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            <c val="#008080">MORPHOLOGICAL PROCESSING</c>
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              <c val="#800080">Erosion and Dilation</c>
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        <phrase>erosion and dilation</phrase>
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              <c val="#800000">dilate8(M, b)</c>
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        <phrase>dilate8 function</phrase>
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            <b>
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                <c val="#800000">erode(M, Melem, r-origin, c-origin, b)</c>
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        <phrase>erode function</phrase>
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      <text use-page-width="false" push-down="false" lock-width="true">
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              <inlineAttr font-style="normal" line-through="false">
                <c val="#800000">dilate(M, Melem, r-origin, c-origin, b)</c>
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        <phrase>dilate function</phrase>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">These functions perform erosion or dilation, wearing away or adding to the edges of an image. The 4 or 8 in the function name indicate whether only orthogonal (4) or all 8 neighbors are considered. Erosion spreads light objects in an image further apart; dilation enlarges light objects, reassembling an image which has become disjointed.</p>
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            <c val="#800000">Arguments</c>
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              <inlineAttr line-through="false">M</inlineAttr>
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          </f>, a structuring element matrix </p>
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              <c val="#008080">• </c>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The functions return a matrix which contains the eroded or dilated image.</p>
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            <c val="#800000">Function Details</c>
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              <c val="#800000">erode4(M, b)</c>
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        <phrase>erode4 function</phrase>
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              <c val="#800000">erode8(M, b)</c>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">These functions scan an image M to find pixels with values less than the intensity threshold <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">b,</inlineAttr></f> the background threshold of the image. In the case of <b>erode4</b>, if any of the background pixel's 4 neighbors have values greater than or equal to <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">b</inlineAttr></f> (that is, they are foreground pixels), those neighbors are set equal to 0. The effect is to nibble away at edges of the foreground image. The same is true of <b>erode8</b>, but the 8-neighborhood of each pixel is considered, instead of the 4-neighborhood.</p>
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            </ml:matrix>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="7" />
    </region>
    <region region-id="1380" left="36" top="743.25" width="183" height="78.75" align-x="120" align-y="786" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:apply>
            <ml:id xml:space="preserve">erode8</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">M</ml:id>
              <ml:real font="0">100</ml:real>
            </ml:sequence>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="4" cols="4">
              <ml:real>0</ml:real>
              <ml:real>80</ml:real>
              <ml:real>0</ml:real>
              <ml:real>56</ml:real>
              <ml:real>30</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>45</ml:real>
              <ml:real>65</ml:real>
              <ml:real>0</ml:real>
              <ml:real>20</ml:real>
            </ml:matrix>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="8" />
    </region>
    <region region-id="1381" left="6" top="834.75" width="331.5" height="28.5" align-x="6" align-y="846" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Now examine the effect this has on an image. Let's use a two-dimensional <b>sinc</b> function.</p>
      </text>
    </region>
    <region region-id="874" left="42" top="892.5" width="93.75" height="39.75" align-x="68.25" align-y="918" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:function>
            <ml:id xml:space="preserve">f</ml:id>
            <ml:boundVars>
              <ml:id xml:space="preserve">x</ml:id>
            </ml:boundVars>
          </ml:function>
          <ml:apply>
            <ml:pow />
            <ml:apply>
              <ml:id xml:space="preserve">sinc</ml:id>
              <ml:apply>
                <ml:div />
                <ml:apply>
                  <ml:mult style="default" />
                  <ml:id xml:space="preserve">x</ml:id>
                  <ml:id xml:space="preserve">π</ml:id>
                </ml:apply>
                <ml:real font="0">2</ml:real>
              </ml:apply>
            </ml:apply>
            <ml:real font="0">2</ml:real>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="9" />
    </region>
    <region region-id="1382" left="6" top="960.75" width="73.5" height="14.25" align-x="6" align-y="972" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Range variables:</p>
      </text>
    </region>
    <region region-id="876" left="42" top="984.75" width="57.75" height="15.75" align-x="71.25" align-y="996" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">imax</ml:id>
          <ml:real font="0">100</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="10" />
    </region>
    <region region-id="877" left="108" top="984.75" width="57.75" height="15.75" align-x="118.5" align-y="996" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">i</ml:id>
          <ml:range>
            <ml:real font="0">0</ml:real>
            <ml:id xml:space="preserve">imax</ml:id>
          </ml:range>
        </ml:define>
      </math>
      <rendering item-idref="11" />
    </region>
    <region region-id="878" left="180" top="984.75" width="59.25" height="15.75" align-x="192" align-y="996" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">j</ml:id>
          <ml:range>
            <ml:real font="0">0</ml:real>
            <ml:id xml:space="preserve">imax</ml:id>
          </ml:range>
        </ml:define>
      </math>
      <rendering item-idref="12" />
    </region>
    <region region-id="1383" left="6" top="1008.75" width="73.5" height="14.25" align-x="6" align-y="1020" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Range of values:</p>
      </text>
    </region>
    <region region-id="1386" left="42" top="1042.5" width="90" height="33.75" align-x="56.25" align-y="1062" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">r</ml:id>
            <ml:id xml:space="preserve">i</ml:id>
          </ml:apply>
          <ml:apply>
            <ml:mult style="default" />
            <ml:parens>
              <ml:apply>
                <ml:minus />
                <ml:id xml:space="preserve">i</ml:id>
                <ml:apply>
                  <ml:div />
                  <ml:id xml:space="preserve">imax</ml:id>
                  <ml:real font="0">2</ml:real>
                </ml:apply>
              </ml:apply>
            </ml:parens>
            <ml:real font="0">.1</ml:real>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="13" />
    </region>
    <region region-id="1385" left="0" top="1092" width="6000" height="6" align-x="0" align-y="1092" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="1384" left="6" top="1098.75" width="60.75" height="14.25" align-x="6" align-y="1110" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Image matrix:</p>
      </text>
    </region>
    <region region-id="1528" left="42" top="1122.75" width="105" height="17.25" align-x="69.75" align-y="1134" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">D</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">i</ml:id>
              <ml:id xml:space="preserve">j</ml:id>
            </ml:sequence>
          </ml:apply>
          <ml:apply>
            <ml:mult style="default" />
            <ml:apply>
              <ml:mult style="default" />
              <ml:real font="0">255</ml:real>
              <ml:apply>
                <ml:id xml:space="preserve">f</ml:id>
                <ml:apply>
                  <ml:indexer />
                  <ml:id xml:space="preserve">r</ml:id>
                  <ml:id xml:space="preserve">i</ml:id>
                </ml:apply>
              </ml:apply>
            </ml:apply>
            <ml:apply>
              <ml:id xml:space="preserve">f</ml:id>
              <ml:apply>
                <ml:indexer />
                <ml:id xml:space="preserve">r</ml:id>
                <ml:id xml:space="preserve">j</ml:id>
              </ml:apply>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="14" />
    </region>
    <region region-id="1387" left="6" top="1152.75" width="167.25" height="14.25" align-x="6" align-y="1164" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Binarize the image, to clarify features:</p>
      </text>
    </region>
    <region region-id="885" left="42" top="1176.75" width="115.5" height="15.75" align-x="60.75" align-y="1188" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">M</ml:id>
          <ml:apply>
            <ml:mult style="default" />
            <ml:real font="0">255</ml:real>
            <ml:apply>
              <ml:id xml:space="preserve">binarize</ml:id>
              <ml:sequence>
                <ml:id xml:space="preserve">D</ml:id>
                <ml:real font="0">4</ml:real>
              </ml:sequence>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="15" />
    </region>
    <region region-id="1482" left="48" top="1206" width="77.25" height="105.75" align-x="85.5" align-y="1206" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <component hide-arguments="false" clsid-buddy="8442da2d-08d7-45ad-a4a6-f28bee6ae06a" item-idref="16" disable-calc="false">
        <inputs>
          <ml:id xml:space="preserve">M</ml:id>
        </inputs>
        <outputs />
      </component>
      <rendering item-idref="17" />
    </region>
    <region region-id="1388" left="6" top="1314.75" width="105.75" height="14.25" align-x="6" align-y="1326" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Now apply the erosion:</p>
      </text>
    </region>
    <region region-id="1076" left="42" top="1338.75" width="42.75" height="15.75" align-x="56.25" align-y="1350" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">b</ml:id>
          <ml:real font="0">180</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="18" />
    </region>
    <region region-id="1393" left="162" top="1338.75" width="90" height="15.75" align-x="177" align-y="1350" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedBIUnit">
          <ml:id xml:space="preserve">F</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">erode4</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">M</ml:id>
              <ml:id xml:space="preserve">b</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="19" />
    </region>
    <region region-id="1389" left="42" top="1362.75" width="91.5" height="15.75" align-x="58.5" align-y="1374" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedBIUnit">
          <ml:id xml:space="preserve">G</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">erode8</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">M</ml:id>
              <ml:id xml:space="preserve">b</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="20" />
    </region>
    <region region-id="1394" left="162" top="1362.75" width="98.25" height="15.75" align-x="187.5" align-y="1374" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">GG</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">erode8</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">G</ml:id>
              <ml:id xml:space="preserve">b</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="21" />
    </region>
    <region region-id="1499" left="42" top="1380" width="77.25" height="105.75" align-x="79.5" align-y="1380" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <component hide-arguments="false" clsid-buddy="8442da2d-08d7-45ad-a4a6-f28bee6ae06a" item-idref="22" disable-calc="false">
        <inputs>
          <ml:id xml:space="preserve">F</ml:id>
        </inputs>
        <outputs />
      </component>
      <rendering item-idref="23" />
    </region>
    <region region-id="1479" left="162" top="1380" width="77.25" height="105.75" align-x="199.5" align-y="1380" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <component hide-arguments="false" clsid-buddy="8442da2d-08d7-45ad-a4a6-f28bee6ae06a" item-idref="24" disable-calc="false">
        <inputs>
          <ml:id xml:space="preserve">G</ml:id>
        </inputs>
        <outputs />
      </component>
      <rendering item-idref="25" />
    </region>
    <region region-id="1480" left="282" top="1380" width="77.25" height="105.75" align-x="319.5" align-y="1380" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <component hide-arguments="false" clsid-buddy="8442da2d-08d7-45ad-a4a6-f28bee6ae06a" item-idref="26" disable-calc="false">
        <inputs>
          <ml:id xml:space="preserve">GG</ml:id>
        </inputs>
        <outputs />
      </component>
      <rendering item-idref="27" />
    </region>
    <region region-id="1395" left="6" top="1512.75" width="306.75" height="42.75" align-x="6" align-y="1524" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Erosion can be used to make a mask of an image for further binary processing. An example is shown in the example <b><u>Machine Quality Control of Corn</u></b>.</p>
      </text>
      <link href="./Corn.xmcd" title="Links to section" popup="false" />
    </region>
    <region region-id="1298" left="0" top="1566" width="4800" height="6" align-x="0" align-y="1566" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="1402" left="6" top="1578.75" width="89.25" height="14.25" align-x="6" align-y="1590" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">
          <b>
            <c val="#800000">Function Details</c>
          </b>
        </p>
      </text>
    </region>
    <region region-id="1399" left="120" top="1578.75" width="68.25" height="14.25" align-x="120" align-y="1590" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">
          <b>
            <inlineAttr font-style="normal">
              <c val="#800000">dilate4(M, b)</c>
            </inlineAttr>
          </b>
        </p>
      </text>
      <indexes>
        <phrase>dilate4 function</phrase>
      </indexes>
    </region>
    <region region-id="1403" left="198" top="1578.75" width="66.75" height="14.25" align-x="198" align-y="1590" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">
          <b>
            <inlineAttr font-style="normal">
              <c val="#800000">dilate8(M, b)</c>
            </inlineAttr>
          </b>
        </p>
      </text>
      <indexes>
        <phrase>dilate8 function</phrase>
      </indexes>
    </region>
    <region region-id="1398" left="6" top="1596.75" width="321" height="114" align-x="6" align-y="1608" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Dilation is the opposite of erosion. The dilation functions find pixels with values less than intensity b, the background threshold of the image. In the case of <b>dilate4</b>, if any of the background pixel's 4 neighbors have values greater than or equal to b (that is, they are foreground pixels), the background pixel is promoted to have value b. The effect is to build up the edges of the foreground image. The same is true of <b>dilate8</b>, but the 8-neighborhood of each pixel is considered, instead of the 4-neighborhood.</p>
      </text>
    </region>
    <region region-id="1408" left="36" top="1716.75" width="42.75" height="15.75" align-x="50.25" align-y="1728" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">d</ml:id>
          <ml:real font="0">255</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="28" />
    </region>
    <region region-id="1409" left="36" top="1740.75" width="95.25" height="15.75" align-x="58.5" align-y="1752" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">D4</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">dilate4</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">M</ml:id>
              <ml:id xml:space="preserve">d</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="29" />
    </region>
    <region region-id="1410" left="150" top="1740.75" width="95.25" height="15.75" align-x="172.5" align-y="1752" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">D8</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">dilate8</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">M</ml:id>
              <ml:id xml:space="preserve">d</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="30" />
    </region>
    <region region-id="1411" left="36" top="1764.75" width="210.75" height="15.75" align-x="70.5" align-y="1776" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">D888</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">dilate8</ml:id>
            <ml:sequence>
              <ml:apply>
                <ml:id xml:space="preserve">dilate8</ml:id>
                <ml:sequence>
                  <ml:apply>
                    <ml:id xml:space="preserve">dilate8</ml:id>
                    <ml:sequence>
                      <ml:id xml:space="preserve">M</ml:id>
                      <ml:id xml:space="preserve">d</ml:id>
                    </ml:sequence>
                  </ml:apply>
                  <ml:id xml:space="preserve">d</ml:id>
                </ml:sequence>
              </ml:apply>
              <ml:id xml:space="preserve">d</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
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      <component hide-arguments="false" clsid-buddy="8442da2d-08d7-45ad-a4a6-f28bee6ae06a" item-idref="32" disable-calc="false">
        <inputs>
          <ml:id xml:space="preserve">D4</ml:id>
        </inputs>
        <outputs />
      </component>
      <rendering item-idref="33" />
    </region>
    <region region-id="1503" left="156" top="1800" width="77.25" height="105.75" align-x="193.5" align-y="1800" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <component hide-arguments="false" clsid-buddy="8442da2d-08d7-45ad-a4a6-f28bee6ae06a" item-idref="34" disable-calc="false">
        <inputs>
          <ml:id xml:space="preserve">D8</ml:id>
        </inputs>
        <outputs />
      </component>
      <rendering item-idref="35" />
    </region>
    <region region-id="1502" left="276" top="1800" width="77.25" height="105.75" align-x="313.5" align-y="1800" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <component hide-arguments="false" clsid-buddy="8442da2d-08d7-45ad-a4a6-f28bee6ae06a" item-idref="36" disable-calc="false">
        <inputs>
          <ml:id xml:space="preserve">D888</ml:id>
        </inputs>
        <outputs />
      </component>
      <rendering item-idref="37" />
    </region>
    <region region-id="1299" left="0" top="1920" width="4800" height="6" align-x="0" align-y="1920" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="1413" left="6" top="1932.75" width="89.25" height="14.25" align-x="6" align-y="1944" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">
          <b>
            <c val="#800000">Function Details</c>
          </b>
        </p>
      </text>
    </region>
    <region region-id="1414" left="6" top="1956.75" width="188.25" height="14.25" align-x="6" align-y="1968" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">
          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr font-style="normal" line-through="false">
                <c val="#800000">erode(M, Melem, r-origin, c-origin, b)</c>
              </inlineAttr>
            </b>
          </f>
        </p>
      </text>
      <indexes>
        <phrase>erode function</phrase>
      </indexes>
    </region>
    <region region-id="1415" left="6" top="1974.75" width="187.5" height="14.25" align-x="6" align-y="1986" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">
          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr font-style="normal" line-through="false">
                <c val="#800000">dilate(M, Melem, r-origin, c-origin, b)</c>
              </inlineAttr>
            </b>
          </f>
        </p>
      </text>
      <indexes>
        <phrase>dilate function</phrase>
      </indexes>
    </region>
    <region region-id="1416" left="6" top="1992.75" width="330.75" height="85.5" align-x="6" align-y="2004" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Instead of considering the 4- or 8-neighborhood of a pixel, <b>erode</b> and <b>dilate</b> use a neighborhood specified by the matrix Melem, with origin given by (r-origin, c-origin). Also, the <b>erode</b> and <b>dilate</b> functions are true morphological binary functions, producing output that is always binarized. (See the <link href="./Graymorp.xmcd" title="Links to section" popup="false"><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" underline="false" line-through="false">section on </inlineAttr></f><b><u>grayscale morphology</u></b></link> for functions that work with grayscale images.) Pixels near the boundary may be invalid in output.</p>
      </text>
    </region>
    <region region-id="1417" left="6" top="2100.75" width="321" height="57" align-x="6" align-y="2112" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Here is a horizontal neighborhood matrix used with the dilate function. The matrix considers the pixel at the origin of the neighborhood matrix, in this example, at (3,0), and the 3 pixels to the left and right of the origin, respectively (recall that matrix indices begin at 0 and not 1).</p>
      </text>
    </region>
    <region region-id="1303" left="30" top="2166.75" width="153.75" height="15.75" align-x="69.75" align-y="2178" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">Melem</ml:id>
          <ml:matrix rows="1" cols="7">
            <ml:real font="0">1</ml:real>
            <ml:real font="0">1</ml:real>
            <ml:real font="0">1</ml:real>
            <ml:real font="0">1</ml:real>
            <ml:real font="0">1</ml:real>
            <ml:real font="0">1</ml:real>
            <ml:real font="0">1</ml:real>
          </ml:matrix>
        </ml:define>
      </math>
      <rendering item-idref="38" />
    </region>
    <region region-id="1304" left="30" top="2190.75" width="165" height="15.75" align-x="54.75" align-y="2202" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">M1</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">dilate</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">M</ml:id>
              <ml:id xml:space="preserve">Melem</ml:id>
              <ml:real font="0">0</ml:real>
              <ml:real font="0">3</ml:real>
              <ml:real font="0">100</ml:real>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="39" />
    </region>
    <region region-id="1517" left="36" top="2214" width="77.25" height="105.75" align-x="73.5" align-y="2214" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <component hide-arguments="false" clsid-buddy="8442da2d-08d7-45ad-a4a6-f28bee6ae06a" item-idref="40" disable-calc="false">
        <inputs>
          <ml:id xml:space="preserve">M</ml:id>
        </inputs>
        <outputs />
      </component>
      <rendering item-idref="41" />
    </region>
    <region region-id="1519" left="150" top="2214" width="77.25" height="105.75" align-x="187.5" align-y="2214" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <component hide-arguments="false" clsid-buddy="8442da2d-08d7-45ad-a4a6-f28bee6ae06a" item-idref="42" disable-calc="false">
        <inputs>
          <ml:apply>
            <ml:mult style="default" />
            <ml:id xml:space="preserve">M1</ml:id>
            <ml:real font="0">255</ml:real>
          </ml:apply>
        </inputs>
        <outputs />
      </component>
      <rendering item-idref="43" />
    </region>
    <region region-id="1419" left="6" top="2334.75" width="311.25" height="42.75" align-x="6" align-y="2346" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Since the neighborhood matrix is a row vector, objects in the image expand horizontally, but are unchanged vertically. There is also a slight displacement of the image from center.</p>
      </text>
    </region>
    <region region-id="1312" left="0" top="2388" width="4800" height="6" align-x="0" align-y="2388" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="1420" left="6" top="2400.75" width="322.5" height="14.25" align-x="6" align-y="2412" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Next, consider erosion with a vertical neighborhood structuring element:</p>
      </text>
    </region>
    <region region-id="1310" left="36" top="2421.75" width="72.75" height="141.75" align-x="75.75" align-y="2496" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">Melem</ml:id>
          <ml:matrix rows="7" cols="1">
            <ml:real font="0">1</ml:real>
            <ml:real font="0">1</ml:real>
            <ml:real font="0">1</ml:real>
            <ml:real font="0">1</ml:real>
            <ml:real font="0">1</ml:real>
            <ml:real font="0">1</ml:real>
            <ml:real font="0">1</ml:real>
          </ml:matrix>
        </ml:define>
      </math>
      <rendering item-idref="44" />
    </region>
    <region region-id="1311" left="126" top="2484.75" width="167.25" height="15.75" align-x="150.75" align-y="2496" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">M1</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">erode</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">M</ml:id>
              <ml:id xml:space="preserve">Melem</ml:id>
              <ml:real font="0">3</ml:real>
              <ml:real font="0">0</ml:real>
              <ml:real font="0">100</ml:real>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="45" />
    </region>
    <region region-id="1526" left="144" top="2514" width="77.25" height="105.75" align-x="181.5" align-y="2514" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <component hide-arguments="false" clsid-buddy="8442da2d-08d7-45ad-a4a6-f28bee6ae06a" item-idref="46" disable-calc="false">
        <inputs>
          <ml:id xml:space="preserve">M</ml:id>
        </inputs>
        <outputs />
      </component>
      <rendering item-idref="47" />
    </region>
    <region region-id="1527" left="264" top="2514" width="77.25" height="105.75" align-x="301.5" align-y="2514" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <component hide-arguments="false" clsid-buddy="8442da2d-08d7-45ad-a4a6-f28bee6ae06a" item-idref="48" disable-calc="false">
        <inputs>
          <ml:apply>
            <ml:mult style="default" />
            <ml:id xml:space="preserve">M1</ml:id>
            <ml:real font="0">255</ml:real>
          </ml:apply>
        </inputs>
        <outputs />
      </component>
      <rendering item-idref="49" />
    </region>
    <region region-id="1421" left="6" top="2628.75" width="325.5" height="57" align-x="6" align-y="2640" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Objects on the image contract vertically, but horizontal size is unchanged. Also, several small objects in the image disappeared due to erosion because their vertical dimension was smaller than the vertical dimension of the structuring element. </p>
      </text>
    </region>
    <region region-id="1322" left="0" top="2694" width="4800" height="6" align-x="0" align-y="2694" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="1426" left="6" top="2706.75" width="319.5" height="57" align-x="6" align-y="2718" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">We can also use the general <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">erode</inlineAttr></b></f> and <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">dilate</inlineAttr></b></f> functions to mimic the behavior of the <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">erode4</inlineAttr></b></f>, <f family="Times New Roman" charset="0"><inlineAttr line-through="false"><b>erode8</b></inlineAttr></f>, <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">dilate4</inlineAttr></b></f>, and <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">dilate8</inlineAttr></b></f> functions, when applied to binary images, by creating the appropriate structuring element. For instance, to mimic <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">erode4</inlineAttr></b></f>, we use:</p>
      </text>
    </region>
    <region region-id="1432" left="36" top="2775.75" width="101.25" height="57.75" align-x="75.75" align-y="2808" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">Melem</ml:id>
          <ml:matrix rows="3" cols="3">
            <ml:real font="0">0</ml:real>
            <ml:real font="0">1</ml:real>
            <ml:real font="0">0</ml:real>
            <ml:real font="0">1</ml:real>
            <ml:real font="0">1</ml:real>
            <ml:real font="0">1</ml:real>
            <ml:real font="0">0</ml:real>
            <ml:real font="0">1</ml:real>
            <ml:real font="0">0</ml:real>
          </ml:matrix>
        </ml:define>
      </math>
      <rendering item-idref="50" />
    </region>
    <region region-id="1430" left="6" top="2838.75" width="45" height="14.25" align-x="6" align-y="2850" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Compare:</p>
      </text>
    </region>
    <region region-id="1433" left="36" top="2862.75" width="106.5" height="15.75" align-x="55.5" align-y="2874" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">e4</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">erode4</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">M</ml:id>
              <ml:real font="0">100</ml:real>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="51" />
    </region>
    <region region-id="1434" left="36" top="2886.75" width="183" height="15.75" align-x="54.75" align-y="2898" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">eg</ml:id>
          <ml:apply>
            <ml:mult style="default" />
            <ml:real font="0">255</ml:real>
            <ml:apply>
              <ml:id xml:space="preserve">erode</ml:id>
              <ml:sequence>
                <ml:id xml:space="preserve">M</ml:id>
                <ml:id xml:space="preserve">Melem</ml:id>
                <ml:real font="0">1</ml:real>
                <ml:real font="0">1</ml:real>
                <ml:real font="0">100</ml:real>
              </ml:sequence>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="52" />
    </region>
    <region region-id="1436" left="6" top="2910.75" width="249.75" height="14.25" align-x="6" align-y="2922" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">They should be identical except on the image boundary:</p>
      </text>
    </region>
    <region region-id="1439" left="30" top="2931.75" width="153" height="50.25" align-x="161.25" align-y="2958" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:apply>
            <ml:summation />
            <ml:lambda>
              <ml:boundVars>
                <ml:id xml:space="preserve">i</ml:id>
              </ml:boundVars>
              <ml:apply>
                <ml:summation />
                <ml:lambda>
                  <ml:boundVars>
                    <ml:id xml:space="preserve">j</ml:id>
                  </ml:boundVars>
                  <ml:apply>
                    <ml:absval />
                    <ml:apply>
                      <ml:minus />
                      <ml:apply>
                        <ml:indexer />
                        <ml:id xml:space="preserve">e4</ml:id>
                        <ml:sequence>
                          <ml:id xml:space="preserve">i</ml:id>
                          <ml:id xml:space="preserve">j</ml:id>
                        </ml:sequence>
                      </ml:apply>
                      <ml:apply>
                        <ml:indexer />
                        <ml:id xml:space="preserve">eg</ml:id>
                        <ml:sequence>
                          <ml:id xml:space="preserve">i</ml:id>
                          <ml:id xml:space="preserve">j</ml:id>
                        </ml:sequence>
                      </ml:apply>
                    </ml:apply>
                  </ml:apply>
                </ml:lambda>
                <ml:bounds>
                  <ml:real font="0">1</ml:real>
                  <ml:real font="0">99</ml:real>
                </ml:bounds>
              </ml:apply>
            </ml:lambda>
            <ml:bounds>
              <ml:real font="0">1</ml:real>
              <ml:real font="0">99</ml:real>
            </ml:bounds>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>0</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="53" />
    </region>
    <region region-id="1437" left="6" top="2988.75" width="93" height="14.25" align-x="6" align-y="3000" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Similarly for <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">dilate8</inlineAttr></b></f>:</p>
      </text>
    </region>
    <region region-id="1440" left="36" top="3009.75" width="101.25" height="57.75" align-x="75.75" align-y="3042" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">Melem</ml:id>
          <ml:matrix rows="3" cols="3">
            <ml:real font="0">1</ml:real>
            <ml:real font="0">1</ml:real>
            <ml:real font="0">1</ml:real>
            <ml:real font="0">1</ml:real>
            <ml:real font="0">1</ml:real>
            <ml:real font="0">1</ml:real>
            <ml:real font="0">1</ml:real>
            <ml:real font="0">1</ml:real>
            <ml:real font="0">1</ml:real>
          </ml:matrix>
        </ml:define>
      </math>
      <rendering item-idref="54" />
    </region>
    <region region-id="1441" left="36" top="3078.75" width="105" height="15.75" align-x="56.25" align-y="3090" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">d8</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">dilate8</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">M</ml:id>
              <ml:real font="0">255</ml:real>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="55" />
    </region>
    <region region-id="1442" left="36" top="3102.75" width="181.5" height="15.75" align-x="55.5" align-y="3114" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">dg</ml:id>
          <ml:apply>
            <ml:mult style="default" />
            <ml:real font="0">255</ml:real>
            <ml:apply>
              <ml:id xml:space="preserve">dilate</ml:id>
              <ml:sequence>
                <ml:id xml:space="preserve">M</ml:id>
                <ml:id xml:space="preserve">Melem</ml:id>
                <ml:real font="0">1</ml:real>
                <ml:real font="0">1</ml:real>
                <ml:real font="0">100</ml:real>
              </ml:sequence>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="56" />
    </region>
    <region region-id="1449" left="6" top="3132.75" width="249.75" height="14.25" align-x="6" align-y="3144" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">They should be identical except on the image boundary:</p>
      </text>
    </region>
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      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:apply>
            <ml:summation />
            <ml:lambda>
              <ml:boundVars>
                <ml:id xml:space="preserve">i</ml:id>
              </ml:boundVars>
              <ml:apply>
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                <ml:lambda>
                  <ml:boundVars>
                    <ml:id xml:space="preserve">j</ml:id>
                  </ml:boundVars>
                  <ml:apply>
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                    <ml:apply>
                      <ml:minus />
                      <ml:apply>
                        <ml:indexer />
                        <ml:id xml:space="preserve">d8</ml:id>
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                          <ml:id xml:space="preserve">i</ml:id>
                          <ml:id xml:space="preserve">j</ml:id>
                        </ml:sequence>
                      </ml:apply>
                      <ml:apply>
                        <ml:indexer />
                        <ml:id xml:space="preserve">dg</ml:id>
                        <ml:sequence>
                          <ml:id xml:space="preserve">i</ml:id>
                          <ml:id xml:space="preserve">j</ml:id>
                        </ml:sequence>
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                    </ml:apply>
                  </ml:apply>
                </ml:lambda>
                <ml:bounds>
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                  <ml:real font="0">99</ml:real>
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            <ml:bounds>
              <ml:real font="0">1</ml:real>
              <ml:real font="0">99</ml:real>
            </ml:bounds>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Note that when you use <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">erode4,</inlineAttr></b></f><sp /><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">erode8,</inlineAttr></b></f><sp /><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">dilate4,</inlineAttr></b></f> and <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">dilate8</inlineAttr></b></f> you must first manually binarize the image at the threshold in order to see the same results as those given by <b><inlineAttr line-through="false"><f family="Times New Roman" charset="0">erode</f></inlineAttr></b> and <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">dilate</inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">,</inlineAttr></f> which binarize automatically. </p>
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