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        <p style="Chapter Head" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">WAVELETS EXTENSION PACK</p>
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            <c val="#000">(x,y,[j,kx,ky]</c>
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            <c val="#000">,s)</c>
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          <c val="#000">These functions compute the value of two-dimensional (2D) wavelets at any scale and position. The arguments to these functions are:</c>
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            <c val="#000">, a real scalar denoting the position along the horizontal axis at which to evaluate the wavelet. </c>
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            <c val="#000">, a real scalar denoting the position along the vertical axis at which to evaluate the wavelet.</c>
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            <c val="#000">, an non-negative integer giving the scale of the wavelet.</c>
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            <c val="#000">, an integer giving the horizontal position of the wavelet.</c>
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            <c val="#000">, an integer giving the vertical position of the wavelet.</c>
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          <b>m</b>
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          <c val="#000">an integer giving the gender of the wavelet along the horizontal axis.</c>
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          <c val="#000">an integer giving the gender of the wavelet along the vertical axis.</c>
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            <c val="#000">, the family to which the wavelet belongs.</c>
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          <c val="#000">, a string argument designating the filter on which the wavelet is based.</c>
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            <b>
              <c val="#008080">• </c>
            </b>
          </f>
          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr line-through="false">
                <c val="#000">ds</c>
              </inlineAttr>
            </b>
          </f>
          <inlineAttr font-weight="normal">
            <c val="#000">, a string denoting either the horizontal, vertical, or diagonal 2D mother wavelet.</c>
          </inlineAttr>
        </p>
      </text>
    </region>
    <region region-id="4210" left="24" top="504" width="321.75" height="43.5" align-x="24" align-y="516" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ffff80" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
          <f size="14">
            <b>
              <c val="#008080">• </c>
            </b>
          </f>
          <b>o</b>
          <inlineAttr font-weight="normal">
            <c val="#000">, (optional), an integer for an orthonormal family, or a 1x2 matrix of integers for a biorthogonal family, giving the offset(s) corresponding to the family </c>
          </inlineAttr>
          <b>
            <c val="#000">f</c>
          </b>
          <inlineAttr font-weight="normal">
            <c val="#000">. If not specified, </c>
          </inlineAttr>
          <c val="#000">
            <b>o </b>
          </c>
          <inlineAttr font-weight="normal">
            <c val="#000">is the default offset for the family </c>
          </inlineAttr>
          <b>
            <c val="#000">f</c>
          </b>
          <inlineAttr font-weight="normal">
            <c val="#000">.</c>
          </inlineAttr>
        </p>
      </text>
    </region>
    <region region-id="4211" left="24" top="558" width="319.5" height="72" align-x="24" align-y="570" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ffff80" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
          <f size="14">
            <b>
              <c val="#008080">• </c>
            </b>
          </f>
          <b>i</b>
          <inlineAttr font-weight="normal">
            <c val="#000">, (optional), zero or one for an orthonormal family, or a 1x2 matrix whose elements are zero or one for a biorthogonal family, specifying negation of the default high pass filter coefficients for the family </c>
          </inlineAttr>
          <b>
            <c val="#000">f</c>
          </b>
          <inlineAttr font-weight="normal">
            <c val="#000">.</c>
          </inlineAttr>
          <inlineAttr font-weight="normal">
            <c val="#000"> Zero specifies no negation; one</c>
          </inlineAttr>
          <inlineAttr font-weight="normal">
            <c val="#000"> specifies negation. If not specified,</c>
          </inlineAttr>
          <inlineAttr font-weight="normal">
            <c val="#000">
              <sp />
            </c>
          </inlineAttr>
          <b>
            <c val="#000">i</c>
          </b>
        </p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
          <inlineAttr font-weight="normal">
            <c val="#000">represents no negation.</c>
          </inlineAttr>
        </p>
      </text>
    </region>
    <region region-id="4217" left="0" top="642" width="6000" height="6" align-x="0" align-y="642" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="4063" left="6" top="654.75" width="332.25" height="28.5" align-x="6" align-y="666" 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">2D wavelets are con<f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">structed by taking the tensor product of a horizontal and a vertical </inlineAttr></f><b><u><link href="./Family1.xmcd" title="Links to section" popup="false">1D wavelet</link></u></b><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">. </inlineAttr></f></p>
      </text>
      <indexes>
        <phrase>1D wavelets, tensor product of</phrase>
        <phrase>tensor product of 1D wavelets</phrase>
      </indexes>
    </region>
    <region region-id="4059" left="6" top="708.75" width="111.75" 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="Heading 1" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Creating Wavelets</p>
      </text>
      <indexes>
        <phrase>creating wavelets</phrase>
      </indexes>
    </region>
    <region region-id="4064" left="24" top="732.75" width="168" height="14.25" align-x="24" align-y="744" 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="0" text-align="inherit" list-style-type="none" tabs="inherit">
          <b>
            <c val="#000">wavelet2d(x,y,j,kx,ky,m,n,f,[o,[i]])</c>
          </b>
        </p>
      </text>
      <indexes>
        <phrase>wavelet2d function</phrase>
      </indexes>
    </region>
    <region region-id="3915" left="6" top="762.75" width="328.5" height="28.5" align-x="6" align-y="774" 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">
          <inlineAttr font-weight="normal">
            <c val="#000">This </c>
          </inlineAttr>
          <inlineAttr font-weight="normal">
            <c val="#000">function creates a rescaled and translated 2D wavelet or wavelet packet.</c>
          </inlineAttr>
        </p>
      </text>
      <indexes>
        <phrase>2D wavelet, rescaled and translated</phrase>
        <phrase>rescaled 2D wavelet</phrase>
        <phrase>translated 2D wavelet</phrase>
      </indexes>
    </region>
    <region region-id="4065" left="6" top="799.5" width="342.75" height="23.25" align-x="165" align-y="816" 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:equal />
          <ml:apply>
            <ml:id xml:space="preserve">wavelet2d</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">x</ml:id>
              <ml:id xml:space="preserve">y</ml:id>
              <ml:id xml:space="preserve">j</ml:id>
              <ml:id xml:space="preserve">kx</ml:id>
              <ml:id xml:space="preserve">ky</ml:id>
              <ml:id xml:space="preserve">m</ml:id>
              <ml:id xml:space="preserve">n</ml:id>
              <ml:id xml:space="preserve">f</ml:id>
            </ml:sequence>
          </ml:apply>
          <ml:apply>
            <ml:mult style="default" />
            <ml:apply>
              <ml:mult style="default" />
              <ml:apply>
                <ml:pow />
                <ml:real>2</ml:real>
                <ml:apply>
                  <ml:neg />
                  <ml:id xml:space="preserve">j</ml:id>
                </ml:apply>
              </ml:apply>
              <ml:apply>
                <ml:id xml:space="preserve" subscript="m">W</ml:id>
                <ml:apply>
                  <ml:minus />
                  <ml:apply>
                    <ml:mult style="auto-select" />
                    <ml:apply>
                      <ml:pow />
                      <ml:real>2</ml:real>
                      <ml:apply>
                        <ml:neg />
                        <ml:id xml:space="preserve">j</ml:id>
                      </ml:apply>
                    </ml:apply>
                    <ml:id xml:space="preserve">x</ml:id>
                  </ml:apply>
                  <ml:id xml:space="preserve">kx</ml:id>
                </ml:apply>
              </ml:apply>
            </ml:apply>
            <ml:apply>
              <ml:id xml:space="preserve" subscript="n">W</ml:id>
              <ml:apply>
                <ml:minus />
                <ml:apply>
                  <ml:mult style="auto-select" />
                  <ml:apply>
                    <ml:pow />
                    <ml:real>2</ml:real>
                    <ml:apply>
                      <ml:neg />
                      <ml:id xml:space="preserve">j</ml:id>
                    </ml:apply>
                  </ml:apply>
                  <ml:id xml:space="preserve">y</ml:id>
                </ml:apply>
                <ml:id xml:space="preserve">ky</ml:id>
              </ml:apply>
            </ml:apply>
          </ml:apply>
        </ml:apply>
      </math>
      <rendering item-idref="6" />
    </region>
    <region region-id="4052" left="6" top="837.75" width="333" height="63" align-x="6" align-y="852" 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">
          <inlineAttr font-weight="normal">
            <c val="#000">where </c>
          </inlineAttr>
          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr line-through="false">
                <c val="#000">W</c>
              </inlineAttr>
            </b>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <b>
              <inlineAttr line-through="false">
                <c val="#000">
                  <sub>n</sub>
                </c>
              </inlineAttr>
            </b>
          </f>
          <f family="Times New Roman" charset="0">
            <inlineAttr line-through="false">
              <b>
                <c val="#000">(x)</c>
              </b>
            </inlineAttr>
          </f>
          <c val="#000">
            <inlineAttr font-weight="normal"> is t</inlineAttr>
          </c>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" line-through="false">
              <c val="#000">he n</c>
            </inlineAttr>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" line-through="false">
              <sup>
                <c val="#000">th</c>
              </sup>
            </inlineAttr>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" line-through="false">
              <c val="#000"> one-</c>
            </inlineAttr>
          </f>
          <inlineAttr font-weight="normal">
            <c val="#000">dimensional (1D) wavelet or wavelet packet of the wavelet family described by the filter </c>
          </inlineAttr>
          <b>
            <inlineAttr line-through="false">
              <c val="#000">
                <f family="Times New Roman" charset="0">f</f>
              </c>
            </inlineAttr>
          </b>
          <f family="Times New Roman" charset="0">
            <inlineAttr font-weight="normal" line-through="false">
              <c val="#000">.</c>
            </inlineAttr>
          </f>
          <inlineAttr font-weight="normal">
            <c val="#000"> The argument </c>
          </inlineAttr>
          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr line-through="false">
                <c val="#000">f</c>
              </inlineAttr>
            </b>
          </f>
          <inlineAttr font-weight="normal">
            <c val="#000"> is usually one of the </c>
          </inlineAttr>
          <link href="./Filters.xmcd" title="Links to section" popup="false">
            <f family="Times New Roman" charset="0">
              <inlineAttr font-weight="normal" underline="false" line-through="false">built-in </inlineAttr>
            </f>
            <u>
              <b>wavelets functions</b>
            </u>
          </link>
          <inlineAttr font-weight="normal">
            <c val="#000"> provided with the </c>
          </inlineAttr>
          <f family="Times New Roman" charset="0">
            <i>
              <inlineAttr font-weight="normal" line-through="false">
                <c val="#000">Wavelets Extension Pack</c>
              </inlineAttr>
            </i>
          </f>
          <inlineAttr font-weight="normal">
            <c val="#000">. </c>
          </inlineAttr>
        </p>
      </text>
    </region>
    <region region-id="4053" left="6" top="912.75" width="323.25" height="71.25" align-x="6" align-y="924" 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="0" text-align="inherit" list-style-type="none" tabs="inherit">
          <c val="#000">When the arguments </c>
          <b>
            <c val="#000">m</c>
          </b>
          <c val="#000"> and </c>
          <b>
            <c val="#000">n</c>
          </b>
          <c val="#000"> equal zero, this function produces </c>
          <c val="#000">father</c>
          <c val="#000"> wavelets. When either </c>
          <b>
            <c val="#000">m</c>
          </b>
          <c val="#000"> or </c>
          <b>
            <c val="#000">n</c>
          </b>
          <c val="#000"> equal one, this function produces </c>
          <c val="#000">mother</c>
          <c val="#000"> wavelets</c>
          <c val="#000080">. </c>The four possible combinations of <b>m</b> and <b>n</b> lead to four different types of 2D wavelets: a father wavelet and three mother wavelets. <b>n</b> and <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">m</inlineAttr></b></f> more generally define a <b><u><link href="./Wpt.xmcd" title="Links to section" popup="false">wavelet packet</link></u></b>.</p>
      </text>
      <indexes>
        <phrase>father wavelets</phrase>
        <phrase>mother wavelets</phrase>
      </indexes>
    </region>
    <region region-id="4025" left="6" top="996.75" width="327" height="28.5" align-x="6" align-y="1008" 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">You can use the <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">wavelet2d</inlineAttr></b></f> function to create a graph of a particular wavelet. For example:</p>
      </text>
      <indexes>
        <phrase>graph of wavelet</phrase>
        <phrase>wavelet, graph of</phrase>
      </indexes>
    </region>
    <region region-id="4066" left="30" top="1038.75" width="84" height="15.75" align-x="48.75" align-y="1050" 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">filt</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">daublet</ml:id>
            <ml:real>12</ml:real>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="7" />
    </region>
    <region region-id="4067" left="162" top="1038.75" width="66.75" height="14.25" align-x="162" align-y="1050" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Wavelet family</p>
      </text>
    </region>
    <region region-id="4068" left="30" top="1062.75" width="28.5" height="15.75" align-x="42" align-y="1074" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">j</ml:id>
          <ml:real>0</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="8" />
    </region>
    <region region-id="4069" left="162" top="1062.75" width="73.5" height="14.25" align-x="162" align-y="1074" 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">Level parameter</p>
      </text>
      <indexes>
        <phrase>level parameter</phrase>
      </indexes>
    </region>
    <region region-id="4070" left="30" top="1086.75" width="36" height="15.75" align-x="49.5" align-y="1098" 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">kx</ml:id>
          <ml:real>0</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="9" />
    </region>
    <region region-id="4071" left="78" top="1086.75" width="36" height="15.75" align-x="97.5" align-y="1098" 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">ky</ml:id>
          <ml:real>0</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="10" />
    </region>
    <region region-id="4072" left="162" top="1086.75" width="98.25" height="14.25" align-x="162" align-y="1098" 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">Translation parameter</p>
      </text>
      <indexes>
        <phrase>translation parameter</phrase>
      </indexes>
    </region>
    <region region-id="4073" left="30" top="1116.75" width="33" height="15.75" align-x="46.5" align-y="1128" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedBIUnit">
          <ml:id xml:space="preserve">m</ml:id>
          <ml:real>0</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="11" />
    </region>
    <region region-id="4074" left="78" top="1116.75" width="30" height="15.75" align-x="91.5" align-y="1128" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">n</ml:id>
          <ml:real>0</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="12" />
    </region>
    <region region-id="4075" left="162" top="1116.75" width="171" height="14.25" align-x="162" align-y="1128" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Wavelet subband</p>
      </text>
      <indexes>
        <phrase>subband, wavelet</phrase>
      </indexes>
    </region>
    <region region-id="4056" left="0" top="1146" width="6000" height="6" align-x="0" align-y="1146" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
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    <region region-id="4024" left="6" top="1158.75" width="84.75" height="14.25" align-x="6" align-y="1170" 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">Graph the wavelet:</p>
      </text>
    </region>
    <region region-id="4076" left="30" top="1182.75" width="52.5" height="15.75" align-x="44.25" 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>
          <ml:id xml:space="preserve">p</ml:id>
          <ml:range>
            <ml:real>0</ml:real>
            <ml:real>39</ml:real>
          </ml:range>
        </ml:define>
      </math>
      <rendering item-idref="13" />
    </region>
    <region region-id="4223" left="96" top="1182.75" width="52.5" height="15.75" align-x="110.25" 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>
          <ml:id xml:space="preserve">q</ml:id>
          <ml:range>
            <ml:real>0</ml:real>
            <ml:real>39</ml:real>
          </ml:range>
        </ml:define>
      </math>
      <rendering item-idref="14" />
    </region>
    <region region-id="4224" left="174" top="1174.5" width="64.5" height="33.75" align-x="192.75" align-y="1194" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">x</ml:id>
            <ml:id xml:space="preserve">p</ml:id>
          </ml:apply>
          <ml:apply>
            <ml:plus />
            <ml:real>-6</ml:real>
            <ml:apply>
              <ml:div />
              <ml:id xml:space="preserve">p</ml:id>
              <ml:real>5</ml:real>
            </ml:apply>
          </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">This wavelet is a 2D <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">father</inlineAttr></f> wavelet since both <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">m</inlineAttr></b></f> and <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">n</inlineAttr></b></f> are zero. Setting only <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">m </inlineAttr></b></f>= 1 gives a "horizontal <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">mother</inlineAttr></f> wavelet," where the x-axis wavelet is a <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">mother</inlineAttr></f> wavelet and the y-axis wavelet is a <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">father</inlineAttr></f> wavelet. Likewise, <b><inlineAttr line-through="false"><f family="Times New Roman" charset="0">n </f></inlineAttr></b>= 1 and <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">m </inlineAttr></b></f>= 0 gives a "vertical <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">mother</inlineAttr></f> wavelet." <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">n </inlineAttr></b></f>= <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">m </inlineAttr></b></f>= 1 gives a "diagonal <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">mother</inlineAttr></f> wavelet."</p>
      </text>
      <indexes>
        <phrase>diagonal mother wavelet</phrase>
        <phrase>horizontal mother wavelet</phrase>
        <phrase>mother wavelet, diagonal</phrase>
        <phrase>mother wavelet, horizontal</phrase>
      </indexes>
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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 <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">father</inlineAttr></f> wavelet is given by the product of two 1D father wavelets, and thus is smooth in both the x and y directions. The horizontal mother wavelet is the product of a 1D horizontal mother and vertical father, so it oscillates in the x-direction and is smooth in the y-direction. This property lets it represent detailed information in only the x-direction. </p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit" />
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Likewise, the vertical mother wavelet represents detail only in the y-direction. The diagonal mother is the product of two 1D mother wavelets, so it oscillates in both directions. It can thus capture detail in both directions simultaneously, i.e. along a diagonal.</p>
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    <region region-id="4083" left="24" top="1746.75" width="177" height="14.25" align-x="24" align-y="1758" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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          <b>
            <c val="#000">dlwavelet2d(x,y,j,kx,ky,m,n,f,[o,[i]])</c>
          </b>
        </p>
      </text>
      <indexes>
        <phrase>dlwavelet2d function</phrase>
      </indexes>
    </region>
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            <c val="#000">This </c>
          </inlineAttr>
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            <c val="#000">function creates a rescaled and translated dual 2D wavelet. </c>
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          <ml:equal />
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                      <ml:real>2</ml:real>
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          <c val="#000">where </c>
          <b>
            <c val="#000">W</c>
          </b>
          <b>
            <c val="#000">
              <sub>n</sub>
            </c>
          </b>
          <b>
            <c val="#000">(x)</c>
          </b>
          <c val="#000"> is t</c>
          <c val="#000">he n</c>
          <c val="#000">
            <sup>th</sup>
          </c>
          <c val="#000">
            <sp />
          </c>
          <c val="#000">1D dual wavelet or dual wavelet packet of the wavelet family described by the filter </c>
          <c val="#000">
            <b>f</b>
          </c>
          <c val="#000">.</c>
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              <c val="#000"> The argument characteristics are the same as those for </c>
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            <b>
              <inlineAttr font-style="normal" line-through="false">
                <c val="#000">wavelet2d</c>
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              <c val="#000">.</c>
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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">We can compute and plot a dual wavelet using <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">dlwavelet2d</inlineAttr></b></f>:</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Biorthogonal wavelet family </p>
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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">Level parameter</p>
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      <math optimize="false" disable-calc="false">
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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">Translation parameters</p>
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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">Wavelet packet, for example n=0 specifies the father wavelet and n=1 specifies the mother wavelet</p>
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              <ml:real>10</ml:real>
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      <pageBreak />
    </region>
    <region region-id="4039" left="6" top="2394.75" width="145.5" height="14.25" align-x="6" align-y="2406" 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="Heading 1" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Mother and Father Wavelets</p>
      </text>
      <indexes>
        <phrase>wavelets, mother and father</phrase>
        <phrase>father wavelets</phrase>
        <phrase>mother wavelets</phrase>
      </indexes>
    </region>
    <region region-id="4038" left="6" top="2418.75" width="328.5" height="28.5" align-x="6" align-y="2430" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">To simplify the process of creating 2D parent wavelets, use the following functions:</p>
      </text>
      <indexes>
        <phrase>2D parent wavelets</phrase>
      </indexes>
    </region>
    <region region-id="3868" left="30" top="2460.75" width="140.25" height="14.25" align-x="30" align-y="2472" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ff8000" 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">
          <b>
            <c val="#000">father2d</c>
          </b>
          <b>
            <c val="#000">(x,y,[j,kx,ky]</c>
          </b>
          <b>
            <c val="#000">,s)</c>
          </b>
        </p>
      </text>
      <indexes>
        <phrase>father2d function</phrase>
      </indexes>
    </region>
    <region region-id="4123" left="180" top="2460.75" width="144.75" height="14.25" align-x="180" align-y="2472" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ff8000" 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">
          <b>
            <c val="#000">mother2d</c>
          </b>
          <b>
            <c val="#000">(x,y,[j,kx,ky]</c>
          </b>
          <b>
            <c val="#000">,s,ds)</c>
          </b>
        </p>
      </text>
      <indexes>
        <phrase>mother2d function</phrase>
      </indexes>
    </region>
    <region region-id="4120" left="30" top="2484.75" width="126.75" height="14.25" align-x="30" align-y="2496" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ff8000" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
          <f family="Times New Roman" charset="0" size="12">
            <b>
              <inlineAttr line-through="false">
                <c val="#000">dlfather2d(x,y,[j,kx,ky]</c>
              </inlineAttr>
            </b>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <b>
              <inlineAttr line-through="false">
                <c val="#000">,s)</c>
              </inlineAttr>
            </b>
          </f>
        </p>
      </text>
      <indexes>
        <phrase>dlfather2d function</phrase>
      </indexes>
    </region>
    <region region-id="4122" left="180" top="2484.75" width="146.25" height="14.25" align-x="180" align-y="2496" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ff8000" 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">
          <b>
            <c val="#000">dlmother2d(x,y,[j,kx,ky]</c>
          </b>
          <b>
            <c val="#000">,s,ds)</c>
          </b>
        </p>
      </text>
      <indexes>
        <phrase>dlmother2d function</phrase>
      </indexes>
    </region>
    <region region-id="4037" left="6" top="2514.75" width="326.25" height="342" align-x="6" align-y="2526" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ff8000" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The function <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">father2d</inlineAttr></b></f><c val="#000"> computes the value at position </c><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><c val="#000">(x,y)</c></inlineAttr></b></f><c val="#000"> of the</c><c val="#000"> father w</c><c val="#000">avelet</c><c val="#000"> of scale </c><f family="Times New Roman" charset="0"><inlineAttr line-through="false"><b><c val="#000">j</c></b></inlineAttr></f><c val="#000"> and centered</c><c val="#000"> at position </c><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><c val="#000">(kx,ky)</c></inlineAttr></b></f><c val="#000"><sp /></c><c val="#000">using the family of wavelets given by </c><b><c val="#000">s</c></b><c val="#000">.</c></p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit" />
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The function <inlineAttr line-through="false"><f family="Times New Roman" charset="0"><b>mother2d</b></f></inlineAttr><inlineAttr font-weight="normal" line-through="false"><c val="#000"><f family="Times New Roman" charset="0" size="12"> computes the values at </f></c></inlineAttr><c val="#000">position </c><b><c val="#000">(x,y)</c></b><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" line-through="false"><c val="#000"> of the</c></inlineAttr></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><c val="#000"> mother</c></inlineAttr></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" line-through="false"><c val="#000"> wavelet</c></inlineAttr></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" line-through="false"><c val="#000"><sp /></c></inlineAttr></f><c val="#000">of scale </c><c val="#000"><b>j</b></c><c val="#000"> and centered</c><c val="#000"> at position </c><c val="#000"><b>(kx,ky)</b></c><c val="#000"><sp /></c><c val="#000">using the family of wavelets given by </c><b><c val="#000">s</c></b><c val="#000">. </c></p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit" />
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The string argument <b>s</b> is the <b><u><link href="./wavefami.xmcd" title="Links to section" popup="false">wavelet family</link></u></b> consisting of a wavelet abbreviation and numbers. The following are valid abbreviations for wavelets: d, daub, daublet, s, symm, symmlet, c, coif, coiflet, bl, bs, </p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Bspline, haar. The numbers which follow must be valid for the wavelet filter specified. For example, some wavelet values for <b>s</b> are "d4" for <c val="#000">daublet(4)</c>, "symm12" for <c val="#000">symmlet(12), "bs33" for Bspline(3,3),</c></p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
          <c val="#000">and "haar" for daublet(2)</c>
          <c val="#008000">.</c>
        </p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit" />
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The arguments <b>j</b>, <b>kx</b>, and <b>ky</b> are optional. If they are omitted, they are assumed to be zero.</p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit" />
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The functions <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">dlfather2d</inlineAttr></b></f> and <b><inlineAttr line-through="false"><f family="Times New Roman" charset="0">dlmother2d</f></inlineAttr></b> create the dual parent wavelets.</p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit" />
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
          <c val="#000">The string argument </c>
          <b>
            <c val="#000">
              <f family="Times New Roman" charset="0">
                <inlineAttr line-through="false">ds</inlineAttr>
              </f>
            </c>
          </b>
          <c val="#000"> is set to </c>"<c val="#000">V</c>"<c val="#000"> for </c><c val="#000">vertical, </c>"<c val="#000">H</c>"<c val="#000"> for horizontal, and </c>"<c val="#000">D</c>"<c val="#000"> for diagonal mother wavelets. </c></p>
      </text>
    </region>
    <region region-id="4219" left="0" top="2862" width="6000" height="6" align-x="0" align-y="2862" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="4040" left="6" top="2874.75" width="327.75" height="42.75" align-x="6" align-y="2886" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">In the following example, different father and mother wavelets are computed using a <b><u><link href="./wavefami.xmcd" title="Links to section" popup="false">Haar wavelet</link></u></b>, also known as a Daubechies 2 ("d2") wavelet.</p>
      </text>
      <indexes>
        <phrase>Daubechies 2 wavelet</phrase>
        <phrase>Haar wavelet</phrase>
      </indexes>
    </region>
    <region region-id="3801" left="30" top="2928.75" width="28.5" height="15.75" align-x="42" align-y="2940" 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:real>0</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="34" />
    </region>
    <region region-id="4183" left="114" top="2928.75" width="36" height="15.75" align-x="133.5" align-y="2940" 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">kx</ml:id>
          <ml:real>0</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="35" />
    </region>
    <region region-id="4184" left="204" top="2928.75" width="36" height="15.75" align-x="223.5" align-y="2940" 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">ky</ml:id>
          <ml:real>0</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="36" />
    </region>
    <region region-id="3804" left="30" top="2958.75" width="39.75" height="15.75" align-x="47.25" align-y="2970" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedBIUnit">
          <ml:id xml:space="preserve">N</ml:id>
          <ml:real>32</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="37" />
    </region>
    <region region-id="4185" left="114" top="2958.75" width="68.25" height="15.75" align-x="128.25" align-y="2970" 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">k</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="38" />
    </region>
    <region region-id="4186" left="204" top="2958.75" width="64.5" height="15.75" align-x="214.5" align-y="2970" 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">l</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="39" />
    </region>
    <region region-id="3807" left="30" top="2986.5" width="72" height="33.75" align-x="48.75" align-y="3006" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">x</ml:id>
            <ml:id xml:space="preserve">k</ml:id>
          </ml:apply>
          <ml:apply>
            <ml:minus />
            <ml:apply>
              <ml:div />
              <ml:id xml:space="preserve">k</ml:id>
              <ml:real>16</ml:real>
            </ml:apply>
            <ml:real>0.5</ml:real>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="40" />
    </region>
    <region region-id="4187" left="204" top="2986.5" width="69" height="33.75" align-x="219.75" align-y="3006" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">y</ml:id>
            <ml:id xml:space="preserve">l</ml:id>
          </ml:apply>
          <ml:apply>
            <ml:minus />
            <ml:apply>
              <ml:div />
              <ml:id xml:space="preserve">l</ml:id>
              <ml:real>16</ml:real>
            </ml:apply>
            <ml:real>0.5</ml:real>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="41" />
    </region>
    <region region-id="3809" left="30" top="3030.75" width="183" height="17.25" align-x="58.5" align-y="3042" 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:indexer />
            <ml:id xml:space="preserve">F</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">k</ml:id>
              <ml:id xml:space="preserve">l</ml:id>
            </ml:sequence>
          </ml:apply>
          <ml:apply>
            <ml:id xml:space="preserve">father2d</ml:id>
            <ml:sequence>
              <ml:apply>
                <ml:indexer />
                <ml:id xml:space="preserve">x</ml:id>
                <ml:id xml:space="preserve">k</ml:id>
              </ml:apply>
              <ml:apply>
                <ml:indexer />
                <ml:id xml:space="preserve">y</ml:id>
                <ml:id xml:space="preserve">l</ml:id>
              </ml:apply>
              <ml:id xml:space="preserve">j</ml:id>
              <ml:id xml:space="preserve">kx</ml:id>
              <ml:id xml:space="preserve">ky</ml:id>
              <ml:str xml:space="preserve">d2</ml:str>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="42" />
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    <region region-id="3810" left="30" top="3060.75" width="219.75" height="22.5" align-x="66" align-y="3072" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve" subscript="V">M</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">k</ml:id>
              <ml:id xml:space="preserve">l</ml:id>
            </ml:sequence>
          </ml:apply>
          <ml:apply>
            <ml:id xml:space="preserve">mother2d</ml:id>
            <ml:sequence>
              <ml:apply>
                <ml:indexer />
                <ml:id xml:space="preserve">x</ml:id>
                <ml:id xml:space="preserve">k</ml:id>
              </ml:apply>
              <ml:apply>
                <ml:indexer />
                <ml:id xml:space="preserve">y</ml:id>
                <ml:id xml:space="preserve">l</ml:id>
              </ml:apply>
              <ml:id xml:space="preserve">j</ml:id>
              <ml:id xml:space="preserve">kx</ml:id>
              <ml:id xml:space="preserve">ky</ml:id>
              <ml:str xml:space="preserve">d2</ml:str>
              <ml:str xml:space="preserve">V</ml:str>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="43" />
    </region>
    <region region-id="3811" left="30" top="3090.75" width="220.5" height="22.5" align-x="66.75" align-y="3102" 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" subscript="H">M</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">k</ml:id>
              <ml:id xml:space="preserve">l</ml:id>
            </ml:sequence>
          </ml:apply>
          <ml:apply>
            <ml:id xml:space="preserve">mother2d</ml:id>
            <ml:sequence>
              <ml:apply>
                <ml:indexer />
                <ml:id xml:space="preserve">x</ml:id>
                <ml:id xml:space="preserve">k</ml:id>
              </ml:apply>
              <ml:apply>
                <ml:indexer />
                <ml:id xml:space="preserve">y</ml:id>
                <ml:id xml:space="preserve">l</ml:id>
              </ml:apply>
              <ml:id xml:space="preserve">j</ml:id>
              <ml:id xml:space="preserve">kx</ml:id>
              <ml:id xml:space="preserve">ky</ml:id>
              <ml:str xml:space="preserve">d2</ml:str>
              <ml:str xml:space="preserve">H</ml:str>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="44" />
    </region>
    <region region-id="3812" left="30" top="3120.75" width="220.5" height="22.5" align-x="66.75" align-y="3132" 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" subscript="D">M</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">k</ml:id>
              <ml:id xml:space="preserve">l</ml:id>
            </ml:sequence>
          </ml:apply>
          <ml:apply>
            <ml:id xml:space="preserve">mother2d</ml:id>
            <ml:sequence>
              <ml:apply>
                <ml:indexer />
                <ml:id xml:space="preserve">x</ml:id>
                <ml:id xml:space="preserve">k</ml:id>
              </ml:apply>
              <ml:apply>
                <ml:indexer />
                <ml:id xml:space="preserve">y</ml:id>
                <ml:id xml:space="preserve">l</ml:id>
              </ml:apply>
              <ml:id xml:space="preserve">j</ml:id>
              <ml:id xml:space="preserve">kx</ml:id>
              <ml:id xml:space="preserve">ky</ml:id>
              <ml:str xml:space="preserve">d2</ml:str>
              <ml:str xml:space="preserve">D</ml:str>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="45" />
    </region>
    <region region-id="4125" left="0" top="3150" width="6000" height="6" align-x="0" align-y="3150" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="4114" left="6" top="3162.75" width="186.75" height="14.25" align-x="6" align-y="3174" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Now plot the father and mother wavelets:</p>
      </text>
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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 horizontal mother wavelet</p>
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        <phrase>mother wavelet graph, horizontal</phrase>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The vertical mother wavelet</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 diagonal mother wavelet</p>
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        <phrase>vertical mother wavelet, graph</phrase>
        <phrase>mother wavelet graph, vertical</phrase>
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        <phrase>mother wavelet graph, diagonal</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">These plots show the oscillation in the horizontal, vertical, and diagonal directions for the different mother wavelets, as well as the smoothness of the father wavelet.</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">Two 2D wavelets are used in applications involving images, matrices, and other 2D data. The properties which make wavelets attractive for analysis of 1D functions hold for 2D functions as well. In particular, wavelets have been found to be very effective for image coding and data compression, since relatively few coefficients are needed to represent the image. </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 plot the dual father wavelet:</p>
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          <c val="#000">As in the </c>
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          <c val="#000">, 2D wavelet packets are a generalization of 2D wavelets. </c>The functions <b>wavelet2d</b>, <b>dlwavelet2</b><b>d</b>, and the functions for father and mother wavelets may also be used to generate 2D wavelet packets. We explore this topic in the section <f family="Times New Roman" charset="0"><b><u><inlineAttr line-through="false"><c val="#000"><link href="./wpt.xmcd" title="Links to section" popup="false">Wavelet Packet Transforms</link></c></inlineAttr></u></b></f><c val="#000"><inlineAttr font-weight="normal" underline="false">. </inlineAttr></c></p>
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