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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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          </f> These functions are defined in the file <b><u><link href="./wavedef.xmcd" title="Links to section" popup="false">wavedef.xmcd</link></u></b> and in this worksheet. They are not part of the <f family="Times New Roman" charset="0"><i><inlineAttr line-through="false">Wavelets Extension Pack</inlineAttr></i></f> DLL.</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">These functions are used to approximate or decompose a two-dimensional (2D) object using the largest coefficients of a discrete wavelet transform. The arguments to the functions are:</p>
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        <p style="Heading 1" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The 2D Wavelet Approximation</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">We have seen how wavelets can be used to represent one-dimensional (1D) functions in <b><u><link href="./Approx1.xmcd" title="Links to The 1D Wavelet Approximation" popup="false">1D wavelet approximation</link></u></b>. The 2D wavelet approximation is a straightforward generalization of the 1D approximation, although the notation gets a lot messier! You can write a 2D function as a sum of 2D wavelets at different scales and locations:</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">Analogous to the 1D approximation, F(x, y) is decomposed into a sum of coarse resolution (level J) smooth coefficients and a sum of fine to coarse resolution (levels 1 to J) detail coefficients. In the 2D approximation, however, there are 3 types of detail coefficients and basis functions: vertical, horizontal, and diagonal. m and n range from 1 to 2<f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><sup>j</sup></inlineAttr></f><f family="Times New Roman" charset="0" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">-1</inlineAttr></f>. The 2D basis functions are generated from the father and mother wavelets<f size="10"><c val="#000080"><sp /></c></f>by scaling and translation: </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 2D wavelet transform coefficients are given by the integrals:</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">In practice, we do not use these formulae. Instead, we use the 2D discrete wavelet transform to generate the coefficients quickly and efficiently. The DWT coefficients are sorted by magnitude and only the largest ones are used for the approximation. </p>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">As an example, the wavelet approximation for the <b>Vbox</b> test image (defined to the right) is computed. The example makes use of the up-sample by 2 utility in the displayed images. The up-sample by 2 operation causes image sizes to increase.</p>
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            <ml:id xml:space="preserve">upsample2</ml:id>
            <ml:id xml:space="preserve">vbox</ml:id>
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        <outputs />
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          <ml:apply>
            <ml:id xml:space="preserve">rows</ml:id>
            <ml:id xml:space="preserve" subscript="mag2">vbox</ml:id>
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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">Next, compute its wavelet approximation of <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">vbox</inlineAttr></f> using one-fourth of the coefficients:</p>
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The function <b><c val="#008080">waveapprox2d</c></b><b><c val="#008080"><sp /></c></b><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000">uses two additional functions, </c></inlineAttr></f><b><c val="#000">shrink</c></b><b><c val="#008080"><sp /></c></b><inlineAttr font-weight="normal" line-through="false"><c val="#000"><f family="Times New Roman" charset="0">and </f></c></inlineAttr><b><c val="#000">wavesort,</c></b><b><c val="#008080"><sp /></c></b><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"><c val="#000">defined below.</c></inlineAttr></f></p>
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        <phrase>waveapprox2d function</phrase>
        <phrase>shrink function</phrase>
        <phrase>wavesort function</phrase>
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            <ml:id xml:space="preserve">shrink</ml:id>
            <ml:boundVars>
              <ml:id xml:space="preserve">WS</ml:id>
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                      <ml:id xml:space="preserve">W</ml:id>
                    </ml:apply>
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                  <ml:apply>
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                      <ml:id xml:space="preserve">W</ml:id>
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            <ml:id xml:space="preserve">upsample2</ml:id>
            <ml:id xml:space="preserve" subscript="a">vbox</ml:id>
          </ml:apply>
        </inputs>
        <outputs />
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    </region>
    <region region-id="4760" left="150" top="2754.75" width="129" height="28.5" align-x="150" align-y="2766" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ff8000" tag="">
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Plot the approximated image</p>
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">up-sampled by 2.</p>
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        <phrase>plot wavelet approximation</phrase>
        <phrase>wavelet approximation, plot</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">Now compute the wavelet approximation using one-third of the coefficients</p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">and plot the result up-sampled by 2:</p>
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              <ml:apply>
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                <ml:real>2</ml:real>
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        </inputs>
        <outputs />
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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 reconstructed image is now indistinguishable from the original. In fact, there is essentially perfect reconstruction of the original image as seen by looking at the approximation error:</p>
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            <ml:real>4.2632564145606011E-13</ml:real>
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        <p style="Heading 1" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">wavedecomp2d(y, J, K, filter)</p>
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        <phrase>wavedecomp2d function</phrase>
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              <c val="#000">The function </c>
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            <c val="#008080">wavedecomp2d(y, J, K, filter) </c>
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              <c val="#000">decomposes the matrix </c>
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                <c val="#000">K</c>
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              <inlineAttr font-weight="normal" line-through="false"> largest wavelet coefficients of a level </inlineAttr>
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                <c val="#000">J</c>
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            </b>
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                <sp />
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                <c val="#000">dwt</c>
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              <c val="#000"> with periodic boundary conditions. It returns a vector of </c>
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              <inlineAttr line-through="false">
                <c val="#000">K</c>
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              <inlineAttr font-weight="normal" line-through="false"> nested matrices. Each of the individual matrices is one level of the wavelet approximation to the original function </inlineAttr>
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            <b>
              <inlineAttr line-through="false">
                <c val="#000">y</c>
              </inlineAttr>
            </b>
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          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" line-through="false">
              <c val="#000">.</c>
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              <c val="#000"> For example, element 0 is the IDWT of the largest wavelet coefficient. Summing the elements in order gives successively better approximations to the original function </c>
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                <c val="#000">y</c>
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            <inlineAttr font-weight="normal" line-through="false">
              <c val="#000">. </c>
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            <b>
              <inlineAttr line-through="false">
                <c val="#008080">wavedecomp2d</c>
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              <f family="Times New Roman" charset="0" size="12"> is </f>
            </c>
          </inlineAttr>defined below and in the file <b><u><link href="./Wavedef.xmcd" title="Links to section" popup="false">wavedef.xmcd</link></u></b>:</p>
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                <ml:localDefine>
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                <ml:localDefine>
                  <ml:apply>
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                      <ml:parens>
                        <ml:apply>
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                  <ml:real>0</ml:real>
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      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">K</ml:id>
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            <ml:id xml:space="preserve">floor</ml:id>
            <ml:apply>
              <ml:mult style="default" />
              <ml:apply>
                <ml:div />
                <ml:real>1</ml:real>
                <ml:real>3</ml:real>
              </ml:apply>
              <ml:apply>
                <ml:pow />
                <ml:id xml:space="preserve">N</ml:id>
                <ml:real>2</ml:real>
              </ml:apply>
            </ml:apply>
          </ml:apply>
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    </region>
    <region region-id="4891" left="36" top="3534.75" width="192.75" height="15.75" align-x="62.25" align-y="3546" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ff8000" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">QQ</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">wavedecomp2d</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">K</ml:id>
              <ml:id xml:space="preserve">filter</ml:id>
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          </ml:apply>
        </ml:define>
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    </region>
    <region region-id="4919" left="0" top="3564" width="6000" height="6" align-x="0" align-y="3564" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
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    <region region-id="5063" left="12" top="3576.75" width="341.25" height="57" align-x="12" align-y="3588" 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">Analogous to the 1D case, the original image can be reconstructed by summing the elements of the matrix generated by the <b><c val="#008080">wavedecomp2d</c></b> function. As seen below, using more elements in the summation will <c val="#000">give successively better approximations to the original function.</c></p>
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    <region region-id="4985" left="138" top="3654.75" width="223.5" height="28.5" align-x="138" align-y="3666" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">K is the number of elements in the approximation </p>
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">vector.</p>
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      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:id xml:space="preserve">K</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>341</ml:real>
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      </math>
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    </region>
    <region region-id="5053" left="198" top="3696.75" width="133.5" height="28.5" align-x="198" align-y="3708" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Use 200 less elements less than K.</p>
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      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">N</ml:id>
          <ml:apply>
            <ml:minus />
            <ml:id xml:space="preserve">K</ml:id>
            <ml:real>200</ml:real>
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    </region>
    <region region-id="5031" left="36" top="3741.75" width="79.5" height="50.25" align-x="51" align-y="3768" 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">Z</ml:id>
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            <ml:summation />
            <ml:lambda>
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                <ml:id xml:space="preserve">n</ml:id>
              </ml:boundVars>
              <ml:apply>
                <ml:indexer />
                <ml:id xml:space="preserve">QQ</ml:id>
                <ml:id xml:space="preserve">n</ml:id>
              </ml:apply>
            </ml:lambda>
            <ml:bounds>
              <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:bounds>
          </ml:apply>
        </ml:define>
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    <region region-id="5054" left="198" top="3756.75" width="81.75" height="14.25" align-x="198" align-y="3768" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Sum the elements.</p>
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      <component hide-arguments="false" clsid-buddy="8442da2d-08d7-45ad-a4a6-f28bee6ae06a" item-idref="47" disable-calc="false">
        <inputs>
          <ml:id xml:space="preserve">Z</ml:id>
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        <outputs />
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      <rendering item-idref="48" />
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    <region region-id="5055" left="198" top="3852.75" width="131.25" height="14.25" align-x="198" align-y="3864" 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>
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      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">N</ml:id>
          <ml:apply>
            <ml:minus />
            <ml:id xml:space="preserve">K</ml:id>
            <ml:real>100</ml:real>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="49" />
    </region>
    <region region-id="5056" left="198" top="3942.75" width="133.5" height="28.5" align-x="198" align-y="3954" 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">Use 100 less elements less than K.</p>
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      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">Z</ml:id>
          <ml:apply>
            <ml:summation />
            <ml:lambda>
              <ml:boundVars>
                <ml:id xml:space="preserve">n</ml:id>
              </ml:boundVars>
              <ml:apply>
                <ml:indexer />
                <ml:id xml:space="preserve">QQ</ml:id>
                <ml:id xml:space="preserve">n</ml:id>
              </ml:apply>
            </ml:lambda>
            <ml:bounds>
              <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:bounds>
          </ml:apply>
        </ml:define>
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