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              <c val="#800080">Useful Mathcad Constructs</c>
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        <phrase>Mathcad constructs</phrase>
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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">In this section we present <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">Mathcad</inlineAttr></f> implementations for a number of operations which can be useful in image processing as well as other applications. The functionality provided here ranges from simple array manipulation techniques to probabilistic and statistical methods. </p>
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            <c val="#800000">Matrix-Vector Conversions</c>
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        <phrase>matrix-vector conversions</phrase>
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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">Here's a way to convert a matrix to a vector by stacking the matrix columns:</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">Now convert a vector to a matrix by filling the matrix column-wise.</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">Start with a vector:</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">Specify the size (<f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">r </inlineAttr></f><f family="Symbol" charset="2"><inlineAttr font-weight="normal" line-through="false">´ </inlineAttr></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">c</inlineAttr></f>) of the desired matrix:</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">Verify the specified matrix size is compatible with the vector length:</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">Fill the array:</p>
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            <ml:str xml:space="preserve">preallocating the matrix speeds the calculation</ml:str>
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                    <ml:minus />
                    <ml:id xml:space="preserve">p</ml:id>
                    <ml:real>1</ml:real>
                  </ml:apply>
                </ml:range>
                <ml:program>
                  <ml:ifThen>
                    <ml:apply>
                      <ml:lessThan />
                      <ml:apply>
                        <ml:plus />
                        <ml:id xml:space="preserve">i</ml:id>
                        <ml:apply>
                          <ml:mult style="default" />
                          <ml:id xml:space="preserve">p</ml:id>
                          <ml:id xml:space="preserve">j</ml:id>
                        </ml:apply>
                      </ml:apply>
                      <ml:apply>
                        <ml:id xml:space="preserve">rows</ml:id>
                        <ml:id xml:space="preserve">V</ml:id>
                      </ml:apply>
                    </ml:apply>
                    <ml:localDefine>
                      <ml:apply>
                        <ml:indexer />
                        <ml:id xml:space="preserve">M</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 />
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                        <ml:apply>
                          <ml:plus />
                          <ml:id xml:space="preserve">i</ml:id>
                          <ml:apply>
                            <ml:mult style="default" />
                            <ml:id xml:space="preserve">p</ml:id>
                            <ml:id xml:space="preserve">j</ml:id>
                          </ml:apply>
                        </ml:apply>
                      </ml:apply>
                    </ml:localDefine>
                  </ml:ifThen>
                  <ml:otherwise>
                    <ml:localDefine>
                      <ml:apply>
                        <ml:indexer />
                        <ml:id xml:space="preserve">M</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:real>0</ml:real>
                    </ml:localDefine>
                  </ml:otherwise>
                </ml:program>
              </ml:for>
            </ml:for>
            <ml:id xml:space="preserve">M</ml:id>
          </ml:program>
        </ml:define>
      </math>
      <rendering item-idref="15" />
    </region>
    <region region-id="843" left="30" top="857.25" width="243.75" height="78.75" align-x="99.75" align-y="900" 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">Pack</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">V</ml:id>
              <ml:id xml:space="preserve">r</ml:id>
              <ml:id xml:space="preserve">c</ml:id>
            </ml:sequence>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="4" cols="6">
              <ml:real>24</ml:real>
              <ml:real>6</ml:real>
              <ml:real>7</ml:real>
              <ml:real>19</ml:real>
              <ml:real>-10</ml:real>
              <ml:real>-7</ml:real>
              <ml:real>125</ml:real>
              <ml:real>78</ml:real>
              <ml:real>92</ml:real>
              <ml:real>0</ml:real>
              <ml:real>-38</ml:real>
              <ml:real>14</ml:real>
              <ml:real>25</ml:real>
              <ml:real>100</ml:real>
              <ml:real>10</ml:real>
              <ml:real>1</ml:real>
              <ml:real>0.1</ml:real>
              <ml:real>0.01</ml:real>
              <ml:real>0.04</ml:real>
              <ml:real>6</ml:real>
              <ml:real>11</ml:real>
              <ml:real>17</ml:real>
              <ml:real>82</ml:real>
              <ml:real>90</ml:real>
            </ml:matrix>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="16" />
    </region>
    <region region-id="669" left="0" top="948" width="6000" height="6" align-x="0" align-y="948" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="460" left="6" top="960.75" width="84" 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="0" text-align="left" list-style-type="none" tabs="inherit">
          <b>
            <c val="#800000">Matrix Statistics</c>
          </b>
        </p>
      </text>
      <indexes>
        <phrase>matrix statistics</phrase>
      </indexes>
    </region>
    <region region-id="461" left="6" top="978.75" width="266.25" height="28.5" align-x="6" align-y="990" 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">Below are expressions for computing basic statistics: mean, variance, and standard deviation of array data. </p>
      </text>
    </region>
    <region region-id="464" left="36" top="1020.75" width="246.75" height="99.75" align-x="54.75" align-y="1074" 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:matrix rows="5" cols="7">
            <ml:real>0.2</ml:real>
            <ml:real>8.1</ml:real>
            <ml:real>-1.9</ml:real>
            <ml:real>3.1</ml:real>
            <ml:real>5.7</ml:real>
            <ml:real>2.9</ml:real>
            <ml:real>1.6</ml:real>
            <ml:real>3.7</ml:real>
            <ml:real>8.1</ml:real>
            <ml:real>-0.5</ml:real>
            <ml:real>1.7</ml:real>
            <ml:real>7.3</ml:real>
            <ml:real>1.9</ml:real>
            <ml:real>10.4</ml:real>
            <ml:real>5.8</ml:real>
            <ml:real>2.1</ml:real>
            <ml:real>3.1</ml:real>
            <ml:real>4.1</ml:real>
            <ml:real>-5.9</ml:real>
            <ml:real>2.6</ml:real>
            <ml:real>-5.3</ml:real>
            <ml:real>5.8</ml:real>
            <ml:real>9.7</ml:real>
            <ml:real>-9.3</ml:real>
            <ml:real>2.7</ml:real>
            <ml:real>1.8</ml:real>
            <ml:real>-2.8</ml:real>
            <ml:real>4.5</ml:real>
            <ml:real>9.0</ml:real>
            <ml:real>-4.5</ml:real>
            <ml:real>14.3</ml:real>
            <ml:real>-3.8</ml:real>
            <ml:real>3.9</ml:real>
            <ml:real>8.3</ml:real>
            <ml:real>2.1</ml:real>
          </ml:matrix>
        </ml:define>
      </math>
      <rendering item-idref="17" />
    </region>
    <region region-id="467" left="6" top="1134.75" width="93" 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>Mean</b> of array data:</p>
      </text>
    </region>
    <region region-id="475" left="36" top="1164.75" width="94.5" height="15.75" align-x="87.75" align-y="1176" 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">mean</ml:id>
            <ml:id xml:space="preserve">M</ml:id>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>2.8714285714285714</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="18" />
    </region>
    <region region-id="474" left="6" top="1194.75" width="108" height="14.25" align-x="6" align-y="1206" 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>Variance</b> of array data:</p>
      </text>
    </region>
    <region region-id="473" left="36" top="1224.75" width="90.75" height="15.75" align-x="78" align-y="1236" 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">var</ml:id>
            <ml:id xml:space="preserve">M</ml:id>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>25.054612244897953</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="19" />
    </region>
    <region region-id="479" left="6" top="1254.75" width="156.75" height="14.25" align-x="6" align-y="1266" 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>Standard deviation</b> of array data:</p>
      </text>
    </region>
    <region region-id="694" left="36" top="1284.75" width="94.5" height="15.75" align-x="87.75" align-y="1296" 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">stdev</ml:id>
            <ml:id xml:space="preserve">M</ml:id>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>5.0054582452456788</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="20" />
    </region>
    <region region-id="670" left="0" top="1314" width="6000" height="6" align-x="0" align-y="1314" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="849" left="6" top="1326.75" width="159" height="14.25" align-x="6" align-y="1338" 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>
            <c val="#800000">Random Noise Generation</c>
          </b>
        </p>
      </text>
      <indexes>
        <phrase>random noise generation</phrase>
      </indexes>
    </region>
    <region region-id="848" left="6" top="1362.75" width="246.75" height="14.25" align-x="6" align-y="1374" 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">
          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr font-style="normal" line-through="false">
                <c val="#800000">Normally (Gaussian) Distributed Random Noise</c>
              </inlineAttr>
            </b>
          </f>
        </p>
      </text>
      <indexes>
        <phrase>noise, Gaussian distributed random</phrase>
        <phrase>noise, normally distributed random</phrase>
      </indexes>
    </region>
    <region region-id="489" left="6" top="1380.75" width="287.25" height="28.5" align-x="6" align-y="1392" 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">For a normal or Gaussian distribution having mean and standard deviation</p>
      </text>
    </region>
    <region region-id="484" left="36" top="1416" width="41.25" height="16.5" align-x="51.75" align-y="1428" 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">μ</ml:id>
          <ml:real>2.3</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="21" />
    </region>
    <region region-id="485" left="120" top="1416" width="42" height="16.5" align-x="136.5" align-y="1428" 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">σ</ml:id>
          <ml:real>1.2</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="22" />
    </region>
    <region region-id="490" left="6" top="1440.75" width="111" height="14.25" align-x="6" align-y="1452" 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">respectively, the function</p>
      </text>
    </region>
    <region region-id="696" left="36" top="1464" width="134.25" height="16.5" align-x="115.5" align-y="1476" 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">rnorm</ml:id>
            <ml:sequence>
              <ml:real>1</ml:real>
              <ml:id xml:space="preserve">μ</ml:id>
              <ml:id xml:space="preserve">σ</ml:id>
            </ml:sequence>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="1" cols="1">
              <ml:real>1.7732350194212667</ml:real>
            </ml:matrix>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="23" />
    </region>
    <region region-id="500" left="6" top="1488.75" width="293.25" height="28.5" align-x="6" align-y="1500" 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">generates a single random deviate. A matrix of such deviates can </p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">then be computed via the following:</p>
      </text>
    </region>
    <region region-id="493" left="36" top="1530.75" width="45.75" height="15.75" align-x="53.25" align-y="1542" 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">N</ml:id>
          <ml:real>100</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="24" />
    </region>
    <region region-id="705" left="120" top="1530" width="32.25" height="16.5" align-x="135.75" align-y="1542" 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">μ</ml:id>
          <ml:real>0</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="25" />
    </region>
    <region region-id="714" left="180" top="1530" width="33" height="16.5" align-x="196.5" align-y="1542" 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">σ</ml:id>
          <ml:real>1</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="26" />
    </region>
    <region region-id="709" left="36" top="1555.5" width="194.25" height="21" align-x="75" align-y="1572" 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">normal</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">Pack</ml:id>
            <ml:sequence>
              <ml:apply>
                <ml:id xml:space="preserve">rnorm</ml:id>
                <ml:sequence>
                  <ml:apply>
                    <ml:pow />
                    <ml:id xml:space="preserve">N</ml:id>
                    <ml:real>2</ml:real>
                  </ml:apply>
                  <ml:id xml:space="preserve">μ</ml:id>
                  <ml:id xml:space="preserve">σ</ml:id>
                </ml:sequence>
              </ml:apply>
              <ml:id xml:space="preserve">N</ml:id>
              <ml:id xml:space="preserve">N</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="27" />
    </region>
    <region region-id="759" left="18" top="1585.5" width="152.25" height="21" align-x="90" align-y="1602" 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">mean</ml:id>
            <ml:id xml:space="preserve">normal</ml:id>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>0.0075961582925657811</ml:real>
          </result>
        </ml:eval>
        <resultFormat>
          <general precision="3" show-trailing-zeros="false" radix="dec" complex-threshold="10" exponential-threshold="3" />
          <matrix display-style="auto" expand-nested-arrays="false" />
          <unit format-units="false" simplify-units="false" />
        </resultFormat>
      </math>
      <rendering item-idref="28" />
    </region>
    <region region-id="760" left="186" top="1590.75" width="114.75" height="15.75" align-x="258" align-y="1602" 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">stdev</ml:id>
            <ml:id xml:space="preserve">normal</ml:id>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>0.99639477546296729</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="29" />
    </region>
    <region region-id="501" left="6" top="1620.75" width="312.75" height="42.75" align-x="6" align-y="1632" 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">All Gaussian noise above the mean, 0, is positive (added to pixel values) and all Gaussian noise below the mean is negative (subtracted from pixel values).</p>
      </text>
    </region>
    <region region-id="502" left="6" top="1680.75" width="207.75" height="14.25" align-x="6" align-y="1692" 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">Exponentially Distributed Random Noise</c>
            </inlineAttr>
          </b>
        </p>
      </text>
      <indexes>
        <phrase>noise, exponentially distributed random</phrase>
      </indexes>
    </region>
    <region region-id="508" left="6" top="1698.75" width="195" height="14.25" align-x="6" align-y="1710" 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">For an exponential distribution having mean</p>
      </text>
    </region>
    <region region-id="504" left="36" top="1722" width="48.75" height="16.5" align-x="53.25" align-y="1734" 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">α</ml:id>
          <ml:real>50.3</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="30" />
    </region>
    <region region-id="509" left="6" top="1746.75" width="51.75" height="14.25" align-x="6" align-y="1758" 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 function</p>
      </text>
    </region>
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      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:apply>
            <ml:id xml:space="preserve">rexp</ml:id>
            <ml:sequence>
              <ml:real>1</ml:real>
              <ml:apply>
                <ml:div />
                <ml:real>1</ml:real>
                <ml:id xml:space="preserve">α</ml:id>
              </ml:apply>
            </ml:sequence>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="1" cols="1">
              <ml:real>26.843190332562678</ml:real>
            </ml:matrix>
          </result>
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    <region region-id="510" left="6" top="1812.75" width="301.5" height="14.25" align-x="6" align-y="1824" 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">returns a random deviate. A matrix of exponential noise is given by:</p>
      </text>
    </region>
    <region region-id="513" left="36" top="1834.5" width="175.5" height="34.5" align-x="66.75" align-y="1854" 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">expo</ml:id>
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            <ml:id xml:space="preserve">Pack</ml:id>
            <ml:sequence>
              <ml:apply>
                <ml:id xml:space="preserve">rexp</ml:id>
                <ml:sequence>
                  <ml:apply>
                    <ml:pow />
                    <ml:id xml:space="preserve">N</ml:id>
                    <ml:real>2</ml:real>
                  </ml:apply>
                  <ml:apply>
                    <ml:div />
                    <ml:real>1</ml:real>
                    <ml:id xml:space="preserve">α</ml:id>
                  </ml:apply>
                </ml:sequence>
              </ml:apply>
              <ml:id xml:space="preserve">N</ml:id>
              <ml:id xml:space="preserve">N</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
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    <region region-id="514" left="36" top="1878.75" width="112.5" height="15.75" align-x="99.75" align-y="1890" 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">mean</ml:id>
            <ml:id xml:space="preserve">expo</ml:id>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>49.918952847165613</ml:real>
          </result>
        </ml:eval>
      </math>
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    <region region-id="515" left="174" top="1878.75" width="112.5" height="15.75" align-x="237.75" align-y="1890" 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">stdev</ml:id>
            <ml:id xml:space="preserve">expo</ml:id>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>49.558293145499867</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="34" />
    </region>
    <region region-id="524" left="0" top="1902" width="6000" height="6" align-x="0" align-y="1902" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="521" left="6" top="1914.75" width="197.25" height="14.25" align-x="6" align-y="1926" 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">Poisson Distributed Random Deviates</c>
            </inlineAttr>
          </b>
        </p>
      </text>
      <indexes>
        <phrase>Poisson distributed random deviates</phrase>
      </indexes>
    </region>
    <region region-id="522" left="6" top="1938.75" width="174" height="14.25" align-x="6" align-y="1950" 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">For a Poisson distribution having mean</p>
      </text>
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      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">λ</ml:id>
          <ml:real>1.2</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="35" />
    </region>
    <region region-id="719" left="6" top="1986.75" width="51.75" height="14.25" align-x="6" align-y="1998" 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 function</p>
      </text>
    </region>
    <region region-id="725" left="36" top="2010" width="94.5" height="16.5" align-x="96.75" align-y="2022" 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">rpois</ml:id>
            <ml:sequence>
              <ml:real>1</ml:real>
              <ml:id xml:space="preserve">λ</ml:id>
            </ml:sequence>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="1" cols="1">
              <ml:real>1</ml:real>
            </ml:matrix>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="36" />
    </region>
    <region region-id="726" left="6" top="2034.75" width="285.75" height="14.25" align-x="6" align-y="2046" 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">returns a random deviate. A matrix of Poisson noise is given by:</p>
      </text>
    </region>
    <region region-id="731" left="36" top="2059.5" width="174" height="21" align-x="73.5" align-y="2076" 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">poisso</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">Pack</ml:id>
            <ml:sequence>
              <ml:apply>
                <ml:id xml:space="preserve">rpois</ml:id>
                <ml:sequence>
                  <ml:apply>
                    <ml:pow />
                    <ml:id xml:space="preserve">N</ml:id>
                    <ml:real>2</ml:real>
                  </ml:apply>
                  <ml:id xml:space="preserve">λ</ml:id>
                </ml:sequence>
              </ml:apply>
              <ml:id xml:space="preserve">N</ml:id>
              <ml:id xml:space="preserve">N</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
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    </region>
    <region region-id="732" left="36" top="2094.75" width="113.25" height="15.75" align-x="106.5" align-y="2106" 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">mean</ml:id>
            <ml:id xml:space="preserve">poisso</ml:id>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>1.2152</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="38" />
    </region>
    <region region-id="733" left="174" top="2094.75" width="113.25" height="15.75" align-x="244.5" align-y="2106" 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">stdev</ml:id>
            <ml:id xml:space="preserve">poisso</ml:id>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>1.0934756330161182</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="39" />
    </region>
    <region region-id="802" left="6" top="2124.75" width="198" height="14.25" align-x="6" align-y="2136" 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>
            <c val="#800000">Locating Elements within an Array</c>
          </b>
        </p>
      </text>
      <indexes>
        <phrase>array, locating elements within</phrase>
      </indexes>
    </region>
    <region region-id="560" left="6" top="2142.75" width="289.5" height="42.75" align-x="6" align-y="2154" 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 expressions that follow determine the location, by specifying indices, of all elements of an array which satisfy a specified condition. </p>
      </text>
    </region>
    <region region-id="561" left="6" top="2196.75" width="147" height="14.25" align-x="6" align-y="2208" 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">For example, start with the array</p>
      </text>
    </region>
    <region region-id="562" left="36" top="2225.25" width="213.75" height="162.75" align-x="55.5" align-y="2310" 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">W</ml:id>
          <ml:matrix rows="8" cols="6">
            <ml:real>5.89</ml:real>
            <ml:real>7.56</ml:real>
            <ml:real>7.45</ml:real>
            <ml:real>1.22</ml:real>
            <ml:real>1.26</ml:real>
            <ml:real>5.83</ml:real>
            <ml:real>3.64</ml:real>
            <ml:real>5.13</ml:real>
            <ml:real>9.57</ml:real>
            <ml:real>1.15</ml:real>
            <ml:real>0.26</ml:real>
            <ml:real>7.36</ml:real>
            <ml:real>2.36</ml:real>
            <ml:real>4.11</ml:real>
            <ml:real>9.93</ml:real>
            <ml:real>3.71</ml:real>
            <ml:real>4.12</ml:real>
            <ml:real>9.33</ml:real>
            <ml:real>0.96</ml:real>
            <ml:real>7.96</ml:real>
            <ml:real>4.74</ml:real>
            <ml:real>3.78</ml:real>
            <ml:real>8.95</ml:real>
            <ml:real>5.78</ml:real>
            <ml:real>4.21</ml:real>
            <ml:real>9.26</ml:real>
            <ml:real>5.41</ml:real>
            <ml:real>7.94</ml:real>
            <ml:real>3.07</ml:real>
            <ml:real>8.14</ml:real>
            <ml:real>9.73</ml:real>
            <ml:real>3.16</ml:real>
            <ml:real>5.52</ml:real>
            <ml:real>5.42</ml:real>
            <ml:real>0.6</ml:real>
            <ml:real>2.91</ml:real>
            <ml:real>5.16</ml:real>
            <ml:real>0.43</ml:real>
            <ml:real>2.71</ml:real>
            <ml:real>9.57</ml:real>
            <ml:real>3.91</ml:real>
            <ml:real>0.15</ml:real>
            <ml:real>9.45</ml:real>
            <ml:real>6.42</ml:real>
            <ml:real>7.97</ml:real>
            <ml:real>5.21</ml:real>
            <ml:real>6.09</ml:real>
            <ml:real>7.17</ml:real>
          </ml:matrix>
        </ml:define>
      </math>
      <rendering item-idref="40" />
    </region>
    <region region-id="738" left="6" top="2400.75" width="325.5" height="42.75" 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="0" text-align="left" list-style-type="none" tabs="inherit">Define a <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">function which returns 1 if a particular value satisfies the desired condition, 0 otherwise. In this example we locate those entries with value between 2 and 4 inclusive. </inlineAttr></f></p>
      </text>
    </region>
    <region region-id="743" left="6" top="2454.75" width="159" height="14.25" align-x="6" align-y="2466" 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">
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr line-through="false">Condition the elements must match:</inlineAttr>
          </f>
        </p>
      </text>
    </region>
    <region region-id="746" left="198" top="2454.75" width="116.25" height="15.75" align-x="226.5" align-y="2466" 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:function>
            <ml:id xml:space="preserve">c</ml:id>
            <ml:boundVars>
              <ml:id xml:space="preserve">x</ml:id>
            </ml:boundVars>
          </ml:function>
          <ml:apply>
            <ml:mult style="default" />
            <ml:parens>
              <ml:apply>
                <ml:greaterOrEqual />
                <ml:id xml:space="preserve">x</ml:id>
                <ml:real>2</ml:real>
              </ml:apply>
            </ml:parens>
            <ml:parens>
              <ml:apply>
                <ml:lessOrEqual />
                <ml:id xml:space="preserve">x</ml:id>
                <ml:real>4</ml:real>
              </ml:apply>
            </ml:parens>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="41" />
    </region>
    <region region-id="747" left="42" top="2478.75" width="59.25" height="15.75" align-x="79.5" align-y="2490" 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">c</ml:id>
            <ml:real>2.5</ml:real>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>1</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="42" />
    </region>
    <region region-id="748" left="138" top="2478.75" width="59.25" height="15.75" align-x="175.5" align-y="2490" 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">c</ml:id>
            <ml:real>0.3</ml:real>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>0</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="43" />
    </region>
    <region region-id="749" left="234" top="2478.75" width="59.25" height="15.75" align-x="271.5" align-y="2490" 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">c</ml:id>
            <ml:real>4.1</ml:real>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>0</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="44" />
    </region>
    <region region-id="833" left="0" top="2508" width="6000" height="6" align-x="0" align-y="2508" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
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    <region region-id="754" left="6" top="2520.75" width="79.5" height="14.25" align-x="6" align-y="2532" 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">
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr line-through="false">Locating function:</inlineAttr>
          </f>
        </p>
      </text>
    </region>
    <region region-id="753" left="42" top="2550.75" width="221.25" height="185.25" align-x="113.25" align-y="2562" 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">Locate</ml:id>
            <ml:boundVars>
              <ml:id xml:space="preserve">M</ml:id>
              <ml:id xml:space="preserve">c</ml:id>
            </ml:boundVars>
          </ml:function>
          <ml:program>
            <ml:localDefine>
              <ml:id xml:space="preserve">i</ml:id>
              <ml:real>0</ml:real>
            </ml:localDefine>
            <ml:localDefine>
              <ml:id xml:space="preserve">L</ml:id>
              <ml:real>0</ml:real>
            </ml:localDefine>
            <ml:for>
              <ml:id xml:space="preserve">m</ml:id>
              <ml:range>
                <ml:real>0</ml:real>
                <ml:apply>
                  <ml:minus />
                  <ml:apply>
                    <ml:id xml:space="preserve">rows</ml:id>
                    <ml:id xml:space="preserve">M</ml:id>
                  </ml:apply>
                  <ml:real>1</ml:real>
                </ml:apply>
              </ml:range>
              <ml:for>
                <ml:id xml:space="preserve">n</ml:id>
                <ml:range>
                  <ml:real>0</ml:real>
                  <ml:apply>
                    <ml:minus />
                    <ml:apply>
                      <ml:id xml:space="preserve">cols</ml:id>
                      <ml:id xml:space="preserve">M</ml:id>
                    </ml:apply>
                    <ml:real>1</ml:real>
                  </ml:apply>
                </ml:range>
                <ml:ifThen>
                  <ml:apply>
                    <ml:equal />
                    <ml:apply>
                      <ml:id xml:space="preserve">c</ml:id>
                      <ml:apply>
                        <ml:indexer />
                        <ml:id xml:space="preserve">M</ml:id>
                        <ml:sequence>
                          <ml:id xml:space="preserve">m</ml:id>
                          <ml:id xml:space="preserve">n</ml:id>
                        </ml:sequence>
                      </ml:apply>
                    </ml:apply>
                    <ml:real>1</ml:real>
                  </ml:apply>
                  <ml:program>
                    <ml:localDefine>
                      <ml:apply>
                        <ml:matcol />
                        <ml:id xml:space="preserve">L</ml:id>
                        <ml:id xml:space="preserve">i</ml:id>
                      </ml:apply>
                      <ml:matrix rows="2" cols="1">
                        <ml:id xml:space="preserve">m</ml:id>
                        <ml:id xml:space="preserve">n</ml:id>
                      </ml:matrix>
                    </ml:localDefine>
                    <ml:localDefine>
                      <ml:id xml:space="preserve">i</ml:id>
                      <ml:apply>
                        <ml:plus />
                        <ml:id xml:space="preserve">i</ml:id>
                        <ml:real>1</ml:real>
                      </ml:apply>
                    </ml:localDefine>
                  </ml:program>
                </ml:ifThen>
              </ml:for>
            </ml:for>
            <ml:id xml:space="preserve">L</ml:id>
          </ml:program>
        </ml:define>
      </math>
      <rendering item-idref="45" />
    </region>
    <region region-id="768" left="36" top="2750.25" width="223.5" height="36.75" align-x="107.25" align-y="2772" 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">Locate</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">W</ml:id>
              <ml:id xml:space="preserve">c</ml:id>
            </ml:sequence>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="2" cols="9">
              <ml:real>0</ml:real>
              <ml:real>5</ml:real>
              <ml:real>3</ml:real>
              <ml:real>4</ml:real>
              <ml:real>4</ml:real>
              <ml:real>1</ml:real>
              <ml:real>4</ml:real>
              <ml:real>3</ml:real>
              <ml:real>5</ml:real>
              <ml:real>2</ml:real>
              <ml:real>6</ml:real>
              <ml:real>0</ml:real>
              <ml:real>6</ml:real>
              <ml:real>4</ml:real>
              <ml:real>7</ml:real>
              <ml:real>1</ml:real>
              <ml:real>7</ml:real>
              <ml:real>3</ml:real>
            </ml:matrix>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="46" />
    </region>
    <region region-id="769" left="6" top="2802.75" width="240.75" height="14.25" align-x="6" align-y="2814" 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">These elements can be highlighted within the array by:</p>
      </text>
    </region>
    <region region-id="767" left="30" top="2837.25" width="248.25" height="162.75" align-x="78.75" align-y="2922" 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:vectorize />
            <ml:apply>
              <ml:mult style="default" />
              <ml:apply>
                <ml:id xml:space="preserve">c</ml:id>
                <ml:id xml:space="preserve">W</ml:id>
              </ml:apply>
              <ml:id xml:space="preserve">W</ml:id>
            </ml:apply>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="8" cols="6">
              <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>3.64</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>2.36</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>3.71</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>3.78</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>3.07</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>3.16</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>2.91</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>2.71</ml:real>
              <ml:real>0</ml:real>
              <ml:real>3.91</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:matrix>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="47" />
    </region>
    <region region-id="834" left="0" top="3012" width="6000" height="6" align-x="0" align-y="3012" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="603" left="6" top="3024.75" width="325.5" height="28.5" align-x="6" align-y="3036" 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">One common application of this construct is to locate occurrences of the maximum (or minimum) array value.</p>
      </text>
    </region>
    <region region-id="604" left="36" top="3060.75" width="69" height="15.75" align-x="54.75" align-y="3072" 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:id xml:space="preserve">max</ml:id>
            <ml:id xml:space="preserve">W</ml:id>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="48" />
    </region>
    <region region-id="605" left="138" top="3060.75" width="54.75" height="15.75" align-x="156" align-y="3072" 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:id xml:space="preserve">M</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>9.93</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="49" />
    </region>
    <region region-id="608" left="6" top="3084.75" width="149.25" height="14.25" align-x="6" align-y="3096" 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">In this case, define the condition: </p>
      </text>
    </region>
    <region region-id="607" left="36" top="3108.75" width="85.5" height="18" align-x="79.5" align-y="3120" 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" subscript="max">c</ml:id>
            <ml:boundVars>
              <ml:id xml:space="preserve">x</ml:id>
            </ml:boundVars>
          </ml:function>
          <ml:apply>
            <ml:equal />
            <ml:id xml:space="preserve">x</ml:id>
            <ml:id xml:space="preserve">M</ml:id>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="50" />
    </region>
    <region region-id="778" left="36" top="3138.75" width="112.5" height="18" align-x="57.75" align-y="3150" 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">ind</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">Locate</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">W</ml:id>
              <ml:id xml:space="preserve" subscript="max">c</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="51" />
    </region>
    <region region-id="793" left="186" top="3128.25" width="59.25" height="36.75" align-x="207" align-y="3150" 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:id xml:space="preserve">ind</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="2" cols="1">
              <ml:real>6</ml:real>
              <ml:real>1</ml:real>
            </ml:matrix>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="52" />
    </region>
    <region region-id="773" left="36" top="3174.75" width="54.75" height="15.75" align-x="54" align-y="3186" 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:id xml:space="preserve">M</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>9.93</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="53" />
    </region>
    <region region-id="794" left="186" top="3174.75" width="96" height="22.5" align-x="245.25" align-y="3186" 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:indexer />
            <ml:id xml:space="preserve">W</ml:id>
            <ml:sequence>
              <ml:apply>
                <ml:indexer />
                <ml:id xml:space="preserve">ind</ml:id>
                <ml:real>0</ml:real>
              </ml:apply>
              <ml:apply>
                <ml:indexer />
                <ml:id xml:space="preserve">ind</ml:id>
                <ml:real>1</ml:real>
              </ml:apply>
            </ml:sequence>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>9.93</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="54" />
    </region>
    <region region-id="620" left="6" top="3204.75" width="309" height="99.75" align-x="6" align-y="3216" 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">In general, pixels satisfying a certain property may reside in a clearly selectable subimage of the original image. In this case or in situations where you are interested only in analyzing a certain subimage, the calculations above can be sped up considerably by visually selecting the appropriate subimage (submatrix) of the given image (matrix), performing the desired search on the subimage, then translating the resulting subimage coordinates to the full image coordinates. </p>
      </text>
    </region>
    <region region-id="799" left="0" top="3318" width="6000" height="6" align-x="0" align-y="3318" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="634" left="6" top="3330.75" width="189.75" height="14.25" align-x="6" align-y="3342" 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">As an example, consider the usual pattern:</p>
      </text>
    </region>
    <region region-id="629" left="36" top="3354.75" width="173.25" height="15.75" align-x="51" align-y="3366" 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">P</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">READ_IMAGE</ml:id>
            <ml:str xml:space="preserve">pattern.bmp</ml:str>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="55" />
    </region>
    <region region-id="846" left="36" top="3374.25" width="174.75" height="126.75" align-x="153" align-y="3396" 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="56" disable-calc="false">
        <inputs>
          <ml:id xml:space="preserve">P</ml:id>
        </inputs>
        <outputs>
          <ml:matrix rows="2" cols="1">
            <ml:id xml:space="preserve">Reg</ml:id>
            <ml:id xml:space="preserve">SubP</ml:id>
          </ml:matrix>
        </outputs>
      </component>
      <rendering item-idref="57" />
    </region>
    <region region-id="635" left="6" top="3516.75" width="315" height="57" align-x="6" align-y="3528" 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">Using a selection rectangle, we select that subimage consisting of the long fuzzy rectangle in the middle, and output it from the Viewer component as matrix SubP. We also output the region coordinates as Reg. </p>
      </text>
    </region>
    <region region-id="652" left="6" top="3576.75" width="296.25" height="14.25" align-x="6" align-y="3588" 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">Next we locate the pixel(s) of minimum value within the subimage.</p>
      </text>
    </region>
    <region region-id="782" left="36" top="3606.75" width="77.25" height="15.75" align-x="52.5" align-y="3618" 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">m</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">min</ml:id>
            <ml:id xml:space="preserve">SubP</ml:id>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="58" />
    </region>
    <region region-id="797" left="150" top="3606.75" width="37.5" height="15.75" align-x="165.75" align-y="3618" 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:id xml:space="preserve">m</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>2</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="59" />
    </region>
    <region region-id="784" left="36" top="3636.75" width="82.5" height="18" align-x="78.75" align-y="3648" 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" subscript="min">c</ml:id>
            <ml:boundVars>
              <ml:id xml:space="preserve">x</ml:id>
            </ml:boundVars>
          </ml:function>
          <ml:apply>
            <ml:equal />
            <ml:id xml:space="preserve">x</ml:id>
            <ml:id xml:space="preserve">m</ml:id>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="60" />
    </region>
    <region region-id="785" left="36" top="3666.75" width="125.25" height="18" align-x="57.75" align-y="3678" 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">ind</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">Locate</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">SubP</ml:id>
              <ml:id xml:space="preserve" subscript="min">c</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="61" />
    </region>
    <region region-id="795" left="180" top="3656.25" width="65.25" height="36.75" align-x="201" align-y="3678" 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:id xml:space="preserve">ind</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="2" cols="1">
              <ml:real>18</ml:real>
              <ml:real>43</ml:real>
            </ml:matrix>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="62" />
    </region>
    <region region-id="787" left="36" top="3708.75" width="37.5" height="15.75" align-x="51.75" align-y="3720" 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:id xml:space="preserve">m</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>2</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="63" />
    </region>
    <region region-id="796" left="180" top="3708.75" width="94.5" height="22.5" align-x="252.75" align-y="3720" 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:indexer />
            <ml:id xml:space="preserve">SubP</ml:id>
            <ml:sequence>
              <ml:apply>
                <ml:indexer />
                <ml:id xml:space="preserve">ind</ml:id>
                <ml:real>0</ml:real>
              </ml:apply>
              <ml:apply>
                <ml:indexer />
                <ml:id xml:space="preserve">ind</ml:id>
                <ml:real>1</ml:real>
              </ml:apply>
            </ml:sequence>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>2</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="64" />
    </region>
    <region region-id="835" left="0" top="3744" width="6000" height="6" align-x="0" align-y="3744" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="664" left="6" top="3756.75" width="315" height="71.25" align-x="6" align-y="3768" 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">To translate these coordinates to those of the original image matrix <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">P</inlineAttr></f>, adjust the row and column coordinates just determined by the region's upper left corner.</p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit" />
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The region is given as (left, top, right, bottom):</p>
      </text>
    </region>
    <region region-id="806" left="36" top="3839.25" width="75.75" height="78.75" align-x="61.5" align-y="3882" 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:id xml:space="preserve">Reg</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="4" cols="1">
              <ml:real>3</ml:real>
              <ml:real>38</ml:real>
              <ml:real>105</ml:real>
              <ml:real>59</ml:real>
            </ml:matrix>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="65" />
    </region>
    <region region-id="659" left="36" top="3930.75" width="55.5" height="17.25" align-x="57.75" align-y="3942" 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">left</ml:id>
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">Reg</ml:id>
            <ml:real>0</ml:real>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="66" />
    </region>
    <region region-id="807" left="132" top="3930.75" width="57" height="17.25" align-x="155.25" align-y="3942" 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">top</ml:id>
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">Reg</ml:id>
            <ml:real>1</ml:real>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="67" />
    </region>
    <region region-id="665" left="6" top="3954.75" width="25.5" height="14.25" align-x="6" align-y="3966" 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">Note:</p>
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