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              <c val="#800080">Filtering in the Fourier Transform Domain</c>
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            <c val="#800000">Function Definitions and Syntax</c>
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              <c val="#800000"> E)</c>
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        <phrase>matconv function</phrase>
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              <c val="#800000">matdeconv(M, E, </c>
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                <c val="#800000">e</c>
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              <c val="#800000">)</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 convolve, or deconvolve, an image with a function, <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">f, or mask, E,</inlineAttr></f> by multiplying the two together in spatial frequency (the Fourier transform domain). Filtering in the frequency domain can result in a variety of smoothing, edge detection, and periodic noise-reduction operations, depending upon the chosen function or mask. It is often mathematically more efficient to filter in the transform domain, or easier to define filters, than by using convolution in the image domain.</p>
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            <c val="#800000">Arguments</c>
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              <inlineAttr line-through="false">M</inlineAttr>
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          </f>, the image matrix</p>
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              <c val="#008080">•</c>
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              <sp />
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          </f>, a function in spatial frequency</p>
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              <f size="14">• </f>
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              <inlineAttr line-through="false">E</inlineAttr>
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          </f>, the frequency-domain mask, and</p>
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          </f>, an error-correction factor for deconvolution routines, to avoid divide-by-zero errors</p>
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        <phrase>deconvolution routines</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">The functions return a matrix containing the convolved image in the image domain.</p>
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          </f>Fourier transforming algorithms in two dimensions return the zero-frequency component in the upper left-hand corner. </p>
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            <c val="#800000">Function Details</c>
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            <b>
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                <c val="#800000">conv(M, f)</c>
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        <phrase>funconv function</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">This function filters an image<b><sp /></b><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">M with a function in spatial frequency. The real-valued function, f, is used to construct a mask, E, the same size as M. M</inlineAttr></f><b><sp /></b>is Fourier-transformed, then multiplied termwise by the mask.</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 mask <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">is constructed by passing arguments to f which are the normalized distance from the upper-left corner of M</inlineAttr></f>. This process continues to the center of the mask, at which point the mask is duplicated in four pieces symmetrically. This process is used because the upper left-hand corner is DC (zero frequency). </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">To see a method for recentering a frequency-domain image, see <f family="Times New Roman" charset="0"><b><u><inlineAttr line-through="false">Centering a Transformed Image</inlineAttr></u></b></f><f family="Times New Roman" charset="0"><b><inlineAttr underline="false" line-through="false">.</inlineAttr></b></f></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">To see how the mask is constructed, use a test image matrix, <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">M.</inlineAttr></f></p>
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            <ml:real>0</ml:real>
            <ml:id xml:space="preserve">colmax</ml:id>
          </ml:range>
        </ml:define>
      </math>
      <rendering item-idref="9" />
    </region>
    <region region-id="383" left="234" top="666.75" width="50.25" height="17.25" align-x="264" align-y="678" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">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:mult style="default" />
            <ml:id xml:space="preserve">i</ml:id>
            <ml:id xml:space="preserve">j</ml:id>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="10" />
    </region>
    <region region-id="378" left="6" top="690.75" width="306.75" height="14.25" align-x="6" align-y="702" 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">Now find the pixel which will be considered the center of the matrix.</p>
      </text>
    </region>
    <region region-id="384" left="36" top="712.5" width="91.5" height="33.75" align-x="48" align-y="732" 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">r</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">ceil</ml:id>
            <ml:apply>
              <ml:div />
              <ml:id xml:space="preserve">rowmax</ml:id>
              <ml:real>2</ml:real>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="11" />
    </region>
    <region region-id="385" left="132" top="712.5" width="88.5" height="33.75" align-x="145.5" align-y="732" 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">c</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">ceil</ml:id>
            <ml:apply>
              <ml:div />
              <ml:id xml:space="preserve">colmax</ml:id>
              <ml:real>2</ml:real>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="12" />
    </region>
    <region region-id="386" left="234" top="720.75" width="33" height="15.75" align-x="245.25" align-y="732" 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">r</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>2</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="13" />
    </region>
    <region region-id="388" left="282" top="720.75" width="34.5" height="15.75" align-x="294.75" align-y="732" 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">c</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>3</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="14" />
    </region>
    <region region-id="393" left="6" top="756.75" width="300.75" height="28.5" align-x="6" align-y="768" 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">Create new range variables which only go from the upper left-hand corner to the "center."</p>
      </text>
    </region>
    <region region-id="391" left="36" top="798.75" width="42" height="15.75" align-x="48" align-y="810" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">I</ml:id>
          <ml:range>
            <ml:real>0</ml:real>
            <ml:id xml:space="preserve">r</ml:id>
          </ml:range>
        </ml:define>
      </math>
      <rendering item-idref="15" />
    </region>
    <region region-id="392" left="132" top="798.75" width="44.25" height="15.75" align-x="144.75" align-y="810" 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">J</ml:id>
          <ml:range>
            <ml:real>0</ml:real>
            <ml:id xml:space="preserve">c</ml:id>
          </ml:range>
        </ml:define>
      </math>
      <rendering item-idref="16" />
    </region>
    <region region-id="394" left="6" top="822.75" width="308.25" height="42.75" align-x="6" align-y="834" 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">Use these, and a specified masking function, to create the upper-left quadrant of the mask. Note that the<f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"> (0, 0) entry is f(0)</inlineAttr></f>, and the<f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"> (r, c)</inlineAttr></f> entry <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">is f(1).</inlineAttr></f></p>
      </text>
    </region>
    <region region-id="395" left="36" top="871.5" width="85.5" height="21" align-x="62.25" align-y="888" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:function>
            <ml:id xml:space="preserve">f</ml:id>
            <ml:boundVars>
              <ml:id xml:space="preserve">x</ml:id>
            </ml:boundVars>
          </ml:function>
          <ml:apply>
            <ml:plus />
            <ml:apply>
              <ml:plus />
              <ml:apply>
                <ml:pow />
                <ml:id xml:space="preserve">x</ml:id>
                <ml:real>2</ml:real>
              </ml:apply>
              <ml:id xml:space="preserve">x</ml:id>
            </ml:apply>
            <ml:real>1</ml:real>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="17" />
    </region>
    <region region-id="398" left="36" top="903" width="121.5" height="46.5" align-x="84.75" align-y="930" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">Mask</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:id xml:space="preserve">f</ml:id>
            <ml:apply>
              <ml:sqrt />
              <ml:apply>
                <ml:div />
                <ml:apply>
                  <ml:plus />
                  <ml:apply>
                    <ml:pow />
                    <ml:id xml:space="preserve">I</ml:id>
                    <ml:real>2</ml:real>
                  </ml:apply>
                  <ml:apply>
                    <ml:pow />
                    <ml:id xml:space="preserve">J</ml:id>
                    <ml:real>2</ml:real>
                  </ml:apply>
                </ml:apply>
                <ml:apply>
                  <ml:plus />
                  <ml:apply>
                    <ml:pow />
                    <ml:id xml:space="preserve">r</ml:id>
                    <ml:real>2</ml:real>
                  </ml:apply>
                  <ml:apply>
                    <ml:pow />
                    <ml:id xml:space="preserve">c</ml:id>
                    <ml:real>2</ml:real>
                  </ml:apply>
                </ml:apply>
              </ml:apply>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="18" />
    </region>
    <region region-id="399" left="36" top="957.75" width="198.75" height="57.75" align-x="69.75" align-y="990" 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">Mask</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="3" cols="4">
              <ml:real>1</ml:real>
              <ml:real>1.3542731750356916</ml:real>
              <ml:real>1.8623925039175369</ml:real>
              <ml:real>1.3542731750356916</ml:real>
              <ml:real>1.5460784241225221</ml:real>
              <ml:real>2.0047890575614269</ml:real>
              <ml:real>1.8623925039175369</ml:real>
              <ml:real>2.0047890575614269</ml:real>
              <ml:real>2.3998491559373516</ml:real>
              <ml:real>2.5243579866455361</ml:real>
              <ml:real>2.6462887885377984</ml:real>
              <ml:real>3</ml:real>
            </ml:matrix>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="19" />
    </region>
    <region region-id="444" left="0" top="1026" width="6000" height="6" align-x="0" align-y="1026" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="405" left="6" top="1038.75" width="328.5" height="14.25" align-x="6" align-y="1050" 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 rest of the <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">Mask</inlineAttr></f> matrix is made of reflections of this upper quadrant.</p>
      </text>
    </region>
    <region region-id="401" left="36" top="1062.75" width="102.75" height="15.75" align-x="49.5" 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>
          <ml:id xml:space="preserve">n</ml:id>
          <ml:range>
            <ml:parens>
              <ml:apply>
                <ml:plus />
                <ml:id xml:space="preserve">r</ml:id>
                <ml:real>1</ml:real>
              </ml:apply>
            </ml:parens>
            <ml:id xml:space="preserve">rowmax</ml:id>
          </ml:range>
        </ml:define>
      </math>
      <rendering item-idref="20" />
    </region>
    <region region-id="589" left="216" top="1062.75" width="102.75" height="15.75" align-x="232.5" 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="WarnRedefinedBIUnit">
          <ml:id xml:space="preserve">m</ml:id>
          <ml:range>
            <ml:parens>
              <ml:apply>
                <ml:plus />
                <ml:id xml:space="preserve">c</ml:id>
                <ml:real>1</ml:real>
              </ml:apply>
            </ml:parens>
            <ml:id xml:space="preserve">colmax</ml:id>
          </ml:range>
        </ml:define>
      </math>
      <rendering item-idref="21" />
    </region>
    <region region-id="591" left="36" top="1092.75" width="155.25" height="17.25" align-x="86.25" align-y="1104" 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:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">Mask</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">n</ml:id>
              <ml:id xml:space="preserve">J</ml:id>
            </ml:sequence>
          </ml:apply>
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">Mask</ml:id>
            <ml:sequence>
              <ml:apply>
                <ml:plus />
                <ml:apply>
                  <ml:minus />
                  <ml:id xml:space="preserve">rowmax</ml:id>
                  <ml:id xml:space="preserve">n</ml:id>
                </ml:apply>
                <ml:real>1</ml:real>
              </ml:apply>
              <ml:id xml:space="preserve">J</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="22" />
    </region>
    <region region-id="590" left="216" top="1092.75" width="152.25" height="17.25" align-x="267" align-y="1104" 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:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">Mask</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">i</ml:id>
              <ml:id xml:space="preserve">m</ml:id>
            </ml:sequence>
          </ml:apply>
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">Mask</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">i</ml:id>
              <ml:apply>
                <ml:plus />
                <ml:apply>
                  <ml:minus />
                  <ml:id xml:space="preserve">colmax</ml:id>
                  <ml:id xml:space="preserve">m</ml:id>
                </ml:apply>
                <ml:real>1</ml:real>
              </ml:apply>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="23" />
    </region>
    <region region-id="449" left="6" top="1116.75" width="147" height="14.25" align-x="6" align-y="1128" 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 full frequency-domain mask:</p>
      </text>
    </region>
    <region region-id="448" left="36" top="1140.75" width="269.25" height="99.75" align-x="69.75" align-y="1194" 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">Mask</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="5" cols="6">
              <ml:real>1</ml:real>
              <ml:real>1.3542731750356916</ml:real>
              <ml:real>1.8623925039175369</ml:real>
              <ml:real>1.8623925039175369</ml:real>
              <ml:real>1.3542731750356916</ml:real>
              <ml:real>1.3542731750356916</ml:real>
              <ml:real>1.5460784241225221</ml:real>
              <ml:real>2.0047890575614269</ml:real>
              <ml:real>2.0047890575614269</ml:real>
              <ml:real>1.5460784241225221</ml:real>
              <ml:real>1.8623925039175369</ml:real>
              <ml:real>2.0047890575614269</ml:real>
              <ml:real>2.3998491559373516</ml:real>
              <ml:real>2.3998491559373516</ml:real>
              <ml:real>2.0047890575614269</ml:real>
              <ml:real>2.5243579866455361</ml:real>
              <ml:real>2.6462887885377984</ml:real>
              <ml:real>3</ml:real>
              <ml:real>3</ml:real>
              <ml:real>2.6462887885377984</ml:real>
              <ml:real>1.8623925039175369</ml:real>
              <ml:real>2.0047890575614269</ml:real>
              <ml:real>2.3998491559373516</ml:real>
              <ml:real>2.3998491559373516</ml:real>
              <ml:real>2.0047890575614269</ml:real>
              <ml:real>1.3542731750356916</ml:real>
              <ml:real>1.5460784241225221</ml:real>
              <ml:real>2.0047890575614269</ml:real>
              <ml:real>2.0047890575614269</ml:real>
              <ml:real>1.5460784241225221</ml:real>
            </ml:matrix>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="24" />
    </region>
    <region region-id="458" left="6" top="1254.75" width="213.75" 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">Transform the image, and find the filtered result:</p>
      </text>
    </region>
    <region region-id="455" left="36" top="1284.75" width="70.5" height="15.75" align-x="61.5" 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:define>
          <ml:id xml:space="preserve">MT</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">cfft</ml:id>
            <ml:id xml:space="preserve">M</ml:id>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="25" />
    </region>
    <region region-id="456" left="138" top="1276.5" width="122.25" height="24" align-x="174.75" 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:define>
          <ml:id xml:space="preserve">output</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">icfft</ml:id>
            <ml:apply>
              <ml:vectorize />
              <ml:apply>
                <ml:mult style="default" />
                <ml:id xml:space="preserve">MT</ml:id>
                <ml:id xml:space="preserve">Mask</ml:id>
              </ml:apply>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="26" />
    </region>
    <region region-id="457" left="36" top="1314.75" width="306" height="99.75" align-x="72" align-y="1368" 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">output</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="5" cols="6">
              <ml:complex>
                <ml:real>-1.220703452483437</ml:real>
                <ml:imag>2.1290418264208983E-16</ml:imag>
              </ml:complex>
              <ml:complex>
                <ml:real>-2.3212349735181639</ml:real>
                <ml:imag>8.2263188198589E-16</ml:imag>
              </ml:complex>
              <ml:complex>
                <ml:real>-3.9576893445519921</ml:real>
                <ml:imag>-6.6208091934274681E-17</ml:imag>
              </ml:complex>
              <ml:complex>
                <ml:real>-5.59414371558582</ml:real>
                <ml:imag>-2.56836785001839E-16</ml:imag>
              </ml:complex>
              <ml:complex>
                <ml:real>-6.6946752366205446</ml:real>
                <ml:imag>5.1418817737159949E-16</ml:imag>
              </ml:complex>
              <ml:complex>
                <ml:real>-1.6558083417813</ml:real>
                <ml:imag>-5.0745730789073371E-17</ml:imag>
              </ml:complex>
              <ml:complex>
                <ml:real>0.81798759237439678</ml:real>
                <ml:imag>1.2854279921033239E-15</ml:imag>
              </ml:complex>
              <ml:complex>
                <ml:real>2.1072652865027708</ml:real>
                <ml:imag>1.3634757067584063E-15</ml:imag>
              </ml:complex>
              <ml:complex>
                <ml:real>3.3965429806311418</ml:real>
                <ml:imag>2.2027382072380366E-15</ml:imag>
              </ml:complex>
              <ml:complex>
                <ml:real>5.87033891478684</ml:real>
                <ml:imag>1.4317132448645981E-15</ml:imag>
              </ml:complex>
              <ml:complex>
                <ml:real>-2.5804098638108419</ml:real>
                <ml:imag>-5.5443785040309558E-16</ml:imag>
              </ml:complex>
              <ml:complex>
                <ml:real>1.6070780638195248</ml:real>
                <ml:imag>-8.4240286276417139E-16</ml:imag>
              </ml:complex>
              <ml:complex>
                <ml:real>3.491880671118154</ml:real>
                <ml:imag>-1.0391588030523734E-15</ml:imag>
              </ml:complex>
              <ml:complex>
                <ml:real>5.37668327841678</ml:real>
                <ml:imag>-6.6805414114766573E-16</ml:imag>
              </ml:complex>
              <ml:complex>
                <ml:real>9.5641712060471491</ml:real>
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              <ml:complex>
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              <ml:complex>
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              <ml:complex>
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          </result>
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    <region region-id="462" left="6" top="1428.75" width="283.5" height="14.25" align-x="6" align-y="1440" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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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">Compare this procedure with the output of the built-in function.</p>
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              <ml:real>-1.6558083417813005</ml:real>
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              <ml:real>2.1072652865027717</ml:real>
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              <ml:real>5.87033891478684</ml:real>
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              <ml:real>9.82450751881284</ml:real>
              <ml:real>16.519157188718982</ml:real>
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              <ml:real>3.9808216103959762</ml:real>
              <ml:real>7.8927347134972283</ml:real>
              <ml:real>11.804647816598481</ml:real>
              <ml:real>20.212989479979296</ml:real>
              <ml:real>-4.8626249422826993</ml:real>
              <ml:real>7.1200441762885376</ml:real>
              <ml:real>13.957689344551991</ml:real>
              <ml:real>20.795334512815444</ml:real>
              <ml:real>32.778003631386682</ml:real>
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    </region>
    <region region-id="463" left="6" top="1590.75" width="330" height="57" align-x="6" align-y="1602" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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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 steps above are provided simply for your understanding of the workings of the function. There is no need to recreate them every time - just use <b>funconv</b>. In general you may need to scale the output matrix to the range [0, 255], since the convolution may expand the image scale.</p>
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      <pageBreak />
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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">
          <b>
            <c val="#800000">Function Details</c>
          </b>
        </p>
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          <b>
            <inlineAttr font-style="normal">
              <c val="#800000">funcdeconv(M, f, </c>
            </inlineAttr>
          </b>
          <f family="Symbol" charset="2">
            <b>
              <inlineAttr font-style="normal">
                <c val="#800000">e</c>
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          </f>
          <b>
            <inlineAttr font-style="normal">
              <c val="#800000">)</c>
            </inlineAttr>
          </b>
        </p>
      </text>
      <indexes>
        <phrase>funcdeconv function</phrase>
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    </region>
    <region region-id="464" left="6" top="1692.75" width="328.5" height="86.25" align-x="6" align-y="1704" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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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 corresponding deconvolution function takes a frequency-domain image and a frequency-domain function and deconvolves them. The argument<b><sp /></b><f family="Symbol" charset="2"><inlineAttr font-weight="normal" line-through="false">e</inlineAttr></f><b><sp /></b>is a small term which is added to all expressions in denominators during all divisions in the routine. Therefore as a starting point you should <i><inlineAttr font-weight="normal">always try a value of 0 for </inlineAttr></i><f family="Symbol" charset="2"><i><inlineAttr font-weight="normal">e</inlineAttr></i></f>; if you use <b>funcdeconv</b> and receive a divide-by-zero error message, try a small value for <f family="Symbol" charset="2"><inlineAttr font-weight="normal" line-through="false">e</inlineAttr></f>.</p>
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              <ml:real>-6.6613381477509392E-16</ml:real>
              <ml:real>4.4408920985006262E-16</ml:real>
              <ml:real>4.4408920985006262E-16</ml:real>
              <ml:real>-2.6645352591003757E-15</ml:real>
              <ml:real>-3.5527136788005009E-15</ml:real>
              <ml:real>1.8884470590877966E-15</ml:real>
              <ml:real>1.0000000000000004</ml:real>
              <ml:real>2.0000000000000013</ml:real>
              <ml:real>3</ml:real>
              <ml:real>4</ml:real>
              <ml:real>4.6698398288848112E-15</ml:real>
              <ml:real>1.999999999999998</ml:real>
              <ml:real>3.9999999999999996</ml:real>
              <ml:real>5.9999999999999982</ml:real>
              <ml:real>7.9999999999999982</ml:real>
              <ml:real>1.9984014443252818E-15</ml:real>
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              <ml:real>5.9999999999999982</ml:real>
              <ml:real>8.9999999999999947</ml:real>
              <ml:real>11.999999999999996</ml:real>
              <ml:real>3.3237659484163163E-15</ml:real>
              <ml:real>3.9999999999999982</ml:real>
              <ml:real>7.9999999999999964</ml:real>
              <ml:real>11.999999999999995</ml:real>
              <ml:real>15.999999999999995</ml:real>
              <ml:real>4.7728910886631426E-15</ml:real>
              <ml:real>4.9999999999999991</ml:real>
              <ml:real>9.9999999999999982</ml:real>
              <ml:real>14.999999999999995</ml:real>
              <ml:real>19.999999999999993</ml:real>
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              <ml:real>0</ml:real>
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              <ml:real>0</ml:real>
              <ml:real>1</ml:real>
              <ml:real>2</ml:real>
              <ml:real>3</ml:real>
              <ml:real>4</ml:real>
              <ml:real>0</ml:real>
              <ml:real>2</ml:real>
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              <ml:real>6</ml:real>
              <ml:real>8</ml:real>
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              <ml:real>6</ml:real>
              <ml:real>9</ml:real>
              <ml:real>12</ml:real>
              <ml:real>0</ml:real>
              <ml:real>4</ml:real>
              <ml:real>8</ml:real>
              <ml:real>12</ml:real>
              <ml:real>16</ml:real>
              <ml:real>0</ml:real>
              <ml:real>5</ml:real>
              <ml:real>10</ml:real>
              <ml:real>15</ml:real>
              <ml:real>20</ml:real>
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      <pageBreak />
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    <region region-id="528" left="6" top="1938.75" width="87" 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">
          <b>
            <c val="#800000">Function Details</c>
          </b>
        </p>
      </text>
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      <text use-page-width="false" push-down="false" lock-width="true">
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          <b>
            <inlineAttr font-style="normal">
              <c val="#800000">matconv(M, E)</c>
            </inlineAttr>
          </b>
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      <indexes>
        <phrase>matconv function</phrase>
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    <region region-id="534" left="204" top="1938" width="102" height="15" align-x="204" 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">
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          <b>
            <inlineAttr font-style="normal">
              <c val="#800000">matdeconv(M, E, </c>
            </inlineAttr>
          </b>
          <f family="Symbol" charset="2">
            <b>
              <inlineAttr font-style="normal">
                <c val="#800000">e</c>
              </inlineAttr>
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          </f>
          <b>
            <inlineAttr font-style="normal">
              <c val="#800000">)</c>
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          </b>
        </p>
      </text>
      <indexes>
        <phrase>matdeconv function</phrase>
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    <region region-id="471" left="6" top="1956.75" width="332.25" height="71.25" align-x="6" align-y="1968" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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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 perform the same frequency-domain multiplication and transforms as the previous two, but they accept a mask as input instead of a function. The dimensions of the mask must be the size as the original image for the function to work; you supply just the upper-left quadrant of this mask.</p>
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      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedBIUnit">
          <ml:id xml:space="preserve">R</ml:id>
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            <ml:id xml:space="preserve">READ_IMAGE</ml:id>
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    <region region-id="478" left="6" top="2082.75" width="304.5" height="14.25" align-x="6" align-y="2094" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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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 dimensions of the mask supplied to the functions are as follows:</p>
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              <ml:real>2</ml:real>
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              <ml:real>2</ml:real>
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              <ml:real>1</ml:real>
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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">You only need to make the top left-hand quadrant of the mask. The convolution function does the rest for you.</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">To recover the original image, we use <b>matdeconv</b>. Since deconvolution potentially involves division by elements with zero intensity, the third argument is added to the denominator of each division to eliminate this problem.</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">To see an example of simple convolution in the image domain with a kernel matrix, see <f family="Times New Roman" charset="0"><b><u><inlineAttr line-through="false">Convolution</inlineAttr></u></b></f>.</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">Experiment with other filter functions, keeping in mind that the upper left-hand corner of the mask corresponds to low frequencies, and the lower right-hand corner corresponds to high frequencies. Try different images.</p>
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