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+} if
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+%%EndResource
+%%BeginResource: procset (Apache XML Graphics EPS ProcSet) 1.0 0
+%%Version: 1.0 0
+%%Copyright: (Copyright 2002-2003 The Apache Software Foundation. License terms: http://www.apache.org/licenses/LICENSE-2.0)
+/BeginEPSF { %def
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+} if
+} if
+} bd
+/EndEPSF { %def
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+} bd
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+%%IncludeResource: font Helvetica
+%%IncludeResource: font Courier-BoldOblique
+%%IncludeResource: font Courier-Oblique
+%%IncludeResource: font Times-Roman
+%%IncludeResource: font Helvetica-BoldOblique
+%%IncludeResource: font Helvetica-Bold
+%%IncludeResource: font Helvetica-Oblique
+%%IncludeResource: font Times-BoldItalic
+%%IncludeResource: font Courier
+%%IncludeResource: font Times-Italic
+%%IncludeResource: font Times-Bold
+%%BeginResource: font CMSY10
+%!PS-AdobeFont-1.0: mwb_cmsy10 1.0
+%%Title: mwb_cmsy10
+%Version: 1.0
+%%CreationDate: Thu Jan 27 10:30:23 2011
+%%Creator: Ben Hinkle
+%Copyright: Copyright (C) 2011 MathWorks, Inc
+% Generated by FontForge 20100429 (http://fontforge.sf.net/)
+%%EndComments
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+ /version (1.0) readonly def
+ /Notice (Copyright \050C\051 2011 MathWorks, Inc) readonly def
+ /FullName (mwb_cmsy10) readonly def
+ /FamilyName (mwb_cmsy10) readonly def
+ /Weight (Medium) readonly def
+ /ItalicAngle -14.035 def
+ /isFixedPitch false def
+end readonly def
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+dup 22/precedesequal put
+dup 23/followsequal put
+dup 24/similar put
+dup 25/approxequal put
+dup 26/propersubset put
+dup 27/propersuperset put
+dup 28/lessmuch put
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+dup 30/precedes put
+dup 31/follows put
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+dup 52/reflexsuperset put
+dup 53/lessequal put
+dup 54/greaterequal put
+dup 55/precedesequal put
+dup 56/followsequal put
+dup 57/similar put
+dup 58/approxequal put
+dup 59/propersubset put
+dup 60/propersuperset put
+dup 61/lessmuch put
+dup 62/greatermuch put
+dup 63/precedes put
+dup 64/follows put
+dup 65/arrowleft put
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+%!PS-AdobeFont-1.0: mwa_cmsy10 1.0
+%%Title: mwa_cmsy10
+%Version: 1.0
+%%CreationDate: Thu Jan 27 10:30:22 2011
+%%Creator: Ben Hinkle
+%Copyright: Copyright (C) 2011 MathWorks, Inc
+% Generated by FontForge 20100429 (http://fontforge.sf.net/)
+%%EndComments
+
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+/PaintType 0 def
+/FontInfo 7 dict dup begin
+ /version (1.0) readonly def
+ /Notice (Copyright \050C\051 2011 MathWorks, Inc) readonly def
+ /FullName (mwa_cmsy10) readonly def
+ /FamilyName (mwa_cmsy10) readonly def
+ /Weight (Medium) readonly def
+ /ItalicAngle -14.035 def
+ /isFixedPitch false def
+end readonly def
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+224.674 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+224.674 374 M
+224.674 313.328 L
+227.16 313.328 L
+227.16 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+227.16 374 M
+227.16 319.115 L
+229.647 319.115 L
+229.647 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+229.647 374 M
+229.647 374 L
+232.133 374 L
+232.133 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+232.133 374 M
+232.133 329.579 L
+234.62 329.579 L
+234.62 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+234.62 374 M
+234.62 334.998 L
+237.106 334.998 L
+237.106 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+237.106 374 M
+237.106 340.033 L
+239.593 340.033 L
+239.593 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+239.593 374 M
+239.593 374 L
+242.079 374 L
+242.079 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+242.079 374 M
+242.079 340.969 L
+244.566 340.969 L
+244.566 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+244.566 374 M
+244.566 344.529 L
+247.052 344.529 L
+247.052 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+247.052 374 M
+247.052 374 L
+249.539 374 L
+249.539 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+249.539 374 M
+249.539 349.593 L
+252.025 349.593 L
+252.025 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+252.025 374 M
+252.025 351.801 L
+254.511 351.801 L
+254.511 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+254.511 374 M
+254.511 374 L
+256.998 374 L
+256.998 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+256.998 374 M
+256.998 354.028 L
+259.484 354.028 L
+259.484 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+259.484 374 M
+259.484 355.363 L
+261.971 355.363 L
+261.971 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+261.971 374 M
+261.971 374 L
+264.457 374 L
+264.457 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+264.457 374 M
+264.457 356.97 L
+266.944 356.97 L
+266.944 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+266.944 374 M
+266.944 356.015 L
+269.43 356.015 L
+269.43 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+269.43 374 M
+269.43 374 L
+271.917 374 L
+271.917 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+271.917 374 M
+271.917 359.097 L
+274.403 359.097 L
+274.403 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+274.403 374 M
+274.403 359.797 L
+276.89 359.797 L
+276.89 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+276.89 374 M
+276.89 360.412 L
+279.376 360.412 L
+279.376 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+279.376 374 M
+279.376 374 L
+281.862 374 L
+281.862 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+281.862 374 M
+281.862 360.863 L
+284.349 360.863 L
+284.349 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+284.349 374 M
+284.349 361.395 L
+286.835 361.395 L
+286.835 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+286.835 374 M
+286.835 374 L
+289.322 374 L
+289.322 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+289.322 374 M
+289.322 361.752 L
+291.808 361.752 L
+291.808 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+291.808 374 M
+291.808 362.056 L
+294.295 362.056 L
+294.295 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+294.295 374 M
+294.295 374 L
+296.781 374 L
+296.781 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+296.781 374 M
+296.781 362.25 L
+299.268 362.25 L
+299.268 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+299.268 374 M
+299.268 362.186 L
+301.754 362.186 L
+301.754 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+301.754 374 M
+301.754 374 L
+304.241 374 L
+304.241 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+304.241 374 M
+304.241 362.027 L
+306.727 362.027 L
+306.727 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+306.727 374 M
+306.727 362.162 L
+309.214 362.162 L
+309.214 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+309.214 374 M
+309.214 374 L
+311.7 374 L
+311.7 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+311.7 374 M
+311.7 362.081 L
+314.186 362.081 L
+314.186 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+314.186 374 M
+314.186 362.201 L
+316.673 362.201 L
+316.673 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+316.673 374 M
+316.673 361.752 L
+319.159 361.752 L
+319.159 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+319.159 374 M
+319.159 374 L
+321.646 374 L
+321.646 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+321.646 374 M
+321.646 362.056 L
+324.132 362.056 L
+324.132 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+324.132 374 M
+324.132 360.949 L
+326.619 360.949 L
+326.619 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+326.619 374 M
+326.619 374 L
+329.105 374 L
+329.105 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+329.105 374 M
+329.105 361.005 L
+331.592 361.005 L
+331.592 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+331.592 374 M
+331.592 360.77 L
+334.078 360.77 L
+334.078 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+334.078 374 M
+334.078 374 L
+336.565 374 L
+336.565 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+336.565 374 M
+336.565 360.108 L
+339.051 360.108 L
+339.051 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+339.051 374 M
+339.051 358.249 L
+341.538 358.249 L
+341.538 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+341.538 374 M
+341.538 374 L
+344.024 374 L
+344.024 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+344.024 374 M
+344.024 357.387 L
+346.51 357.387 L
+346.51 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+346.51 374 M
+346.51 354.944 L
+348.997 354.944 L
+348.997 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+348.997 374 M
+348.997 374 L
+351.483 374 L
+351.483 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+351.483 374 M
+351.483 350.01 L
+353.97 350.01 L
+353.97 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+353.97 374 M
+353.97 340.344 L
+356.456 340.344 L
+356.456 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+356.456 374 M
+356.456 323.861 L
+358.943 323.861 L
+358.943 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+358.943 374 M
+358.943 374 L
+361.429 374 L
+361.429 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+361.429 374 M
+361.429 298.699 L
+363.916 298.699 L
+363.916 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+363.916 374 M
+363.916 262.439 L
+366.402 262.439 L
+366.402 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+366.402 374 M
+366.402 374 L
+368.889 374 L
+368.889 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+368.889 374 M
+368.889 189.718 L
+371.375 189.718 L
+371.375 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+371.375 374 M
+371.375 148.852 L
+373.861 148.852 L
+373.861 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+373.861 374 M
+373.861 374 L
+376.348 374 L
+376.348 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+376.348 374 M
+376.348 59.844 L
+378.834 59.844 L
+378.834 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+378.834 374 M
+378.834 103.716 L
+381.321 103.716 L
+381.321 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+381.321 374 M
+381.321 374 L
+383.807 374 L
+383.807 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+383.807 374 M
+383.807 156.491 L
+386.294 156.491 L
+386.294 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+386.294 374 M
+386.294 160.26 L
+388.78 160.26 L
+388.78 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+388.78 374 M
+388.78 374 L
+391.267 374 L
+391.267 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+391.267 374 M
+391.267 270.475 L
+393.753 270.475 L
+393.753 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+393.753 374 M
+393.753 251.039 L
+396.24 251.039 L
+396.24 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+396.24 374 M
+396.24 289.382 L
+398.726 289.382 L
+398.726 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+398.726 374 M
+398.726 374 L
+401.212 374 L
+401.212 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+401.212 374 M
+401.212 281.713 L
+403.699 281.713 L
+403.699 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+403.699 374 M
+403.699 344.122 L
+406.185 344.122 L
+406.185 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+406.185 374 M
+406.185 374 L
+408.672 374 L
+408.672 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+408.672 374 M
+408.672 341.917 L
+411.158 341.917 L
+411.158 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+411.158 374 M
+411.158 360.106 L
+413.645 360.106 L
+413.645 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+413.645 374 M
+413.645 374 L
+416.131 374 L
+416.131 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+416.131 374 M
+416.131 355.858 L
+418.618 355.858 L
+418.618 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+418.618 374 M
+418.618 374 L
+421.104 374 L
+421.104 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+421.104 374 M
+421.104 374 L
+423.591 374 L
+423.591 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+423.591 374 M
+423.591 374 L
+426.077 374 L
+426.077 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+426.077 374 M
+426.077 374 L
+428.564 374 L
+428.564 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+428.564 374 M
+428.564 374 L
+431.05 374 L
+431.05 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+431.05 374 M
+431.05 374 L
+433.536 374 L
+433.536 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+433.536 374 M
+433.536 374 L
+436.023 374 L
+436.023 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+436.023 374 M
+436.023 374 L
+438.509 374 L
+438.509 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+438.509 374 M
+438.509 374 L
+440.996 374 L
+440.996 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+440.996 374 M
+440.996 374 L
+443.482 374 L
+443.482 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+443.482 374 M
+443.482 374 L
+445.969 374 L
+445.969 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+445.969 374 M
+445.969 374 L
+448.455 374 L
+448.455 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+448.455 374 M
+448.455 374 L
+450.942 374 L
+450.942 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+450.942 374 M
+450.942 374 L
+453.428 374 L
+453.428 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+453.428 374 M
+453.428 374 L
+455.915 374 L
+455.915 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+455.915 374 M
+455.915 374 L
+458.401 374 L
+458.401 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+458.401 374 M
+458.401 374 L
+460.888 374 L
+460.888 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+460.888 374 M
+460.888 374 L
+463.374 374 L
+463.374 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+463.374 374 M
+463.374 374 L
+465.86 374 L
+465.86 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+465.86 374 M
+465.86 374 L
+468.347 374 L
+468.347 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+468.347 374 M
+468.347 374 L
+470.833 374 L
+470.833 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
+10.0 ML
+N
+73 374 M
+73 373.794 L
+75.486 373.794 L
+75.486 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
+10.0 ML
+N
+75.486 374 M
+75.486 373.784 L
+77.973 373.784 L
+77.973 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
+10.0 ML
+N
+77.973 374 M
+77.973 373.765 L
+80.459 373.765 L
+80.459 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
+10.0 ML
+N
+80.459 374 M
+80.459 374 L
+82.946 374 L
+82.946 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
+10.0 ML
+N
+82.946 374 M
+82.946 373.762 L
+85.432 373.762 L
+85.432 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
+10.0 ML
+N
+85.432 374 M
+85.432 373.787 L
+87.919 373.787 L
+87.919 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
+10.0 ML
+N
+87.919 374 M
+87.919 374 L
+90.405 374 L
+90.405 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
+10.0 ML
+N
+90.405 374 M
+90.405 373.716 L
+92.892 373.716 L
+92.892 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
+10.0 ML
+N
+92.892 374 M
+92.892 373.674 L
+95.378 373.674 L
+95.378 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
+10.0 ML
+N
+95.378 374 M
+95.378 374 L
+97.865 374 L
+97.865 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
+10.0 ML
+N
+97.865 374 M
+97.865 373.593 L
+100.351 373.593 L
+100.351 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
+10.0 ML
+N
+100.351 374 M
+100.351 373.606 L
+102.838 373.606 L
+102.838 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
+10.0 ML
+N
+102.838 374 M
+102.838 374 L
+105.324 374 L
+105.324 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
+10.0 ML
+N
+105.324 374 M
+105.324 373.566 L
+107.81 373.566 L
+107.81 374 L
+S
+GR
+GS
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+%%Pages: 1
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%emph=[2]{word1,word2}, emphstyle=[2]{style},
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+\newcommand{\mli}[1]{\mathit{#1}} % Multi-letter identifier.
+
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\pagestyle{fancy}
\end{center}
\end{titlepage}
+%%%%%%%%%%%%%%%%%%
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-
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+%%%%%%%%%%%%%%%%%%
% Page contenant le résumé et les mots-clefs.
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\begin{center}
\textbf{Résumé}
\end{center}
-Cet article a pour but la description et la réalisation d'un méthode automatique permettant le calcul de la parasitémie de sang infecté par la malaria à partir d'images agrandies cent fois.
+Cet article a pour but la description et la réalisation d'une méthode automatique permettant le calcul de la parasitémie de sang infecté par la malaria à partir d'images microscopiques.
\vspace{5cm}
\newpage
-
-
-
-
\tableofcontents
\newpage
-
+%%%%%%%%%%%%%%%%%%
\section{Introduction}
+% TODO : utiliser \mli{} pour tous les variables à plus d'un caractère.
+
Le but de ce projet est de dénombrer de manière automatisée les globules rouges infectés par la malaria. Le nombre total de globules rouges est également recensé afin de pouvoir établir un taux d'infection. Cela est réalisé à partir d'images microscopiques fournies par Dr. Guy Prod'hom du laboratoire de parasitologie du \emph{CHUV} à Lausanne.
Pour ce faire, nous nous basons sur l'article \emph{Analysis of infected blood cell images using morphological operators}\cite{DiRuberto-Analysis-infected-blood-morphological} qui décrit une approche principalement basée sur des opérations morphologiques afin d'établir la parasitémie d'images d'échantillons de sang infecté par la malaria.
-L'ensemble du processus décrit ci après est implémenté sous \emph{MATLAB}, le code est fournit en annexe. Des résultats de sa mise en oeuvre sur plusieurs images de sang infecté sont présentés en fin d'article.
+L'ensemble du processus décrit ci-après est implémenté sous \emph{MATLAB} : le code est fourni en annexe. Des résultats de sa mise en œuvre sur plusieurs images de sang infecté sont présentés en fin d'article.
-<TODO> : expliquer est montrer à quoi ressemble un parasite
+\subsection{Images}
-\section{Approche générale}
+Les images initiales correspondent à des photographies de sang infecté agrandies 100 fois. Un colorant est utilisé afin de faire ressortir les parasites avec une teinte particulière. Les éléments principaux composant les images sont : les globules rouges (figure~\ref{fig:medical-globules-rouges}), les globules blancs (figure~\ref{fig:medical-globules-blancs}), les plaquettes (figure~\ref{fig:medical-plaquettes}) et les différentes formes du parasite décrit ci-après et montré par la figure~\ref{fig:medical-plasmodium}.
+
+\begin{figure}[htbp]
+ \centering
+ \begin{subfigure}[t]{0.3\textwidth}
+ \includegraphics[width=\linewidth]{img/medical_globules_rouges.jpg}
+ \caption{Globules rouges (\emph{érythrocyte})}
+ \label{fig:medical-globules-rouges}
+ \end{subfigure}
+ ~
+ \begin{subfigure}[t]{0.3\textwidth}
+ \includegraphics[width=\linewidth]{img/medical_globules_blancs.jpg}
+ \caption{Globules blancs (\emph{leucocyte})}
+ \label{fig:medical-globules-blancs}
+ \end{subfigure}
+ ~
+ \begin{subfigure}[t]{0.3\textwidth}
+ \includegraphics[width=\linewidth]{img/medical_plaquettes.jpg}
+ \caption{Plaquette (\emph{thrombocyte})}
+ \label{fig:medical-plaquettes}
+ \end{subfigure}
+ \caption{Les éléments principaux composant le sang}
+ \label{fig:medical-elements-sang}
+\end{figure}
+
+Dans le cycle de vie du parasite \emph{Plasmodium}, responsable de la malaria, celui-ci passe par une étape d'alimentation active (trophozoïte) où il va se loger à l'intérieur des globules rouges. Puis il va subir la schizogonie (reproduction asexuée) et va se développer en schizonte. Ces étapes sont montrées par la figure~\ref{fig:medical-plasmodium}.
+
+Notre but est de dénombrer les globules rouges sains ainsi que ceux infectés par des trophozoïtes immatures en début de croissance, comme montré par la figure~\ref{fig:medical-plasmodium_1}.
+
+\begin{figure}[htbp]
+ \centering
+ \begin{subfigure}[t]{0.2\textwidth}
+ \includegraphics[width=\linewidth]{img/medical_plasmodium_1.jpg}
+ \caption{Trophozoïte immature (anneau)}
+ \label{fig:medical-plasmodium_1}
+ \end{subfigure}
+ ~
+ \begin{subfigure}[t]{0.2\textwidth}
+ \includegraphics[width=\linewidth]{img/medical_plasmodium_2.jpg}
+ \caption{Maturation du trophozoïte}
+ \label{fig:medical-plasmodium_2}
+ \end{subfigure}
+ ~
+ \begin{subfigure}[t]{0.2\textwidth}
+ \includegraphics[width=\linewidth]{img/medical_plasmodium_3.jpg}
+ \caption{Trophozoïte mature}
+ \label{fig:medical-plasmodium_3}
+ \end{subfigure}
+ ~
+ \begin{subfigure}[t]{0.2\textwidth}
+ \includegraphics[width=\linewidth]{img/medical_plasmodium_4.jpg}
+ \caption{Schizonte}
+ \label{fig:medical-plasmodium_4}
+ \end{subfigure}
+ \caption{Les différents stades de l'infection d'un globule rouge}
+ \label{fig:medical-plasmodium}
+\end{figure}
+
+
+\section{Outils utilisés}
-Le processus complet est montré par la figure \ref{fig:processusComplet}. Les différents chiffres correspondent aux étapes décrites ci après.
+Cette section aborde la description des différents outils morphologiques ainsi que les logiciels qui ont permis de mettre en pratique le processus.
-L'image de départ correspond à une photographie de sang infecté agrandie 100 fois. Un colorant est utilisé afin de faire ressortir les parasites avec une teinte particulière.
+\subsection{Outils morphologique}
-La première étape (\ding{192}) consiste à détecter les éléments colorés comprenant les parasites, les globules blanc et les plaquettes. Les parasites peuvent se trouver à l'extérieure de cellules, par exemple groupés sous la forme de schizontes.
+* mmdil
+* mmero
+* mmopen
+* mmclose
+* mmareaopen (areaOpen)
+* mmareaclose
+* mminfrec reconstruction par dilatation
+
+* gamma\_lambda à la place de areaOpen
+
+
+\subsection{Outils logiciels}
+
+<TODO> Description de la structuration du projet MATLAB, du toolbox morphologique, etc...
+
+
+\section{Approche générale}
+
+Le processus complet est illustré par la figure \ref{fig:processusComplet}. Les différents chiffres correspondent aux étapes décrites ci après.
+
+La première étape (\ding{192}) consiste à détecter les éléments qui ont été spécifiquement colorés comprenant les parasites, les globules blancs et les plaquettes. Les parasites peuvent se trouver à l'extérieur de cellules, par exemple groupés sous la forme de schizontes.
Les deuxième et troisième étapes (\ding{193} et \ding{194}) consistent à chercher et à extraire les globules blancs, respectivement les schizontes.
La quatrième étape (\ding{195}) va segmenter les globules rouges.
-La cinquième étape (\ding{196}) consiste à reconstruire les globules rouges parasités à partir des parasites marqués de la première étape (\ding{192}) et de la ségmentation de la quatrième étape (\ding{195}) .
+La cinquième étape (\ding{196}) consiste à reconstruire les globules rouges parasités à partir des parasites marqués de la première étape (\ding{192}) et de la segmentation de la quatrième étape (\ding{195}) .
La sixième et dernière étape va compter le nombre de globules rouges total et de globules rouges infectés afin de calculer le rapport entre ces deux valeurs.
\end{figure}
-\section{Outils utilisés}
-
-<TODO> Description de la structuration du projet MATLAB, du toolbox morphologique, etc...
-
\section{Détail du processus}
-Le processus est actuellement réalisé sur des images réduites de 40 \% à l'aide d'une interpolation bicubique par rapport à la taille originale afin d'augmenter la vitesse de l'ensemble du processus. La taille des images traitées après réduction est de 1167 x 1556. Voir la figure \ref{fig:imgRGB}.
+Le processus est actuellement réalisé sur des images réduites de 40~\% à l'aide d'une interpolation bicubique par rapport à la taille originale afin d'augmenter la vitesse de l'ensemble du processus. La taille des images traitées après réduction est de 1167 x 1556. Voir la figure \ref{fig:imgRGB}.
-La qualité du résultat final n'est pas beaucoup affectée par diminution de la taille des images, sur l'ensemble des douze images présentées à la section~\ref{}, nous obtenons un taux d'erreur moyen de 17.9~\% pour les images originales contre 18.2~\% pour les images réduites. Le taux mesuré de référence est de 16.9 \%. Le temps moyen est de 485~s pour les images originales contre 78~s pour les images réduites.
+La qualité du résultat final n'est pas beaucoup affectée par diminution de la taille des images : sur l'ensemble des douze images présentées à la section~\ref{resultats}, nous obtenons un taux d'erreur moyen de 17.9~\% pour les images originales contre 18.2~\% pour les images réduites. Le taux mesuré de référence est de 16.9 \%. Le temps moyen est de 485~s pour les images originales contre 78~s pour les images réduites.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\linewidth]{img/imgRGB.png}
- \caption{Image originale en entrée ($imgRGB$)}
+ \caption{Image originale en entrée ($\mli{imgRGB}$)}
\label{fig:imgRGB}
\end{figure}
-Tout au long de cette description nous lions certains résultats d'opération à des noms de variables issue du code \emph{MATLAB} (fourni en annexe). Ceci afin de pouvoir facilement retrouver le détail de ces opérations dans le code et de facilité le suivit de l'enchaînement des opérations.
+Tout au long de cette description nous lions certains résultats d'opération à des noms de variables issues du code \emph{MATLAB} (fourni en annexe). Ceci afin de pouvoir facilement retrouver le détail de ces opérations dans le code et de faciliter le suivi de l'enchaînement des opérations.
<TODO> Caractériser les images en entrée + mettre une image en figure (sachant que l'ensemble du processus y est très sensible).
-<TODO> faire des mesures précises sur des images plus grandes afin de voir s'il y a beaucoup d'imprécisions liées à la réduction, également mettre en évidence le temps de calcul ?
\subsection{Détection des parasites}
+\label{detection-parasites}
Cette opération a pour but de marquer les éléments colorés qui correspondent aux parasites, schizontes et globules blancs. Nous utilisons ici les composantes \emph{teinte} et \emph{saturation} de l'image.
\subsubsection{Filtrage préliminaire}
-Il est nécessaire pour la suite du traitement d'avoir les intensités les plus élevés pour les éléments colorés pour les deux composantes teinte et valeur. Pour se faire nous allons utiliser le négatif des deux composantes. De plus la composante \emph{teinte} est translatée de $100 / 255$ (141°) afin que les différents éléments de l'image, à savoir le fond, les cellules et les parties colorées, ne soient pas proche des valeurs extremes, voir l'histogramme montré par la figure \ref{fig:histogram_imgH}. $imgRGB \rightarrow (imgH, imgS)$.
+Il est nécessaire, pour la suite du traitement, d'avoir les intensités les plus élevées pour les éléments colorés pour les deux composantes teinte et valeur. Pour se faire nous allons utiliser le négatif des deux composantes. De plus la composante \emph{teinte} est translatée de $100 / 255$ (141°) afin que les différents éléments de l'image, à savoir le fond, les cellules et les parties colorées, ne soient pas proches des valeurs extrêmes, voir l'histogramme montré par la figure \ref{fig:histogram_imgH}. $\mli{imgRGB} \rightarrow (\mli{imgH}, \mli{imgS})$.
-Les histogrammes des deux images $imgH$ et $imgS$ sont montrés par les figures \ref{fig:histogram_imgH} et \ref{fig:histogram_imgS}. L'histogramme de la teinte montre un premier sommet vers 80 correspondant au fond et un deuxième sommet vers 88 correspondant aux cellules, les éléments colorés se trouvent aux environs de 120. L'histogramme de la saturation montre un premier sommet vers 128 correspondant aux cellules et un deuxième sommet vers 150 correspondant au fond, les éléments colorés se trouvent aux environs de 200.
+Les histogrammes des deux images $\mli{imgH}$ et $\mli{imgS}$ sont montrés par les figures \ref{fig:histogram_imgH} et \ref{fig:histogram_imgS}. L'histogramme de la teinte montre un premier sommet vers 80 correspondant au fond et un deuxième sommet vers 88 correspondant aux cellules, Les éléments colorés se trouvent aux environs de 120. L'histogramme de la saturation montre un premier sommet vers 128 correspondant aux cellules et un deuxième sommet vers 150 correspondant au fond : les éléments colorés se trouvent aux environs de 200.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\linewidth]{img/histogram_imgH.eps}
- \caption{Histogramme de la teinte ($imgH$)}
+ \caption{Histogramme de la teinte ($\mli{imgH}$)}
\label{fig:histogram_imgH}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\linewidth]{img/histogram_imgS.eps}
- \caption{Histogramme de la saturation ($imgS$)}
+ \caption{Histogramme de la saturation ($\mli{imgS}$)}
\label{fig:histogram_imgS}
\end{figure}
-Afin de gommer le bruit et de rendre le fond plus homogène et plus sombre, un filtre médian 5 x 5 suivi d'une fermeture par aire de 1000 sont appliqués à $imgH$ et $imgS$. $imgH \rightarrow imgFiltered\{1\}, imgS \rightarrow imgFiltered\{2\}$. Les figures \ref{fig:imgHFiltered} et \ref{fig:imgSFiltered} montrent les composantes \emph{teinte} et \emph{saturation} après filtrage.
+Afin de gommer le bruit et de rendre le fond plus homogène, un filtre médian 5 x 5 suivi d'une fermeture par aire de 1000 sont appliqués à $\mli{imgH}$ et $\mli{imgS}$. $\mli{imgH} \rightarrow \mli{imgFiltered\{1\}}, \mli{imgS} \rightarrow \mli{imgFiltered\{2\}}$. Les figures \ref{fig:imgHFiltered} et \ref{fig:imgSFiltered} montrent les composantes \emph{teinte} et \emph{saturation} après filtrage.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\linewidth]{img/imgHFiltered.jpg}
- \caption{Composante \emph{teinte} après filtrage ($imgFiltered\{1\}$) représentée en fausse couleur}
+ \caption{Composante \emph{teinte} après filtrage ($\mli{imgFiltered\{1\}}$) représentée en fausse couleur}
\label{fig:imgHFiltered}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\linewidth]{img/imgSFiltered.jpg}
- \caption{Composante \emph{saturation} après filtrage ($imgFiltered\{2\}$) représentée en fausse couleur}
+ \caption{Composante \emph{saturation} après filtrage ($\mli{imgFiltered\{2\}}$) représentée en fausse couleur}
\label{fig:imgSFiltered}
\end{figure}
\subsubsection{Granulométrie}
-L'objectif ici est de déterminer le rayon moyenne des globules rouges (souvent abrégé \emph{RBC} pour \emph{red blood cell}). Pour ce faire, nous réalisons une succession de fermetures à l'aide d'un élément structurant de forme octogonale sur la composante \emph{saturation} ($imgFiltered\{2\}$). Nous évitons ici un élément structurant ayant la forme d'un disque pour des raisons de performance, une fermeture avec ce dernier demandant d'effectuer beaucoup plus de calculs. Une ouverture par aire de 1000 est appliquée au préalable sur la composante \emph{saturation} afin de niveler l'intensité des cellules.
+L'objectif ici est de déterminer le rayon moyen des globules rouges (souvent abrégé \emph{RBC} pour \emph{red blood cell}). Pour ce faire, nous réalisons une succession de fermetures à l'aide d'un élément structurant de forme octogonale sur la composante \emph{saturation} ($\mli{imgFiltered\{2\}}$). Il est à noter que les globules rouges ont une intensité moindre que le fond (voir la figure~\ref{imgSFiltered}), c'est pour cela que nous utilisons une fermeture et non une ouverture. Nous évitons ici un élément structurant ayant la forme d'un disque pour des raisons de performance, une fermeture avec ce dernier demandant d'effectuer beaucoup plus de calculs. Une ouverture par aire de 1000 est appliquée au préalable sur la composante \emph{saturation} afin de niveler l'intensité des cellules.
\begin{sloppypar} % Pour éviter que certaines formules inline débordent dans la marge.
-Nous partons d'un rayon initial de un pour arriver à un rayon maximal de $width(imgRGB) / 35$. La rayon maximal est définit en fonction de la largeur de l'image ce qui permet de ne pas être dépendant de sa résolution. À chaque itération une fermeture est effectuée suivi du calcul du volume relatif de l'image : $volume(imgClosed) \rightarrow A, 1 - A / volImg \rightarrow N$ où $volImg$ est le volume de $imgFiltered\{2\}$. La différence de volume est calculé pour chaque itération $i$ comme suit : $N_{i + 1} - N_i$.
+Nous partons d'un rayon initial de un pour arriver à un rayon maximal de $\mli{width}(\mli{imgRGB}) / 35$. Le rayon maximal est défini en fonction de la largeur de l'image, ce qui permet de ne pas être dépendant de sa résolution. À chaque itération une fermeture est effectuée suivi du calcul du volume relatif de l'image : $\mli{volume}(\mli{imgClosed}) \rightarrow A, 1 - A / \mli{volImg} \rightarrow N$ où $\mli{volImg}$ est le volume de $\mli{imgFiltered\{2\}}$. La différence de volume est calculée pour chaque itération $i$ comme suit : $\lvert N_{i + 1} - N_i \rvert$.
\end{sloppypar}
-Nous définissions le rayon moyen des globules rouges ($RBCRadius$) comme étant le rayon de l'élément structurant correspondant à la valeur maximale parmi les différences de volume. Le rayon moyen du noyau des parasites est impossible à extraire des différences de volume, nous le calculons comme étant un cinquième de la taille des globules rouges : $RBCRadius / 5 \rightarrow nucleiRadius$. La figure \ref{fig:patternSpectrum} montre le graphique de la distribution des différences. Dans ce cas $RBCRadius$ est égal à 35 et $nucleiRadius$ à 7.
+Nous définissons le rayon moyen des globules rouges ($\mli{RBCRadius}$) comme étant le rayon de l'élément structurant correspondant à la valeur maximale parmi les différences de volume. Le rayon moyen du noyau des parasites est impossible à extraire des différences de volume : nous le calculons comme étant un cinquième de la taille des globules rouges : $\mli{RBCRadius} / 5 \rightarrow \mli{nucleiRadius}$. La figure \ref{fig:patternSpectrum} montre le graphique de la distribution des différences. Dans ce cas $\mli{RBCRadius}$ est égal à 35 et $\mli{nucleiRadius}$ à 7.
\begin{figure}[htbp]
\centering
\subsubsection{Extrêmes régionaux}
-Pour la recherche des extrêmes régionaux nous utilisons la fonction \texttt{mmregmax}. La figure \ref{fig:regmax_sample} montre un exemple d'application de cette fonction à l'aide d'un élément structurant carré de taille 5.
+Pour la recherche des extrêmes régionaux nous utilisons la fonction \texttt{mmregmax}.
+
+<TODO> : Formaliser mmregmax et dire que ça ne correspond pas à la définition du bouquin.
+
+La figure \ref{fig:regmax_sample} montre un exemple d'application de cette fonction à l'aide d'un élément structurant carré de taille 5.
\begin{figure}[htbp]
\centering
\begin{subfigure}[b]{0.38\textwidth}
\includegraphics[width=\linewidth]{img/regmax_sample.eps}
- \caption{Image à laquelle est appliquée la fonction \texttt{mmregmax}}
+ \caption{Image à laquelle est appliquée la fonction \texttt{mmregmax}. Les pixels noires valent zéro}
\label{fig:regmax_sample_original}
\end{subfigure}
~
\begin{subfigure}[b]{0.15\textwidth}
\includegraphics[width=\linewidth]{img/regmax_sample_se.eps}
- \caption{L'élément structurant}
+ \caption{L'élément structurant : un carré de taille 5}
\label{fig:regmax_sample_se}
\end{subfigure}
~
\begin{subfigure}[b]{0.38\textwidth}
\includegraphics[width=\linewidth]{img/regmax_sample_result.eps}
- \caption{Le résultat de l'application de \texttt{mmregmax}}
+ \caption{Le résultat de l'application de \texttt{mmregmax}. Les pixels noires valent zéro}
\label{fig:regmax_sample_result}
\end{subfigure}
\caption{Example d'application de la fonction \texttt{mmregmax}}
\label{fig:regmax_sample}
\end{figure}
-Afin d'identifier les éléments colorés, une recherche des maximaux régionaux est réalisée sur les deux composantes \emph{teinte} et \emph{saturation} à l'aide d'un élément structurant octogonal de rayon $RBCRadius$. Les deux résultats sont ensuite dilatés de $nucleiRadius$ puis combinés avec une intersection.
+Afin d'identifier les éléments colorés, une recherche des maximaux régionaux est réalisée sur les deux composantes \emph{teinte} et \emph{saturation} à l'aide d'un élément structurant octogonal de rayon $\mli{RBCRadius}$. Les deux résultats sont ensuite dilatés de $\mli{nucleiRadius}$ puis combinés avec une intersection.
{\setlength{\abovedisplayskip}{0pt}
\begin{flalign}
- M\{1\} &= regmax(imgFiltered\{1\}, RBCRadius) &\\
- M\{2\} &= regmax(imgFiltered\{2\}, RBCRadius) &\\
- MHS &= (M\{1\} \oplus nucleiRadius) \cap (M\{2\} \oplus nucleiRadius) &
+ \mli{M\{1\}} &= \mli{regmax}(\mli{imgFiltered\{1\}}, \mli{RBCRadius}) &\\
+ \mli{M\{2\}} &= \mli{regmax}(\mli{imgFiltered\{2\}}, \mli{RBCRadius}) &\\
+ \mli{MHS} &= (\mli{M\{1\}} \oplus \mli{nucleiRadius}) \cap (\mli{M\{2\}} \oplus \mli{nucleiRadius}) &
\end{flalign}
-Nous allons ensuite seuiller les deux composantes puis combiner les résultats avec une intersection. Le seuil de chaque image, ${\mu}H$ et ${\mu}S$, est déterminé comme étant la moyenne des valeurs de l'image désignée par le masque $MHS$.
+Nous allons ensuite seuiller les deux composantes puis combiner les résultats avec une intersection. Le seuil de chaque image, $\mli{{\mu}H}$ et ${\mu}S$, est déterminé comme étant la moyenne des valeurs de l'image désignée par le masque $\mli{MHS}$.
{\setlength{\abovedisplayskip}{0pt}
\begin{flalign}
- {\mu}H &= \frac{\displaystyle\sum_{p \in MHS} H(p)}{sum(MHS)} & où~H &= imgFiltered\{1\} &\\
- {\mu}S &= \frac{\displaystyle\sum_{p \in MHS} S(p)}{sum(MHS)} & où~S &= imgFiltered\{2\} &
+ \mli{MHS}_{vol} &= \displaystyle\sum_{p \in \mli{MHS}}p &\\
+ \mli{{\mu}H} &= \frac{\displaystyle\sum_{p \in \mli{MHS}} H(p)}{\mli{MHS}_{vol}} & où~H &= \mli{imgFiltered\{1\}} &\\
+ \mli{{\mu}S} &= \frac{\displaystyle\sum_{p \in \mli{MHS}} S(p)}{\mli{MHS}_{vol}} & où~S &= \mli{imgFiltered\{2\}} &
\end{flalign}
-La teinte et la saturation sont seuillés à l'aide ${\mu}H$, respectivement ${\mu}S$. Nous obtenons alors les images $TH$ et $TS$ que nous combinons comme suit pour obtenir $THS$. L'ouverture par aire permet d'éliminer les toutes petites traces qui représentent dans la majorité des cas du bruit. La figure \ref{fig:THS} montre $THS$ en blanc imprimé sur l'image d'entrée.
+La teinte et la saturation sont seuillées à l'aide des valeurs $\mli{{\mu}H}$, respectivement $\mli{{\mu}S}$. Nous obtenons alors les images $\mli{TH}$ et $\mli{TS}$ que nous combinons comme suit pour obtenir $\mli{THS}$. L'ouverture par aire permet d'éliminer les toutes petites traces qui représentent dans la majorité des cas du bruit. La figure \ref{fig:THS} montre $\mli{THS}$ en blanc imprimé sur l'image d'entrée.
{\setlength{\abovedisplayskip}{0pt}
\begin{flalign}
- THS &= areaOpen(TH \cap TS, 10) &
+ \mli{THS} &= \mli{areaOpen}(\mli{TH} \cap \mli{TS}, 10) &
\end{flalign}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\linewidth]{img/THS.jpg}
- \caption{Les éléments colorés mis en évidence ($THS$)}
+ \caption{Les éléments colorés mis en évidence ($\mli{THS}$)}
\label{fig:THS}
\end{figure}
\subsection{Détection des globules blancs}
-Cette opération utilise l'image de marquage des éléments colorés ($THS$) ainsi que la taille des globules rouges ($RBCRadius$). Les globules blancs sont tout d'abord marqués à l'aide d'une érosion utilisant un élément structurant octogonal dont la taille correspond à 50 \% de celle des globules rouges ($0.5 \cdot RBCRadius$). Puis reconstruction par dilatation des éléments colorés en utilisant le marqueur précédent permet de ne garder que les globules blanc.
+Cette opération utilise l'image de marquage des éléments colorés ($\mli{THS}$) ainsi que la taille des globules rouges ($\mli{RBCRadius}$). Les globules blancs sont tout d'abord marqués à l'aide d'une érosion utilisant un élément structurant octogonal dont la taille correspond à 50 \% de celle des globules rouges ($0.5 \cdot \mli{RBCRadius}$). Puis reconstruction par dilatation des éléments colorés en utilisant le marqueur précédent permet de ne garder que les globules blancs.
-Finalement l'on lisse le résultat en réalisant une fermeture suivit d'une ouverture afin de fermer les petits trous qu'il pourrait rester. Le résultat est appelé $WBC$ et est montré par la figure \ref{fig:WBC}.
+Finalement l'on lisse le résultat en réalisant une fermeture suivie d'une ouverture afin de fermer les petits trous qui pourraient rester. Le résultat est appelé $\mli{WBC}$ et est montré par la figure \ref{fig:WBC}.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\linewidth]{img/WBC.jpg}
- \caption{Le gradient des globules blancs imprimé sur l'image originale ($WBC$)}
+ \caption{Le gradient des globules blancs imprimé sur l'image originale ($\mli{WBC}$)}
\label{fig:WBC}
\end{figure}
\subsection{Détection des schizontes}
-Pour détecter les schizontes, nous utilisons l'image de marquage des éléments colorés ($THS$) ainsi que la taille des globules rouges ($RBCRadius$). Nous allons ici plus loin que l'article et créons un graphe non-orienté dont les arcs représentent une \emph{distance de Hausdorff} égale ou plus petite à 1.3 fois le rayon des globules rouges entre deux sous-ensembles. Nous cherchons alors les sous-ensembles connexes de plus de $n$ éléments et les marquons comme étant des schizontes. Dans notre cas $n$ est égal à 6. Il est important de ne pas utiliser un nombre trop bas car des cellules poly-infectées pourraient être considérées comme des schizontes, la figure \ref{fig:poly-et-schizonte} illustre cela.
+Pour détecter les schizontes, nous utilisons l'image de marquage des éléments colorés ($\mli{THS}$) ainsi que la taille des globules rouges ($\mli{RBCRadius}$). Nous allons ici plus loin que l'article et créons un graphe non-orienté dont les arcs représentent une \emph{distance de Hausdorff} égale ou plus petite à 1.3 fois le rayon des globules rouges entre deux sous-ensembles. Nous cherchons alors les sous-ensembles connexes de plus de $n$ éléments et les marquons comme étant des schizontes. Dans notre cas $n$ est égal à 6. Il est important de ne pas utiliser un nombre trop bas car des cellules poly-infectées pourraient être considérées comme des schizontes : la figure \ref{fig:poly-et-schizonte} illustre cela.
\begin{figure}[htbp]
\centering
\label{fig:poly-et-schizonte}
\end{figure}
-Pour savoir si deux sous-ensembles, $A$ et $B$, sont à une \emph{distance de Hausdorff} égale ou plus petite que $k$, il faut que $B \subseteq A \oplus SE_k$ et que $A \subseteq B \oplus SE_k$ où $SE_k$ est un élément structurant ayant la forme d'un disque de diamètre $k$.
+Pour savoir si deux sous-ensembles, $A$ et $B$, sont à une \emph{distance de Hausdorff} égale ou plus petite que $k$, il faut que $B \subseteq A \oplus \mli{SE_k}$ et que $A \subseteq B \oplus \mli{SE_k}$ où $\mli{SE_k}$ est un élément structurant ayant la forme d'un disque de diamètre $k$.
\subsection{Segmentation des globules rouges}
+\label{segmentation-globules-rouges}
-Le but ici est de séparer les globules rouges, nous utilisons l'image initiale ($imgRGB$), le rayon des globules rouges ($RBCRadius$), les globules blancs ($WBC$) et les schizontes ($schizonts$). Pour ce faire nous utilisons dans un premier temps une segmentation par \emph{watersheds} à l'aide d'une transformée de distance. Nous séparons ensuite les segments en deux groupes, un groupe dont chaque élément représente une unique cellule et un autre groupe dont chaque élément représente un composé de cellules. Les cellules des éléments composés sont ensuite séparées, le résultat est unies aux cellules uniques et les éléments touchant les bords sont enlevé afin d'obtenir le résultat final.
+Le but ici est de séparer les globules rouges : nous utilisons l'image initiale ($\mli{imgRGB}$), le rayon des globules rouges ($\mli{RBCRadius}$), les globules blancs ($\mli{WBC}$) et les schizontes ($\mli{schizonts}$). Pour ce faire nous utilisons dans un premier temps une segmentation par \emph{watersheds} à l'aide d'une transformée de distance. Nous séparons ensuite les segments en deux groupes, un groupe dont chaque élément représente une unique cellule et un autre groupe dont chaque élément représente un composé de cellules. Les cellules des éléments composés sont ensuite séparées : le résultat est uni aux cellules uniques et les éléments touchant les bords sont enlevés afin d'obtenir le résultat final.
-Durant le processus, nous avons besoin de définir le rayon minimum d'un globule rouge ainsi que son aire. Le rayon minimum est définit comme étant 75~\% du rayon nominal comme montré ci dessous.
+Durant le processus, nous avons besoin de définir le rayon minimum d'un globule rouge ainsi que son aire. Le rayon minimum est défini comme étant 75~\% du rayon nominal, comme montré ci dessous.
{\setlength{\abovedisplayskip}{0pt}
\begin{flalign}
- RBCRadiusMin &= \lfloor 0.75 \cdot RBCRadius \rfloor &\\
- RBCRadiusMinArea &= \pi \cdot RBCRadiusMin^2 &
+ \mli{RBCRadiusMin} &= \lfloor 0.75 \cdot \mli{RBCRadius} \rfloor &\\
+ \mli{RBCRadiusMinArea} &= \pi \cdot \mli{RBCRadiusMin}^2 &
\end{flalign}
\subsubsection{Filtrage préliminaire}
-Nous utilisons ici la composantes verte de l'image à laquelle nous appliquons une ouverture par aire avec une zone de taille $RBCRadiusMinArea$. Les globules blancs ($WBC$) ainsi que les schizontes ($schizonts$) sont \og~soustraits~\fg de l'image, le résultat est nommé $grayFlat$ comme montré par la figure~\ref{fig:segmentation-grayflat}.
+Nous utilisons ici la composante verte de l'image à laquelle nous appliquons une ouverture par aire avec une zone de taille $\mli{RBCRadiusMinArea}$. Les globules blancs ($\mli{WBC}$) ainsi que les schizontes ($\mli{schizonts}$) sont \og{}soustraits\fg{} de l'image : le résultat est nommé $\mli{grayFlat}$ comme montré par la figure~\ref{fig:segmentation-grayflat}.
{\setlength{\abovedisplayskip}{0pt}
\begin{flalign}
- grayFlat &= areaOpen(green(imgRGB)) \setminus WBC \setminus schizonts
+ \mli{grayFlat} &= \mli{areaOpen}(\mli{green}(\mli{imgRGB})) \setminus \mli{WBC} \setminus \mli{schizonts} &
\end{flalign}
-La figure~\ref{fig:segmentation-histogramme-grayflat} montre l'histogramme de $grayFlat$. Un \emph{seuillage d'Otsu} lui est appliqué suivit d'une ouverture par un disque de rayon $RBCRadiusMin$ afin d'adoucir les contours et d'enlever tout les éléments plus petits qu'un globules rouge.
+La figure~\ref{fig:segmentation-histogramme-grayflat} montre l'histogramme de $\mli{grayFlat}$. Un \emph{seuillage d'Otsu} lui est appliqué suivi d'une ouverture par un disque de rayon $\mli{RBCRadiusMin}$ afin d'adoucir les contours et d'enlever tous les éléments plus petits qu'un globule rouge.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\linewidth]{img/segmentation_input_grayflat.jpg}
- \caption{Composante verte dont l'intensité des cellules a été homogénéisée à l'aide d'une ouverture par aire, les globules blancs et les schizontes ont été peints en blanc ($grayFlat$)}
+ \caption{Composante verte dont l'intensité des cellules a été homogénéisée à l'aide d'une ouverture par aire. Les globules blancs et les schizontes ont été peints en blanc ($\mli{grayFlat}$)}
\label{fig:segmentation-grayflat}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\linewidth]{img/histogram_segmentation_grayFlat.eps}
- \caption{Histogramme de la composante verte de $imgRGB$ après l'application d'une ouverture par aire et de la \og{}suppression\fg{} des globules blancs et des schizontes}
+ \caption{Histogramme de la composante verte de $\mli{imgRGB}$ après l'application d'une ouverture par aire et de la \og{}suppression\fg{} des globules blancs et des schizontes}
\label{fig:segmentation-histogramme-grayflat}
\end{figure}
\subsubsection{Calcul des distances et \emph{watersheds}}
-Après le seuillage des globules rouge certaines régions comprennent deux ou plusieurs globules rouges qu'il faut séparer. Pour se faire nous appliquons une transformé de distance (\texttt{mmdist}) qui permet de donner une distance à chaque pixel blanc. Cette distance correspond à la plus courte distance euclidienne entre le pixel blanc et un pixel noir. Nous cherchons ensuite les maximaux régionaux de l'image de distance en utilisant un élément structurant octogonal dont le rayon est la moitié de $RBCRadius$. Une dilaté d'un disque de rayon $RBCRadius / 7$ est alors appliqué aux maximaux pour fusionner d'éventuels zones très proches. L'algorithme \emph{watersheds} est appliqué au négatif des distances en utilisant les maximaux dilatés comme marqueurs. Finalement l'intersection entre le négatif des \emph{watersheds} et de $thresholdOpened$ correspond au résultat. La figure~\ref{fig:segmentation-watershed} montre les étapes de ce processus.
+Après le seuillage des globules rouges, certaines régions comprennent deux ou plusieurs globules rouges qu'il faut séparer. Pour se faire nous appliquons une transformée de distance (\texttt{mmdist}) qui permet de donner une distance à chaque pixel blanc. Cette distance correspond à la plus courte distance euclidienne entre le pixel blanc et un pixel noir. Nous cherchons ensuite les maximaux régionaux de l'image de distance en utilisant un élément structurant octogonal dont le rayon est la moitié de $\mli{RBCRadius}$. Une dilatée d'un disque de rayon $\mli{RBCRadius} / 7$ est alors appliquée aux maximaux pour fusionner d'éventuelles zones très proches. L'algorithme \emph{watersheds} est appliqué au négatif des distances en utilisant les maximaux dilatés comme marqueurs. Finalement l'intersection entre le négatif des \emph{watersheds} et de $\mli{thresholdOpened}$ correspond au résultat. La figure~\ref{fig:segmentation-watershed} montre les étapes de ce processus.
\begin{figure}[htbp]
\centering
\begin{subfigure}[t]{0.3\textwidth}
\includegraphics[width=\linewidth]{img/segmentation_watershed_1.jpg}
- \caption{Image binaire initiale ($thresholdOpened$)}
+ \caption{Image binaire initiale ($\mli{thresholdOpened}$)}
\label{fig:segmentation-watershed-1}
\end{subfigure}
~
\end{figure}
-
-
-%\begin{framed}
-%$i$ : l'image seuillée
-%\begin{enumerate}
-% \item d <-
-%\end{enumerate}
-%\end{framed}
-
\subsubsection{Identification et traitement des cellules composés}
-Afin d'identifier les zones composées de plusieurs cellules nous calculons le rapport grand-axe / petit-axe pour chacune des zones, si le rapport d'une zone est plus grand ou égal à 1.3 alors la zone est définit comme étant composée.
+Afin d'identifier les zones composées de plusieurs cellules nous calculons le rapport grand-axe / petit-axe pour chacune des zones : si le rapport d'une zone est plus grand ou égal à 1.3, alors la zone est défini comme étant composée.
-La figure~\ref{fig:segmentation-composites} montre le processus de séparation d'une zone composée de plusieurs cellules, ce processus est décrit ci après.
+La figure~\ref{fig:segmentation-composites} montre le processus de séparation d'une zone composée de plusieurs cellules : ce processus est décrit ci-après.
-Une ouverture à l'aide d'un disque de rayon $RBCRadiusMin$ est appliquée à $gray$ en utilisant les zones composées comme masque. Le résultat est montrée par la figure \ref{fig:segmentation-composites-3}. Un gradient morphologique suivit d'une binarisation est réaliser pour obtenir l'image de la figure \ref{fig:segmentation-composites-4}. Les trous d'une aire plus petite ou égale à la moitié de $RBCRadiusMinArea$ sont remplis avec une ouverture par aire puis un amincissement est appliqué. Nous remplissons ensuite les trous de l'amincissement est mettons à zéro les pixels marqués par l'amincissement. Nous érodons le résultat afin d'éviter que les cellules composées ne touchent les cellules uniques lors de l'union qui aura lieu après. Finalement une ouverture par aire de valeur $RBCRadiusMinArea / 5$ est effectué pour enlever d'éventuels artefacts comme montré par la figure~\ref{fig:soustraction-thinning}.
+Une ouverture à l'aide d'un disque de rayon $\mli{RBCRadiusMin}$ est appliquée à $\mli{gray}$ en utilisant les zones composées comme masque. Le résultat est montré par la figure \ref{fig:segmentation-composites-3}. Un gradient morphologique suivi d'une binarisation est réalisée pour obtenir l'image de la figure \ref{fig:segmentation-composites-4}. Les trous d'une aire plus petite ou égale à la moitié de $\mli{RBCRadiusMinArea}$ sont remplis avec une ouverture par aire puis un amincissement est appliqué. Nous remplissons ensuite les trous de l'amincissement et mettons à zéro les pixels marqués par celui-ci. Nous érodons le résultat afin d'éviter que les cellules composées ne touchent les cellules uniques lors de l'union qui aura lieu après. Finalement une ouverture par aire de valeur $\mli{RBCRadiusMinArea} / 5$ est effectuée pour enlever d'éventuels artefacts, comme montré par la figure~\ref{fig:soustraction-thinning}.
% L'exemple provient de 1401063467-0007-schizonte.png (x: 900, y: 50).
\begin{figure}[htbp]
\centering
\begin{subfigure}[t]{0.3\textwidth}
\includegraphics[width=\linewidth]{img/segmentation_composites_1.jpg}
- \caption{Composante verte ($gray$)}
+ \caption{Composante verte ($\mli{gray}$)}
\label{fig:segmentation-composites-1}
\end{subfigure}
\begin{subfigure}[t]{0.3\textwidth}
\includegraphics[width=\linewidth]{img/segmentation_composites_2.jpg}
- \caption{Image binaire des éléments composés, obtenue à partir de $grayFlat$}
+ \caption{Image binaire des éléments composés, obtenue à partir de $\mli{grayFlat}$}
\label{fig:segmentation-composites-2}
\end{subfigure}
~
~
\begin{subfigure}[t]{0.3\textwidth}
\includegraphics[width=\linewidth]{img/segmentation_composites_7.jpg}
- \caption{Fermeture par aire puis mise à zéro des pixels marqué par \emph{(f)}}
+ \caption{Fermeture par aire puis mise à zéro des pixels marqués par \emph{(f)}}
\label{fig:segmentation-composites-7}
\end{subfigure}
\caption{Processus de séparation des cellules composées}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\linewidth]{img/splitting_composites.jpg}
- \caption{Résultat de l'amincissement soustrait de son remplissage. De petits artefacts, ici marqués en jaune, restent présents. S'ils ne sont pas enlevés, ceux ci seront comptés comme des cellules à part entière lors de la labellisation finale}
+ \caption{Résultat de l'amincissement soustrait de son remplissage. De petits artefacts, ici marqués en jaune, restent présents. S'ils ne sont pas enlevés, ceux-ci seront comptés comme des cellules à part entière lors de la labellisation finale}
\label{fig:soustraction-thinning}
\end{figure}
\subsubsection{Résultat final}
-Les deux résultats précédents, c'est à dire les cellules uniques et les zones composées de plusieurs cellules sont unies. Les zones touchant les bords sont supprimé afin d'obtenir les résultat final, son gradient imprimé sur l'image initiale est montré par la figure~\ref{fig:segmentation_finale}. Le résultat est nommé $redCells$.
+Les deux résultats précédents, c'est-à-dire les cellules uniques et les zones composées de plusieurs cellules sont unies. Les zones touchant les bords sont supprimées afin d'obtenir le résultat final. Son gradient imprimé sur l'image initiale est montré par la figure~\ref{fig:segmentation_finale}. Le résultat est nommé $\mli{redCells}$.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\linewidth]{img/segmentation_finale.jpg}
- \caption{Le gradient du résultat de la segmentation des globules rouges imprimé sur l'image initiale ($redCells$)}
+ \caption{Le gradient du résultat de la segmentation des globules rouges imprimé sur l'image initiale ($\mli{redCells}$)}
\label{fig:segmentation_finale}
\end{figure}
\subsection{Identification des globules rouges infectés}
-À partir du résultat de la segmentation des globules rouges obtenue à la section~\ref{} et des éléments colorés obtenus à la section~\ref{} nous pouvons reconstruire les cellules infectés par une succession de dilatation des éléments colorés puis de masquage par les globules rouges. Ceci est réalisé à l'aide de la fonction \texttt{mminfrec}. Le résultat est nommé $infectedRedCells$.
+À partir du résultat de la segmentation des globules rouges obtenu à la section~\ref{segmentation-globules-rouges} et des éléments colorés obtenus à la section~\ref{detection-parasites} nous pouvons reconstruire les cellules infectées par une succession de dilatations des éléments colorés puis de masquages par les globules rouges. Ceci est réalisé à l'aide de la fonction \texttt{mminfrec}. Le résultat est nommé $\mli{infectedRedCells}$.
La figure~\ref{fig:globules_rouges_infectes} montre les globules rouges infectés.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\linewidth]{img/globules_rouges_infectes.jpg}
- \caption{Le gradient des globules rouges imprimé sur l'image originale ($infectedRedCells$)}
+ \caption{Le gradient des globules rouges imprimé sur l'image originale ($\mli{infectedRedCells}$)}
\label{fig:globules_rouges_infectes}
\end{figure}
\subsection{Calcul du taux d'infection}
-Les deux images $redCells$ et $infectedRedCells$ sont labellisées afin d'obtenir le nombre totale de globules rouges ($nbRedCells$) et le nombre de globules rouges infecté ($nbInfectedRedCells$). Le pourcentage d'infection est le rapport de ces deux valeurs : $100 \cdot nbInfectedRedCells / nbRedCells$.
+Les deux images $\mli{redCells}$ et $\mli{infectedRedCells}$ sont labellisées afin d'obtenir le nombre total de globules rouges ($\mli{nbRedCells}$) et le nombre de globules rouges infectés ($\mli{nbInfectedRedCells}$). Le pourcentage d'infection est le rapport de ces deux valeurs : $100 \cdot \mli{nbInfectedRedCells} / \mli{nbRedCells}$.
\section{Résultats}
+\label{resultats}
\subsection{Qualité des images en entrée}
-<TODO>
+La première étape, décrite à la section~\ref{detection-parasites}, qui consiste à marquer les éléments colorés est très sensible à la qualité de l'image initiale. Suivant les réglages du microscope les éléments colorés peuvent ressortir avec une plus grande ou une plus faible saturation par rapport au fond et aux globules rouges. De plus la teinte des éléments colorés peuvent être similaire à la celle du fond : La figure~\ref{fig:problemes-teinte} montre une image dont la teinte est inexploitable.
+
+\begin{figure}[htbp]
+ \centering
+ \begin{subfigure}[t]{0.4\textwidth}
+ \includegraphics[width=\linewidth]{img/problemes_teinte_input.jpg}
+ \caption{Image en entrée}
+ \label{fig:problemes-teinte-input}
+ \end{subfigure}
+ ~
+ \begin{subfigure}[t]{0.4\textwidth}
+ \includegraphics[width=\linewidth]{img/problemes_teinte_composante.jpg}
+ \caption{Composante teinte}
+ \label{fig:problemes-teinte-composante}
+ \end{subfigure}
+
+ \begin{subfigure}[t]{0.6\textwidth}
+ \includegraphics[width=\linewidth]{img/histogram_probleme_teinte.eps}
+ \caption{Histogramme de la composante teinte de l'image complète. Les éléments colorés se situent entre l'intensité 190 et 200 (mis en évidence en rouge) et se confondent avec le fond}
+ \label{fig:problemes-teinte-histogramme}
+ \end{subfigure}
+ \caption{}
+ \label{fig:problemes-teinte}
+\end{figure}
+
\subsection{Méthode}
-Les résultats se basent sur l'analyse d'un ensemble d'images provenant d'un même patient, le taux d'infection devrait être homogène entre les différentes images. Pour mesurer la qualité de nos détections, un dénombrement réalisé à la main est effectué afin de refléter la réalité, puis celui-ci est comparé à nos détections automatisées. La différence entre le résultat issue de la détection et la réalité est alors calculée et nommé \emph{erreur}.
+Les résultats se basent sur l'analyse d'un ensemble d'images provenant d'un même patient, le taux d'infection devrait être homogène entre les différentes images. Pour mesurer la qualité de nos détections, un dénombrement réalisé à la main est effectué, puis celui-ci est comparé à nos détections automatisées. La différence entre le résultat issu de la détection et la réalité est alors calculée et nommée \emph{erreur}.
Finalement, nous calculons les moyennes et les écarts types des erreurs, et des deux taux, celui de référence et celui détecté.
\subsection{Échantillon n°1412151257}
-Les tableaux \ref{tab:resultat60} et \ref{tab:resultat100} montrent les résultats du processus appliqué à des images réduite de 40~\%, respectivement à des images originales.
+Les tableaux \ref{tab:resultats60} et \ref{tab:resultats100} montrent les résultats du processus appliqué à des images réduites de 40~\%, respectivement à des images originales.
\begin{table}[htbp]
\setlength{\tabcolsep}{3pt}
\centering
-\begin{tabular}{ r || r | r | r || r | r | r || r || r}
+\begin{tabular}{ r || r | r | r || r | r | r || r || r }
& \multicolumn{3}{ c| }{Mesuré} & \multicolumn{3}{ c| }{Détecté} & & \\ \cline{2-8}
N° & Total & Infectés & Taux & Total & Infectés & Taux & Erreur & t~[s] \\ \cline{1-9}
0013 & 150 & 22 & 14.7~\% & 153 & 23 & 15.0~\% & 2.5~\% & 68 \\
\begin{table}[htbp]
\setlength{\tabcolsep}{3pt}
\centering
-\begin{tabular}{ r || r | r | r || r | r | r || r || r}
+\begin{tabular}{ r || r | r | r || r | r | r || r || r }
& \multicolumn{3}{ c| }{Mesuré} & \multicolumn{3}{ c| }{Détecté} & & \\ \cline{2-8}
N° & Total & Infectés & Taux & Total & Infectés & Taux & Erreur & t~[s] \\ \cline{1-9}
0013 & 150 & 22 & 14.7~\% & 153 & 22 & 14.4~\% & -2.0~\% & 429 \\
\label{tab:resultats100}
\end{table}
-Le tableau ci dessous sont les résultat du processus appliqué aux images originales.
-
-
+Le taux moyen d'infection réel est de 17.3~\%, le taux détecté avec des images réduites
+\begin{table}[htbp]
+\setlength{\tabcolsep}{3pt}
+\centering
+\begin{tabular}{ l | r | r | r }
+ & Réel & 40~\% & 100~\% \\ \cline{2-4}
+ Taux de cellules infectées moyen détecté & 17.3~\% & 18.2~\% & 17.9~\% \\
+ Écart type sur le taux de cellules infectées moyen détecté & 2.9~\% & 3.1~\% & 3.0~\% \\ \cline{1-4}
+ Erreur moyenne & & 5.3~\% & 3.7~\% \\
+ Écart type sur l'erreur & & 11.9~\% & 11.4~\% \\ \cline{1-4}
+ Temps moyen & & 78~s & 485~s
+\end{tabular}
+\caption{Résultats résumant les tables \ref{tab:resultats60} et \ref{tab:resultats100}. La colonne 40~\% correspond aux images réduites. La colonne 100~\% correspond aux images originales}
+\label{tab:resultats}
+\end{table}
-<TODO> : Analyses les résultats
\section{Conclusion}
-<TODO>Travaux futurs ?
+<TODO>
+* Travaux futurs ?
+* Gain de vitesse et de précision par rapport aux méthodes actuellement utilisées.
\bibliographystyle{plain}
\bibliography{main}
imgV = uint8(255 - 255 * imgHSV(:, :, 3));
imwrite(imgV, '../output/imgV.png')
+ % Uncomment one of the following line to display the histogram of the
+ % hue or saturation component.
% histogram(imgH, 'BinLimits', [70, 130], 'FaceColor', 'black', 'EdgeColor', 'black', 'FaceAlpha', 1);
% histogram(imgS, 'BinLimits', [100, 210], 'FaceColor', 'black', 'EdgeColor', 'black', 'FaceAlpha', 1);
% p. 136: median filter to smooth the noise and
- % area closing to enhance the bright objects and make flatter, darker and cleaner the image background.
+ % area closing to enhance the bright objects and make flatter and cleaner the image background.
medianFilterWindow = [5, 5];
imgFiltered{1} = mmareaclose(medfilt2(imgH, medianFilterWindow), 1000); % Hue.
imgFiltered{2} = mmareaclose(medfilt2(imgS, medianFilterWindow), 1000); % Saturation.
%% Parameters
+interactive = true;
defaultDir = '../imgs_corrMK/';
-if exist('imagePath', 'var')
- defaultPath = imagePath;
-else
- defaultPath = defaultDir;
-end
-
-% imagePathInput = ...
-% inputdlg('Enter the path to the input image: ', 'Sample', ...
-% [1 100], {defaultPath});
-
-imagePathInput = {
- [defaultDir '13.05.2015/1412151257-100x-Teinte30-Saturation3-0013.tif']
- [defaultDir '13.05.2015/1412151257-100x-Teinte30-Saturation3-0014.tif']
- [defaultDir '22.05.2015/1412151257-100x-avecfiltre-lum8-676ms-Teinte0-Saturation0- SatMachine150-0036.png']
- [defaultDir '22.05.2015/1412151257-100x-avecfiltre-lum8-676ms-Teinte0-Saturation0- SatMachine150-0038.png']
- [defaultDir '22.05.2015/1412151257-100x-avecfiltre-lum8-676ms-Teinte0-Saturation0- SatMachine150-0039.png']
- [defaultDir '22.05.2015/1412151257-100x-avecfiltre-lum8-676ms-Teinte0-Saturation0- SatMachine150-0040.png']
- [defaultDir '22.05.2015/1412151257-100x-avecfiltre-lum8-676ms-Teinte0-Saturation0- SatMachine150-0041.png']
- [defaultDir '22.05.2015/1412151257-100x-avecfiltre-lum8-676ms-Teinte0-Saturation0- SatMachine150-0042.png']
- [defaultDir '22.05.2015/1412151257-100x-avecfiltre-lum8-676ms-Teinte0-Saturation0- SatMachine150-0043.png']
- [defaultDir '22.05.2015/1412151257-100x-avecfiltre-lum8-676ms-Teinte0-Saturation0- SatMachine150-0044.png']
- [defaultDir '22.05.2015/1412151257-100x-avecfiltre-lum8-676ms-Teinte0-Saturation0- SatMachine150-0045.png']
- [defaultDir '22.05.2015/1412151257-100x-avecfiltre-lum8-676ms-Teinte0-Saturation0- SatMachine150-0046.png']
- };
-
-
-scaleFactor = 1;
+scaleFactor = 0.6;
invertSaturation = true;
-
+
+if interactive
+ if exist('imagePath', 'var')
+ defaultPath = imagePath;
+ else
+ defaultPath = defaultDir;
+ end
+
+ imagePathInput = ...
+ inputdlg('Enter the path to the input image: ', 'Sample', ...
+ [1 100], {defaultPath});
+else
+ imagePathInput = {
+ [defaultDir '13.05.2015/1412151257-100x-Teinte30-Saturation3-0013.tif']
+ [defaultDir '13.05.2015/1412151257-100x-Teinte30-Saturation3-0014.tif']
+ [defaultDir '22.05.2015/1412151257-100x-avecfiltre-lum8-676ms-Teinte0-Saturation0- SatMachine150-0036.png']
+ [defaultDir '22.05.2015/1412151257-100x-avecfiltre-lum8-676ms-Teinte0-Saturation0- SatMachine150-0038.png']
+ [defaultDir '22.05.2015/1412151257-100x-avecfiltre-lum8-676ms-Teinte0-Saturation0- SatMachine150-0039.png']
+ [defaultDir '22.05.2015/1412151257-100x-avecfiltre-lum8-676ms-Teinte0-Saturation0- SatMachine150-0040.png']
+ [defaultDir '22.05.2015/1412151257-100x-avecfiltre-lum8-676ms-Teinte0-Saturation0- SatMachine150-0041.png']
+ [defaultDir '22.05.2015/1412151257-100x-avecfiltre-lum8-676ms-Teinte0-Saturation0- SatMachine150-0042.png']
+ [defaultDir '22.05.2015/1412151257-100x-avecfiltre-lum8-676ms-Teinte0-Saturation0- SatMachine150-0043.png']
+ [defaultDir '22.05.2015/1412151257-100x-avecfiltre-lum8-676ms-Teinte0-Saturation0- SatMachine150-0044.png']
+ [defaultDir '22.05.2015/1412151257-100x-avecfiltre-lum8-676ms-Teinte0-Saturation0- SatMachine150-0045.png']
+ [defaultDir '22.05.2015/1412151257-100x-avecfiltre-lum8-676ms-Teinte0-Saturation0- SatMachine150-0046.png']
+ };
+end
+
%% Main
for k = 1:length(imagePathInput)
+++ /dev/null
-% The method used by the function SegmentationO
-classdef SegmentationMethod
- enumeration
- % Segmentation of red blood cells with a distance transform following by applying
- % the watershed algorithm.
- WatershedByDistanceTransform,
-
- % Segmentation of red blood celles with morphological gradient following by applying
- % the watershed algorithm.
- WatershedByMorphologicalGradient
- end
-end
-
filepath_img = [imagesFolder, '/', filepath];
img = imread(filepath_img);
end
-
%% Display
img = a;
+gridColor = 'white';
+drawNumbers = true;
+drawAxisLabels = true;
+drawCenter = false; % For SE.
imagesc(img)
% Print each pixel value except the zero values.
-for i = 1:size(img, 1)
- for j = 1:size(img, 2)
- v = img(i, j);
- if v ~= 0
- text(j-0.1, i, num2str(v), 'fontsize', 18);
+imageMin = min(min(img));
+imageMax = max(max(img));
+if drawNumbers
+ for i = 1:size(img, 1)
+ for j = 1:size(img, 2)
+ v = img(i, j);
+ if v ~= 0
+ color = 'black';
+ if v < (imageMax - imageMin) / 2
+ color = 'white';
+ end
+ text(j-0.1, i, num2str(v), 'fontsize', 18, 'color', color);
+ end
end
end
end
colormap(gray);
axis image
-
+s
grid on
ax = gca;
ax.XTick = linspace(0.5, 5.5, 6);
ax.YTick = linspace(0.5, 6.5, 7);
-ax.GridColor = 'white';
+ax.GridColor = gridColor;
ax.GridAlpha = 1;
ax.LineWidth = 1;
ax.TickDir = 'out';
ax.XTickLabel = [];
ax.YTickLabel = [];
-% Draw vertical pixel labels.
-for i = 1:size(img, 1)
- text(0, i, num2str(i), 'fontsize', 13);
+if drawAxisLabels
+ % Draw vertical pixel labels.
+ for i = 1:size(img, 1)
+ text(0, i, num2str(i), 'fontsize', 13);
+ end
+
+ % Draw horizontal pixel labels.
+ for j = 1:size(img, 2)
+ text(j, size(img, 1)+0.9, num2str(j), 'fontsize', 13);
+ end
end
-
-% Draw horizontal pixel labels.
-for j = 1:size(img, 2)
- text(j, size(img, 1)+0.9, num2str(j), 'fontsize', 13);
+
+if drawCenter
+ text(2.9, 3, '+', 'fontsize', 25);
end
projects / malaria.git / commitdiff
