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            <c val="#008080">EDGE FINDERS</c>
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            <b>
              <c val="#800080">Convolution Edge Finders</c>
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        <phrase>convolution edge finders</phrase>
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            <c val="#800000">Function Definitions and Syntax</c>
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              <c val="#800000">difedge(M)</c>
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              <c val="#800000">sobel(M)</c>
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              <c val="#800000">prewitt(M)</c>
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              <c val="#800000">roberts(M)</c>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">These functions detect edges in an image, <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">M</inlineAttr></f>, by convolving various kernels in sequence over the matrix. Edge detection is used to enhance image readability for certain types of features that depend on boundaries. </p>
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          <b>
            <c val="#800000">Arguments</c>
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              <inlineAttr line-through="false">M</inlineAttr>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The functions return a matrix containing the detected edges. The perimeter of the image is untreated, since the kernels do not fully overlap there.</p>
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          <b>
            <c val="#800000">Function Details</c>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">This function finds edge<f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">s in M by convolving M</inlineAttr></f> with the following two kernels and then combining the result by finding the gradient magnitude elementwise between the two convolution matrices using the kernels:</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The output equals</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">where <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">C1 and C2</inlineAttr></f><b><sp /></b>are the convolution matrices.</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The concept here is similar to that used in <b><u>Laplacian (Differential) Edge Finders</u></b>. Each of the kernels above takes a directional differential (<f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">K1 is horizontal, and K2</inlineAttr></f> is vertical). The subsequent sum of squares makes the process directionally independent.</p>
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        <phrase>Laplacian (differential) edge finders</phrase>
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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">The output of the edge finders is compared visually at the bottom of the page.</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr font-style="normal" line-through="false">
                <c val="#800000">freichen(M)</c>
              </inlineAttr>
            </b>
          </f>
        </p>
      </text>
      <indexes>
        <phrase>freichen function</phrase>
      </indexes>
    </region>
    <region region-id="478" left="6" top="696.75" width="212.25" height="14.25" align-x="6" align-y="708" 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="0" text-align="left" list-style-type="none" tabs="inherit">This function convolves<b><sp /></b><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">M</inlineAttr></f><b><sp /></b>with the two kernels</p>
      </text>
    </region>
    <region region-id="476" left="36" top="717" width="150.75" height="60" align-x="59.25" align-y="750" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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          <ml:id xml:space="preserve">K1</ml:id>
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              <ml:apply>
                <ml:plus />
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                <ml:apply>
                  <ml:sqrt />
                  <ml:real>2</ml:real>
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            </ml:apply>
            <ml:matrix rows="3" cols="3">
              <ml:real>1</ml:real>
              <ml:apply>
                <ml:sqrt />
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              <ml:real>1</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>-1</ml:real>
              <ml:apply>
                <ml:neg />
                <ml:apply>
                  <ml:sqrt />
                  <ml:real>2</ml:real>
                </ml:apply>
              </ml:apply>
              <ml:real>-1</ml:real>
            </ml:matrix>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="11" />
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    <region region-id="477" left="192" top="715.5" width="155.25" height="62.25" align-x="215.25" align-y="750" 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">K2</ml:id>
          <ml:apply>
            <ml:mult style="default" />
            <ml:apply>
              <ml:div />
              <ml:real>1</ml:real>
              <ml:apply>
                <ml:plus />
                <ml:real>2</ml:real>
                <ml:apply>
                  <ml:sqrt />
                  <ml:real>2</ml:real>
                </ml:apply>
              </ml:apply>
            </ml:apply>
            <ml:matrix rows="3" cols="3">
              <ml:real>-1</ml:real>
              <ml:real>0</ml:real>
              <ml:real>1</ml:real>
              <ml:apply>
                <ml:neg />
                <ml:apply>
                  <ml:sqrt />
                  <ml:real>2</ml:real>
                </ml:apply>
              </ml:apply>
              <ml:real>0</ml:real>
              <ml:apply>
                <ml:sqrt />
                <ml:real>2</ml:real>
              </ml:apply>
              <ml:real>-1</ml:real>
              <ml:real>0</ml:real>
              <ml:real>1</ml:real>
            </ml:matrix>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="12" />
    </region>
    <region region-id="547" left="36" top="786.75" width="90.75" height="15.75" align-x="63.75" align-y="798" 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">Rfre</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">freichen</ml:id>
            <ml:id xml:space="preserve">R</ml:id>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="13" />
    </region>
    <region region-id="475" left="6" top="810.75" width="337.5" height="57" align-x="6" align-y="822" 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="0" text-align="left" list-style-type="none" tabs="inherit">The <b>freichen</b> edge detector is useful for images in which there are a variety of intensity levels defining the various edges in the picture. These kernels show a greater sensitivity to the relative pixel values, independent of their brightness.</p>
      </text>
    </region>
    <region region-id="520" left="6" top="870.75" width="348" height="14.25" align-x="6" align-y="882" 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">The output of the edge finders is compared visually at the bottom of the page.</p>
      </text>
    </region>
    <region region-id="142" left="0" top="900" width="6000" height="6" align-x="0" align-y="900" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="430" left="6" top="912.75" width="45.75" height="14.25" align-x="6" align-y="924" 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="0" text-align="left" list-style-type="none" tabs="inherit">
          <b>
            <inlineAttr font-style="normal">
              <c val="#800000">sobel(M)</c>
            </inlineAttr>
          </b>
        </p>
      </text>
      <indexes>
        <phrase>sobel function</phrase>
      </indexes>
    </region>
    <region region-id="460" left="6" top="930.75" width="212.25" height="14.25" align-x="6" align-y="942" 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="0" text-align="left" list-style-type="none" tabs="inherit">This function convolves <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">M</inlineAttr></f> with the two kernels</p>
      </text>
    </region>
    <region region-id="465" left="36" top="951.75" width="104.25" height="57.75" align-x="59.25" align-y="984" 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">K1</ml:id>
          <ml:apply>
            <ml:mult style="default" />
            <ml:apply>
              <ml:div />
              <ml:real>1</ml:real>
              <ml:real>4</ml:real>
            </ml:apply>
            <ml:matrix rows="3" cols="3">
              <ml:real>1</ml:real>
              <ml:real>2</ml:real>
              <ml:real>1</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>-1</ml:real>
              <ml:real>-2</ml:real>
              <ml:real>-1</ml:real>
            </ml:matrix>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="14" />
    </region>
    <region region-id="471" left="162" top="951.75" width="117.75" height="57.75" align-x="185.25" align-y="984" 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">K2</ml:id>
          <ml:apply>
            <ml:mult style="default" />
            <ml:apply>
              <ml:div />
              <ml:real>1</ml:real>
              <ml:real>4</ml:real>
            </ml:apply>
            <ml:matrix rows="3" cols="3">
              <ml:real>-1</ml:real>
              <ml:real>0</ml:real>
              <ml:real>1</ml:real>
              <ml:real>-2</ml:real>
              <ml:real>0</ml:real>
              <ml:real>2</ml:real>
              <ml:real>-1</ml:real>
              <ml:real>0</ml:real>
              <ml:real>1</ml:real>
            </ml:matrix>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="15" />
    </region>
    <region region-id="548" left="36" top="1020.75" width="84" height="15.75" align-x="68.25" align-y="1032" 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">Rsob</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">sobel</ml:id>
            <ml:id xml:space="preserve">R</ml:id>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="16" />
    </region>
    <region region-id="472" left="6" top="1044.75" width="316.5" height="42.75" align-x="6" align-y="1056" 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="0" text-align="left" list-style-type="none" tabs="inherit">The <b>sobel</b> kernel gives a more uniform edge detection than <b>difedge</b>, although it still gives increased weight to the orthogonal pixels over the diagonal pixels.</p>
      </text>
    </region>
    <region region-id="523" left="6" top="1092.75" width="348" height="14.25" align-x="6" align-y="1104" 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">The output of the edge finders is compared visually at the bottom of the page.</p>
      </text>
    </region>
    <region region-id="511" left="0" top="1122" width="4800" height="6" align-x="0" align-y="1122" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="454" left="6" top="1134.75" width="56.25" height="14.25" align-x="6" align-y="1146" 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="0" text-align="left" list-style-type="none" tabs="inherit">
          <b>
            <inlineAttr font-style="normal">
              <c val="#800000">prewitt(M)</c>
            </inlineAttr>
          </b>
        </p>
      </text>
      <indexes>
        <phrase>prewitt function</phrase>
      </indexes>
    </region>
    <region region-id="453" left="6" top="1152.75" width="212.25" 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="0" text-align="left" list-style-type="none" tabs="inherit">This function convolves <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">M</inlineAttr></f><b><sp /></b>with the two kernels</p>
      </text>
    </region>
    <region region-id="550" left="36" top="1173.75" width="104.25" height="57.75" align-x="59.25" align-y="1206" 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">K1</ml:id>
          <ml:apply>
            <ml:mult style="default" />
            <ml:apply>
              <ml:div />
              <ml:real>1</ml:real>
              <ml:real>3</ml:real>
            </ml:apply>
            <ml:matrix rows="3" cols="3">
              <ml:real>1</ml:real>
              <ml:real>1</ml:real>
              <ml:real>1</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>-1</ml:real>
              <ml:real>-1</ml:real>
              <ml:real>-1</ml:real>
            </ml:matrix>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="17" />
    </region>
    <region region-id="452" left="162" top="1173.75" width="117.75" height="57.75" align-x="185.25" align-y="1206" 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">K2</ml:id>
          <ml:apply>
            <ml:mult style="default" />
            <ml:apply>
              <ml:div />
              <ml:real>1</ml:real>
              <ml:real>3</ml:real>
            </ml:apply>
            <ml:matrix rows="3" cols="3">
              <ml:real>-1</ml:real>
              <ml:real>0</ml:real>
              <ml:real>1</ml:real>
              <ml:real>-1</ml:real>
              <ml:real>0</ml:real>
              <ml:real>1</ml:real>
              <ml:real>-1</ml:real>
              <ml:real>0</ml:real>
              <ml:real>1</ml:real>
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          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="18" />
    </region>
    <region region-id="549" left="36" top="1236.75" width="90" height="15.75" align-x="66.75" align-y="1248" 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">Rpre</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">prewitt</ml:id>
            <ml:id xml:space="preserve">R</ml:id>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="19" />
    </region>
    <region region-id="451" left="6" top="1260.75" width="282.75" height="28.5" align-x="6" align-y="1272" 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="0" text-align="left" list-style-type="none" tabs="inherit">The <b>prewitt</b> kernel considers the orthogonal and diagonal pixel differentials equally.</p>
      </text>
    </region>
    <region region-id="524" left="6" top="1290.75" width="288.75" height="28.5" align-x="6" align-y="1302" 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">The output of the edge finders is compared visually at the bottom of the page.</p>
      </text>
    </region>
    <region region-id="513" left="0" top="1326" width="4800" height="6" align-x="0" align-y="1326" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="437" left="6" top="1338.75" width="63.75" height="14.25" align-x="6" align-y="1350" 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="0" text-align="left" list-style-type="none" tabs="inherit">
          <b>
            <inlineAttr font-style="normal">
              <c val="#800000">roberts(M)</c>
            </inlineAttr>
          </b>
        </p>
      </text>
      <indexes>
        <phrase>roberts function</phrase>
      </indexes>
    </region>
    <region region-id="438" left="6" top="1356.75" width="212.25" height="14.25" align-x="6" align-y="1368" 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="0" text-align="left" list-style-type="none" tabs="inherit">This function convolves <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">M</inlineAttr></f><b><sp /></b>with the two kernels</p>
      </text>
    </region>
    <region region-id="552" left="36" top="1377.75" width="91.5" height="57.75" align-x="59.25" align-y="1410" 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">K1</ml:id>
          <ml:matrix rows="3" cols="3">
            <ml:real>0</ml:real>
            <ml:real>0</ml:real>
            <ml:real>0</ml:real>
            <ml:real>0</ml:real>
            <ml:real>1</ml:real>
            <ml:real>0</ml:real>
            <ml:real>-1</ml:real>
            <ml:real>0</ml:real>
            <ml:real>0</ml:real>
          </ml:matrix>
        </ml:define>
      </math>
      <rendering item-idref="20" />
    </region>
    <region region-id="442" left="162" top="1377.75" width="91.5" height="57.75" align-x="185.25" align-y="1410" 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">K2</ml:id>
          <ml:matrix rows="3" cols="3">
            <ml:real>-1</ml:real>
            <ml:real>0</ml:real>
            <ml:real>0</ml:real>
            <ml:real>0</ml:real>
            <ml:real>1</ml:real>
            <ml:real>0</ml:real>
            <ml:real>0</ml:real>
            <ml:real>0</ml:real>
            <ml:real>0</ml:real>
          </ml:matrix>
        </ml:define>
      </math>
      <rendering item-idref="21" />
    </region>
    <region region-id="551" left="36" top="1440.75" width="91.5" height="15.75" align-x="67.5" align-y="1452" 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">Rrob</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">roberts</ml:id>
            <ml:id xml:space="preserve">R</ml:id>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="22" />
    </region>
    <region region-id="443" left="6" top="1464.75" width="297" height="42.75" align-x="6" align-y="1476" 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="0" text-align="left" list-style-type="none" tabs="inherit">The <b>roberts</b> kernel considers only the diagonal pixel differentials, which emphasizes corners more clearly but can blur together small horizontal or vertical features.</p>
      </text>
      <indexes>
        <phrase>roberts kernel</phrase>
      </indexes>
    </region>
    <region region-id="525" left="6" top="1512.75" width="315.75" height="28.5" 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="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The output of the edge finders is compared visually at the bottom of the page.</p>
      </text>
    </region>
    <region region-id="555" left="6" top="1548.75" width="141.75" height="14.25" align-x="6" align-y="1560" 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">
          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr line-through="false">
                <c val="#800000">Comparison of Edge Finders</c>
              </inlineAttr>
            </b>
          </f>
        </p>
      </text>
    </region>
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      <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 have calculated output of the <f family="Times New Roman" charset="0"><b><inlineAttr font-style="normal" line-through="false">difedge, freichen, sobel, prewitt</inlineAttr></b></f><inlineAttr font-style="normal">,</inlineAttr> and <f family="Times New Roman" charset="0"><b><inlineAttr font-style="normal" line-through="false">roberts</inlineAttr></b></f> edge finders, and stored the output as <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">Rdif, Rfre, Rsob, Rpre</inlineAttr></b></f>, and <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">Rrob</inlineAttr></b></f>. Use the images below to compare various edge-finding routines visually. You may wish to adjust the contrast.</p>
      </text>
      <indexes>
        <phrase>difedge function</phrase>
        <phrase>freichen function</phrase>
        <phrase>sobel function</phrase>
        <phrase>prewitt function</phrase>
        <phrase>roberts function</phrase>
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