]],
  "-distributed data\n"
}], "Title"],

Cell["\<\
Stephen P Luttrell
Pattern Processing Principles
SP4 division, RSRE
Malvern, WORCS, WR14 3PS\
\>", "Author"],

Cell["\<\
This paper appeared as SP4 Research Note, No. 102, 31 October 1989.\
\>", "Text"],

Cell["Copyright \[Copyright] Controller HMSO, London, 1989.", "Text"],

Cell[TextData[{
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  "\n\nWe derive an ",
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  " [",
  ButtonBox["1",
    ButtonData:>"Ref:DempsterLairdRubin1977",
    ButtonStyle->"Hyperlink"],
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  ButtonBox["2",
    ButtonData:>"Ref:DelvesPrydeLuttrell1988",
    ButtonStyle->"Hyperlink"],
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  ButtonBox["3",
    ButtonData:>"Ref:Luttrell1989",
    ButtonStyle->"Hyperlink"],
  "]. We extend the technique by incorporating prior knowledge of the \
properties of ",
  Cell[BoxData[
      FormBox[
        StyleBox["K",
          FontSlant->"Plain"], TraditionalForm]]],
  "-distributed data, and obtain a simple modification to the basic \
super-resolution algorithm"
}], "Abstract"],

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    ButtonStyle->"Hyperlink"],
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  ButtonBox["6",
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  "]. The algorithm was originally developed heuristically, but was then \
partially justified from a Bayesian viewpoint [",
  ButtonBox["3",
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    ButtonStyle->"Hyperlink"],
  "]. In this paper we shall present an improved justification by appealing \
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problems ",
  "[",
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    ButtonData:>"Ref:DempsterLairdRubin1977",
    ButtonStyle->"Hyperlink"],
  "]",
  ". We concern ourselves with the inverse cross section problem rather than \
the inverse object field problem, because the former is more closely related \
to the problem of interpreting data. The solution of the inverse object field \
problem is used only as an intermediate step in solving the inverse cross \
section problem."
}], "Text"],

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  "A fully Bayesian treatment of super-resolution must incorporate prior \
knowledge of the underlying cross section - this was done only heuristically \
in [",
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  ButtonBox["8",
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  "]. We find that the iterative super-resolution algorithm needs only a \
slight modification in order to incorporate the effects of this type of prior \
knowledge."
}], "Text"],

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In this section we shall solve the problem of recovering from a set of image \
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\>", "Text"],

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First of all let us solve the inverse object field problem. Introduce \
probability density functions (PDF) to transform into the language Bayesian \
calculus\
\>", "Text"],

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Solve the inverse object field problem by introducing Bayes theorem in the \
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\>", "Text"],

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Solve the inverse cross section problem by introducing Bayes theorem in the \
form\
\>", "Text"],

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  Cell[BoxData[
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Any situation in which the underlying cross section is ",
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  "-distributed, and which is illuminated coherently to produce a scattered \
field (whose intensity statistics are ",
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          FontSlant->"Plain"], TraditionalForm]]],
  "-distributed), and which is coherently imaged using a linear imaging \
system (i.e. obeying the principle of superposition), is a candidate for the \
super-resolution method that we have presented in this paper."
}], "Text"]
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Cell["References", "Section"],

Cell[TextData[{
  "[1] Dempster A P, Laird N M and Rubin D B, 1977, Maximum likelihood from \
incomplete data via the EM algorithm, ",
  StyleBox["J. Roy. Statist. Soc. Ser. B",
    FontSlant->"Italic"],
  ", ",
  StyleBox["39",
    FontWeight->"Bold"],
  ", 1-37"
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        URL[ "http://www.luttrell.org.uk/papers/sr_alg/sr_alg.nb"], None},
      Active->True,
      ButtonStyle->"Hyperlink"],
    " "
  }]],
  "[2] Delves L M, Pryde G C and Luttrell S P, 1988, A super-resolution \
algorithm for SAR images, ",
  StyleBox["Inverse Problems",
    FontSlant->"Italic"],
  ", ",
  StyleBox["4",
    FontWeight->"Bold"],
  ", 681-703"
}], "Reference",
  CellTags->"Ref:DelvesPrydeLuttrell1988"],

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      Active->True,
      ButtonStyle->"Hyperlink"],
    " "
  }]],
  "[3] Luttrell S P, 1989, The inverse cross section problem for complex \
data, ",
  StyleBox["Inverse Problems",
    FontSlant->"Italic"],
  ", ",
  StyleBox["5",
    FontWeight->"Bold"],
  ", 35-50"
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    " ",
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      ButtonData:>{
        URL[ "http://www.luttrell.org.uk/papers/ble/ble.nb"], None},
      Active->True,
      ButtonStyle->"Hyperlink"],
    " "
  }]],
  "[4] Luttrell S P, 1985, Prior knowledge and object reconstruction using \
the best linear estimate technique, ",
  StyleBox["Opt. Acta",
    FontSlant->"Italic"],
  ", ",
  StyleBox["32",
    FontWeight->"Bold"],
  ", 703-716"
}], "Reference",
  CellTags->"Ref:Luttrell1985"],

Cell[TextData[{
  Cell[TextData[{
    " ",
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      ButtonData:>{
        URL[ "http://www.luttrell.org.uk/papers/priorsar/priorsar.nb"], None},
      
      Active->True,
      ButtonStyle->"Hyperlink"],
    " "
  }]],
  "[5] Luttrell S P and Oliver C J, 1986, Prior knowledge in synthetic \
aperture radar processing, ",
  StyleBox["J. Phys. D: Appl. Phys.",
    FontSlant->"Italic"],
  ", ",
  StyleBox["19",
    FontWeight->"Bold"],
  ", 333-356"
}], "Reference",
  CellTags->"Ref:LuttrellOliver1986"],

Cell[TextData[{
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    " ",
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      ButtonData:>{
        URL[ "http://www.luttrell.org.uk/papers/bayessup/bayessup.nb"], None},
      
      Active->True,
      ButtonStyle->"Hyperlink"],
    " "
  }]],
  "[6] Luttrell S P, 1990, The theory of Bayesian super-resolution of \
coherent images: a review, ",
  StyleBox["Int. J. Remote Sensing",
    FontSlant->"Italic"],
  ", to appear"
}], "Reference",
  CellTags->"Ref:Luttrell1990"],

Cell[TextData[{
  "[7] Jakeman B and Pusey P N, 1976, A model for non-Rayleigh sea echo, ",
  StyleBox["IEEE Trans. Antenn. Propag.",
    FontSlant->"Italic"],
  ", ",
  StyleBox["24",
    FontWeight->"Bold"],
  "(6), 806-814"
}], "Reference",
  CellTags->"Ref:JakemanPusey1976"],

Cell[TextData[{
  "[8] Ward K D, 1981, Compound representation of high resolution sea \
clutter, ",
  StyleBox["Electronics Letters",
    FontSlant->"Italic"],
  ", ",
  StyleBox["17",
    FontWeight->"Bold"],
  "(16), 561-563"
}], "Reference",
  CellTags->"Ref:Ward1981"]
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(*******************************************************************
End of Mathematica Notebook file.
*******************************************************************)