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      <imagefile filename="h2n51.png" lenght="1845" width="64" height="33" option="border=true;" dataRenderXmlLength="0" renderExt=".png" OLDwidht="64" OLDheight="33" select_rotation="0°">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</imagefile>
      <input>
        <e type="operand">N</e>
        <e type="operand">rgb</e>
        <e type="operator" args="2">-</e>
      </input>
    </image>
  </region>
  <region id="14" left="18" top="1260" width="323" height="33" color="#000000" bgColor="#80ffff" fontSize="12">
    <text lang="eng">
      <p bold="true">"Fourier zoom" works well</p>
    </text>
  </region>
  <region id="15" left="378" top="1260" width="245" height="29" color="#000000" bgColor="#ffffff" fontSize="8">
    <math fractionType="auto" decimalPlaces="6">
      <input>
        <e type="operand">rgb</e>
        <e type="function" preserve="true" args="1">cols</e>
        <e type="operand">rgb</e>
        <e type="function" preserve="true" args="1">rows</e>
        <e type="operand">1</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="4">mat</e>
      </input>
      <result action="numeric">
        <e type="operand">64</e>
        <e type="operand">33</e>
        <e type="operand">1</e>
        <e type="operand">2</e>
        <e type="function" preserve="true" args="4">mat</e>
      </result>
    </math>
  </region>
  <region id="16" top="1341" color="#000000" bgColor="#ffffff">
    <area collapsed="true">
      <title lang="eng">
        <p>     Pad(M,nc,nr), Scale(M,lo,hi)     </p>
      </title>
    </area>
    <region id="17" left="18" top="1368" width="382" height="194" border="true" color="#000000" bgColor="#ffffff" fontSize="8">
      <math optimize="2" fractionType="auto" decimalPlaces="6">
        <description active="true" position="Top" lang="eng">
          <p>A robust construct of the Matrix Scale function</p>
        </description>
        <input>
          <e type="operand">M</e>
          <e type="operand">Low</e>
          <e type="operand">High</e>
          <e type="function" args="3">Scale</e>
          <e type="operand">min</e>
          <e type="operand">M</e>
          <e type="function" preserve="true" args="1">Min</e>
          <e type="operator" args="2">:</e>
          <e type="operand">max</e>
          <e type="operand">M</e>
          <e type="function" preserve="true" args="1">Max</e>
          <e type="operator" args="2">:</e>
          <e type="operand">maxmin</e>
          <e type="operand">max</e>
          <e type="operand">min</e>
          <e type="operator" args="2">-</e>
          <e type="bracket">(</e>
          <e type="operand">10</e>
          <e type="operand">12</e>
          <e type="operator" args="1">-</e>
          <e type="operator" args="2">^</e>
          <e type="operator" args="2">+</e>
          <e type="operator" args="2">:</e>
          <e type="operand">HighLow</e>
          <e type="operand">High</e>
          <e type="operand">Low</e>
          <e type="operator" args="2">-</e>
          <e type="bracket">(</e>
          <e type="operator" args="2">:</e>
          <e type="operand">i</e>
          <e type="operand">1</e>
          <e type="operand">M</e>
          <e type="function" preserve="true" args="1">rows</e>
          <e type="function" preserve="true" args="2">range</e>
          <e type="operand">j</e>
          <e type="operand">1</e>
          <e type="operand">M</e>
          <e type="function" preserve="true" args="1">cols</e>
          <e type="function" preserve="true" args="2">range</e>
          <e type="operand">N</e>
          <e type="operand">i</e>
          <e type="operand">j</e>
          <e type="function" preserve="true" args="3">el</e>
          <e type="operand">M</e>
          <e type="operand">i</e>
          <e type="operand">j</e>
          <e type="function" preserve="true" args="3">el</e>
          <e type="operand">min</e>
          <e type="operator" args="2">-</e>
          <e type="operand">maxmin</e>
          <e type="operator" args="2">/</e>
          <e type="operator" args="2">:</e>
          <e type="function" preserve="true" args="3">for</e>
          <e type="function" preserve="true" args="3">for</e>
          <e type="operand">HighLow</e>
          <e type="operand">N</e>
          <e type="operator" args="2">*</e>
          <e type="operand">Low</e>
          <e type="operator" args="2">+</e>
          <e type="operand">6</e>
          <e type="operand">1</e>
          <e type="function" preserve="true" args="8">line</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region id="18" left="423" top="1422" width="73" height="26" color="#000000" bgColor="#ffffff" fontSize="8">
      <math fractionType="auto" decimalPlaces="6">
        <input>
          <e type="operand">s</e>
          <e type="function" preserve="true" args="1">mwfix</e>
        </input>
      </math>
    </region>
    <region id="19" left="504" top="1422" width="90" height="26" color="#000000" bgColor="#ffffff" fontSize="8">
      <math fractionType="auto" decimalPlaces="6">
        <input>
          <e type="operand">s</e>
          <e type="function" preserve="true" args="1">mwround</e>
        </input>
      </math>
    </region>
    <region id="20" left="18" top="1593" width="590" height="407" border="true" color="#000000" bgColor="#ebebeb" fontSize="8">
      <math fractionType="fraction">
        <input>
          <e type="operand">M</e>
          <e type="operand">nc</e>
          <e type="operand">nr</e>
          <e type="function" args="3">Pad</e>
          <e type="operand">A</e>
          <e type="operand">M</e>
          <e type="function" preserve="true" args="1">mwfft2</e>
          <e type="operator" args="2">:</e>
          <e type="operand" style="string">insert 'nc' cols in 'A' </e>
          <e type="operand">i</e>
          <e type="operand">1</e>
          <e type="operand">A</e>
          <e type="function" preserve="true" args="1">rows</e>
          <e type="function" preserve="true" args="2">range</e>
          <e type="operand">j</e>
          <e type="operand">1</e>
          <e type="operand">nc</e>
          <e type="function" preserve="true" args="2">range</e>
          <e type="operand">vert</e>
          <e type="operand">i</e>
          <e type="operand">j</e>
          <e type="function" preserve="true" args="3">el</e>
          <e type="operand">0</e>
          <e type="operator" args="2">:</e>
          <e type="function" preserve="true" args="3">for</e>
          <e type="function" preserve="true" args="3">for</e>
          <e type="operand">a</e>
          <e type="operand">A</e>
          <e type="operand">1</e>
          <e type="operand">A</e>
          <e type="function" preserve="true" args="1">rows</e>
          <e type="operand">1</e>
          <e type="operand">A</e>
          <e type="function" preserve="true" args="1">cols</e>
          <e type="operand">2</e>
          <e type="operator" args="2">/</e>
          <e type="function" preserve="true" args="1">mwfloor</e>
          <e type="function" preserve="true" args="5">submatrix</e>
          <e type="operator" args="2">:</e>
          <e type="operand">b</e>
          <e type="operand">A</e>
          <e type="operand">1</e>
          <e type="operand">A</e>
          <e type="function" preserve="true" args="1">rows</e>
          <e type="operand">A</e>
          <e type="function" preserve="true" args="1">cols</e>
          <e type="operand">2</e>
          <e type="operator" args="2">/</e>
          <e type="function" preserve="true" args="1">mwfloor</e>
          <e type="operand">1</e>
          <e type="operator" args="2">+</e>
          <e type="operand">A</e>
          <e type="function" preserve="true" args="1">cols</e>
          <e type="function" preserve="true" args="5">submatrix</e>
          <e type="operator" args="2">:</e>
          <e type="operand">S</e>
          <e type="operand">a</e>
          <e type="operand">vert</e>
          <e type="operand">b</e>
          <e type="function" preserve="true" args="3">augment</e>
          <e type="function" preserve="true" args="1">transpose</e>
          <e type="operator" args="2">:</e>
          <e type="operand" style="string">insert 'nr' rows in the transpose 'S' </e>
          <e type="operand">i</e>
          <e type="operand">1</e>
          <e type="operand">S</e>
          <e type="function" preserve="true" args="1">rows</e>
          <e type="function" preserve="true" args="2">range</e>
          <e type="operand">j</e>
          <e type="operand">1</e>
          <e type="operand">nr</e>
          <e type="function" preserve="true" args="2">range</e>
          <e type="operand">horiz</e>
          <e type="operand">i</e>
          <e type="operand">j</e>
          <e type="function" preserve="true" args="3">el</e>
          <e type="operand">0</e>
          <e type="operator" args="2">:</e>
          <e type="function" preserve="true" args="3">for</e>
          <e type="function" preserve="true" args="3">for</e>
          <e type="operand">a</e>
          <e type="operand">S</e>
          <e type="operand">1</e>
          <e type="operand">S</e>
          <e type="function" preserve="true" args="1">rows</e>
          <e type="operand">1</e>
          <e type="operand">S</e>
          <e type="function" preserve="true" args="1">cols</e>
          <e type="operand">2</e>
          <e type="operator" args="2">/</e>
          <e type="function" preserve="true" args="1">mwfloor</e>
          <e type="function" preserve="true" args="5">submatrix</e>
          <e type="operator" args="2">:</e>
          <e type="operand">b</e>
          <e type="operand">S</e>
          <e type="operand">1</e>
          <e type="operand">S</e>
          <e type="function" preserve="true" args="1">rows</e>
          <e type="operand">S</e>
          <e type="function" preserve="true" args="1">cols</e>
          <e type="operand">2</e>
          <e type="operator" args="2">/</e>
          <e type="function" preserve="true" args="1">mwfloor</e>
          <e type="operand">1</e>
          <e type="operator" args="2">+</e>
          <e type="operand">S</e>
          <e type="function" preserve="true" args="1">cols</e>
          <e type="function" preserve="true" args="5">submatrix</e>
          <e type="operator" args="2">:</e>
          <e type="operand">S</e>
          <e type="operand">a</e>
          <e type="operand">horiz</e>
          <e type="operand">b</e>
          <e type="function" preserve="true" args="3">augment</e>
          <e type="function" preserve="true" args="1">transpose</e>
          <e type="operator" args="2">:</e>
          <e type="operand">11</e>
          <e type="operand">1</e>
          <e type="function" preserve="true" args="13">line</e>
          <e type="operator" args="2">:</e>
        </input>
      </math>
    </region>
    <region id="21" top="1998" color="#000000" bgColor="#ffffff">
      <area terminator="true" />
    </region>
  </region>
  <region id="22" left="9" top="2034" width="96" height="26" color="#000000" bgColor="#ebebc8" fontSize="8">
    <math fractionType="auto" decimalPlaces="6">
      <input>
        <e type="operand">t0</e>
        <e type="operand">1</e>
        <e type="function" preserve="true" args="1">time</e>
        <e type="operator" args="2">:</e>
      </input>
    </math>
  </region>
  <region id="23" left="9" top="2061" width="221" height="24" color="#800000" bgColor="#ffffff" fontSize="8">
    <text lang="eng">
      <p bold="true">3. Conclude the  zoom demo</p>
    </text>
  </region>
  <region id="24" left="9" top="2088" width="214" height="24" color="#000000" bgColor="#ffffff" fontSize="8">
    <math fractionType="auto" decimalPlaces="6">
      <input>
        <e type="operand" style="string">practical zoom limit</e>
        <e type="operand">2</e>
        <e type="operator" args="2">≤</e>
      </input>
    </math>
  </region>
  <region id="25" left="9" top="2115" width="202" height="111" color="#000000" bgColor="#ebebc8" fontSize="8">
    <math optimize="2" fractionType="auto" decimalPlaces="6">
      <description active="false" position="Top" lang="eng">
        <p>Group the recovered rgb</p>
      </description>
      <input>
        <e type="operand">S</e>
        <e type="operand">rgb</e>
        <e type="operand">32</e>
        <e type="operand">16</e>
        <e type="function" args="3">Pad</e>
        <e type="operator" args="2">:</e>
        <e type="operand">iVrgb</e>
        <e type="operand">S</e>
        <e type="function" preserve="true" args="1">mwifft2</e>
        <e type="operator" args="2">:</e>
        <e type="operand">RGB</e>
        <e type="operand">iVrgb</e>
        <e type="function" preserve="true" args="1">mwabs</e>
        <e type="function" preserve="true" args="1">eval</e>
        <e type="operator" args="2">:</e>
        <e type="operand">s</e>
        <e type="operand">RGB</e>
        <e type="operand">0</e>
        <e type="operand">255</e>
        <e type="function" args="3">Scale</e>
        <e type="operator" args="2">:</e>
        <e type="operand">zoom</e>
        <e type="operand">s</e>
        <e type="function" preserve="true" args="1">mwround</e>
        <e type="operator" args="2">:</e>
        <e type="operand">5</e>
        <e type="operand">1</e>
        <e type="function" preserve="true" args="7">line</e>
      </input>
    </math>
  </region>
  <region id="26" left="216" top="2115" width="303" height="99" color="#000000" bgColor="#ffffff" fontSize="8">
    <math fractionType="auto" decimalPlaces="6">
      <input>
        <e type="operand" style="string">Insert/center ncols/nrows</e>
        <e type="operand" style="string">inverse Fourier</e>
        <e type="operand" style="string">absolute of the inversed matrix </e>
        <e type="operand" style="string">scale to rgb grayscale</e>
        <e type="operand" style="string">round OFF the zoomed image </e>
        <e type="operand">5</e>
        <e type="operand">1</e>
        <e type="function" preserve="true" args="7">line</e>
      </input>
    </math>
  </region>
  <region id="27" left="9" top="2232" width="106" height="57" color="#000000" bgColor="#ffffff" fontSize="8">
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        <e type="operand">rgb</e>
      </input>
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    <math fractionType="auto" decimalPlaces="0">
      <input>
        <e type="operand">1</e>
        <e type="function" preserve="true" args="1">time</e>
        <e type="operand">t0</e>
        <e type="operator" args="2">-</e>
      </input>
      <result action="numeric">
        <e type="operand">275</e>
        <e type="operand" style="unit">s</e>
        <e type="operator" args="2">*</e>
      </result>
    </math>
  </region>
  <region id="30" left="9" top="2394" width="224" height="24" color="#000000" bgColor="#80ffff" fontSize="8">
    <text lang="eng">
      <p bold="true">Obtain image from a matrix</p>
    </text>
  </region>
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    <text lang="eng">
      <p bold="true">1. Define the functions</p>
    </text>
  </region>
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    <math fractionType="auto" decimalPlaces="6">
      <input>
        <e type="operand">M</e>
        <e type="function" args="1">cfft2d</e>
        <e type="operand">M</e>
        <e type="function" preserve="true" args="1">mwfft2</e>
        <e type="operator" args="2">:</e>
      </input>
    </math>
  </region>
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      <input>
        <e type="operand">M</e>
        <e type="function" args="1">icfft2d</e>
        <e type="operand">M</e>
        <e type="function" preserve="true" args="1">mwifft2</e>
        <e type="operator" args="2">:</e>
      </input>
    </math>
  </region>
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    <math evaluate="false" fractionType="auto" decimalPlaces="6">
      <input>
        <e type="operand">M</e>
        <e type="function" preserve="true" args="1">abs</e>
        <e type="operand">M</e>
        <e type="function" preserve="true" args="1">mwabs</e>
        <e type="function" preserve="true" args="1">eval</e>
        <e type="operator" args="2">:</e>
      </input>
    </math>
  </region>
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    <text lang="eng">
      <p bold="true">2. Fourier 2D Procedure</p>
    </text>
  </region>
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    <math fractionType="auto" decimalPlaces="6">
      <input>
        <e type="operand">V</e>
        <e type="operand">rgb</e>
        <e type="function" args="1">cfft2d</e>
        <e type="operator" args="2">:</e>
      </input>
    </math>
  </region>
  <region id="37" left="9" top="2592" width="121" height="26" color="#000000" bgColor="#ffffff" fontSize="8">
    <math fractionType="auto" decimalPlaces="6">
      <input>
        <e type="operand">iV</e>
        <e type="operand">V</e>
        <e type="function" args="1">icfft2d</e>
        <e type="operator" args="2">:</e>
      </input>
    </math>
  </region>
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    <math fractionType="auto" decimalPlaces="6">
      <input>
        <e type="operand">Fo</e>
        <e type="operand">iV</e>
        <e type="function" preserve="true" args="1">mwround</e>
        <e type="operator" args="2">:</e>
      </input>
    </math>
  </region>
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