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        <p style="Chapter Head" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">WAVELETS EXTENSION PACK</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Wavelets have been found particularly useful for compressing digitized fingerprints. Digital fingerprints are created at a resolution of 500 pixels per inch and 256 gray scale levels (8 bits). As a result, a typical uncompressed fingerprint image requires roughly half a megabyte of storage. Wavelet methods were selected as part of an FBI standard for compression of fingerprint images. This section analyzes a fingerprint image using the approach in the FBI standard.</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">For example, begin by importing data in a sample fingerprint image from fingerpr.txt.</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">This fingerprint image is a 512<f family="Arial" charset="0" size="10">x</f>512 matrix, rotated 90 degrees. First, undo the rotation:</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The proposed FBI standard is based on a specific wavelet packet transform with 64 subbands, using a symmlet(8) wavelet. The FBI basis is included below: </p>
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        <inputs />
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          <ml:id xml:space="preserve">T</ml:id>
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          </ml:apply>
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            <ml:real>64</ml:real>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Extract the FBI basis from the table T and plot it:</p>
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              <ml:real>6</ml:real>
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            <ml:id xml:space="preserve">basis_display2d</ml:id>
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            <ml:id xml:space="preserve">Concentration</ml:id>
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            <ml:localDefine>
              <ml:id xml:space="preserve">totalEnergy</ml:id>
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                <ml:mult style="default" />
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                <ml:id xml:space="preserve">w</ml:id>
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        <ml:eval>
          <ml:apply>
            <ml:id xml:space="preserve">rows</ml:id>
            <ml:id xml:space="preserve">FBIbasis</ml:id>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>64</ml:real>
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          <ml:apply>
            <ml:id xml:space="preserve">cols</ml:id>
            <ml:id xml:space="preserve">FBIbasis</ml:id>
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    <region region-id="3658" left="6" top="1422.75" width="311.25" height="28.5" align-x="6" align-y="1434" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Now subtract the mean from the image, and create a level-5 wavelet packet table:</p>
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      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve" subscript="mean">F</ml:id>
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            <ml:id xml:space="preserve">mean</ml:id>
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        <ml:define>
          <ml:id xml:space="preserve">F1</ml:id>
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            <ml:minus />
            <ml:id xml:space="preserve">F</ml:id>
            <ml:id xml:space="preserve" subscript="mean">F</ml:id>
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        <ml:define warning="WarnRedefinedBIUnit">
          <ml:id xml:space="preserve">J</ml:id>
          <ml:real>5</ml:real>
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      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve" subscript="wpt">F1</ml:id>
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            <ml:id xml:space="preserve">wpt</ml:id>
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              <ml:id xml:space="preserve">F1</ml:id>
              <ml:real>5</ml:real>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">For a 256x256 pixel image, setting the level J=5 gives an 8x8 block size.</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Next, compare the ability of the DWT and the Best Basis WPT to the FBI WPT to compress the image energy into a few coefficients.</p>
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        <phrase>FBI WPT to compress image</phrase>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The FBI Basis</p>
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            <ml:id xml:space="preserve">basis_display2d</ml:id>
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            <ml:id xml:space="preserve">dwt</ml:id>
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                            <ml:id xml:space="preserve">ii</ml:id>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The DWT basis</p>
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                    <ml:id xml:space="preserve">nrm</ml:id>
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                <ml:real>2</ml:real>
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    <region region-id="3591" left="246" top="1872" width="105" height="128.25" align-x="294" align-y="1872" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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            <ml:id xml:space="preserve">basis_display2d</ml:id>
            <ml:id xml:space="preserve" subscript="dwt">b</ml:id>
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              <ml:id xml:space="preserve">entropy</ml:id>
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    <region region-id="3472" left="30" top="1932.75" width="192.75" height="18" align-x="63.75" align-y="1944" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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    <region region-id="3473" left="30" top="1962.75" width="144.75" height="18" align-x="63.75" align-y="1974" 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="BBE">C</ml:id>
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            <ml:id xml:space="preserve">Concentration</ml:id>
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    </region>
    <region region-id="3645" left="0" top="2004" width="6000" height="6" align-x="0" align-y="2004" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
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    <region region-id="3498" left="6" top="2010.75" width="190.5" height="28.5" align-x="6" align-y="2022" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Next, notice how much of the energy is captured in the top 5% of the coefficients.</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The best basis</p>
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            <ml:id xml:space="preserve">basis_display2d</ml:id>
            <ml:id xml:space="preserve">BestBasis</ml:id>
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            <ml:apply>
              <ml:id xml:space="preserve">cols</ml:id>
              <ml:id xml:space="preserve" subscript="reduced">F</ml:id>
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      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve" subscript="keep">P</ml:id>
          <ml:apply>
            <ml:mult style="auto-select" />
            <ml:real>5</ml:real>
            <ml:id xml:space="preserve">%</ml:id>
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            <ml:id xml:space="preserve">floor</ml:id>
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              <ml:mult style="default" />
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              <ml:id xml:space="preserve">N</ml:id>
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            <ml:real>3276</ml:real>
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                <ml:id xml:space="preserve">ceil</ml:id>
                <ml:apply>
                  <ml:div />
                  <ml:id xml:space="preserve" subscript="keep">N</ml:id>
                  <ml:real>500</ml:real>
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              </ml:apply>
              <ml:real>500</ml:real>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Now compare the concentration obtained with each of the 3 transforms:</p>
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    </region>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">The horizontal lines illustrate the concentration for 5% of the coefficients, as marked by the vertical line. The solid lines are the concentration of the DWT, FBI, and Best Basis as a function of the number of coefficients.</p>
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    <region region-id="3374" left="6" top="2772.75" width="339.75" height="57" align-x="6" align-y="2784" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">You can also plot the mean-squared error for the three bases, which is defined as (1 - Concentration), and plot it against the compression ratio. The compression ratio is the total number of coefficients divided by the number of kept coefficients.</p>
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      <indexes>
        <phrase>compression ratio</phrase>
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              <ml:apply>
                <ml:id xml:space="preserve">ceil</ml:id>
                <ml:apply>
                  <ml:div />
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                  <ml:real>100</ml:real>
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              <ml:apply>
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                <ml:apply>
                  <ml:mult style="default" />
                  <ml:real>2</ml:real>
                  <ml:apply>
                    <ml:div />
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                    <ml:real>100</ml:real>
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                </ml:apply>
              </ml:apply>
            </ml:sequence>
            <ml:apply>
              <ml:minus />
              <ml:id xml:space="preserve">N</ml:id>
              <ml:real>3</ml:real>
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        <ml:define>
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          <ml:apply>
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        <ml:define>
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve" subscript="fbi">MSE</ml:id>
            <ml:id xml:space="preserve">i3</ml:id>
          </ml:apply>
          <ml:apply>
            <ml:minus />
            <ml:real>1</ml:real>
            <ml:apply>
              <ml:indexer />
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        <ml:define>
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve" subscript="dwt">MSE</ml:id>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Now you can reconstruct the fingerprints from the three transforms using only the top 5% of the coefficients.</p>
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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">Reconstruction of the Fingerprint Image</p>
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        <phrase>fingerprint image, reconstruction of</phrase>
        <phrase>reconstruction of fingerprint image</phrase>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Original fingerprint</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">WPT with the FBI basis</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">WPT with the best basis</p>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">DWT is more smudged than the WPT transforms. The area near the center of the whorl is much clearer than the DWT. The higher oscillation indices in the WPT capture the ripples more effectively, and after all, a fingerprint is an collection of ripples.</p>
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