(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 1976503, 29270]*) (*NotebookOutlinePosition[ 1977782, 29309]*) (* CellTagsIndexPosition[ 1977738, 29305]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Erosion, dilation and related operators", "Title"], Cell["Mariusz Jankowski", "Author"], Cell["\<\ Department of Electrical Engineering University of Southern Maine Portland, Maine, USA\ \>", "TextAboutAuthor"], Cell["mjankowski@usm.maine.edu", "Program"], Cell[TextData[{ "This paper will present implementation details of an important set of \ numerical operators in the ", StyleBox["Digital Image Processing", FontSlant->"Italic"], " application package. These include the erode and dilate operators and \ two related recursive operators that are central to morphological \ reconstruction algorithms which are currently being developed for a future \ release candidate." }], "Abstract"], Cell["\<\ Erode and dilate are the fundamental operators of mathematical \ morphology, a theory for the analysis of spatial structure. The methods of \ mathematical morphology make possible a large number of very powerful image \ analysis techniques and therefore these operators and their implementations \ are of great theoretical and practical interest to many involved in image \ processing and analysis.\ \>", "Abstract"], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell["\<\ Mathematical morphology is a powerful tool for geometrical shape \ analysis and description. It was originaly developed by G. Matheron and J. \ Serra for the purpose of analysing binary images, thus a set formalism was \ used with simple operations such as intersection, union, or translation. It \ was subsequently extended to integer and real-valued signals and images and \ has evolved into a powerful image analysis tool.\ \>", "Text"], Cell["\<\ This paper presents in some detail two fundamental morphological \ operators: dilation and erosion. Most morphological algorithms are based on \ these two primitive operators, in particular all the additional operators \ discussed in this paper, including open, close, top-hat, and reconstruction \ by dilation. It is of considerable interest therefore to explore effective \ implementations of these operators.\ \>", "Text"], Cell[TextData[{ "The paper is organized as follows. First, the basic definitions of binary \ erosion and dilation are given. These are based on the concepts of sets and \ their translations. The ", StyleBox["Mathematica", FontSlant->"Italic"], " implementations follow immediately, but due to their computational \ inefficiency other algorithms will be developed and presented. The section \ concludes with a comparison of timing data for several different versions of \ the operators. In the next section, grayscale erosion and dilation operators \ are presented and several realizations are compared. Finally, in the last \ section a brief introduction to morphological reconstruction is given with \ details of reconstruction by geodesic dilation." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Binary erosion and dilation", "Section"], Cell[CellGroupData[{ Cell["Set-theoretic formulation", "Subsection"], Cell["\<\ The binary operators of erosion and dilation are typically defined \ using the concept of a set description of a binary image. A set description \ of a binary image is simply a list of integer pairs representing the \ positions of the 1-valued samples. Thus we have the following correspondence \ between a binary image (in matrix format) and a set where a coordinate system \ centered on the upper-left sample was used. \ \>", "Text"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", GridBox[{ {"0", "0", "0", "0"}, {"1", "1", "1", "0"}, {"0", "0", "1", "0"}, {"0", "0", "0", "0"} }], ")"}], " ", "\[Rule]", " ", StyleBox[\({{1, 0}, {1, 1}, {1, 2}, {2, 2}}\), "MR"], " "}], TraditionalForm]]]], "NumberedEquation"], Cell["\<\ When displayed graphically the origin shifts to the lower-left \ position as shown in Figure 1.\ \>", "Text"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .2735 %%ImageSize: 352 97 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0 0.002849 -0.002849 0.002849 [ [ 0 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Binary image and its set description.", \ "NumberedFigureCaption"], Cell[TextData[{ "The definitions of dilation and erosion are typically formulated using the \ concept of a set translation and a set reflection. The translation of a set \ \[Alpha] by a point (or vector) ", Cell[BoxData[ \(TraditionalForm\`x\)]], ", denoted ", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_x\)]], ", is defined by " }], "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_x\ = \ {a + \(x\ : \ a \[Element] \ \[Alpha]\)}\)]]], "NumberedEquation"], Cell[TextData[{ "The reflection of a set \[Alpha], denoted ", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\&_\)]], ", is defined" }], "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`\(\[Alpha]\&\[Vee]\)\ = \ {\(-\(a\ : \ a \[Element] \ \[Alpha]\)\)\ }\)]]], "NumberedEquation"], Cell[TextData[{ "Binary dilation of set \[Alpha] by set \[Beta], denoted here \[Alpha] \ \[CirclePlus] \[Beta], is the set union of all translations of set \[Alpha] \ by elements of set \[Beta] or equivalently the set of all positions of the \ reflected set ", Cell[BoxData[ \(TraditionalForm\`\(\[Beta]\&\[Vee]\)\)]], " for which it intersects with, or \"hits\", set \[Alpha]. The set \[Beta] \ is commonly called the structuring element and plays a similar role to a \ finite impulse response filter in linear signals and systems theory." }], "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`\[Alpha]\[CirclePlus]\[Beta]\ = \ \(\(\[Union]\+\(b\ \ \[Epsilon]\ \[Beta]\)\[Alpha]\_b\)\ = \ {x\ : \ \(\[Beta]\&\[Vee]\)\_x \ \[Intersection] \[Alpha]\ \[NotEqual] 0}\)\)]]], "NumberedEquation"], Cell["Here are two example sets.", "Text"], Cell[BoxData[ \(\((\[Alpha] = {{1, 0}, {1, 1}, {1, 2}, {2, 2}}; \[IndentingNewLine]\[Beta] = {{0, 0}, {0, 1}, {1, 1}};)\)\)], "Input"], Cell["\<\ This gives the dilation of set \[Alpha] by the structuring element \ \[Beta].\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Union @@ Table[\[Alpha][\([j]\)] + \[Beta][\([i]\)], {i, Length[\[Beta]]}, {j, Length[\[Alpha]]}]\)], "Input"], Cell[BoxData[ \({{1, 0}, {1, 1}, {1, 2}, {1, 3}, {2, 1}, {2, 2}, {2, 3}, {3, 3}}\)], "Output"] }, Open ]], Cell["\<\ The erosion of set \[Alpha] by set \[Beta], denoted here \[Alpha] \ \[CircleMinus] \[Beta], is the set intersection of all negative translations \ of set \[Alpha] by elements of set \[Beta] or equivalently the set of all \ positions for which set \[Beta] is a subset of \[Alpha] (i.e., it \"fits\" \ inside \[Alpha]).\ \>", "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`\[Alpha]\[CircleMinus]\[Beta]\ = \ \ \(\(\[Intersection]\+\(b\ \[Epsilon]\ \[Beta]\)\[Alpha]\_\(-b\)\)\ = \ {x\ \ : \ \[Beta]\_x \[Subset] \ \[Alpha]}\)\)]]], "NumberedEquation"], Cell[CellGroupData[{ Cell[BoxData[ \(Intersection @@ Table[\[Alpha][\([j]\)] - \[Beta][\([i]\)], {i, Length[\[Beta]]}, {j, Length[\[Alpha]]}]\)], "Input"], Cell[BoxData[ \({{1, 1}}\)], "Output"] }, Open ]], Cell[TextData[{ "The set descriptions of binary erosion and dilation have straightforward \ implementations in ", StyleBox["Mathematica", FontSlant->"Italic"], " but unfortunately they are terribly inefficient. " }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Other implementations", "Subsection"], Cell[TextData[{ "For purposes of signal and image processing it is preferable to consider \ formulations of erosion and dilation in terms of arrays instead of sets. \ Interestingly this leads to realizations of the operators that are strikingly \ similar to convolution and correlation. Recall that linear convolution of a \ one-dimensional (1D) signal ", Cell[BoxData[ \(TraditionalForm\`f\)]], " with filter ", Cell[BoxData[ \(TraditionalForm\`h\)]], " is defined." }], "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`\((f* h)\) \((x)\)\ = \ \(\[Sum]\+\(y \[Element] \ \[ScriptCapitalD]\_h\)\%\ \)\(f(x - y)\)\ \(h(y)\)\)]]], "NumberedEquation"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalD]\_h\)]], " denotes the region of support of filter ", Cell[BoxData[ \(TraditionalForm\`h\)]], ". The dilation of a signal ", Cell[BoxData[ \(TraditionalForm\`f\)]], " with structuring element ", Cell[BoxData[ \(TraditionalForm\`s\)]], " may likewise be written in a convolution-like form." }], "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`\((f\[CirclePlus]\ s)\) \((x)\)\ = \(\(\[Union]\+\(y \[Element] \ \[ScriptCapitalD]\_s\)f(x - y)\)\(\ \)\(\[Intersection]\)\(s( y)\)\(\ \)\)\)]]], "NumberedEquation"], Cell[TextData[{ "where symbols \[Union] and \[Intersection] are the logical OR (union) and \ logical AND (intersection) operators, respectively. This process of scanning \ a small kernel over a large data array and computing a sort of inner product \ of the corresponding samples suggests an implementation using ", StyleBox["PartitionMap", "MR"], ". Here matrices ", StyleBox["A", "MR"], " and ", StyleBox["B", "MR"], " represent the sets \[Alpha] and \[Beta], respectively and ", StyleBox["rB", "MR"], " is the reflection of ", StyleBox["B", "MR"], ". " }], "Text"], Cell[BoxData[{ \(\(A = {{0, 0, 0, 0}, {1, 1, 1, 0}, {0, 0, 1, 0}, {0, 0, 0, 0}};\)\), "\[IndentingNewLine]", \(\(B = {{0, 0, 0}, {0, 1, 1}, {0, 0, 1}};\)\), "\[IndentingNewLine]", \(\(rB = {{1, 0, 0}, {1, 1, 0}, {0, 0, 0}};\)\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(PartitionMap[BitOr @@ BitAnd[Flatten[#], Flatten[rB]] &, A, {3, 3}, {1, 1}, 2, 0] // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0", "0", "0"}, {"1", "1", "1", "1"}, {"0", "1", "1", "1"}, {"0", "0", "0", "1"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[TextData[{ "Alternatively, this convolution-like computation may be implemented with \ function ", StyleBox["ListConvolve", "MR"], " by replacing the default operators ", StyleBox["Times", "MR"], " and ", StyleBox["Plus", "MR"], " by ", StyleBox["BitAnd", "MR"], " and ", StyleBox["BitOr", "MR"], ". This ", "leads immediately to the following realization of binary dilation." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ListConvolve[B, A, 2, 0, BitAnd, BitOr]\ // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0", "0", "0"}, {"1", "1", "1", "1"}, {"0", "1", "1", "1"}, {"0", "0", "0", "1"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[TextData[{ "Unfortunately, neither one of these two implementations is sufficiently \ fast for typical applications. There are two more possible solutions. The \ first is based on the equivalence of logical ", "\[OpenCurlyDoubleQuote]", "and\[CloseCurlyDoubleQuote] (", StyleBox["BitAnd", "MR"], ") and binary multiplication (", StyleBox["Times", "MR"], "). It is easy to see that for binary-valued signal ", Cell[BoxData[ \(TraditionalForm\`f\)]], " and structuring element ", Cell[BoxData[ \(TraditionalForm\`s\)]], " the following relation between linear convolution, denoted by ", Cell[BoxData[ \(TraditionalForm\`*\)]], " and binary dilation holds. " }], "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`\((f\[CirclePlus]\ s)\) \((x)\)\ \ = \ T(\ \ \((f*s)\) \((x)\)\ )\)]], ", where ", Cell[BoxData[ FormBox[ RowBox[{\(T(x)\), " ", "=", " ", TagBox[ StyleBox[ RowBox[{"{", GridBox[{ {\(\(0\)\(,\)\), \(0 \[LessEqual] x < 1\)}, {\(\(1\)\(,\)\), "else"} }]}], ShowAutoStyles->False], (#&)]}], TraditionalForm]]] }], "NumberedEquation"], Cell["Thus we get", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Threshold[ListConvolve[B, A, 2, 0], 1] // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0", "0", "0"}, {"1", "1", "1", "1"}, {"0", "1", "1", "1"}, {"0", "0", "0", "1"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[TextData[{ "This indeed gives the best possible realization with ", StyleBox["Mathematica", FontSlant->"Italic"], "'s built-in functions. The last remaining alternative is to code the \ dilation and erosion algorithms in a compiled language such as Java or C++ \ and use ", StyleBox["Mathematica", FontSlant->"Italic"], "'s linking technologies to execute such code. This has been demonstrated \ to provide clear benefits under certain conditions [1]. 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oooo0?ooo`<000000`3oool2000000<0oooo0P0000030?ooo`8000000P3oool5000000<0oooo0`00 00020?ooo`030000003oool0oooo00H000000P3oool300000080oooo1P0000030?ooo`030000003o ool0oooo00L000000`3oool2000000030?ooo`3clo<0lo?c0180lo?c000B0??clbP000000`3oool7 000000@0oooo1@0000050?ooo`L000001`3oool2000000P0oooo1@0000080?ooo`@000002@3oool5 000000`0lo?cN000000<0??clbP000000`3oool7000000@0oooo1@0000050?ooo`L000001`3oool2 000000P0oooo1@0000080?ooo`@000002@3oool5000001@0lo?c000B0??clgL00000303clo=h0000 00`0lo?cM`00000D0??cl`004P3cloU0??cl`00o`3clo>U0??cl`00\ \>"], ImageRangeCache->{{{180, 599}, {243.375, 144.375}} -> {-1.55752, 1.11482, \ 0.00784987, 0.00784987}, {{198.375, 317.812}, {241, 146.688}} -> {-315.577, \ 225.878, 1.59081, 1.59081}, {{329.75, 449.188}, {241, 146.688}} -> {-524.569, \ 225.878, 1.59081, 1.59081}, {{461.125, 580.562}, {241, 146.688}} -> \ {-733.561, 225.878, 1.59081, 1.59081}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Timing", "Subsection"], Cell["\<\ In this section I measure and compare timing results for several \ realizations of binary dilation and three different image dimensions. All \ results can be compared to the speed of linear convolution which sets the \ processing standard for linear filtering of signals and images. Here I define \ all the realizations.\ \>", "Text"], Cell[BoxData[ \(\((method1 := PartitionMap[BitOr @@ BitAnd[Flatten[#], Flatten[B]] &, A, {3, 3}, {1, 1}, 2, 0]\ ; \n method2 := ListConvolve[B, A, 2, 0, BitAnd, BitOr]\ ; \n method3 := UnitStep[ListConvolve[B, A, 2, 0] - 1]\ ; \n method4 := obj@dilate[A, B]\ ; \[IndentingNewLine]convolve := ListConvolve[B, A, 2, 0]\ ;)\)\)], "Input"], Cell["\<\ This executes the code, collects the timing data and present the \ results in a table.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Block[{A, B}, \((B = Table[1, {3}, {3}]; \[IndentingNewLine]A = Table[Random[ Integer], {256}, {256}]; \[IndentingNewLine]t256 = {time[ method1], time[method2], time[method3], time[method4], time[convolve]}; \[IndentingNewLine]A = Table[Random[ Integer], {512}, {512}]; \[IndentingNewLine]t512 = {time[ method1], time[method2], time[method3], time[method4], time[convolve]}; \[IndentingNewLine]A = Table[Random[ Integer], {1024}, {1024}]; \[IndentingNewLine]t1K = {"\", time[method2], time[method3], time[method4], time[convolve]}; \[IndentingNewLine]TableForm[ Join[{{"\< \>", "\<256\[Times]256\>", "\<512\[Times]512\>", "\<1024\ \[Times]1024\>"}}, Transpose[{{"\", "\", "\", \ "\", "\"}, t256, t512, t1K}]]])\)]\)], "Input"], Cell[BoxData[ TagBox[GridBox[{ {"\<\" \"\>", "\<\"256\[Times]256\"\>", "\<\"512\[Times]512\"\>", "\ \<\"1024\[Times]1024\"\>"}, {"\<\"method 1\"\>", "1.015625`7.458278376154941", "4.078125`8.062005526850367", "\<\"n/a\"\>"}, {"\<\"method 2\"\>", "0.703125`7.599607528951411", "2.8125`7.900637524615393", "11.125`8.497845013148943"}, {"\<\"method 3\"\>", "0.01708984375`7.490463059526343", "0.0771484375`7.542992110802528", "0.3046875`7.537459622202565"}, {"\<\"method 4\"\>", "0.01904296875`7.537459622202565", "0.0888671875`7.604406411833179", "0.208984375`7.674748797197297"}, {"\<\"convolve\"\>", "0.016357421875`7.471439822212913", "0.0703125`7.502697515943353", "0.27734375`7.496623368231161"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[TextData[{ "Note that the timing results for method 4 include the time to transfer the \ data over the Java link, which at an average rate of approximately 10 MB/sec \ requires about 0.1 sec for a ", Cell[BoxData[ \(TraditionalForm\`1024\^2\)]], " image. These results imply that while method 3 is a suitable algorithm \ for evaluating binary erosion and dilation, a kernel version of method 4 (in \ C/C++ ) has the potential of increasing speed by roughly a factor of 2 by \ eliminating the data transfer time." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Grayscale erosion and dilation", "Section"], Cell["\<\ Grayscale morphology extends the morphological operators to the \ domain of integer or real-valued signals defined on a Cartesian grid. The \ definitions of binary morphology extend naturally to the domain of digital \ grayscale signals with translation, reflection, and inversion defined as in \ linear processing while intersection and union become point-wise minimum and \ maximum operators, respectively. Therefore we have the following definition \ of grayscale dilation.\ \>", "Text", TextJustification->1], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`f\[CirclePlus]s\ = \ \(\( \[Or] \+\(x\ \[Epsilon]\ \ \[ScriptCapitalD]\_s\)\)\(f\_x\)\(\ \)\)\)]]], "NumberedEquation"], Cell["\<\ where the symbol \[Or] denotes a point-wise maximum. Thus the \ dilation of a grayscale signal or image is a point-wise maximum of a series \ of translations defined by the shape of the structuring element. This \ defintion implies a flat structuring element. In the more general case of a \ nonflat structuring element and again, for sake of simplicity assuming 1D \ signals, we have\ \>", "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`\((f\[CirclePlus]s)\) \((x)\)\ = \ \(\( \[Or] \+\(y\ \ \[Epsilon]\ \[ScriptCapitalD]\_s\)\)\((f(x - y)\ + \ s(y)\ )\)\(\ \)\)\)]]], "NumberedEquation"], Cell[TextData[{ "This yields the following ", StyleBox["Mathematica", FontSlant->"Italic"], " realizations." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ListConvolve[B, A, 2, Min[A], Plus, Max]\ // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "1", "1", "1"}, {"2", "2", "2", "2"}, {"1", "2", "2", "2"}, {"1", "1", "1", "2"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[TextData[{ "Here is the equivalent ", StyleBox["PartitionMap", "MR"], " implemenentation. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(PartitionMap[Max[rB + #] &, A, {3, 3}, {1, 1}, 2, 0] // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "1", "1", "1"}, {"2", "2", "2", "2"}, {"1", "2", "2", "2"}, {"1", "1", "1", "2"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["\<\ For a flat structuring element the dilation operator may be \ simplified as follows. This scans the image with a rectangular window and \ returns the maximum of the samples within the window. Note the difference in \ the result due to a change to the structuring element.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(PartitionMap[Max, A, {3, 3}, {1, 1}, 2, 0] // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "1", "1", "1"}, {"1", "1", "1", "1"}, {"1", "1", "1", "1"}, {"0", "1", "1", "1"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["\<\ Erosion and dilation operators are seldom used by themselves. Two \ well-known combinations of the operators result in the so-called open (\ \[EmptySmallCircle]) and close (\[FilledSmallCircle]) operators.\ \>", "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`f\ \[EmptySmallCircle]\ s\ = \ \((\((f\ \ \[CircleMinus]\ s)\)\ \[CirclePlus]\ s)\)\)]]], "NumberedEquation"], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`f\ \[FilledSmallCircle]\ s\ = \ \((\((f\ \ \[CirclePlus]\ s)\)\ \[CircleMinus]\ s\ )\)\)]]], "NumberedEquation"], Cell["\<\ Openings and closings are typically used to suppress structures \ that cannot contain the structuring element, peaks in the case of openings \ and valleys in the case of closings. Here is an example of opening the books \ image with flat structuring elements of increasing dimensions. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(DisplayTogetherArray[\[IndentingNewLine]\(Graphics[ GrayscaleOpen[books, #], PlotRange \[Rule] {0, 255}] &\) /@ {Table[1, {5}, {5}], Table[1, {10}, {10}], Table[1, {25}, {25}]}, ImageSize \[Rule] {420, 100}];\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .24671 %%ImageSize: 420 100 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.31746 0.00587406 0.31746 [ [ 0 0 0 0 ] [ 1 .24671 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 .24671 L 0 .24671 L closepath clip newpath % Start of sub-graphic p 0.0238095 0.00587406 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The top-hat by opening operation, defined ", Cell[BoxData[ \(TraditionalForm\`f\ - \ f\ \[EmptySmallCircle]\ g\)]], ", is a very useful background normalization operator. Here is an example, \ which shows the original image, the result of opening with a large \ structuring element, and the tophat transformation." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(se = Table[0, {25}, {25}];\)\), "\[IndentingNewLine]", \(\(DisplayTogetherArray[{Graphics[books], Graphics[GrayscaleOpen[books, se], PlotRange \[Rule] {0, 255}], Graphics[books - GrayscaleOpen[books, se]]}, ImageSize \[Rule] {420, 100}];\)\)}], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .24671 %%ImageSize: 420 100 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.31746 0.00587406 0.31746 [ [ 0 0 0 0 ] [ 1 .24671 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 .24671 L 0 .24671 L 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1.59081, 1.59081}, {{329.75, 449.188}, {459.125, \ 364.812}} -> {-524.569, 572.872, 1.59081, 1.59081}, {{461.125, 580.562}, \ {459.125, 364.812}} -> {-733.561, 572.872, 1.59081, 1.59081}}] }, Open ]], Cell[CellGroupData[{ Cell["Timing", "Subsection"], Cell["\<\ In this section I measure the timing data for four different \ evaluations of grayscale dilation and several different image \ dimensions.\ \>", "Text"], Cell[BoxData[ \(\((Clear[method1, method2, method3, method4]; \n method1 := ListConvolve[B, A, 2, Min[A], Plus, Max]\ \ ; \n method2 := PartitionMap[Max[rB + #] &, A, {3, 3}, {1, 1}, 2, 0]\ ; \n method3 := PartitionMap[Max, A, {3, 3}, {1, 1}, 2, 0]\ ; \n method4 := GrayscaleDilate[A, B]\ ;)\)\)], "Input"], Cell["\<\ This evaluates the code and presents the results in a table. For \ comparison, the timing of linear convolution is also shown.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Block[{A, B}, \((B = Table[1, {3}, {3}]; \[IndentingNewLine]A = Table[Random[ Integer], {256}, {256}]; \[IndentingNewLine]t256 = {time[ method1], time[method2], time[method3], time[method4], time[convolve]}; \[IndentingNewLine]A = Table[Random[ Integer], {512}, {512}]; \[IndentingNewLine]t512 = {time[ method1], time[method2], time[method3], time[method4], time[convolve]}; \[IndentingNewLine]A = Table[Random[ Integer], {1024}, {1024}]; \[IndentingNewLine]t1K = {"\", \ "\", time[method3], time[method4], time[convolve]}; \[IndentingNewLine]TableForm[ Join[{{"\< \>", "\<256\[Times]256\>", "\<512\[Times]512\>", "\<1024\ \[Times]1024\>"}}, Transpose[{{"\", "\", "\", \ "\", "\"}, t256, t512, t1K}]]])\)]\)], "Input"], Cell[BoxData[ TagBox[GridBox[{ {"\<\" \"\>", "\<\"256\[Times]256\"\>", "\<\"512\[Times]512\"\>", "\ \<\"1024\[Times]1024\"\>"}, {"\<\"method 1\"\>", "0.6328125`7.553850038390737", "2.5`7.8494850021680085", "\<\"n/a\"\>"}, {"\<\"method 2\"\>", "0.5`7.451544993495973", "2.046875`7.762636315167852", "\<\"n/a\"\>"}, {"\<\"method 3\"\>", "0.06005859375`7.735270130951486", "0.2421875`7.738786704674322", "1.015625`7.458278376154941"}, {"\<\"method 4\"\>", "0.091796875`7.6184928731117845", "0.359375`7.609152846857642", "0.8828125`7.698443462995504"}, {"\<\"convolution\"\>", "0.0166015625`7.477873932218323", "0.0703125`7.502697515943353", "0.28515625`7.508687879632542"} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[TextData[{ "It is clear from the timing data presented here that grayscale \ morphological operators present processing challenges. Fortunately, this \ issue has been addressed in a number of articles in the image processing and \ mathematical computing literature. In two important articles van Herk [2] and \ independently Gil and Werman [3] proposed a one-dimensional algorithm with \ computational complexity which is largely independent of the length of the \ structuring element. The algorithm breaks the signal into segments whose \ length matches the size of the structuring element, and computes minima \ forwards and backwards. A second step combines the results to produce the \ final result with a fixed cost of three comparisons per sample. Other authors \ [4, 5] have proposed additional improvements that hold the potential of an \ additional speedup by a factor of 2. It is clearly of interest to investigate \ the ", StyleBox["Mathematica", FontSlant->"Italic"], " and Java/C++ realizations of these algorithms." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Morphological reconstruction", "Section"], Cell["\<\ Reconstruction from markers is a very important category of \ morphological operators as many image processing tasks have a natural \ formulation in terms of these operators. Unlike the fundamental operators \ presented thus far which take as inputs an image and a structuring element, \ reconstruction requires two input images. During reconstruction some \ fundamental operator (for example, dilation or erosion) is applied repeatedly \ to a so-called marker image, while the result of each operation is \ constrained by the other of the two images called the mask.\ \>", "Text"], Cell[TextData[{ "Here I present an example of reconstruction by geodesic dilation for \ binary images. A sequence of ", Cell[BoxData[ \(TraditionalForm\`n\)]], " conditional dilations is known as a size-", StyleBox["n", FontSlant->"Italic"], " geodesic dilation where a conditional dilation is defined as the \ intersection of the mask image ", Cell[BoxData[ \(TraditionalForm\`g\)]], " with a dilation of the marker image ", Cell[BoxData[ \(TraditionalForm\`f\)]], "[6]." }], "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`\((f\ \(\[CirclePlus]\_g\)\ s)\)\ = \ \((f\ \[CirclePlus]\ s)\)\ \[Intersection] \ g\)]]], "NumberedEquation"], Cell[TextData[{ "Reconstruction of image ", Cell[BoxData[ \(TraditionalForm\`g\)]], " from the marker image ", Cell[BoxData[ \(TraditionalForm\`f\)]], " is accomplished by repeating the conditional dilations to stability, \ until the result no longer changes. " }], "Text"], Cell[TextData[Cell[BoxData[ \(TraditionalForm\`\(R\_g\^\[CirclePlus]\)( f)\ = \ \(\((f\ \(\[CirclePlus]\_g\)\ s)\)\^n\ = \ \((\(\((\((f\ \(\ \[CirclePlus]\_g\)\ s)\)\ \(\[CirclePlus]\_g\)\ s)\)\ ... \)\ \(\[CirclePlus]\_g\)\ s)\)\)\)]]], "NumberedEquation"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`n\)]], " is such that ", Cell[BoxData[ \(TraditionalForm\`\((f\ \(\[CirclePlus]\_g\)\ s)\)\^n\ = \ \((f\ \(\ \[CirclePlus]\_g\)\ s)\)\^\(n + 1\)\)]], ". " }], "Text"], Cell[TextData[{ "Here are a marker image, ", Cell[BoxData[ FormBox[ RowBox[{"f", " ", "=", RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0", "1", "0", "0"}, {"0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}]}], TraditionalForm]]], " and a mask image, ", Cell[BoxData[ FormBox[ RowBox[{"g", "=", RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "1", "1", "0", "0"}, {"0", "1", "1", "0", "0"}, {"1", "1", "1", "0", "1"}, {"0", "1", "0", "0", "1"}, {"0", "0", "0", "1", "1"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}]}], TraditionalForm]]], "." }], "Text"], Cell[TextData[{ "This shows the results of three consecutive conditional dilations of the \ marker image with a flat structuring element of dimensions ", Cell[BoxData[ \(TraditionalForm\`3\[Times]3\)]], " at which point further conditional dilations do not change the result." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ RowBox[{\(\((f\ \(\[CirclePlus]\_g\)\ s)\)\^1\), " ", "=", " ", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "1", "1", "0", "0"}, {"0", "1", "1", "0", "0"}, {"0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{\(\((f\ \(\[CirclePlus]\_g\)\ s)\)\^2\), " ", "=", " ", RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "1", "1", "0", "0"}, {"0", "1", "1", "0", "0"}, {"1", "1", "1", "0", "0"}, {"0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}]}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{\(\((f\ \(\[CirclePlus]\_g\)\ s)\)\^3\), " ", "=", " ", TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "1", "1", "0", "0"}, {"0", "1", "1", "0", "0"}, {"1", "1", "1", "0", "0"}, {"0", "1", "0", "0", "0"}, {"0", "0", "0", "0", "0"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]}], TraditionalForm]]], "." }], "Text"], Cell[TextData[{ "The ", StyleBox["Mathematica", FontSlant->"Italic"], " implementation of reconstruction by geodesic dilation is straightforward. \ Here are two image arrays." }], "Text"], Cell[BoxData[{ \(\(A = {{0, 0, 1, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}};\)\), "\[IndentingNewLine]", \(\(B = {{0, 1, 1, 0, 0}, {0, 1, 1, 0, 0}, {1, 1, 1, 0, 1}, {0, 1, 0, 0, 1}, {0, 0, 0, 1, 1}};\)\)}], "Input"], Cell[TextData[{ "The following one-liner computes the reconstruction of ", StyleBox["B", "MR"], " from marker image ", StyleBox["A", "MR"], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(FixedPoint[B\ BinaryDilate[#, Table[1, {3}, {3}]] &, A] // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "1", "1", "0", "0"}, {"0", "1", "1", "0", "0"}, {"1", "1", "1", "0", "0"}, {"0", "1", "0", "0", "0"}, {"0", "0", "0", "0", "0"} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell[TextData[{ "Clearly, evaluating the reconstruction could be time-consuming as the \ number of iterations is image dependent. Fortunately, efficient algorithms \ requiring a limited number of scans of the marker image are known [7]. ", StyleBox["Mathematica", FontSlant->"Italic"], " implementations of these are currently under development. Here is an \ example of region filling using reconstruction by geodesic dilation. " }], "Text"], Cell[BoxData[{ \(\(mask = Threshold[Round[ToGrayLevel[cells]], 173];\)\), "\[IndentingNewLine]", \(\(se = Table[1, {3}, {3}];\)\)}], "Input"], Cell["\<\ This defines a marker image, an all-zero image with a 1-valued seed \ placed anywhere within the background region of the original image. This seed \ defines the starting location of a flooding process that fills the background \ region.\ \>", "Text"], Cell[BoxData[{ \(\(mark = \((0\ mask)\)\[LeftDoubleBracket]1\[RightDoubleBracket];\)\), \ "\n", \(\(mark\[LeftDoubleBracket]230, 194\[RightDoubleBracket] = 1;\)\), "\n", \(\(tmp = ToGrayLevel[ FixedPoint[ mask\[LeftDoubleBracket]1\[RightDoubleBracket]\ BinaryDilate[#, se] &, mark]];\)\)}], "Input"], Cell["This shows the result.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(DisplayTogetherArray[{Graphics[cells], Graphics[mask], Graphics[tmp]}, ImageSize \[Rule] {420, 100}];\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .22911 %%ImageSize: 420 100 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 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