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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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          <b>best_basis</b> creates the best packet basis from a packet table <b>T</b> and an additive cost function <b>f</b>. <b>basis_display2d</b> is used to display a 2D packet basis matrix. The arguments to these 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 Best Basis Algorithm</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">This function creates a best basis for selecting a wavelet packet transform from the wavelet packet table<f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"><sp /></inlineAttr></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">T</inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">.</inlineAttr></f> A basis is "best" if it minimizes the cost function <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">f</inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">. </inlineAttr></f>Sample cost functions are:</p>
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          <b>thresh(x)</b>
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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">Keep only those coefficients of x which are larger than some threshold. The default threshold is .5, but it should be modified to reflect the range of the data.</p>
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          <b>entropy(x)</b>
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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">where nrm is equal to the square root of the sum of the squares of the data values.</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">Coifman and Wickerhauser developed the "best basis" algorithm for selecting optimal bases (that is, transforms) from wavelet packet tables. The best basis algorithm automatically adapts the transform to best match the characteristics of the signal. The best basis algorithm finds the wavelet packet transform <f family="Times New Roman" charset="0"><inlineAttr font-style="normal" line-through="false">W</inlineAttr></f> that minimizes an additive cost function E:</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">where I is the set of index pairs (j,b) of the basis vectors in the transform W. </p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">For this example, a wavelet packet table for a linear chirp is created. <b>entropy(x) </b><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">is</inlineAttr></f><b><sp /></b><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">used as the</inlineAttr></f> cost function.</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">You can extract the basis with the function <b><u><link href="./Iwpt.xmcd" title="Links to section" popup="false">extract basis</link></u></b><b>.</b></p>
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                <c val="#000"> iwpt</c>
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            <c val="#000"> function:</c>
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        <phrase>iwpt function</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">How well did it work? </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">As should be the case with any valid basis, there is perfect reconstruction with the basis.</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">Try this with the level basis, for comparison</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">And with the DWT basis, also for comparison</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">Some Other Best Basis Cost Functions</p>
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        <phrase>best basis cost functions, other</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 entropy function is not the only cost function which can be used. The <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">abs(x)</inlineAttr></b></f> function returns the absolute value of the magnitude of the vector. The <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">thresh(x)</inlineAttr></b></f> function returns 1 if x &gt; 0.5 and zero otherwise. This value may be modified to suit the application. For example:</p>
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              <inlineAttr line-through="false">basis_display2d(b)</inlineAttr>
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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 function creates a matrix for graphically displaying two-dimensional (2D) best basis <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">b</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">The following illustration of 2D wavelet packet transforms uses the <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">Vbox</inlineAttr></b></f> function (defined to the right) with the 3-level wavelet transform.</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">As in the 1D case, there are many possible 2D wavelet packet transforms. For wavelet packets, you can split a coefficient block into separate blocks or merge separate blocks into a single block. Fortunately, the Coifman and Wickerhauser best basis algorithm extends to two dimensions.</p>
      </text>
      <indexes>
        <phrase>2D wavelet packet transforms</phrase>
        <phrase>coefficient block</phrase>
        <phrase>packet transforms, 2D wavelet</phrase>
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      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve" subscript="bbe">b</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">best_basis</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">T</ml:id>
              <ml:id xml:space="preserve">entropy</ml:id>
            </ml:sequence>
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    </region>
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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">To visualize the basis, use the<f family="Times New Roman" charset="0"><inlineAttr line-through="false"><sp /></inlineAttr></f><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">basis_display2d</inlineAttr></b></f><f family="Times New Roman" charset="0"><inlineAttr line-through="false"> fu</inlineAttr></f>nction: </p>
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      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">Q</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">basis_display2d</ml:id>
            <ml:id xml:space="preserve" subscript="bbe">b</ml:id>
          </ml:apply>
        </ml:define>
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    </region>
    <region region-id="5058" left="6" top="3192.75" width="329.25" height="28.5" align-x="6" align-y="3204" 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">Then press <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">[Ctrl] T</inlineAttr></b></f> on the keyboard or 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 to display the matrix.</p>
      </text>
    </region>
    <region region-id="5004" left="12" top="3240" width="97.5" height="126" align-x="60" align-y="3240" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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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">Since it is sometimes difficult to interpret a basis display, there is another basis display function which may make it a little easier. Below are three examples which select different parts of the basis to show what basis components are covering which part of the transform table. </p>
      </text>
    </region>
    <region region-id="4867" left="0" top="3438" width="6000" height="6" align-x="0" align-y="3438" 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">The function <b><c val="#008080">BasisDisplayPacket</c></b><f family="Times New Roman" charset="0"><b><inlineAttr line-through="false"><c val="#008080">2d</c></inlineAttr></b></f> is defined to the right of this region and</p>
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">in the file <b><u><link href="./Wavedef.xmcd" title="Link to wavedef.mcd" popup="false">wavedef.xmcd</link></u></b>.</p>
      </text>
      <indexes>
        <phrase>basisdisplaypacket2d function</phrase>
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          <ml:function>
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            <ml:boundVars>
              <ml:id xml:space="preserve">b</ml:id>
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                        <ml:apply>
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                          <ml:apply>
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      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">Q</ml:id>
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              <ml:id xml:space="preserve" subscript="bbe">b</ml:id>
              <ml:id xml:space="preserve">level</ml:id>
              <ml:id xml:space="preserve">row</ml:id>
              <ml:id xml:space="preserve">col</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">There are other bases besides the best basis. The <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">wavebs</inlineAttr></b></f> function returns a basis which corresponds to the DWT. The <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">create_level</inlineAttr></b></f> function returns a basis which is a specific level.</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 that you have selected the basis, you can extract it with the function <b><u><link href="./Iwpt.xmcd" title="Links to section" popup="false">extract_basis</link></u></b><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">,</inlineAttr></f> reconstruct it, and compute the reconstruction error.</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">As should be the case with any valid basis, there is perfect reconstruction using the best basis.</p>
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