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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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+[0.75 0 0 0.75 0 0] CT
+N
+92.892 374 M
+92.892 373.674 L
+95.378 373.674 L
+95.378 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+95.378 374 M
+95.378 374 L
+97.865 374 L
+97.865 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+97.865 374 M
+97.865 373.593 L
+100.351 373.593 L
+100.351 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+100.351 374 M
+100.351 373.606 L
+102.838 373.606 L
+102.838 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+102.838 374 M
+102.838 374 L
+105.324 374 L
+105.324 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+105.324 374 M
+105.324 373.566 L
+107.81 373.566 L
+107.81 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+107.81 374 M
+107.81 373.062 L
+110.297 373.062 L
+110.297 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+110.297 374 M
+110.297 374 L
+112.783 374 L
+112.783 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+112.783 374 M
+112.783 373.206 L
+115.27 373.206 L
+115.27 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+115.27 374 M
+115.27 373.267 L
+117.756 373.267 L
+117.756 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+117.756 374 M
+117.756 367.561 L
+120.243 367.561 L
+120.243 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+120.243 374 M
+120.243 374 L
+122.729 374 L
+122.729 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+122.729 374 M
+122.729 373.414 L
+125.216 373.414 L
+125.216 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+125.216 374 M
+125.216 373.289 L
+127.702 373.289 L
+127.702 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+127.702 374 M
+127.702 374 L
+130.189 374 L
+130.189 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+130.189 374 M
+130.189 373.084 L
+132.675 373.084 L
+132.675 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+132.675 374 M
+132.675 372.853 L
+135.161 372.853 L
+135.161 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+135.161 374 M
+135.161 374 L
+137.648 374 L
+137.648 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+137.648 374 M
+137.648 372.459 L
+140.134 372.459 L
+140.134 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+140.134 374 M
+140.134 371.898 L
+142.621 371.898 L
+142.621 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+142.621 374 M
+142.621 374 L
+145.107 374 L
+145.107 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+145.107 374 M
+145.107 371.261 L
+147.594 371.261 L
+147.594 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+147.594 374 M
+147.594 370.398 L
+150.08 370.398 L
+150.08 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+150.08 374 M
+150.08 374 L
+152.567 374 L
+152.567 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+152.567 374 M
+152.567 369.058 L
+155.053 369.058 L
+155.053 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+155.053 374 M
+155.053 367.579 L
+157.54 367.579 L
+157.54 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+157.54 374 M
+157.54 366.493 L
+160.026 366.493 L
+160.026 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+160.026 374 M
+160.026 374 L
+162.512 374 L
+162.512 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+162.512 374 M
+162.512 365.021 L
+164.999 365.021 L
+164.999 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+164.999 374 M
+164.999 361.706 L
+167.485 361.706 L
+167.485 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+167.485 374 M
+167.485 374 L
+169.972 374 L
+169.972 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+169.972 374 M
+169.972 355.478 L
+172.458 355.478 L
+172.458 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+172.458 374 M
+172.458 343.897 L
+174.945 343.897 L
+174.945 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+174.945 374 M
+174.945 374 L
+177.431 374 L
+177.431 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+177.431 374 M
+177.431 328.619 L
+179.918 328.619 L
+179.918 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+179.918 374 M
+179.918 308.779 L
+182.404 308.779 L
+182.404 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+182.404 374 M
+182.404 374 L
+184.891 374 L
+184.891 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+184.891 374 M
+184.891 285.477 L
+187.377 285.477 L
+187.377 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+187.377 374 M
+187.377 258.97 L
+189.864 258.97 L
+189.864 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+189.864 374 M
+189.864 374 L
+192.35 374 L
+192.35 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+192.35 374 M
+192.35 241.796 L
+194.836 241.796 L
+194.836 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+194.836 374 M
+194.836 229.808 L
+197.323 229.808 L
+197.323 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+197.323 374 M
+197.323 225.613 L
+199.809 225.613 L
+199.809 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+199.809 374 M
+199.809 374 L
+202.296 374 L
+202.296 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+202.296 374 M
+202.296 230.192 L
+204.782 230.192 L
+204.782 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+204.782 374 M
+204.782 243.295 L
+207.269 243.295 L
+207.269 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+207.269 374 M
+207.269 374 L
+209.755 374 L
+209.755 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+209.755 374 M
+209.755 247.786 L
+212.242 247.786 L
+212.242 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+212.242 374 M
+212.242 261.54 L
+214.728 261.54 L
+214.728 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+214.728 374 M
+214.728 374 L
+217.215 374 L
+217.215 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+217.215 374 M
+217.215 293.356 L
+219.701 293.356 L
+219.701 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+219.701 374 M
+219.701 299.968 L
+222.188 299.968 L
+222.188 374 L
+cp
+f
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+N
+222.188 374 M
+222.188 374 L
+224.674 374 L
+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
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+cp
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+cp
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+[0.75 0 0 0.75 0 0] CT
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+383.807 374 L
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+cp
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+cp
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+[0.75 0 0 0.75 0 0] CT
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+cp
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+[0.75 0 0 0.75 0 0] CT
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+cp
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+cp
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+cp
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+[0.75 0 0 0.75 0 0] CT
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+cp
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+cp
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+cp
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+cp
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+408.672 374 L
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+cp
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+cp
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+cp
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+416.131 374 L
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+cp
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+cp
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+421.104 374 L
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+cp
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+423.591 374 L
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+cp
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+426.077 374 L
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+cp
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+428.564 374 L
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+cp
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+431.05 374 L
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+cp
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+433.536 374 L
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+cp
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+436.023 374 L
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+cp
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+438.509 374 L
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+cp
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+440.996 374 L
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+cp
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+443.482 374 L
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+cp
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+445.969 374 L
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+cp
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+448.455 374 L
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+cp
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+cp
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+453.428 374 L
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+cp
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+455.915 374 L
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+cp
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+455.915 374 L
+458.401 374 L
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+cp
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+460.888 374 L
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+cp
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+463.374 374 L
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+cp
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+465.86 374 L
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+cp
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+468.347 374 L
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+cp
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+470.833 374 L
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+cp
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+87.919 374 L
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+87.919 374 L
+90.405 374 L
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+97.865 374 L
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+105.324 374 L
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+112.783 374 L
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+2 setlinecap
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+122.729 374 L
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+130.189 374 L
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+S
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+137.648 374 L
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+S
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+2 setlinecap
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+142.621 374 L
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+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
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+S
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+GS
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+2 setlinecap
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+150.08 374 L
+S
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+152.567 374 L
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+GS
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+2 setlinecap
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+GS
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+162.512 374 L
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+S
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+GS
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+2 setlinecap
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+167.485 374 L
+169.972 374 L
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+GS
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+GR
+GS
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+2 setlinecap
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+174.945 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
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+174.945 374 M
+174.945 374 L
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+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
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+179.918 374 L
+S
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+GS
+[0.75 0 0 0.75 0 0] CT
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+182.404 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
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+182.404 374 L
+184.891 374 L
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+S
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+GS
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+2 setlinecap
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+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
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+189.864 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
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+189.864 374 M
+189.864 374 L
+192.35 374 L
+192.35 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
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+192.35 241.796 L
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+194.836 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
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+194.836 229.808 L
+197.323 229.808 L
+197.323 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
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+197.323 374 M
+197.323 225.613 L
+199.809 225.613 L
+199.809 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
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+199.809 374 M
+199.809 374 L
+202.296 374 L
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+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
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+202.296 374 M
+202.296 230.192 L
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+204.782 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
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+204.782 243.295 L
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+207.269 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
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+207.269 374 M
+207.269 374 L
+209.755 374 L
+209.755 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
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+209.755 247.786 L
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+212.242 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
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+212.242 261.54 L
+214.728 261.54 L
+214.728 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
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+214.728 374 M
+214.728 374 L
+217.215 374 L
+217.215 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
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+217.215 293.356 L
+219.701 293.356 L
+219.701 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
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+N
+219.701 374 M
+219.701 299.968 L
+222.188 299.968 L
+222.188 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
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+N
+222.188 374 M
+222.188 374 L
+224.674 374 L
+224.674 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
+10.0 ML
+N
+224.674 374 M
+224.674 313.328 L
+227.16 313.328 L
+227.16 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
+10.0 ML
+N
+227.16 374 M
+227.16 319.115 L
+229.647 319.115 L
+229.647 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
+10.0 ML
+N
+229.647 374 M
+229.647 374 L
+232.133 374 L
+232.133 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
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+N
+232.133 374 M
+232.133 329.579 L
+234.62 329.579 L
+234.62 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
+10.0 ML
+N
+234.62 374 M
+234.62 334.998 L
+237.106 334.998 L
+237.106 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
+10.0 ML
+N
+237.106 374 M
+237.106 340.033 L
+239.593 340.033 L
+239.593 374 L
+S
+GR
+GS
+[0.75 0 0 0.75 0 0] CT
+2 setlinecap
+10.0 ML
+N
+239.593 374 M
+239.593 374 L
+242.079 374 L
+242.079 374 L
+S
+GR
+GS
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+ /UnderlineThickness get Ts mul /Tt exch def
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+%%Version: 1.0 0
+%%Copyright: (Copyright 2002-2003 The Apache Software Foundation. License terms: http://www.apache.org/licenses/LICENSE-2.0)
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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 7/minusplus put
+dup 8/circleplus put
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+dup 10/circlemultiply put
+dup 11/circledivide put
+dup 12/circledot put
+dup 13/circlecopyrt put
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+dup 15/bullet put
+dup 16/equivasymptotic put
+dup 17/equivalence put
+dup 18/reflexsubset put
+dup 19/reflexsuperset put
+dup 20/lessequal put
+dup 21/greaterequal put
+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
+dup 29/greatermuch put
+dup 30/precedes put
+dup 31/follows put
+dup 32/arrowleft put
+dup 127/spade put
+dup 128/arrowleft put
+dup 33/minus put
+dup 34/periodcentered put
+dup 35/multiply put
+dup 36/asteriskmath put
+dup 37/divide put
+dup 38/diamondmath put
+dup 39/plusminus put
+dup 40/minusplus put
+dup 41/circleplus put
+dup 42/circleminus put
+dup 43/circlemultiply put
+dup 44/circledivide put
+dup 45/circledot put
+dup 46/circlecopyrt put
+dup 47/openbullet put
+dup 48/bullet put
+dup 49/equivasymptotic put
+dup 50/equivalence put
+dup 51/reflexsubset put
+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
+dup 66/spade put
+readonly def
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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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+/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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+%%Trailer
+%%Pages: 1
+%%EOF
\usepackage{upquote}
\usepackage{color}
\usepackage[usenames,dvipsnames]{xcolor}
-\usepackage{epstopdf}
\usepackage{fancyhdr}
\usepackage{courier}
\usepackage{pifont}
\begin{center}
\textbf{Résumé}
\end{center}
-Cet article a pour but la description et la mise en oeuvre d'une méthode automatique permettant l'établissement de la parasitémie de la malaria d'un patient à partir d'une ou plusieurs images microscopiques d'un frottis sanguin.
+Cet article a pour but la description et la mise en oeuvre d'une méthode automatique permettant l'établissement de la parasitémie d'un patient infecté par la malaria. Cette méthode se base sur une ou plusieurs images microscopiques d'un frottis sanguin.
\vspace{5cm}
% TODO: Future dans l'intro?
-Le but de ce projet est d'établir une méthode complète et non-supervisée afin de dénombrer les erythrocytes et de les classer en deux catégories de cellules à savoir les saines et les infectées. Celle ci sera ensuite implémentée sous la forme d'un logiciel qui puisse être facilement utilisé par une personne avec un minimum de connaissance en informatique.
+Le but de ce projet est d'établir une méthode complète et non-supervisée afin de dénombrer les érythrocytes et de les classer en deux catégories de cellules à savoir les saines et les infectées. Cette méthode sera ensuite implémentée sous la forme d'un logiciel qui puisse être facilement utilisé par une personne avec un minimum de connaissance en informatique.
-Cette réalisation est faite en partenariat avec le laboratoire de parasitologie du CHUV
+Cette réalisation est faite en partenariat avec le Dr. Guy Prod'hom de l'institut de microbiologie du \emph{CHUV} à Lausanne. % Ajouter du blabla
-Images 100x
-Images 50x
+% Problème du comptage manuel
-CHUV
+% \subsection{Images sources}
-\subection{Parasitémie de la malaria}
+% Mentionner certains problèmes comme les débris ou les plaquettes
-% Parlé du cycle de développement, à quel stade de développement on compte les rbc infecté, etc..
-
-
-\subsection{Images}
-
-Les images initiales correspondent à des photographies de sang infecté agrandies 100 fois. Une \emph{coloration de May-Grünwald Giemsa} est utilisée 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 (figure~\ref{fig:medical-plasmodium}).
+Les images à disposition correspondent à des photographies de sang infecté agrandies 50 fois. Une \emph{coloration de May-Grünwald Giemsa} est utilisée 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 (figure~\ref{fig:medical-plasmodium}).
\begin{figure}[htbp]
\centering
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}
\label{fig:medical-plasmodium}
\end{figure}
+%\subsection{Parasitémie de la malaria}
-\section{Outils utilisés}
-
-Cette section aborde brièvement la description des différents outils morphologiques ainsi que les logiciels qui ont permis de mettre en pratique le processus.
-
-
-\subsection{Outils morphologiques}
-
-Une grande partie du processus décrit dans ce document s'appuie sur la morphologie mathématique\footnote{Georges Matheron et Jean Serra - 1964}\cite{Talbot-Morphologie}\cite{Manzanera-Morphologie}, ici appliquée au traitement d'images.
-
-Nous passons en revue un certain nombre d'opérateurs qui sont utilisés dans le processus. Pour chacun d'entre eux, nous donnons une brève explication, une définition formelle et la fonction correspondante utilisée dans \emph{MATLAB}.
-
-
-\subsubsection{Définitions de base}
-
-Nous définissions tout d'abord l'image $E$ comme étant un sous-ensemble à deux dimensions de $\mathbb{Z}$ puis $B$ comme un sous-ensemble de $E$ que nous nommons \emph{élément structurant}. $B_x$ correspond à $B$ translaté de $x$ dont $x$ appartient à $E$. $\breve{B}$ est l'élément symétrique de $B$, si l'élément $B$ est symétrique, alors $\breve{B} = B$.
-
-{\setlength{\abovedisplayskip}{0pt}
-\begin{flalign*}
- E &= \mathbb{Z}^2 & \\
- B_x &= \{ b + x \mid b \in B \} & \\
- \breve{B} &= \{ -b \mid b \in B \} &
-\end{flalign*}
-
-\subsubsection{Dilatation et érosion}
-
-La dilatation est définie comme la somme de \emph{Minkowski} et nommée $\delta$. L'érosion est nommée $\epsilon$.
-
-{\setlength{\abovedisplayskip}{0pt}
-\begin{flalign*}
- \delta_B(X) &= X \oplus B = \{ x \mid \breve{B}_x \cap{X} \neq \emptyset \} &\\
- \epsilon_B(X) &= X \ominus B = \{x \mid B_x \subset X \} &
-\end{flalign*}
-
-Les deux fonctions \emph{MATLAB} associées sont \texttt{mmdil} et \texttt{mmero}.
-
-
-\subsubsection{Ouverture et fermeture}
-
-L'ouverture et la fermeture sont définies à partir de dilatation et d'érosion et sont nommées $\gamma$, respectivement $\phi$.
-
-{\setlength{\abovedisplayskip}{0pt}
-\begin{flalign*}
- \gamma_B(X) &= X \circ B = \delta_B (\epsilon_B (X)) &\\
- \phi_B(X) &= X \bullet B = \epsilon_B (\delta_B (X)) &
-\end{flalign*}
-
-Les deux fonctions \emph{MATLAB} associées sont \texttt{mmopen} et \texttt{mmclose}.
-
+L'objectif 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}. Ce stade est appelé \emph{ring stage} car les parasites ont une forme d'anneau. Cet anneau, montré en détail par la figure~\ref{fig:noyau-cytoplasme}, comprend un noyau ainsi qu'un cytoplasme l'entourant.
-\subsubsection{Ouverture et fermeture par aire}
-
-L'ouverture et la fermeture par aire sont nommées $\gamma_{\lambda{}B}$ et $\phi_{\lambda{}B}$. Ces opérateurs permettent de raboter les bosses, respectivement de combler les trous ayant une aire égale ou inférieure à $\lambda$ et dont chaque élément est connexe à au moins un autre par l'élément structurant $B$.
-
-La figure~\ref{fig:fermeture-par-aire} illustre cela en montrant une vue en deux dimensions de deux fermetures ayant pour valeur $\lambda_1$ et $\lambda_2$.
+La parasitémie est établit en pourcentage comme étant le rapport entre nombre d'érythrocytes infecté et le nombre total d'érythrocyte. Le group d'experts \emph{UK NEQAS} (\emph{United Kingdom National External Quality Assessment Service}) recommande de considéré un minimum de 1000 erythrocytes. Le \emph{CDC} (\emph{Centers for Disease Control and Prevention}) recommande quant à lui au moins 500 érythrocytes si la parasitémie est supérieure à 10~\% et au moins 2000 érythrocytes si la parasitémie est inférieure à 0.1~\%.
\begin{figure}[htbp]
\centering
- \includegraphics[width=0.5\linewidth]{img/schema_fermeture_par_aire.eps}
- \caption{Vue en deux dimensions de deux fermetures par aire ayant des valeurs de $\lambda$ différentes}
- \label{fig:fermeture-par-aire}
+ \includegraphics[width=0.5\linewidth]{img/RBC_noyau_cytoplasme.pdf}
+ \caption{Détail des deux parties constituantes d'un trophozoïte immature}
+ \label{fig:noyau-cytoplasme}
\end{figure}
-Les deux fonctions \emph{MATLAB} associées sont \texttt{mmareaopen} et \texttt{mmareaclose}.
-
-
-\subsubsection{Reconstruction par dilatation}
-
-Une reconstruction par dilatation est réalisée sur une image $G$ à partir d'un marqueur $F$ et d'un élément structurant $B$. Elle consiste à l'application successive de dilatations sur $F$ à l'aide de $B$ en ne gardant que les éléments appartenant à $G$ jusqu'à idempotence.
-
-{\setlength{\abovedisplayskip}{0pt}
-\begin{flalign*}
- F_0 &= F &\\
- F_{n+1} &= (F_n \oplus B) \cap G &
-\end{flalign*}
-
-La fonction \emph{MATLAB} associée est \texttt{mminfrec}.
-
-
-\subsubsection{Ligne de partage des eaux}
-
-La ligne de partage des eaux (\emph{watersheds} en anglais) consiste à considérer une image comme une surface en trois dimensions où les valeurs d'intensité correspondent à des hauteurs puis à remplir progressivement les creux avec de l'eau jusqu'à que toute l'image soit entièrement remplie. Lorsque deux bassins se rejoignent pendant la montée des eaux, une digue est construite. L'ensemble des digues représente alors le résultat final.
-
-Dans notre cas, nous utilisons des marqueurs pour définir à partir de quels creux l'eau est injectée.
-
-La fonction \emph{MATLAB} associée est \texttt{mmcwatershed}.
-
-
-\subsubsection{Maximaux régionaux}
-
-Les maximaux régionaux d'une image sont définis comme suit. La première ligne correspond à la définition de la connectivité des éléments de $M$. $B$ est un élément structurant.
-
-{\setlength{\abovedisplayskip}{0pt}
-\begin{flalign*}
- \begin{cases}
- \forall p \in M & \text{si } f(p) = t, \quad \exists p' \in M, \quad p' \neq p, \quad p' \in (p \oplus B) \\
- \forall p \in (M \oplus B) \setminus M & \text{si } f(p) < t
- \end{cases}&&
-\end{flalign*}
-
-La fonction \emph{MATLAB} associée est \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.
+Certains artefacts comme des débris ou des plaquettes peuvent venir se supperposer aux érythrocyte et être confondus avec des parasites. Ces cas peuvent être relativement fréquents et altérer significativement la parasitémie si comptés comme étant des cas positifs d'infection. La figure~\ref{fig:RBC-artefact} montre quelques exemple de faux positifs potentiels.
\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}. Les pixels noires valent zéro}
- \label{fig:regmax_sample_original}
+ \begin{subfigure}[t]{0.2\textwidth}
+ \includegraphics[width=\linewidth]{img/RBC-plaquette.jpg}
+ \caption{Thrombocyte}
+ \label{fig:RBC-artefact-plaquette}
\end{subfigure}
~
- \begin{subfigure}[b]{0.15\textwidth}
- \includegraphics[width=\linewidth]{img/regmax_sample_se.eps}
- \caption{L'élément structurant : un carré de taille 5 symétrique}
- \label{fig:regmax_sample_se}
+ \begin{subfigure}[t]{0.2\textwidth}
+ \includegraphics[width=\linewidth]{img/RBC-debris-1.jpg}
+ \caption{Débris}
+ \label{fig:RBC-artefact-debris-1}
\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}. Les pixels noires valent zéro}
- \label{fig:regmax_sample_result}
+ \begin{subfigure}[t]{0.2\textwidth}
+ \includegraphics[width=\linewidth]{img/RBC-debris-2.jpg}
+ \caption{Débris}
+ \label{fig:RBC-artefact-debris-2}
+ \end{subfigure}
+ ~
+ \begin{subfigure}[t]{0.2\textwidth}
+ \includegraphics[width=\linewidth]{img/RBC-debris-3.jpg}
+ \caption{Débris}
+ \label{fig:RBC-artefact-debris-3}
\end{subfigure}
- \caption{Exemple d'application de la fonction \texttt{mmregmax}. Les deux pixels de valeur 5 forment une zone connexe : celle-ci est écartée car la dilaté du pixel de valeur 8 par l'élément structurant contient un des pixels de valeur 5}
- \label{fig:regmax_sample}
+ \caption{Artefacts se supperposant à des érytrocytes}
+ \label{fig:RBC-artefact}
\end{figure}
-\subsection{Outils logiciels}
-
-L'implémentation est réalisée à l'aide de \emph{MATLAB 2015a}. Les différentes fonctions morphologiques (préfixées par \texttt{mm}) proviennent de la bibliothèque nommée \emph{SDC Morphology Toolbox for MATLAB}\footnote{\url{http://mmorph.com/}}.
-
-\section{Approche générale}
-Le processus complet est illustré par la figure \ref{fig:processusComplet}. Les différents chiffres correspondent aux étapes décrites ci-après. Il y a deux phases principales : le traitement des éléments colorés (étapes 1 à 3) et la segmentation des globules rouges (étapes 4 et 5).
+\section{Aperçu du processus}
-La première étape (\ding{192}) consiste à détecter les éléments qui ont été 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 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.
+La figure~\ref{fig:processusComplet} montre le processus complet du traitement d'une image de frottis sanguin. Celui ci peut-être divisé en trois groupe de traitement à savoir la ségmentation des érytrocytes, la ségmentation des parasites et finalement la classification des cellules.
\begin{figure}[htbp]
\centering
- \includegraphics[width=1\linewidth]{img/schema_processus.eps}
+ \includegraphics[width=0.8\linewidth]{img/schema_processus_2.pdf}
\caption{Le processus complet.}
\label{fig:processusComplet}
\end{figure}
+Dans le premier groupe, l'on va chercher à construire des ellipses qui correspondent le mieux aux bords des érythrocytes. Pour ce faire une estimation du rayon moyen est réalisée par granulométrie à l'aide d'une succession de fermeture par aire. La construction des ellipses utilise les pixels des bords des erytrocytes ainsi que son gradient, la méthode \emph{RANSAC} permet de créer un grand nombre d'ellipses candidates auquel un score est attribué. Finalement les ellipses sont élaguée en supprimant celles chevauchées par d'autres ellipses sont le score est supérieur.
-\section{Détail du processus}
+Dans le deuxième groupe, les zones colorées et sombre vont être extraites en comparant les valeurs d'intensité de l'image avec la moyenne d'intensité des éléments de l'avant-plan. Ces zones sombres correspondent aux leucocytes, aux plaquettes et à certains débris. Une fermeture morphoplogique, dont la taille de l'élément structurant est calculé en fonction du rayon moyen, va permettre de mettre en évidence le cytoplasme des parasites. Les noyaux sont, quant à eux, mis en évidence à l'aide d'une fermeture par aire.
-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 troisième groupe correspond à la mise en commun des informations du premier et du deuxième groupe afin de définir les erytrocytes et de les classer. La parasitémie est alors établit en calculant le rapport entre la population d'érythrocytes infecté et celle saine.
-La qualité du résultat final n'est pas très affectée par la 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 ($\mli{imgRGB}$)}
- \label{fig:imgRGB}
-\end{figure}
-
-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.
-
-\subsection{Détection des parasites}
-\label{detection-parasites}
+\section{Détail de la méthode}
-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.
+% filtrage préliminaire (taille d'un érythrocyte: 7.5 \micro~m)
-\subsubsection{Filtrage préliminaire}
+Les filtres gaussiens appliquées initialement vont permettre de supprimer une partie du bruit haute-fréquence. L'écart type de chaque filtre est choisi en fonction de la taille des objets. La taille d'un érytrocyte est d'environ 7.5~\textmu{m} et la taille d'un parasite d'environ 2.5~\textmu{m}. Des écarts types de 0.2~\textmu{m} et 0.15~\textmu{m} sont choisis pour pour la ségmentation des érytrocytes respectivement la ségmentation des parasites.
-Il est nécessaire, pour la suite du traitement, d'avoir les intensités les plus élevées pour les éléments colorés pour les deux composantes \emph{teinte} et \emph{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 $\mli{imgH}$ 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 ($\mli{imgH}$)}
- \label{fig:histogram_imgH}
-\end{figure}
-
-\begin{figure}[htbp]
- \centering
- \includegraphics[width=0.8\linewidth]{img/histogram_imgS.eps}
- \caption{Histogramme de la saturation ($\mli{imgS}$)}
- \label{fig:histogram_imgS}
-\end{figure}
+\subsection{Identification des érytrocytes}
-Afin de gommer le bruit et de rendre le fond plus homogène, un filtre médian 5 x 5 puis 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.
+Le but de cette étape est d'identifier les érythrocytes. Le profil de ceux-ci est un disque biconcave et apparait de manière plus ou moins déformée sur les photographies. Ces déformations les font prendre une forme plus ou moins elliptique dont le rapport entre le grand axe et le petit axe n'excède pas 1.6 ($\pm 23~\%$ d'un rayon moyen).
-\begin{figure}[htbp]
- \centering
- \includegraphics[width=0.8\linewidth]{img/imgHFiltered.jpg}
- \caption{Composante \emph{teinte} après filtrage ($\mli{imgFiltered\{1\}}$) représentée en fausse couleur}
- \label{fig:imgHFiltered}
-\end{figure}
+Le résultat de cette recherche sera une liste d'ellipses pouvant se recouvrir partiellement. Les ellipses sont décrites par les paramètres montrés par la figure~\ref{fig:ellipse-parametres}.
\begin{figure}[htbp]
\centering
- \includegraphics[width=0.8\linewidth]{img/imgSFiltered.jpg}
- \caption{Composante \emph{saturation} après filtrage ($\mli{imgFiltered\{2\}}$) représentée en fausse couleur}
- \label{fig:imgSFiltered}
+ \includegraphics[width=0.5\linewidth]{img/ellipse-parametres.pdf}
+ \caption{Paramètre d'une ellipse, $\alpha \in [0, \pi[$ est l'angle d'inclinaison du grand axe $a$, $a \geqslant b$ > 0}
+ \label{fig:ellipse-parametres}
\end{figure}
\subsubsection{Granulométrie}
-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{fig: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 1 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 (spectre granulométrique) est calculée pour chaque itération $i$ comme suit : $\lvert N_{i + 1} - N_i \rvert$.
-\end{sloppypar}
-
-Nous définissons le rayon moyen des globules rouges ($\mli{RBCRadius}$) comme étant le rayon de l'élément structurant correspondant à la valeur maximale parmi le spectre granulométrique. 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 d'un spectre granulométrique. Dans ce cas $\mli{RBCRadius}$ est égal à 35 et $\mli{nucleiRadius}$ à 7.
-
-\begin{figure}[htbp]
- \centering
- \includegraphics[width=0.8\linewidth]{img/pattern_spectrum_granulometry.eps}
- \caption{Spectre granulométrique}
- \label{fig:patternSpectrum}
-\end{figure}
-
-\subsubsection{Segmentation des éléments colorés}
-
-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*}
- \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, $\mli{{\mu}H}$ 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*}
- \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 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*}
- \mli{THS} &= \gamma_{10}(\mli{TH} \cap \mli{TS}) &
-\end{flalign*}
-
-\begin{figure}[htbp]
- \centering
- \includegraphics[width=0.8\linewidth]{img/THS.jpg}
- \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 ($\mli{THS}$) 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 une reconstruction par dilatation des éléments colorés en utilisant le marqueur précédent permet de ne garder que les globules blancs.
-
-Finalement nous lissons 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 ($\mli{WBC}$)}
- \label{fig:WBC}
-\end{figure}
+L'objectif est de déterminer le rayon moyen des érytrocytes. Pour ce faire nous allons appliquer une série de fermetures par aire et calculer les différences des sommes des intensités de chaque pixel. Chaque aire est calculée comme étant celui d'un cercle de rayon donné. Les rayons des aires vont être compris autour d'un rayon estimé à partir de la résolution donnée initialement. Les bornes inférieure et supérieure correspondent au rayon estimé minoré de 50~\% respectivement au rayon estimé majoré de 50~\%.
+La fermeture suppose ici que les erythrocytes on un niveau d'intensité plus bas que le fond, ce qui est le cas pour la composante verte dans toutes les images étudiées.
-\subsection{Détection des schizontes}
-
-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.
+Deux autres approches on également été essayées, à savoir la fermeture morphologique par un élément structurant en forme de disque et par un élément structurant de forme octogonale. Un des problèmes de ces deux approches est qu'elles ont tendance à sous-estimer légèrement le rayon moyen dans le cas d'éléments elliptiques. La figure~\ref{fig:granulometrie-ellipse-cercle-closing-comparaison} montre à gauche le résultat d'une granulometrie par aire (le cercle et l'ellipse ont la même aire) et à droite le résultat d'une granulometrie morphologique. Le rayon $r$ est plus proche de la moyenne entre les deux rayons de l'ellipse $a$ et $b$ que le rayon $r'$. Il est à noter que dans cet exemple $a = 2b$ et que donc $r = \sqrt{2}b = 1.41b$ ce qui n'est pas très éloigné de la moyenne $(a+b)/2 = (2+1)/2 = 1.5$.
\begin{figure}[htbp]
\centering
- \begin{subfigure}[t]{0.3\textwidth}
- \includegraphics[width=\linewidth]{img/poly-infected_cell.jpg}
- \caption{Globule rouge poly-infecté, quatre sous-ensembles}
- \label{fig:regmax_sample_original}
+ \begin{subfigure}[t]{0.4\textwidth}
+ \includegraphics[width=\linewidth]{img/granulometrie-ellipse-cercle-closing-area.pdf}
+ \caption{Cercle dont la surface est égale à celle de l'ellipse}
+ \label{fig:granulometrie-ellipse-cercle-closing-area}
\end{subfigure}
~
- \begin{subfigure}[t]{0.3\textwidth}
- \includegraphics[width=\linewidth]{img/schizonte.jpg}
- \caption{Un schizonte, sept sous-ensembles}
- \label{fig:schizonte}
- \end{subfigure}
- \caption{Comparaison entre le nombre de sous-ensembles d'une cellule poly-infectée et d'un schizonte}
- \label{fig:poly-et-schizonte}
+ \begin{subfigure}[t]{0.4\textwidth}
+ \includegraphics[width=\linewidth]{img/granulometrie-ellipse-cercle-closing-morpho.pdf}
+ \caption{Cercle dont \emph{r} est égale au petit rayon \emph{a} de l'ellipse}
+ \label{fig:granulometrie-ellipse-cercle-closing-morpho}
+ \end{subfigure}
+ \caption{Comparaison du rayon d'un cercle trouvé par fermeture par aire d'une ellipse (à gauche) et celui trouvé par une fermeture morphologique (à droite)}
+ \label{fig:granulometrie-ellipse-cercle-closing-comparaison}
\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 $B \subseteq A \oplus \mli{SE_k}$ 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 ($\mli{imgRGB}$), 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 ligne de partage des eaux à l'aide d'une transformée en 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 défini comme étant 75~\% du rayon nominal, comme montré ci-dessous.
-
-{\setlength{\abovedisplayskip}{0pt}
-\begin{flalign*}
- \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 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}.
+Un autre problème de la granulométrie à l'aide de fermetures morphologique utilisant un disque comme élément structurant est sa complexité algorithmique quadratique en fonction du rayon du disque comme le montre la figure~\ref{fig:spectre-granulométrique-morpho-cercle}. La figure~\ref{fig:spectre-granulométrique-morpho-cercle} montre un exemple de progression du temps de calcul en fonction du rayon. Une solution à ce problème est l'utilisation d'une approximation du cercle, l'octogone, dont la dilaté ou l'érodé par un autre octogone crée un nouvel octogone. L'associativité de l'erosion ou de la dilatation, montré par l'équation~\ref{eq:dilatation-associatif}, permet la décomposition d'une dilatation ($e_{1} \oplus e_{2}$) en plusieurs dilatations successives ($\oplus~e_{1}$ puis $\oplus~e_{2}$). Cela permet de réduire la complexité algorithmique et ainsi de gagner du temps.
{\setlength{\abovedisplayskip}{0pt}
-\begin{flalign*}
- \mli{grayFlat} &= \gamma_{\mli{RBCRadiusMinArea}}(\mli{green}(\mli{imgRGB})) \setminus \mli{WBC} \setminus \mli{schizonts} &
-\end{flalign*}
-
-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 ($\mli{grayFlat}$)}
- \label{fig:segmentation-grayflat}
-\end{figure}
+\begin{flalign} \label{eq:dilatation-associatif}
+ f' = f \oplus (e_{1} \oplus e_{2}) = (f \oplus e_{1}) \oplus e_{2} &
+\end{flalign}
-\begin{figure}[htbp]
- \centering
- \includegraphics[width=0.8\linewidth]{img/histogram_segmentation_grayFlat.eps}
- \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}
+L'implémentation utilisée ici, \emph{OpenCV}, ne réalise pas, à priori, cette optimisation lorsque l'élément structurant peut être décomposé. De ce fait le temps de calcul, montré par la figure~\ref{fig:spectre-granulométrique-morpho-octogone}, ne se différencie pas de l'utilisation d'un disque. De plus, l'octogone produit un spectre granulométrique (figure~\ref{fig:spectre-granulométrique-morpho-octogone}) moins harmonieux que pour les deux autres granulométries.
+Concernant les mesures du temps de calcul avec des images réelles, l'on constate que la fermeture par aire est beaucoup plus rapide : 8.4~s (figure~\ref{fig:temps-aire}) que la fermeture morphologique : 54~s (figures~\ref{fig:spectre-granulométrique-morpho-cercle} et~\ref{fig:spectre-granulométrique-morpho-octogone}). La première fermeture par aire, pour le premier rayon, prend un peu moins de 5~s puis les suivantes environ 100~ms. Cela s'explique par la recherche initiale des minima puis par l'agrandissement ceux-ci jusqu'à l'aire correspondant au premier rayon. Le détail de cet algorithme se trouve à la section~\ref{} % TODO: référence.
-\subsubsection{Calcul des distances et ligne de partage des eaux}
-
-Après le seuillage des globules rouges, certaines régions comprennent deux ou plusieurs globules rouges qu'il faut séparer. Pour ce faire nous appliquons une transformée en 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 en 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 de recherche de la ligne de partage des eaux est appliqué au négatif des distances en utilisant les maximaux dilatés comme marqueurs. Finalement l'intersection entre le négatif de la ligne de partage des eaux et de $\mli{thresholdOpened}$ correspond au résultat. La figure~\ref{fig:segmentation-watershed} montre les étapes de ce processus.
+Il faut remarquer que le spectre granulométrique par aire comporte un deuxième sommet au rayon 44 qui correspond à certains amas de cellules. Il est nécessaire d'éviter de traiter des images comprenant trop de cellules se touchant. En pratique, ce cas ne survient pas car l'image est alors trop peu lisible pour être exploitée.
\begin{figure}[htbp]
- \centering
- \begin{subfigure}[t]{0.3\textwidth}
- \includegraphics[width=\linewidth]{img/segmentation_watershed_1.jpg}
- \caption{Image binaire initiale ($\mli{thresholdOpened}$)}
- \label{fig:segmentation-watershed-1}
- \end{subfigure}
- ~
- \begin{subfigure}[t]{0.3\textwidth}
- \includegraphics[width=\linewidth]{img/segmentation_watershed_2.jpg}
- \caption{Superposition de trois images : les distances, les maximaux et sa dilaté}
- \label{fig:segmentation-watershed-3}
+ \centering
+ \begin{subfigure}[t]{0.48\textwidth}
+ \includegraphics[width=1\linewidth]{img/Granulometry-pattern-spectrum-area.pdf}
+ \caption{Spectre granulométrique d'une succession de fermetures par aire}
+ \label{fig:spectre-granulométrique-aire}
\end{subfigure}
- ~
- \begin{subfigure}[t]{0.3\textwidth}
- \includegraphics[width=\linewidth]{img/segmentation_watershed_3.jpg}
- \caption{Image après découpe de l'image initiale avec le résultat de la ligne de partage des eaux}
- \label{fig:segmentation-watershed-3}
+ ~
+ \begin{subfigure}[t]{0.48\textwidth}
+ \includegraphics[width=1\linewidth]{img/Granulometry-time-area.pdf}
+ \caption{Temps en milliseconde pour chaque fermeture. Temps total : 8.4~s}
+ \label{fig:temps-aire}
\end{subfigure}
- \caption{Wastershed par transformé en distance}
- \label{fig:segmentation-watershed}
+ \caption{Spectre granulométrique et temps en milliseconde pour chacune des fermetures par aire. Réalisé sur une image 2592 \texttimes 1944 contenant environ 700 érythrocytes}
+ \label{fig:Granulometry-area}
\end{figure}
-
-\subsubsection{Identification et traitement des cellules composées}
-
-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 définie 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.
-
-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 sont réalisés 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 ($\mli{gray}$)}
- \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 $\mli{grayFlat}$}
- \label{fig:segmentation-composites-2}
+ \begin{subfigure}[t]{0.48\textwidth}
+ \includegraphics[width=1\linewidth]{img/Granulometry-pattern-spectrum-morpho-circle.pdf}
+ \caption{Spectre granulométrique d'une succession de fermetures morphologiques avec un disque}
+ \label{fig:spectre-granulométrique-morpho-cercle}
\end{subfigure}
- ~
- \begin{subfigure}[t]{0.3\textwidth}
- \includegraphics[width=\linewidth]{img/segmentation_composites_3.jpg}
- \caption{Ouverture de \emph{(a)} en utilisant \emph{(b)} comme masque}
- \label{fig:segmentation-composites-3}
- \end{subfigure}
~
- \begin{subfigure}[t]{0.3\textwidth}
- \includegraphics[width=\linewidth]{img/segmentation_composites_4.jpg}
- \caption{Gradient morphologique}
- \label{fig:segmentation-composites-4}
- \end{subfigure}
-
- \begin{subfigure}[t]{0.3\textwidth}
- \includegraphics[width=\linewidth]{img/segmentation_composites_5.jpg}
- \caption{Remplissage des trous}
- \label{fig:segmentation-composites-5}
- \end{subfigure}
- ~
- \begin{subfigure}[t]{0.3\textwidth}
- \includegraphics[width=\linewidth]{img/segmentation_composites_6.jpg}
- \caption{Amincissement}
- \label{fig:segmentation-composites-6}
+ \begin{subfigure}[t]{0.48\textwidth}
+ \includegraphics[width=1\linewidth]{img/Granulometry-time-morpho-circle.pdf}
+ \caption{Temps en milliseconde pour chaque fermeture. Temps total : 54~s}
+ \label{fig:temps-morpho-cercle}
\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 marqués par \emph{(f)}}
- \label{fig:segmentation-composites-7}
- \end{subfigure}
- \caption{Processus de séparation des cellules composées}
- \label{fig:segmentation-composites}
+ \caption{Spectre granulométrique et temps en milliseconde pour chacune des fermetures morphologique avec un disque comme élément structurant. Réalisé sur une image 2592 \texttimes 1944 contenant environ 700 érythrocytes}
+ \label{fig:Granulometry-morpho-circle}
\end{figure}
\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}
- \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 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 ($\mli{redCells}$)}
- \label{fig:segmentation_finale}
+ \begin{subfigure}[t]{0.48\textwidth}
+ \includegraphics[width=1\linewidth]{img/Granulometry-pattern-spectrum-morpho-octagon.pdf}
+ \caption{Spectre granulométrique d'une succession de fermetures morphologiques avec un octogone}
+ \label{fig:spectre-granulométrique-morpho-octogone}
+ \end{subfigure}
+ ~
+ \begin{subfigure}[t]{0.48\textwidth}
+ \includegraphics[width=1\linewidth]{img/Granulometry-time-morpho-octagon.pdf}
+ \caption{Temps en milliseconde pour chaque fermeture. Temps total : 54~s}
+ \label{fig:temps-morpho-octogone}
+ \end{subfigure}
+ \caption{Spectre granulométrique et temps en milliseconde pour chacune des fermetures morphologique avec un octogone comme élément structurant. Réalisé sur une image 2592 \texttimes 1944 contenant environ 700 érythrocytes}
+ \label{fig:Granulometry-morpho-octagon}
\end{figure}
-\subsection{Identification des globules rouges infectés}
+\subsubsection{Calcul du gradient}
-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 $\mli{infectedRedCells}$.
+\subsubsection{Recherche des bords des érythrocytes}
-La figure~\ref{fig:globules_rouges_infectes} montre les globules rouges infectés.
+\subsubsection{Recherche d'ellipses}
-\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 ($\mli{infectedRedCells}$)}
- \label{fig:globules_rouges_infectes}
-\end{figure}
+% RANSAC + scoring + pruning
+\subsection{Segmentation des parasites}
-\subsection{Calcul du taux d'infection}
+\subsubsection{Extraction des zones colorées sombres}
-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 : $100 \cdot \mli{nbInfectedRedCells} / \mli{nbRedCells}$.
+\subsubsection{Segmentation des cytoplasmes}
+\subsubsection{Segmentation des noyaux}
-\section{Résultats}
-\label{resultats}
+\subsection{Classification des cellules}
-\subsection{Qualité des images en entrée}
+\subsubsection{Suppression des cellules non-conformes}
-La première étape, décrite à la section~\ref{detection-parasites} et qui consiste à marquer les éléments colorés, est très sensible à la qualité de l'image initiale. Suivant les réglages du microscope et du logiciel de capture associé, 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 peut être similaire à la celle du fond : la figure~\ref{fig:problemes-teinte} montre une image dont la teinte est inexploitable.
+\subsubsection{Attribution d'une classe}
-\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 \emph{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}
+\section{Implémentation}
-\subsection{Méthode}
-Les résultats se basent sur l'analyse d'un ensemble d'images provenant d'un même patient : le taux d'infection devrait donc 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}
+\section{Résultats}
-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.
+% Comparaison avec MA et le PA.
-\begin{table}[htbp]
-\setlength{\tabcolsep}{3pt}
-\centering
-\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 \\
- 0014 & 132 & 18 & 13.6~\% & 131 & 18 & 13.7~\% & 0.8~\% & 65 \\
- 0036 & 133 & 20 & 15.0~\% & 135 & 21 & 15.6~\% & 3.4~\% & 70 \\
- 0038 & 116 & 19 & 16.4~\% & 117 & 19 & 16.2~\% & -0.9~\% & 68 \\
- 0039 & 136 & 26 & 19.1~\% & 137 & 27 & 19.7~\% & 3.1~\% & 79 \\
- 0040 & 128 & 19 & 14.8~\% & 128 & 18 & 14.1~\% & -5.3~\% & 78 \\
- 0041 & 120 & 26 & 21.7~\% & 121 & 27 & 22.3~\% & 3.0~\% & 89 \\
- 0042 & 135 & 26 & 19.3~\% & 144 & 30 & 20.8~\% & 8.2~\% & 82 \\
- 0043 & 128 & 20 & 15.6~\% & 132 & 27 & 20.5~\% & 30.9~\% & 80 \\
- 0044 & 133 & 25 & 18.8~\% & 135 & 25 & 18.5~\% & -1.5~\% & 86 \\
- 0045 & 136 & 22 & 16.2~\% & 136 & 28 & 20.6~\% & 27.3~\% & 84 \\
- 0046 & 127 & 29 & 22.8~\% & 128 & 27 & 21.1~\% & -7.6~\% & 85
-\end{tabular}
-\caption{Résultats sur les images réduites de 40~\%}
-\label{tab:resultats60}
-\end{table}
-
-
-\begin{table}[htbp]
-\setlength{\tabcolsep}{3pt}
-\centering
-\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 \\
- 0014 & 132 & 18 & 13.6~\% & 136 & 19 & 14.0~\% & 2.5~\% & 462 \\
- 0036 & 133 & 20 & 15.0~\% & 135 & 21 & 15.6~\% & 3.4~\% & 431 \\
- 0038 & 116 & 19 & 16.4~\% & 118 & 19 & 16.1~\% & -1.7~\% & 447 \\
- 0039 & 136 & 26 & 19.1~\% & 136 & 27 & 19.9~\% & 3.8~\% & 463 \\
- 0040 & 128 & 19 & 14.8~\% & 131 & 18 & 13.7~\% & -7.4~\% & 458 \\
- 0041 & 120 & 26 & 21.7~\% & 122 & 27 & 22.1~\% & 2.1~\% & 526 \\
- 0042 & 135 & 26 & 19.3~\% & 142 & 27 & 19.0~\% & -1.3~\% & 483 \\
- 0043 & 128 & 20 & 15.6~\% & 128 & 25 & 19.5~\% & 25.0~\% & 515 \\
- 0044 & 133 & 25 & 18.8~\% & 137 & 25 & 18.2~\% & -2.9~\% & 505 \\
- 0045 & 136 & 22 & 16.2~\% & 144 & 30 & 20.8~\% & 28.8~\% & 538 \\
- 0046 & 127 & 29 & 22.8~\% & 126 & 27 & 21.4~\% & -6.2~\% & 557
-\end{tabular}
-\caption{Résultats sur les images originales (non-réduites)}
-\label{tab:resultats100}
-\end{table}
-
-Le taux moyen d'infection réel est de 17.3~\% et le taux détecté avec des images réduites de 40~\% est de 18.2~\%. Ces données sont présentées dans le tableau~\ref{tab:resultats}.
-
-\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}
\section{Conclusion}
-Par la mise en pratique de la méthode décrite dans \emph{Analysis of infected blood cell images using morphological operators}\cite{DiRuberto-Analysis-infected-blood-morphological} nous avons pu démontrer sa pertinence et son efficacité. Quelques modifications ont dû être apportées telles que les critères de détection des schizontes ou encore l'utilisation d'une transformée en distance pour la définition des marqueurs de l'algorithme de la ligne de partage des eaux lors de la segmentation des globules rouges.
-
-De plus, une des difficultés majeures a été d'obtenir une image initiale ayant des composantes \emph{teinte} et \emph{saturation} satisfaisantes. Cette première étape de détection des éléments colorés reste très dépendante de la qualité et des caractéristiques de l'image d'entrée.
-
-Néanmoins, une fois de bons réglages effectués sur le système de capture d'image du microscope, les résultats du processus, en terme de taux d'infection, correspondent à ce qui est attendu. Sur le cas analysé (n°1412151257), à partir de douze captures, nous obtenons une différence de taux de 5.3~\% pour les images réduites et de 3.7~\% pour les images non-réduites.
-
-
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