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      <link href="./Manip.xmcd" title="Links to Image Manipulation " popup="false" />
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    <region region-id="554" left="66" top="6.75" width="138.75" height="14.25" align-x="66" align-y="18" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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          <b>
            <c val="#008080">IMAGE MANIPULATION</c>
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            <b>
              <c val="#800080">Function and Level Mapping</c>
            </b>
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        <phrase>function and level Mapping</phrase>
        <phrase>level Mapping, function and</phrase>
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            <c val="#800000">Function Definitions and Syntax</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 map a function of intensity, or a vector of new intensity levels, onto an existing image. Function and level maps specify a uniform transformation on intensity levels in an image or image transform. They can also be used like equalization, to optimize the display of an image.</p>
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          <b>
            <c val="#800000">Arguments</c>
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              <inlineAttr line-through="false">M</inlineAttr>
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          </f>, the image matrix</p>
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            <c val="#008080">
              <f size="14">• </f>
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          </f>, the function to perform at each entry, or vec, a vector of 256 levels used to map the original intensities.</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">Each function returns an image matrix which contains the specified intensity transformation.</p>
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            <c val="#800000">Function Details</c>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Replaces <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">M</inlineAttr></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" line-through="false"><sub>ij</sub></inlineAttr></f><f family="Times New Roman" charset="0" size="8"><inlineAttr font-weight="normal" line-through="false"><sub><sp /></sub></inlineAttr></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">by f(M</inlineAttr></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" line-through="false"><sub>ij</sub></inlineAttr></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">).</inlineAttr></f><sp /></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">Construct a test 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">Now define a function in <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">Mathcad.</inlineAttr></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">The function map performs this function on every element of the matrix.</p>
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              <ml:real>576</ml:real>
              <ml:real>1296</ml:real>
              <ml:real>0</ml:real>
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              <ml:real>1296</ml:real>
              <ml:real>2916</ml:real>
              <ml:real>0</ml:real>
              <ml:real>576</ml:real>
              <ml:real>2304</ml:real>
              <ml:real>5184</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">Note that this function performs the same task as using Mathcad's vectorize operator:</p>
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              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>36</ml:real>
              <ml:real>144</ml:real>
              <ml:real>324</ml:real>
              <ml:real>0</ml:real>
              <ml:real>144</ml:real>
              <ml:real>576</ml:real>
              <ml:real>1296</ml:real>
              <ml:real>0</ml:real>
              <ml:real>324</ml:real>
              <ml:real>1296</ml:real>
              <ml:real>2916</ml:real>
              <ml:real>0</ml:real>
              <ml:real>576</ml:real>
              <ml:real>2304</ml:real>
              <ml:real>5184</ml:real>
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      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The <b>vectorize</b> operator used here is available by typing <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">[Ctrl][</inlineAttr></b></f><b>-]</b>.</p>
      </text>
    </region>
    <region region-id="459" left="6" top="792.75" width="178.5" height="14.25" align-x="6" align-y="804" 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">Now try the same function on an image:</p>
      </text>
    </region>
    <region region-id="448" left="36" top="816.75" width="161.25" height="15.75" align-x="51.75" align-y="828" 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">R</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">READ_IMAGE</ml:id>
            <ml:str xml:space="preserve">arcs.bmp</ml:str>
          </ml:apply>
        </ml:define>
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      <rendering item-idref="13" />
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      <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">Change the function to see different maps. Try some of those listed below the definition.</p>
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                <ml:real>160</ml:real>
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          <ml:real>255</ml:real>
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            <ml:apply>
              <ml:neg />
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                <ml:real>10</ml:real>
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              <ml:div />
              <ml:apply>
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              <ml:div />
              <ml:apply>
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                <ml:id xml:space="preserve">x</ml:id>
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              <ml:real>160</ml:real>
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          </ml:apply>
        </ml:apply>
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      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">fmap</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">Re</ml:id>
            <ml:apply>
              <ml:id xml:space="preserve">funmap</ml:id>
              <ml:sequence>
                <ml:id xml:space="preserve">R</ml:id>
                <ml:id xml:space="preserve">f</ml:id>
              </ml:sequence>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="18" />
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          <ml:id xml:space="preserve">R</ml:id>
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        <outputs />
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      <component hide-arguments="false" clsid-buddy="8442da2d-08d7-45ad-a4a6-f28bee6ae06a" item-idref="21" disable-calc="false">
        <inputs>
          <ml:id xml:space="preserve">fmap</ml:id>
        </inputs>
        <outputs />
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    <region region-id="610" left="0" top="1236" width="6000" height="6" align-x="0" align-y="1236" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="578" left="6" top="1248.75" width="88.5" height="14.25" align-x="6" align-y="1260" 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>
    </region>
    <region region-id="621" left="126" top="1248.75" width="90.75" height="14.25" align-x="126" align-y="1260" 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">levelmap(M, vec)</c>
            </inlineAttr>
          </b>
        </p>
      </text>
      <indexes>
        <phrase>levelmap function</phrase>
      </indexes>
    </region>
    <region region-id="477" left="6" top="1266.75" width="339.75" height="71.25" align-x="6" align-y="1278" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">This function replaces the intensities in the image <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">M by th</inlineAttr></f>e intensity levels given by a vector,<b><sp /></b><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">vec. For example, the 100th element in vec gives the new level for pixels with intensity 100. The entries of M must be between 0 and the length of vec minus 1. So, for example, if we wish to</inlineAttr></f> compensate an image with a squared intensity scale, we'd create the vector</p>
      </text>
    </region>
    <region region-id="581" left="36" top="1356.75" width="56.25" height="15.75" align-x="48" align-y="1368" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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    <region region-id="481" left="108" top="1348.5" width="138" height="33.75" align-x="135.75" align-y="1368" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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            <ml:id xml:space="preserve">r</ml:id>
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          <ml:apply>
            <ml:mult style="default" />
            <ml:real>255</ml:real>
            <ml:apply>
              <ml:id xml:space="preserve">log</ml:id>
              <ml:apply>
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                <ml:apply>
                  <ml:mult style="default" />
                  <ml:apply>
                    <ml:div />
                    <ml:id xml:space="preserve">r</ml:id>
                    <ml:real>255</ml:real>
                  </ml:apply>
                  <ml:real>9</ml:real>
                </ml:apply>
                <ml:real>1</ml:real>
              </ml:apply>
            </ml:apply>
          </ml:apply>
        </ml:define>
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    <region region-id="483" left="6" top="1536.75" width="133.5" height="57" align-x="6" align-y="1548" 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">Here are some other typical intensity compensation scales for better screen and printing resolution:</p>
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              <ml:id xml:space="preserve">r</ml:id>
              <ml:real>255</ml:real>
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          </ml:apply>
        </ml:apply>
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    <region region-id="492" left="240" top="1535.25" width="62.25" height="39" align-x="259.5" align-y="1560" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:apply>
          <ml:mult style="default" />
          <ml:real>255</ml:real>
          <ml:apply>
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              <ml:apply>
                <ml:div />
                <ml:id xml:space="preserve">r</ml:id>
                <ml:real>255</ml:real>
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            <ml:real>2</ml:real>
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        </ml:apply>
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    <region region-id="493" left="168" top="1581.75" width="63" height="36.75" align-x="187.5" align-y="1614" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:apply>
          <ml:mult style="default" />
          <ml:real>255</ml:real>
          <ml:apply>
            <ml:pow />
            <ml:real>10</ml:real>
            <ml:apply>
              <ml:div />
              <ml:apply>
                <ml:minus />
                <ml:id xml:space="preserve">r</ml:id>
                <ml:real>255</ml:real>
              </ml:apply>
              <ml:real>200</ml:real>
            </ml:apply>
          </ml:apply>
        </ml:apply>
      </math>
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    <region region-id="487" left="6" top="1626.75" width="307.5" height="42.75" align-x="6" align-y="1638" 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">A particular function map or lookup table might be provided with a monitor or sensor, which maps irregularities caused by the display to their correct values.</p>
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    </region>
    <region region-id="498" left="6" top="1674.75" width="47.25" height="14.25" align-x="6" align-y="1686" 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">Recall that</p>
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      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:id xml:space="preserve">M</ml:id>
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            <ml:matrix rows="4" cols="5">
              <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>6</ml:real>
              <ml:real>12</ml:real>
              <ml:real>18</ml:real>
              <ml:real>0</ml:real>
              <ml:real>12</ml:real>
              <ml:real>24</ml:real>
              <ml:real>36</ml:real>
              <ml:real>0</ml:real>
              <ml:real>18</ml:real>
              <ml:real>36</ml:real>
              <ml:real>54</ml:real>
              <ml:real>0</ml:real>
              <ml:real>24</ml:real>
              <ml:real>48</ml:real>
              <ml:real>72</ml:real>
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          </result>
        </ml:eval>
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      <pageBreak />
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      <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">So that</p>
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      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:apply>
            <ml:id xml:space="preserve">levelmap</ml:id>
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              <ml:id xml:space="preserve">M</ml:id>
              <ml:id xml:space="preserve">vec</ml:id>
            </ml:sequence>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="4" cols="5">
              <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>21.271666242674257</ml:real>
              <ml:real>39.108443373550131</ml:real>
              <ml:real>54.466948007649592</ml:real>
              <ml:real>0</ml:real>
              <ml:real>39.108443373550131</ml:real>
              <ml:real>67.952585307209972</ml:real>
              <ml:real>90.815287739837657</ml:real>
              <ml:real>0</ml:real>
              <ml:real>54.466948007649592</ml:real>
              <ml:real>90.815287739837657</ml:real>
              <ml:real>118.13589702407012</ml:real>
              <ml:real>0</ml:real>
              <ml:real>67.952585307209972</ml:real>
              <ml:real>109.7562219395268</ml:real>
              <ml:real>140.0326303192854</ml:real>
            </ml:matrix>
          </result>
        </ml:eval>
      </math>
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    </region>
    <region region-id="504" left="6" top="1908.75" width="180.75" height="14.25" align-x="6" align-y="1920" 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">See the effect this has on our test image.</p>
      </text>
    </region>
    <region region-id="585" left="36" top="1932.75" width="117.75" height="15.75" align-x="64.5" align-y="1944" 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">level</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">levelmap</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">R</ml:id>
              <ml:id xml:space="preserve">vec</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
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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">Try changing the mapping vector to one of the other suggested functions. Does it do what you expect? In some sense, this is the same as <b>funmap</b>, if you structure the mapping vector correctly.</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">For another example of a function which maps levels in an image, see <b><u><link href="./Equalize.xmcd" title="Links to section" popup="false">Equalization</link></u></b>. This function maps levels so that the cumulative histogram is linear.</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">Choosing and creating level maps is an important science for many applications. For a few examples of what can be accomplished, see <b><u><link href="./Mormaps.xmcd" title="Links to section" popup="false">Mapping Image Intensities</link></u></b>.</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">A function map might be considered a kind of lookup table which can be generated based on knowledge of the way an image was formed, and can correct for variations across an image. See the example <b><u><link href="./Destripe.xmcd" title="Links to section" popup="false">Removing Scanner Stripes</link></u></b>.</p>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Note that <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">levelmap</inlineAttr></b></f> does not use the built-in variable ORIGIN, which specifies the starting index of an array for other functions. </p>
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