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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">T</c>
          <c val="#000">wo-dimensional (2D) wavelet transform</c>
          <c val="#000">s are computed with the same functions used to compute one-dimensional (1D) wavelet transforms.<sp count="2" /></c>
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          <c val="#000"> is a function defined in this file and </c>
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          <c val="#000"> DLL. </c>
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            <c val="#000"> a matrix of data to be transformed. </c>
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            <c val="#000"> the family to which the wavelet belongs.</c>
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            <c val="#000">no negation; one</c>
          </inlineAttr>
          <c val="#000">
            <inlineAttr font-weight="normal"> specifies negation. If not specified,</inlineAttr>
          </c>
          <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="4439" left="6" top="504.75" width="382.5" height="28.5" align-x="6" align-y="516" 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">Each of the <b>dwt</b> functions return a matrix containing the transform of the input matrix. </p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
          <b>
            <c val="#008080">MaxDWTLevel</c>
          </b>
          <c val="#000"> returns an integer.</c>
        </p>
      </text>
    </region>
    <region region-id="4166" left="0" top="546" width="6000" height="6" align-x="0" align-y="546" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="4293" left="6" top="558.75" width="148.5" height="14.25" align-x="6" align-y="570" 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">2D Wavelet Transform</p>
      </text>
    </region>
    <region region-id="4425" left="24" top="582.75" width="76.5" height="14.25" align-x="24" align-y="594" 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">dwt(a,n,f,[o,[i]])</c>
          </b>
        </p>
      </text>
      <indexes>
        <phrase>dwt function</phrase>
      </indexes>
    </region>
    <region region-id="4386" left="468" top="582.75" width="210.75" height="407.25" align-x="534.75" align-y="594" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:function>
            <ml:id xml:space="preserve">Vbox</ml:id>
            <ml:boundVars>
              <ml:id xml:space="preserve">nrow</ml:id>
            </ml:boundVars>
          </ml:function>
          <ml:program>
            <ml:for>
              <ml:id xml:space="preserve">i</ml:id>
              <ml:range>
                <ml:real>0</ml:real>
                <ml:apply>
                  <ml:minus />
                  <ml:id xml:space="preserve">nrow</ml:id>
                  <ml:real>1</ml:real>
                </ml:apply>
              </ml:range>
              <ml:for>
                <ml:id xml:space="preserve">j</ml:id>
                <ml:range>
                  <ml:real>0</ml:real>
                  <ml:apply>
                    <ml:minus />
                    <ml:id xml:space="preserve">nrow</ml:id>
                    <ml:real>1</ml:real>
                  </ml:apply>
                </ml:range>
                <ml:localDefine>
                  <ml:apply>
                    <ml:indexer />
                    <ml:id xml:space="preserve">box</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:real>255</ml:real>
                </ml:localDefine>
              </ml:for>
            </ml:for>
            <ml:localDefine>
              <ml:id xml:space="preserve">lb</ml:id>
              <ml:apply>
                <ml:div />
                <ml:id xml:space="preserve">nrow</ml:id>
                <ml:real>8</ml:real>
              </ml:apply>
            </ml:localDefine>
            <ml:for>
              <ml:id xml:space="preserve">i</ml:id>
              <ml:range>
                <ml:id xml:space="preserve">lb</ml:id>
                <ml:apply>
                  <ml:mult style="default" />
                  <ml:id xml:space="preserve">lb</ml:id>
                  <ml:real>7</ml:real>
                </ml:apply>
              </ml:range>
              <ml:program>
                <ml:for>
                  <ml:id xml:space="preserve">j</ml:id>
                  <ml:range>
                    <ml:id xml:space="preserve">lb</ml:id>
                    <ml:apply>
                      <ml:mult style="default" />
                      <ml:id xml:space="preserve">lb</ml:id>
                      <ml:real>3</ml:real>
                    </ml:apply>
                  </ml:range>
                  <ml:localDefine>
                    <ml:apply>
                      <ml:indexer />
                      <ml:id xml:space="preserve">box</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:real>0</ml:real>
                  </ml:localDefine>
                </ml:for>
                <ml:localDefine>
                  <ml:apply>
                    <ml:indexer />
                    <ml:id xml:space="preserve">box</ml:id>
                    <ml:sequence>
                      <ml:id xml:space="preserve">i</ml:id>
                      <ml:apply>
                        <ml:mult style="default" />
                        <ml:id xml:space="preserve">lb</ml:id>
                        <ml:real>7</ml:real>
                      </ml:apply>
                    </ml:sequence>
                  </ml:apply>
                  <ml:real>0</ml:real>
                </ml:localDefine>
                <ml:for>
                  <ml:id xml:space="preserve">j</ml:id>
                  <ml:range>
                    <ml:id xml:space="preserve">lb</ml:id>
                    <ml:apply>
                      <ml:mult style="default" />
                      <ml:id xml:space="preserve">lb</ml:id>
                      <ml:real>2</ml:real>
                    </ml:apply>
                  </ml:range>
                  <ml:localDefine>
                    <ml:apply>
                      <ml:indexer />
                      <ml:id xml:space="preserve">box</ml:id>
                      <ml:sequence>
                        <ml:id xml:space="preserve">j</ml:id>
                        <ml:id xml:space="preserve">i</ml:id>
                      </ml:sequence>
                    </ml:apply>
                    <ml:real>0</ml:real>
                  </ml:localDefine>
                </ml:for>
                <ml:localDefine>
                  <ml:apply>
                    <ml:indexer />
                    <ml:id xml:space="preserve">box</ml:id>
                    <ml:sequence>
                      <ml:id xml:space="preserve">i</ml:id>
                      <ml:apply>
                        <ml:mult style="default" />
                        <ml:id xml:space="preserve">lb</ml:id>
                        <ml:real>3</ml:real>
                      </ml:apply>
                    </ml:sequence>
                  </ml:apply>
                  <ml:real>0</ml:real>
                </ml:localDefine>
                <ml:localDefine>
                  <ml:apply>
                    <ml:indexer />
                    <ml:id xml:space="preserve">box</ml:id>
                    <ml:sequence>
                      <ml:id xml:space="preserve">i</ml:id>
                      <ml:apply>
                        <ml:mult style="default" />
                        <ml:id xml:space="preserve">lb</ml:id>
                        <ml:real>4</ml:real>
                      </ml:apply>
                    </ml:sequence>
                  </ml:apply>
                  <ml:real>0</ml:real>
                </ml:localDefine>
                <ml:localDefine>
                  <ml:apply>
                    <ml:indexer />
                    <ml:id xml:space="preserve">box</ml:id>
                    <ml:sequence>
                      <ml:id xml:space="preserve">i</ml:id>
                      <ml:apply>
                        <ml:mult style="default" />
                        <ml:id xml:space="preserve">lb</ml:id>
                        <ml:real>5</ml:real>
                      </ml:apply>
                    </ml:sequence>
                  </ml:apply>
                  <ml:real>0</ml:real>
                </ml:localDefine>
                <ml:localDefine>
                  <ml:apply>
                    <ml:indexer />
                    <ml:id xml:space="preserve">box</ml:id>
                    <ml:sequence>
                      <ml:id xml:space="preserve">i</ml:id>
                      <ml:apply>
                        <ml:mult style="default" />
                        <ml:id xml:space="preserve">lb</ml:id>
                        <ml:real font="0">6</ml:real>
                      </ml:apply>
                    </ml:sequence>
                  </ml:apply>
                  <ml:real>0</ml:real>
                </ml:localDefine>
                <ml:localDefine>
                  <ml:apply>
                    <ml:indexer />
                    <ml:id xml:space="preserve">box</ml:id>
                    <ml:sequence>
                      <ml:id xml:space="preserve">i</ml:id>
                      <ml:id xml:space="preserve">i</ml:id>
                    </ml:sequence>
                  </ml:apply>
                  <ml:real>0</ml:real>
                </ml:localDefine>
                <ml:localDefine>
                  <ml:apply>
                    <ml:indexer />
                    <ml:id xml:space="preserve">box</ml:id>
                    <ml:sequence>
                      <ml:id xml:space="preserve">i</ml:id>
                      <ml:apply>
                        <ml:minus />
                        <ml:apply>
                          <ml:plus />
                          <ml:id xml:space="preserve">i</ml:id>
                          <ml:id xml:space="preserve">lb</ml:id>
                        </ml:apply>
                        <ml:real>1</ml:real>
                      </ml:apply>
                    </ml:sequence>
                  </ml:apply>
                  <ml:real>127</ml:real>
                </ml:localDefine>
                <ml:localDefine>
                  <ml:apply>
                    <ml:indexer />
                    <ml:id xml:space="preserve">box</ml:id>
                    <ml:sequence>
                      <ml:id xml:space="preserve">i</ml:id>
                      <ml:apply>
                        <ml:minus />
                        <ml:apply>
                          <ml:mult style="default" />
                          <ml:real>7</ml:real>
                          <ml:id xml:space="preserve">lb</ml:id>
                        </ml:apply>
                        <ml:id xml:space="preserve">i</ml:id>
                      </ml:apply>
                    </ml:sequence>
                  </ml:apply>
                  <ml:real>127</ml:real>
                </ml:localDefine>
              </ml:program>
            </ml:for>
            <ml:id xml:space="preserve">box</ml:id>
          </ml:program>
        </ml:define>
      </math>
      <rendering item-idref="5" />
    </region>
    <region region-id="4307" left="6" top="606.75" width="327.75" height="45.75" align-x="6" align-y="618" 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">
          <c val="#000">
            <inlineAttr font-weight="normal" line-through="false">
              <f family="Times New Roman" charset="0" size="12">This function computes t</f>
            </inlineAttr>
          </c>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" line-through="false">
              <c val="#000">he </c>
            </inlineAttr>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr line-through="false">
              <b>
                <c val="#000">n-</c>
              </b>
            </inlineAttr>
          </f>
          <inlineAttr font-weight="normal" line-through="false">
            <c val="#000">
              <f family="Times New Roman" charset="0" size="12">level</f>
            </c>
          </inlineAttr>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" line-through="false">
              <c val="#000"> discrete wavelet transform of the matrix </c>
            </inlineAttr>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <b>
              <inlineAttr line-through="false">
                <c val="#000">a</c>
              </inlineAttr>
            </b>
          </f>
          <inlineAttr font-weight="normal" line-through="false">
            <c val="#000">
              <f family="Times New Roman" charset="0" size="12"> using the wavelet </f>
            </c>
          </inlineAttr>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" font-style="normal" line-through="false">
              <c val="#000">filter </c>
            </inlineAttr>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <b>
              <inlineAttr font-style="normal" line-through="false">
                <c val="#000">f</c>
              </inlineAttr>
            </b>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" line-through="false">
              <c val="#000">. </c>
            </inlineAttr>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <c val="#000">
              <inlineAttr font-weight="normal" line-through="false">Note that rows(</inlineAttr>
            </c>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <b>
              <inlineAttr line-through="false">
                <c val="#000">a</c>
              </inlineAttr>
            </b>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" line-through="false">
              <c val="#000">)/</c>
            </inlineAttr>
          </f>
          <c val="#000">2</c>
          <f family="Times New Roman" charset="0" size="12">
            <b>
              <inlineAttr font-style="normal" underline="false" line-through="false">
                <c val="#000">
                  <sup>n</sup>
                </c>
              </inlineAttr>
            </b>
          </f>
          <inlineAttr font-weight="normal" line-through="false">
            <c val="#000">
              <f family="Times New Roman" charset="0" size="12"> and cols(</f>
            </c>
          </inlineAttr>
          <f family="Times New Roman" charset="0" size="12">
            <b>
              <inlineAttr line-through="false">
                <c val="#000">a</c>
              </inlineAttr>
            </b>
          </f>
          <inlineAttr font-weight="normal" line-through="false">
            <c val="#000">
              <f family="Times New Roman" charset="0" size="12">)/</f>
            </c>
          </inlineAttr>
          <c val="#000">2</c>
          <f family="Times New Roman" charset="0" size="12">
            <b>
              <inlineAttr font-style="normal" underline="false" line-through="false">
                <c val="#000">
                  <sup>n</sup>
                </c>
              </inlineAttr>
            </b>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" line-through="false">
              <c val="#000"> must be integers</c>
            </inlineAttr>
          </f>
          <c val="#000">. </c>
        </p>
      </text>
    </region>
    <region region-id="4308" left="6" top="660.75" width="328.5" height="28.5" align-x="6" align-y="672" 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 example below shows the transform of the <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">V</inlineAttr></b></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">box</inlineAttr></b></f> function (defined to the right) computed using the Daubechies 4 ("d4") wavelet.</p>
      </text>
      <indexes>
        <phrase>Daubechies 4 wavelet</phrase>
        <phrase>Vbox function</phrase>
      </indexes>
    </region>
    <region region-id="4561" left="18" top="702.75" width="45.75" height="15.75" align-x="35.25" align-y="714" 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>128</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="6" />
    </region>
    <region region-id="4548" left="138" top="702.75" width="85.5" height="15.75" align-x="168.75" align-y="714" 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">vbox</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">Vbox</ml:id>
            <ml:id xml:space="preserve">N</ml:id>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="7" />
    </region>
    <region region-id="4562" left="18" top="738" width="97.5" height="126" align-x="66" align-y="738" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <component hide-arguments="false" clsid-buddy="8442da2d-08d7-45ad-a4a6-f28bee6ae06a" item-idref="8" disable-calc="false">
        <inputs>
          <ml:id xml:space="preserve">vbox</ml:id>
        </inputs>
        <outputs />
      </component>
      <rendering item-idref="9" />
    </region>
    <region region-id="4550" left="138" top="774.75" width="142.5" height="42.75" align-x="138" align-y="786" 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">Plot the vbox test image using</p>
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">the picture operator. (Choose <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">Picture</inlineAttr></b></f> from the <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">Insert</inlineAttr></b></f> menu.)</p>
      </text>
      <indexes>
        <phrase>picture operator</phrase>
      </indexes>
    </region>
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      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedBIFunction">
          <ml:id xml:space="preserve">wave</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">daublet</ml:id>
            <ml:real>4</ml:real>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="10" />
    </region>
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      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedBIUnit">
          <ml:id xml:space="preserve">J</ml:id>
          <ml:real>3</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="11" />
    </region>
    <region region-id="4767" left="192" top="876.75" width="117" height="15.75" align-x="208.5" align-y="888" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">w</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">dwt</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">vbox</ml:id>
              <ml:id xml:space="preserve">J</ml:id>
              <ml:id xml:space="preserve">wave</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="12" />
    </region>
    <region region-id="4768" left="18" top="906.75" width="183" height="84" align-x="69" align-y="918" 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="WarnRedefinedBIFunction">
          <ml:function>
            <ml:id xml:space="preserve">scale</ml:id>
            <ml:boundVars>
              <ml:id xml:space="preserve">M</ml:id>
            </ml:boundVars>
          </ml:function>
          <ml:program>
            <ml:localDefine>
              <ml:id xml:space="preserve">myMax</ml:id>
              <ml:apply>
                <ml:id xml:space="preserve">max</ml:id>
                <ml:id xml:space="preserve">M</ml:id>
              </ml:apply>
            </ml:localDefine>
            <ml:localDefine>
              <ml:id xml:space="preserve">myMin</ml:id>
              <ml:apply>
                <ml:id xml:space="preserve">min</ml:id>
                <ml:id xml:space="preserve">M</ml:id>
              </ml:apply>
            </ml:localDefine>
            <ml:apply>
              <ml:vectorize />
              <ml:apply>
                <ml:mult style="default" />
                <ml:apply>
                  <ml:div />
                  <ml:apply>
                    <ml:minus />
                    <ml:id xml:space="preserve">M</ml:id>
                    <ml:id xml:space="preserve">myMin</ml:id>
                  </ml:apply>
                  <ml:apply>
                    <ml:minus />
                    <ml:id xml:space="preserve">myMax</ml:id>
                    <ml:id xml:space="preserve">myMin</ml:id>
                  </ml:apply>
                </ml:apply>
                <ml:real>255</ml:real>
              </ml:apply>
            </ml:apply>
          </ml:program>
        </ml:define>
      </math>
      <rendering item-idref="13" />
    </region>
    <region region-id="4771" left="30" top="1008" width="97.5" height="126" align-x="78" align-y="1008" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <component hide-arguments="false" clsid-buddy="8442da2d-08d7-45ad-a4a6-f28bee6ae06a" item-idref="14" disable-calc="false">
        <inputs>
          <ml:apply>
            <ml:minus />
            <ml:real font="0">255</ml:real>
            <ml:apply>
              <ml:id xml:space="preserve">scale</ml:id>
              <ml:id xml:space="preserve">w</ml:id>
            </ml:apply>
          </ml:apply>
        </inputs>
        <outputs />
      </component>
      <rendering item-idref="15" />
    </region>
    <region region-id="4470" left="144" top="1068.75" width="167.25" height="14.25" align-x="144" align-y="1080" 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">Plot the 2D DWT of the <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">vbox</inlineAttr></f> image.</p>
      </text>
    </region>
    <region region-id="4483" 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 />
    </region>
    <region region-id="4531" left="6" top="1164.75" width="330.75" height="128.25" align-x="6" align-y="1176" 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 coarse scale reproduction of the original image appears in the upper left corner. This region contains the smooth scale coefficients. In the <b><u><c val="#000"><link href="./Fingerpr.xmcd" title="Links to section" popup="false">Fingerprint Identification</link></c></u></b> discussion, this property is used to rapidly decimate an image. The level 1 horizontal components of the image are in the lower left corner. <inlineAttr font-weight="normal">V</inlineAttr><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">box</inlineAttr></f> does not have a lot of horizontal edges, so this region does not reveal very much content. The level 1 vertical components of the image are in the upper right corner. <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">vbox</inlineAttr></f> is rich with vertical components. The level 1 diagonal components of the image appear in the lower right corner of the DWT.</p>
      </text>
    </region>
    <region region-id="4322" left="6" top="1308.75" width="225" height="14.25" align-x="6" align-y="1320" 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">The 2D Inverse Discrete Wavelet Transform</p>
      </text>
      <indexes>
        <phrase>wavelet transform, 2D inverse discrete</phrase>
        <phrase>2D inverse discrete wavelet transform</phrase>
      </indexes>
    </region>
    <region region-id="4566" left="24" top="1332.75" width="79.5" height="14.25" align-x="24" align-y="1344" 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">idwt(a,n,f,[o,[i]])</c>
          </b>
        </p>
      </text>
      <indexes>
        <phrase>idwt function</phrase>
      </indexes>
    </region>
    <region region-id="4532" left="6" top="1356.75" width="328.5" height="74.25" align-x="6" align-y="1368" 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">
          <c val="#000">
            <inlineAttr font-weight="normal" line-through="false">
              <f family="Times New Roman" charset="0" size="12">This function computes the</f>
            </inlineAttr>
          </c>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" line-through="false">
              <c val="#000">
                <sp />
              </c>
            </inlineAttr>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <b>
              <inlineAttr line-through="false">
                <c val="#000">n-</c>
              </inlineAttr>
            </b>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" line-through="false">
              <c val="#000">level</c>
            </inlineAttr>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" line-through="false">
              <c val="#000"> inverse wavelet transform of the matrix </c>
            </inlineAttr>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <b>
              <inlineAttr line-through="false">
                <c val="#000">a</c>
              </inlineAttr>
            </b>
          </f>
          <inlineAttr font-weight="normal" line-through="false">
            <c val="#000">
              <f family="Times New Roman" charset="0" size="12"> using the wavelet </f>
            </c>
          </inlineAttr>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" line-through="false">
              <c val="#000">filter </c>
            </inlineAttr>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <b>
              <inlineAttr line-through="false">
                <c val="#000">f</c>
              </inlineAttr>
            </b>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" line-through="false">
              <c val="#000">. Again, </c>
            </inlineAttr>
          </f>
          <c val="#000">rows(</c>
          <f family="Times New Roman" charset="0">
            <inlineAttr line-through="false">
              <c val="#000">
                <b>a</b>
              </c>
            </inlineAttr>
          </f>
          <c val="#000">)/</c>
          <c val="#000">2</c>
          <f family="Times New Roman" charset="0" size="12">
            <c val="#000">
              <b>
                <sup>
                  <inlineAttr font-style="normal" underline="false" line-through="false">n</inlineAttr>
                </sup>
              </b>
            </c>
          </f>
          <c val="#000"> and cols(</c>
          <f family="Times New Roman" charset="0">
            <b>
              <inlineAttr line-through="false">
                <c val="#000">a</c>
              </inlineAttr>
            </b>
          </f>
          <c val="#000">)/</c>
          <c val="#000">2</c>
          <f family="Times New Roman" charset="0" size="12">
            <c val="#000">
              <sup>
                <inlineAttr font-style="normal" underline="false" line-through="false">
                  <b>n</b>
                </inlineAttr>
              </sup>
            </c>
          </f>
          <c val="#000"> must be integers. </c>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" line-through="false">
              <c val="#000">The level </c>
            </inlineAttr>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr line-through="false">
              <b>
                <c val="#000">n</c>
              </b>
            </inlineAttr>
          </f>
          <inlineAttr font-weight="normal" line-through="false">
            <f family="Times New Roman" charset="0" size="12"> must be the same as the level of the </f>
          </inlineAttr>
          <f family="Times New Roman" charset="0" size="12">
            <b>
              <inlineAttr line-through="false">dwt</inlineAttr>
            </b>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" line-through="false"> which first computed the transform. </inlineAttr>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr line-through="false">
              <b>
                <c val="#000">idwt</c>
              </b>
            </inlineAttr>
          </f>
          <inlineAttr font-weight="normal" line-through="false">
            <c val="#000">
              <f family="Times New Roman" charset="0" size="12"> can be used to recover the original image from the </f>
            </c>
          </inlineAttr>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" line-through="false">
              <c val="#000">dwt</c>
            </inlineAttr>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" line-through="false">
              <c val="#000">.</c>
            </inlineAttr>
          </f>
        </p>
      </text>
    </region>
    <region region-id="4567" left="24" top="1440.75" width="122.25" height="18" align-x="57.75" align-y="1452" 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" subscript="r">vbox</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">idwt</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">w</ml:id>
              <ml:id xml:space="preserve">J</ml:id>
              <ml:id xml:space="preserve">wave</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="16" />
    </region>
    <region region-id="4497" left="192" top="1440.75" width="138" height="14.25" align-x="192" align-y="1452" 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">Reconstruct the original image.</p>
      </text>
    </region>
    <region region-id="4572" left="24" top="1470" width="97.5" height="128.25" align-x="72" align-y="1470" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <component hide-arguments="false" clsid-buddy="8442da2d-08d7-45ad-a4a6-f28bee6ae06a" item-idref="17" disable-calc="false">
        <inputs>
          <ml:id xml:space="preserve" subscript="r">vbox</ml:id>
        </inputs>
        <outputs />
      </component>
      <rendering item-idref="18" />
    </region>
    <region region-id="4496" left="192" top="1506.75" width="131.25" height="14.25" align-x="192" align-y="1518" 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">Plot the reconstructed image.</p>
      </text>
    </region>
    <region region-id="4573" left="24" top="1605.75" width="219.75" height="70.5" align-x="77.25" align-y="1662" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:function>
            <ml:id xml:space="preserve">RMS</ml:id>
            <ml:boundVars>
              <ml:id xml:space="preserve">M</ml:id>
            </ml:boundVars>
          </ml:function>
          <ml:apply>
            <ml:div />
            <ml:apply>
              <ml:sqrt />
              <ml:apply>
                <ml:summation />
                <ml:lambda>
                  <ml:boundVars>
                    <ml:id xml:space="preserve">i</ml:id>
                  </ml:boundVars>
                  <ml:apply>
                    <ml:summation />
                    <ml:lambda>
                      <ml:boundVars>
                        <ml:id xml:space="preserve">j</ml:id>
                      </ml:boundVars>
                      <ml:apply>
                        <ml:pow />
                        <ml:parens>
                          <ml:apply>
                            <ml:indexer />
                            <ml:id xml:space="preserve">M</ml:id>
                            <ml:sequence>
                              <ml:id xml:space="preserve">i</ml:id>
                              <ml:id xml:space="preserve">j</ml:id>
                            </ml:sequence>
                          </ml:apply>
                        </ml:parens>
                        <ml:real>2</ml:real>
                      </ml:apply>
                    </ml:lambda>
                    <ml:bounds>
                      <ml:real>0</ml:real>
                      <ml:apply>
                        <ml:minus />
                        <ml:apply>
                          <ml:id xml:space="preserve">cols</ml:id>
                          <ml:id xml:space="preserve">M</ml:id>
                        </ml:apply>
                        <ml:real>1</ml:real>
                      </ml:apply>
                    </ml:bounds>
                  </ml:apply>
                </ml:lambda>
                <ml:bounds>
                  <ml:real>0</ml:real>
                  <ml:apply>
                    <ml:minus />
                    <ml:apply>
                      <ml:id xml:space="preserve">rows</ml:id>
                      <ml:id xml:space="preserve">M</ml:id>
                    </ml:apply>
                    <ml:real>1</ml:real>
                  </ml:apply>
                </ml:bounds>
              </ml:apply>
            </ml:apply>
            <ml:apply>
              <ml:mult style="default" />
              <ml:apply>
                <ml:id xml:space="preserve">rows</ml:id>
                <ml:id xml:space="preserve">M</ml:id>
              </ml:apply>
              <ml:apply>
                <ml:id xml:space="preserve">cols</ml:id>
                <ml:id xml:space="preserve">M</ml:id>
              </ml:apply>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="19" />
    </region>
    <region region-id="4574" left="24" top="1687.5" width="189.75" height="23.25" align-x="128.25" align-y="1704" 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">RMS</ml:id>
            <ml:apply>
              <ml:minus />
              <ml:id xml:space="preserve">vbox</ml:id>
              <ml:id xml:space="preserve" subscript="r">vbox</ml:id>
            </ml:apply>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>1.4639027567995484E-15</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="20" />
    </region>
    <region region-id="4760" left="216" top="1692.75" width="153.75" height="14.25" align-x="216" align-y="1704" 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">Compute the reconstruction error.</p>
      </text>
    </region>
    <region region-id="4415" left="0" top="1728" width="6000" height="6" align-x="0" align-y="1728" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="4427" left="6" top="1740.75" width="183" height="14.25" align-x="6" align-y="1752" 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">Maximum Wavelet Transform Level</p>
      </text>
      <indexes>
        <phrase>wavelet transform level, maximum</phrase>
        <phrase>maximum wavelet transform level</phrase>
      </indexes>
    </region>
    <region region-id="4428" left="24" top="1770.75" width="93.75" height="14.25" align-x="24" align-y="1782" 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">
          <f family="Times New Roman" charset="0" size="12">
            <b>
              <inlineAttr line-through="false">
                <c val="#008080">MaxDWTLevel(a)</c>
              </inlineAttr>
            </b>
          </f>
        </p>
      </text>
      <indexes>
        <phrase>MaxDWTLevel function</phrase>
      </indexes>
    </region>
    <region region-id="4369" left="6" top="1794.75" width="338.25" height="42.75" align-x="6" align-y="1806" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">This function returns the largest valid <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">level</inlineAttr></f> of the DWT for the matrix <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">a</inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">. </inlineAttr></f>This function is defined in the file <b><u><link href="./Wavedef.xmcd" title="Links to section" popup="false">wavedef.xmcd</link></u></b> and below for convenience.</p>
      </text>
    </region>
    <region region-id="4505" left="6" top="1854.75" width="531.75" height="144" align-x="104.25" align-y="1866" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:function>
            <ml:id xml:space="preserve">MaxDWTLevel</ml:id>
            <ml:boundVars>
              <ml:id xml:space="preserve">A</ml:id>
            </ml:boundVars>
          </ml:function>
          <ml:program>
            <ml:localDefine>
              <ml:id xml:space="preserve">lev</ml:id>
              <ml:real>0</ml:real>
            </ml:localDefine>
            <ml:localDefine>
              <ml:id xml:space="preserve">r</ml:id>
              <ml:apply>
                <ml:id xml:space="preserve">rows</ml:id>
                <ml:id xml:space="preserve">A</ml:id>
              </ml:apply>
            </ml:localDefine>
            <ml:localDefine>
              <ml:id xml:space="preserve">c</ml:id>
              <ml:apply>
                <ml:id xml:space="preserve">cols</ml:id>
                <ml:id xml:space="preserve">A</ml:id>
              </ml:apply>
            </ml:localDefine>
            <ml:while>
              <ml:apply>
                <ml:mult style="default" />
                <ml:parens>
                  <ml:apply>
                    <ml:equal />
                    <ml:apply>
                      <ml:div />
                      <ml:id xml:space="preserve">r</ml:id>
                      <ml:apply>
                        <ml:pow />
                        <ml:real>2</ml:real>
                        <ml:id xml:space="preserve">lev</ml:id>
                      </ml:apply>
                    </ml:apply>
                    <ml:apply>
                      <ml:id xml:space="preserve">floor</ml:id>
                      <ml:apply>
                        <ml:plus />
                        <ml:apply>
                          <ml:div />
                          <ml:id xml:space="preserve">r</ml:id>
                          <ml:apply>
                            <ml:pow />
                            <ml:real>2</ml:real>
                            <ml:id xml:space="preserve">lev</ml:id>
                          </ml:apply>
                        </ml:apply>
                        <ml:real>0.5</ml:real>
                      </ml:apply>
                    </ml:apply>
                  </ml:apply>
                </ml:parens>
                <ml:parens>
                  <ml:apply>
                    <ml:plus />
                    <ml:apply>
                      <ml:mult style="default" />
                      <ml:parens>
                        <ml:apply>
                          <ml:notEqual />
                          <ml:id xml:space="preserve">c</ml:id>
                          <ml:real>1</ml:real>
                        </ml:apply>
                      </ml:parens>
                      <ml:parens>
                        <ml:apply>
                          <ml:equal />
                          <ml:apply>
                            <ml:div />
                            <ml:id xml:space="preserve">c</ml:id>
                            <ml:apply>
                              <ml:pow />
                              <ml:real>2</ml:real>
                              <ml:id xml:space="preserve">lev</ml:id>
                            </ml:apply>
                          </ml:apply>
                          <ml:apply>
                            <ml:id xml:space="preserve">floor</ml:id>
                            <ml:apply>
                              <ml:plus />
                              <ml:apply>
                                <ml:div />
                                <ml:id xml:space="preserve">c</ml:id>
                                <ml:apply>
                                  <ml:pow />
                                  <ml:real>2</ml:real>
                                  <ml:id xml:space="preserve">lev</ml:id>
                                </ml:apply>
                              </ml:apply>
                              <ml:real>0.5</ml:real>
                            </ml:apply>
                          </ml:apply>
                        </ml:apply>
                      </ml:parens>
                    </ml:apply>
                    <ml:parens>
                      <ml:apply>
                        <ml:equal />
                        <ml:id xml:space="preserve">c</ml:id>
                        <ml:real>1</ml:real>
                      </ml:apply>
                    </ml:parens>
                  </ml:apply>
                </ml:parens>
              </ml:apply>
              <ml:localDefine>
                <ml:id xml:space="preserve">lev</ml:id>
                <ml:apply>
                  <ml:plus />
                  <ml:id xml:space="preserve">lev</ml:id>
                  <ml:real>1</ml:real>
                </ml:apply>
              </ml:localDefine>
            </ml:while>
            <ml:localDefine>
              <ml:id xml:space="preserve">lev</ml:id>
              <ml:apply>
                <ml:minus />
                <ml:id xml:space="preserve">lev</ml:id>
                <ml:real>1</ml:real>
              </ml:apply>
            </ml:localDefine>
          </ml:program>
        </ml:define>
      </math>
      <rendering item-idref="21" />
    </region>
    <region region-id="4499" left="30" top="2028.75" width="127.5" height="15.75" align-x="42.75" align-y="2040" 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:apply>
            <ml:id xml:space="preserve">MaxDWTLevel</ml:id>
            <ml:id xml:space="preserve">vbox</ml:id>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="22" />
    </region>
    <region region-id="4500" left="186" top="2028.75" width="145.5" height="28.5" align-x="186" 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">Compute the maximum DWT level for the vbox image.</p>
      </text>
      <indexes>
        <phrase>vbox image, maximum DWT level</phrase>
      </indexes>
    </region>
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