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                <c val="#008080">SECTION 6 - ADDITIONAL TRANSFORMS</c>
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        <phrase>transforms, additional</phrase>
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        <phrase>walsh function</phrase>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Calculates a vector <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">w</inlineAttr></b></f> of sign information for use in the Walsh transform. </p>
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          <b>n</b>, order of the walsh function, implying 2<sup>n</sup> data points, given by an integer 0 and 12, inclusive</p>
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          <b>k</b>, the index of the transformed value, an integer between 0 and 2<sup>n</sup><f size="8"><sup><sp /></sup></f>- 1, inclusive</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">Returns a vector of length 2<sup>n</sup>, the kth Walsh function of order n, each of whose elements is +1 or -1. Given a data vector <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">v</inlineAttr></f>, the Walsh transform is a vector of the same length whose <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">k</inlineAttr></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><sup>th</sup></inlineAttr></f> component has the form </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">Any two <i>different</i> Walsh functions of the <i>same</i> order are orthogonal, that is, their dot product is <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">0. For example:</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">For example, the Walsh functions of order 2 are</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">These are <i>all</i> the Walsh functions of order 2. Note what happens when an index is requested outside the range <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">0 . . . 2</inlineAttr></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" line-through="false"><sup>n</sup></inlineAttr></f><f family="Times New Roman" charset="0" size="8"><inlineAttr font-weight="normal" line-through="false"><sup><sp /></sup></inlineAttr></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">- 1 </inlineAttr></f></p>
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            <ml:id xml:space="preserve">walsh</ml:id>
            <ml:sequence>
              <ml:real>2</ml:real>
              <ml:real>4</ml:real>
            </ml:sequence>
          </ml:apply>
        </ml:eval>
      </math>
      <rendering item-idref="12" />
    </region>
    <region region-id="21" left="0" top="696" width="6000" height="6" align-x="0" align-y="696" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="427" left="6" top="708.75" width="188.25" height="14.25" align-x="6" align-y="720" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr line-through="false">
                <c val="#800000">Example - 5th-Order Walsh Functions</c>
              </inlineAttr>
            </b>
          </f>
        </p>
      </text>
      <indexes>
        <phrase>Walsh functions, 5th-order</phrase>
        <phrase>functions, Walsh, 5th-order</phrase>
      </indexes>
    </region>
    <region region-id="245" left="6" top="726.75" width="299.25" height="57" align-x="6" align-y="738" 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 pattern of <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">1s and -1</inlineAttr></f>s for a set of Walsh functions can be displayed as the columns of a matrix, plotting the result as a patch plot. Here are the Walsh functions of order 5, for which there are 32 functions. Note that the resulting array is its own transpose. </p>
      </text>
    </region>
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      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">i</ml:id>
          <ml:range>
            <ml:real>0</ml:real>
            <ml:real>31</ml:real>
          </ml:range>
        </ml:define>
      </math>
      <rendering item-idref="13" />
    </region>
    <region region-id="249" left="108" top="794.25" width="90" height="20.25" align-x="138" 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:apply>
            <ml:matcol />
            <ml:id xml:space="preserve">W</ml:id>
            <ml:id xml:space="preserve">i</ml:id>
          </ml:apply>
          <ml:apply>
            <ml:id xml:space="preserve">walsh</ml:id>
            <ml:sequence>
              <ml:real>5</ml:real>
              <ml:id xml:space="preserve">i</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="14" />
    </region>
    <region region-id="251" left="36" top="828" width="181.5" height="210" align-x="126" align-y="828" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <component hide-arguments="false" clsid-buddy="62e334b8-fdda-4b2b-9bb4-48def95f2af1" item-idref="15" disable-calc="false">
        <inputs>
          <ml:id xml:space="preserve">W</ml:id>
        </inputs>
        <outputs />
      </component>
      <rendering item-idref="16" />
    </region>
    <region region-id="252" left="0" top="1044" width="6000" height="6" align-x="0" align-y="1044" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="256" left="6" top="1056.75" width="300" height="57" align-x="6" align-y="1068" 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">Sometimes it is desirable to reorder the Walsh function vectors so that they are arranged in order of <i>sequency</i>, that is, in order of the number of changes of sign. Here's how you could do this for the case pictured above:</p>
      </text>
    </region>
    <region region-id="257" left="36" top="1128.75" width="50.25" height="15.75" align-x="48" align-y="1140" 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">j</ml:id>
          <ml:range>
            <ml:real>0</ml:real>
            <ml:real>30</ml:real>
          </ml:range>
        </ml:define>
      </math>
      <rendering item-idref="17" />
    </region>
    <region region-id="29" left="108" top="1125.75" width="123.75" height="38.25" align-x="138" align-y="1140" 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:function>
            <ml:id xml:space="preserve">S</ml:id>
            <ml:boundVars>
              <ml:id xml:space="preserve">v</ml:id>
            </ml:boundVars>
          </ml:function>
          <ml:apply>
            <ml:summation />
            <ml:lambda>
              <ml:boundVars>
                <ml:id xml:space="preserve">j</ml:id>
              </ml:boundVars>
              <ml:parens>
                <ml:apply>
                  <ml:lessThan />
                  <ml:apply>
                    <ml:mult style="default" />
                    <ml:apply>
                      <ml:indexer />
                      <ml:id xml:space="preserve">v</ml:id>
                      <ml:id xml:space="preserve">j</ml:id>
                    </ml:apply>
                    <ml:apply>
                      <ml:indexer />
                      <ml:id xml:space="preserve">v</ml:id>
                      <ml:apply>
                        <ml:plus />
                        <ml:id xml:space="preserve">j</ml:id>
                        <ml:real>1</ml:real>
                      </ml:apply>
                    </ml:apply>
                  </ml:apply>
                  <ml:real>0</ml:real>
                </ml:apply>
              </ml:parens>
            </ml:lambda>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="18" />
    </region>
    <region region-id="258" left="36" top="1173" width="89.25" height="25.5" align-x="93" 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:define warning="WarnRedefinedBIUnit">
          <ml:apply>
            <ml:matcol />
            <ml:id xml:space="preserve">R</ml:id>
            <ml:apply>
              <ml:id xml:space="preserve">S</ml:id>
              <ml:apply>
                <ml:matcol />
                <ml:id xml:space="preserve">W</ml:id>
                <ml:id xml:space="preserve">i</ml:id>
              </ml:apply>
            </ml:apply>
          </ml:apply>
          <ml:apply>
            <ml:matcol />
            <ml:id xml:space="preserve">W</ml:id>
            <ml:id xml:space="preserve">i</ml:id>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="19" />
    </region>
    <region region-id="259" left="6" top="1206.75" width="277.5" height="28.5" align-x="6" align-y="1218" 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">Here's a plot of the array <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">R, which contains the columns of W</inlineAttr></f><f family="MS Serif" charset="0"><sp /></f>reordered by sequency:</p>
      </text>
    </region>
    <region region-id="260" left="36" top="1248" width="157.5" height="186" align-x="114" align-y="1248" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <component hide-arguments="false" clsid-buddy="62e334b8-fdda-4b2b-9bb4-48def95f2af1" item-idref="20" disable-calc="false">
        <inputs>
          <ml:id xml:space="preserve">R</ml:id>
        </inputs>
        <outputs />
      </component>
      <rendering item-idref="21" />
    </region>
    <region region-id="33" left="0" top="1446" width="6000" height="6" align-x="0" align-y="1446" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="261" left="6" top="1458.75" width="109.5" height="14.25" align-x="6" align-y="1470" 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">The Walsh Transform</c>
            </inlineAttr>
          </b>
        </p>
      </text>
      <indexes>
        <phrase>Walsh transform</phrase>
        <phrase>transform, Walsh</phrase>
      </indexes>
    </region>
    <region region-id="262" left="6" top="1476.75" width="306" height="28.5" align-x="6" align-y="1488" 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">Given the <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">walsh</inlineAttr></b></f> function, it is easy to compute the one-dimensional Walsh transform of a data vector with length which is a power of 2. </p>
      </text>
    </region>
    <region region-id="263" left="6" top="1512.75" width="144.75" height="14.25" align-x="6" align-y="1524" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Here is a sample data sequence:</p>
      </text>
    </region>
    <region region-id="267" left="36" top="1536.75" width="30" height="15.75" align-x="49.5" align-y="1548" show-border="false" show-highlight="true" 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:real>6</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="22" />
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    <region region-id="268" left="84" top="1531.5" width="39" height="21" align-x="101.25" align-y="1548" 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:apply>
            <ml:pow />
            <ml:real>2</ml:real>
            <ml:id xml:space="preserve">n</ml:id>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="23" />
    </region>
    <region region-id="269" left="144" top="1536.75" width="64.5" height="15.75" align-x="154.5" align-y="1548" 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">i</ml:id>
          <ml:range>
            <ml:real>0</ml:real>
            <ml:apply>
              <ml:minus />
              <ml:id xml:space="preserve">N</ml:id>
              <ml:real>1</ml:real>
            </ml:apply>
          </ml:range>
        </ml:define>
      </math>
      <rendering item-idref="24" />
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    <region region-id="270" left="36" top="1563.75" width="175.5" height="34.5" align-x="51.75" align-y="1584" 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">v</ml:id>
            <ml:id xml:space="preserve">i</ml:id>
          </ml:apply>
          <ml:apply>
            <ml:plus />
            <ml:apply>
              <ml:plus />
              <ml:real>1</ml:real>
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                <ml:id xml:space="preserve">sin</ml:id>
                <ml:apply>
                  <ml:div />
                  <ml:apply>
                    <ml:mult style="default" />
                    <ml:apply>
                      <ml:mult style="default" />
                      <ml:real>2</ml:real>
                      <ml:id xml:space="preserve">π</ml:id>
                    </ml:apply>
                    <ml:id xml:space="preserve">i</ml:id>
                  </ml:apply>
                  <ml:id xml:space="preserve">N</ml:id>
                </ml:apply>
              </ml:apply>
            </ml:apply>
            <ml:apply>
              <ml:id xml:space="preserve">sin</ml:id>
              <ml:apply>
                <ml:mult style="default" />
                <ml:real>3</ml:real>
                <ml:apply>
                  <ml:div />
                  <ml:apply>
                    <ml:mult style="default" />
                    <ml:apply>
                      <ml:mult style="default" />
                      <ml:real>2</ml:real>
                      <ml:id xml:space="preserve">π</ml:id>
                    </ml:apply>
                    <ml:id xml:space="preserve">i</ml:id>
                  </ml:apply>
                  <ml:id xml:space="preserve">N</ml:id>
                </ml:apply>
              </ml:apply>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="25" />
    </region>
    <region region-id="271" left="36" top="1614" width="185.25" height="145.5" align-x="36" align-y="1614" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <plot disable-calc="false" item-idref="26" />
      <rendering item-idref="27" />
    </region>
    <region region-id="42" left="0" top="1770" width="6000" height="6" align-x="0" align-y="1770" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="275" left="6" top="1782.75" width="303.75" height="14.25" align-x="6" align-y="1794" 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 Walsh transform of the vector<f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"> v </inlineAttr></f>can be computed pointwise by</p>
      </text>
    </region>
    <region region-id="273" left="36" top="1804.5" width="111" height="33.75" align-x="63" align-y="1824" 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">Wv</ml:id>
            <ml:id xml:space="preserve">i</ml:id>
          </ml:apply>
          <ml:apply>
            <ml:mult style="default" />
            <ml:apply>
              <ml:mult style="default" />
              <ml:apply>
                <ml:div />
                <ml:real>1</ml:real>
                <ml:id xml:space="preserve">N</ml:id>
              </ml:apply>
              <ml:id xml:space="preserve">v</ml:id>
            </ml:apply>
            <ml:apply>
              <ml:id xml:space="preserve">walsh</ml:id>
              <ml:sequence>
                <ml:id xml:space="preserve">n</ml:id>
                <ml:id xml:space="preserve">i</ml:id>
              </ml:sequence>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="28" />
    </region>
    <region region-id="277" left="6" top="1848.75" width="255" height="14.25" align-x="6" align-y="1860" 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">Here's a plot of the transform, formatted as a step graph:</p>
      </text>
      <indexes>
        <phrase>step graph</phrase>
        <phrase>graph, step</phrase>
      </indexes>
    </region>
    <region region-id="285" left="6" top="1872" width="320.25" height="145.5" align-x="6" align-y="1872" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <plot disable-calc="false" item-idref="29" />
      <rendering item-idref="30" />
    </region>
    <region region-id="278" left="6" top="2028.75" width="284.25" height="28.5" align-x="6" align-y="2040" 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 that the <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">0</inlineAttr></f>th element of the transform is the average of the observed data:</p>
      </text>
    </region>
    <region region-id="283" left="36" top="2064.75" width="51" height="17.25" align-x="65.25" 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:eval>
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">Wv</ml:id>
            <ml:real>0</ml:real>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>1</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="31" />
    </region>
    <region region-id="284" left="108" top="2064.75" width="68.25" height="15.75" align-x="154.5" 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:eval>
          <ml:apply>
            <ml:id xml:space="preserve">mean</ml:id>
            <ml:id xml:space="preserve">v</ml:id>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>0.99999999999999978</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="32" />
    </region>
    <region region-id="50" left="0" top="2088" width="6000" height="6" align-x="0" align-y="2088" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="286" left="6" top="2100.75" width="304.5" height="42.75" align-x="6" align-y="2112" 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 more compact way to compute the transform is to premultiply the data by a square matrix like the one we generated above with a Walsh function in each column:</p>
      </text>
    </region>
    <region region-id="287" left="6" top="2148.75" width="132.75" height="14.25" align-x="6" align-y="2160" 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">Square transformation matrix:</p>
      </text>
    </region>
    <region region-id="323" left="36" top="2174.25" width="96" height="20.25" align-x="72.75" align-y="2190" 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:matcol />
            <ml:id xml:space="preserve">WT</ml:id>
            <ml:id xml:space="preserve">i</ml:id>
          </ml:apply>
          <ml:apply>
            <ml:id xml:space="preserve">walsh</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">n</ml:id>
              <ml:id xml:space="preserve">i</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="33" />
    </region>
    <region region-id="320" left="6" top="2202.75" width="108" height="14.25" align-x="6" align-y="2214" 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 of the data <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">v</inlineAttr></f>:</p>
      </text>
    </region>
    <region region-id="324" left="36" top="2224.5" width="78" height="33.75" align-x="60.75" align-y="2244" 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">Wv</ml:id>
          <ml:apply>
            <ml:mult style="default" />
            <ml:apply>
              <ml:mult style="default" />
              <ml:apply>
                <ml:div />
                <ml:real>1</ml:real>
                <ml:id xml:space="preserve">N</ml:id>
              </ml:apply>
              <ml:id xml:space="preserve">WT</ml:id>
            </ml:apply>
            <ml:id xml:space="preserve">v</ml:id>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="34" />
    </region>
    <region region-id="292" left="6" top="2268.75" width="300.75" height="28.5" align-x="6" align-y="2280" 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 advantage to this method is that any Walsh transform of order 5 can be conducted using the matrix <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">WT</inlineAttr></f>.</p>
      </text>
    </region>
    <region region-id="293" left="6" top="2304.75" width="297.75" height="28.5" align-x="6" align-y="2316" 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 inverse Walsh transform is given by the same formula without the <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">1/N</inlineAttr></f> scale factor.</p>
      </text>
    </region>
    <region region-id="297" left="6" top="2340.75" width="84" height="14.25" align-x="6" align-y="2352" 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">Inverse transform: </p>
      </text>
    </region>
    <region region-id="298" left="36" top="2364.75" width="65.25" height="15.75" align-x="52.5" align-y="2376" 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">V</ml:id>
          <ml:apply>
            <ml:mult style="default" />
            <ml:id xml:space="preserve">WT</ml:id>
            <ml:id xml:space="preserve">Wv</ml:id>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="35" />
    </region>
    <region region-id="296" left="6" top="2388.75" width="307.5" height="28.5" align-x="6" align-y="2400" 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">We compute some values to verify that the transform has recovered the original data <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">v</inlineAttr></f>:</p>
      </text>
    </region>
    <region region-id="299" left="36" top="2424.75" width="63.75" height="17.25" align-x="57" align-y="2436" 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">V</ml:id>
            <ml:real>3</ml:real>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>2.063295130617198</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="36" />
    </region>
    <region region-id="306" left="120" top="2424.75" width="69" height="17.25" align-x="146.25" align-y="2436" 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">V</ml:id>
            <ml:real>25</ml:real>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>2.5163145485119989</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="37" />
    </region>
    <region region-id="311" left="216" top="2424.75" width="75.75" height="17.25" align-x="242.25" align-y="2436" 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">V</ml:id>
            <ml:real>57</ml:real>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>-0.51631454851200065</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="38" />
    </region>
    <region region-id="312" left="306" top="2436.75" width="49.5" height="14.25" align-x="306" align-y="2448" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">and so on</p>
      </text>
    </region>
    <region region-id="303" left="36" top="2448.75" width="60.75" height="17.25" align-x="54" align-y="2460" 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">v</ml:id>
            <ml:real>3</ml:real>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>2.0632951306171994</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="39" />
    </region>
    <region region-id="309" left="120" top="2448.75" width="66" height="17.25" align-x="143.25" align-y="2460" 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">v</ml:id>
            <ml:real>25</ml:real>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>2.516314548512</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="40" />
    </region>
    <region region-id="313" left="216" top="2448.75" width="72.75" height="17.25" align-x="239.25" align-y="2460" 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">v</ml:id>
            <ml:real>57</ml:real>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>-0.51631454851200065</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="41" />
    </region>
    <region region-id="68" left="0" top="2472" width="6000" height="6" align-x="0" align-y="2472" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="325" left="6" top="2484.75" width="296.25" height="45.75" align-x="6" align-y="2496" 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 two-dimensional Walsh transform is best computed using the matrix multiplication form. If <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">X</inlineAttr></f> is a square <f family="Times New Roman" charset="0" size="12">(</f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">2</inlineAttr></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" line-through="false"><sup>n</sup></inlineAttr></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" line-through="false"><sp /></inlineAttr></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">x</inlineAttr></f><inlineAttr font-weight="normal" line-through="false"><f family="Times New Roman" charset="0" size="12"><sp /></f></inlineAttr><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">2</inlineAttr></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" line-through="false"><sup>n</sup></inlineAttr></f><f family="Times New Roman" charset="0" size="12">)</f> matrix, the two-dimensional Walsh transform is given by</p>
      </text>
    </region>
    <region region-id="326" left="36" top="2536.5" width="66" height="39" align-x="51.75" align-y="2556" 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:apply>
            <ml:div />
            <ml:real>1</ml:real>
            <ml:apply>
              <ml:pow />
              <ml:real>2</ml:real>
              <ml:id xml:space="preserve">n</ml:id>
            </ml:apply>
          </ml:apply>
          <ml:parens>
            <ml:apply>
              <ml:mult style="default" />
              <ml:apply>
                <ml:mult style="default" />
                <ml:id xml:space="preserve">W</ml:id>
                <ml:id xml:space="preserve">X</ml:id>
              </ml:apply>
              <ml:id xml:space="preserve">W</ml:id>
            </ml:apply>
          </ml:parens>
        </ml:apply>
      </math>
      <rendering item-idref="42" />
    </region>
    <region region-id="327" left="6" top="2580.75" width="333.75" height="14.25" align-x="6" align-y="2592" 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">where <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">W</inlineAttr></f> is the matrix whose columns are the Walsh functions of order <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">n</inlineAttr></f>.</p>
      </text>
    </region>
    <region region-id="328" left="6" top="2610.75" width="57" height="14.25" align-x="6" align-y="2622" 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</p>
      </text>
    </region>
    <region region-id="329" left="36" top="2634.75" width="30" height="15.75" align-x="49.5" align-y="2646" show-border="false" show-highlight="true" 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">n</ml:id>
          <ml:real>5</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="43" />
    </region>
    <region region-id="330" left="84" top="2629.5" width="39" height="21" align-x="101.25" align-y="2646" 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">N</ml:id>
          <ml:apply>
            <ml:pow />
            <ml:real>2</ml:real>
            <ml:id xml:space="preserve">n</ml:id>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="44" />
    </region>
    <region region-id="333" left="150" top="2634.75" width="64.5" height="15.75" align-x="160.5" align-y="2646" 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">i</ml:id>
          <ml:range>
            <ml:real>0</ml:real>
            <ml:apply>
              <ml:minus />
              <ml:id xml:space="preserve">N</ml:id>
              <ml:real>1</ml:real>
            </ml:apply>
          </ml:range>
        </ml:define>
      </math>
      <rendering item-idref="45" />
    </region>
    <region region-id="335" left="234" top="2634.75" width="66" height="15.75" align-x="246" align-y="2646" 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">j</ml:id>
          <ml:range>
            <ml:real>0</ml:real>
            <ml:apply>
              <ml:minus />
              <ml:id xml:space="preserve">N</ml:id>
              <ml:real>1</ml:real>
            </ml:apply>
          </ml:range>
        </ml:define>
      </math>
      <rendering item-idref="46" />
    </region>
    <region region-id="338" left="6" top="2658.75" width="172.5" height="14.25" align-x="6" align-y="2670" 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">Construct an image to be transformed:</p>
      </text>
    </region>
    <region region-id="343" left="36" top="2679.75" width="219.75" height="34.5" align-x="63.75" align-y="2700" 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">X</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:minus />
            <ml:apply>
              <ml:plus />
              <ml:apply>
                <ml:mult style="default" />
                <ml:apply>
                  <ml:id xml:space="preserve">sin</ml:id>
                  <ml:apply>
                    <ml:div />
                    <ml:apply>
                      <ml:mult style="default" />
                      <ml:apply>
                        <ml:mult style="default" />
                        <ml:real>4</ml:real>
                        <ml:id xml:space="preserve">π</ml:id>
                      </ml:apply>
                      <ml:id xml:space="preserve">i</ml:id>
                    </ml:apply>
                    <ml:id xml:space="preserve">N</ml:id>
                  </ml:apply>
                </ml:apply>
                <ml:apply>
                  <ml:id xml:space="preserve">cos</ml:id>
                  <ml:apply>
                    <ml:div />
                    <ml:apply>
                      <ml:mult style="default" />
                      <ml:apply>
                        <ml:mult style="default" />
                        <ml:real>6</ml:real>
                        <ml:id xml:space="preserve">π</ml:id>
                      </ml:apply>
                      <ml:id xml:space="preserve">j</ml:id>
                    </ml:apply>
                    <ml:id xml:space="preserve">N</ml:id>
                  </ml:apply>
                </ml:apply>
              </ml:apply>
              <ml:apply>
                <ml:id xml:space="preserve">rnd</ml:id>
                <ml:real>1</ml:real>
              </ml:apply>
            </ml:apply>
            <ml:real>.5</ml:real>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="47" />
    </region>
    <region region-id="346" left="6" top="2724.75" width="140.25" height="14.25" align-x="6" align-y="2736" 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">Build the transformation matrix:</p>
      </text>
    </region>
    <region region-id="344" left="36" top="2744.25" width="89.25" height="20.25" align-x="66" align-y="2760" 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:matcol />
            <ml:id xml:space="preserve">W</ml:id>
            <ml:id xml:space="preserve">i</ml:id>
          </ml:apply>
          <ml:apply>
            <ml:id xml:space="preserve">walsh</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">n</ml:id>
              <ml:id xml:space="preserve">i</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="48" />
    </region>
    <region region-id="340" left="6" top="2772.75" width="132.75" height="14.25" align-x="6" align-y="2784" 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 Walsh transform of <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">X</inlineAttr></f> is: </p>
      </text>
    </region>
    <region region-id="345" left="36" top="2794.5" width="106.5" height="39" align-x="63.75" align-y="2814" 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">WX</ml:id>
          <ml:apply>
            <ml:mult style="default" />
            <ml:apply>
              <ml:div />
              <ml:real>1</ml:real>
              <ml:apply>
                <ml:pow />
                <ml:id xml:space="preserve">N</ml:id>
                <ml:real>2</ml:real>
              </ml:apply>
            </ml:apply>
            <ml:parens>
              <ml:apply>
                <ml:mult style="default" />
                <ml:apply>
                  <ml:mult style="default" />
                  <ml:id xml:space="preserve">W</ml:id>
                  <ml:id xml:space="preserve">X</ml:id>
                </ml:apply>
                <ml:id xml:space="preserve">W</ml:id>
              </ml:apply>
            </ml:parens>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="49" />
    </region>
    <region region-id="83" left="0" top="2838" width="6000" height="6" align-x="0" align-y="2838" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="350" left="6" top="2850" width="181.5" height="210" align-x="96" align-y="2850" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <component hide-arguments="false" clsid-buddy="62e334b8-fdda-4b2b-9bb4-48def95f2af1" item-idref="50" disable-calc="false">
        <inputs>
          <ml:id xml:space="preserve">X</ml:id>
        </inputs>
        <outputs />
      </component>
      <rendering item-idref="51" />
    </region>
    <region region-id="352" left="198" top="2850" width="181.5" height="210" align-x="288" align-y="2850" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <component hide-arguments="false" clsid-buddy="62e334b8-fdda-4b2b-9bb4-48def95f2af1" item-idref="52" disable-calc="false">
        <inputs>
          <ml:id xml:space="preserve">WX</ml:id>
        </inputs>
        <outputs />
      </component>
      <rendering item-idref="53" />
    </region>
    <region region-id="353" left="6" top="3066.75" width="294" height="28.5" align-x="6" align-y="3078" 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">Analogously to the 1D case, the <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">(0, 0)</inlineAttr></f> element of the transform is the average of the data</p>
      </text>
    </region>
    <region region-id="355" left="36" top="3114.75" width="80.25" height="17.25" align-x="79.5" align-y="3126" 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">WX</ml:id>
            <ml:sequence>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
            </ml:sequence>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
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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 inverse 2D Walsh transform is identical to the forward transform without the <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">1/N</inlineAttr></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" line-through="false"><sup>2</sup></inlineAttr></f> term.</p>
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        <phrase>inverse 2-D Walsh transform</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">As a filtering example, let's see what happens to the image above if we save only the "lower frequency" information along the left and top edges of the transform:</p>
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