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            <c val="#008080">IMAGE FILTERING AND RESTORATION</c>
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            <b>
              <c val="#800080">Removing Scanner Stripes</c>
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          <b>
            <i>Contributed by</i>
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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">Dr. Michael Wegener</p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Defense Science and Technology Organization</p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Electronics and Surveillance Research Laboratory</p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Land, Space and Optoelectronics Division</p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Salisbury, South Australia 5108</p>
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          <b>
            <c val="#800000">Introduction</c>
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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 many cases, visible and infrared images are obtained by using multisensor scanners. Some of these scanners are serial-parallel scanning devices such as the LANDSAT Multispectral Scanner (MSS), or the UK Common Module Thermal Imager (TICM). Scenes recorded by these devices tend to show scanner stripes caused by slightly different gains and biases of the individual detector-amplifier combinations, and due to minimal variations in the composition of each detector element's material.</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">While striping induced by differences in the gains and biases of the individual detector-amplifier combinations can generally be described by a linear function, and removed by a two-point calibration, variations in the composition of the various detector elements induce higher order nonlinearities that cannot be removed easily. A probability-based model described by B. K. P. Horn in [1] and enhanced by M. Wegener in [2] will be used to compensate for these nonlinear variations in sensor output.</p>
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        <phrase>Horn, B. K. P.</phrase>
        <phrase>Wegener, M.</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">Examine an image which has been artificially striped.</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">Notice the subtle horizontal striping in the second image. This makes it particularly difficult to tell which stripes are artifacts and which are actual tracks on the ground in the image.</p>
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          <b>
            <c val="#800000">Destriping by Histogram Matching</c>
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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 destriping algorithm is based on the assumption that the radiance incident on a detector can be represented by a random variable, <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">x</inlineAttr></f>, with probability density function (pdf) <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">p(x)</inlineAttr></f>. The detector output signal is then determined by the radiance incident on the detector and the nonlinear detector response, and is conveniently described by another random variable, <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">y</inlineAttr></f>. The individual detector-amplifier combination's <f family="Times New Roman" charset="0"><inlineAttr line-through="false"><b>cumulative probability functions (cpf)</b></inlineAttr></f>, which are required to perform the destriping operation, are found by dividing the cumulative histograms of the individual detector-amplifier combinations by the number of pixels recorded by each detector-amplifier combination.</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">To achieve the destriping, cpfs of the individual detector-amplifier combinations are compared with that for a hypothetical average detector element: a detector element whose response is equal to the average response of all detectors making up the entire detector array. This process is called (cumulative) <b><inlineAttr line-through="false"><f family="Times New Roman" charset="0">histogram matching</f></inlineAttr></b>. For each individual detector, we first need to find the value of the detector's individual cpf corresponding to a given pixel value (gray level). This value is then used to find the gray level that would have been produced by the hypothetical average detector. The hypothetical gray level is then used to replace the current gray value.</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 easiest and fastest way to implement the destriping algorithm is by the use of look-up tables. First, the cpfs for the individual sensors must be calculated.</p>
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        <phrase>cpf, cumulative probablility functions</phrase>
        <phrase>cumulative probablility functions (cpf)</phrase>
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            <c val="#800000">Calculate cpf for Each Detector</c>
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          <b>
            <c val="#008080">Note</c>
          </b>: For this example we assume that there were four detector elements scanning the scene in a parallel-serial pattern, producing 256 shades of gray.</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">Initialize number of detector elements:</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">Initialize number of gray levels:</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">Divide the image up into four subimages, each of which was recorded by one of the individual sensor elements:</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">Every fourth row was recorded by the same detector element.</p>
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          <ml:apply>
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              <ml:apply>
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                <ml:real>3</ml:real>
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    <region region-id="370" left="0" top="1536" width="6000" height="6" align-x="0" align-y="1536" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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    <region region-id="242" left="6" top="1548.75" width="196.5" height="14.25" align-x="6" align-y="1560" 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">Calculate the histograms for each subimage:</p>
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          <ml:id xml:space="preserve">H1</ml:id>
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              <ml:id xml:space="preserve">levels</ml:id>
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        <ml:define warning="WarnRedefinedBIFunction">
          <ml:id xml:space="preserve">H2</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">To build the cumulative histograms for each subimage, initialize the first entry with the contents of the first bin of the histogram:</p>
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      <math optimize="false" disable-calc="false">
        <ml:define>
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            <ml:real>0</ml:real>
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          <ml:apply>
            <ml:indexer />
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        </ml:define>
      </math>
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    <region region-id="273" left="108" top="1656.75" width="57.75" height="17.25" align-x="135.75" align-y="1668" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:apply>
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            <ml:real>0</ml:real>
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          <ml:apply>
            <ml:indexer />
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            <ml:real>0</ml:real>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="27" />
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    <region region-id="274" left="180" top="1656.75" width="57.75" height="17.25" align-x="207.75" align-y="1668" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
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          <ml:apply>
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            <ml:real>0</ml:real>
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        </ml:define>
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    <region region-id="275" left="252" top="1656.75" width="57.75" height="17.25" align-x="279.75" align-y="1668" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
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          <ml:apply>
            <ml:indexer />
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    <region region-id="259" left="6" top="1686.75" width="306" height="28.5" align-x="6" align-y="1698" 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">Adding subsequent entries in the four regular histograms creates the four cumulative histograms required for the matching process:</p>
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        <ml:define warning="WarnRedefinedUDScalar">
          <ml:apply>
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          <ml:apply>
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            <ml:apply>
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        <ml:define warning="WarnRedefinedUDScalar">
          <ml:apply>
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            <ml:id xml:space="preserve">p</ml:id>
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          <ml:apply>
            <ml:plus />
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              <ml:indexer />
              <ml:id xml:space="preserve">H4</ml:id>
              <ml:id xml:space="preserve">p</ml:id>
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            <ml:apply>
              <ml:indexer />
              <ml:id xml:space="preserve">C4</ml:id>
              <ml:apply>
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                <ml:real>1</ml:real>
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    <region region-id="271" left="6" top="1806.75" width="198" height="14.25" align-x="6" align-y="1818" 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">Convert the cumulative histograms into cpfs:</p>
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              <ml:real>1</ml:real>
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            <ml:apply>
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            <ml:apply>
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    <region region-id="285" left="36" top="1911" width="96" height="36.75" align-x="69.75" align-y="1932" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">cpf3</ml:id>
            <ml:id xml:space="preserve">k</ml:id>
          </ml:apply>
          <ml:apply>
            <ml:div />
            <ml:apply>
              <ml:indexer />
              <ml:id xml:space="preserve">C3</ml:id>
              <ml:id xml:space="preserve">k</ml:id>
            </ml:apply>
            <ml:apply>
              <ml:indexer />
              <ml:id xml:space="preserve">C3</ml:id>
              <ml:apply>
                <ml:minus />
                <ml:id xml:space="preserve">levels</ml:id>
                <ml:real>1</ml:real>
              </ml:apply>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="38" />
    </region>
    <region region-id="368" left="156" top="1911" width="96" height="36.75" align-x="189.75" align-y="1932" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">cpf4</ml:id>
            <ml:id xml:space="preserve">k</ml:id>
          </ml:apply>
          <ml:apply>
            <ml:div />
            <ml:apply>
              <ml:indexer />
              <ml:id xml:space="preserve">C4</ml:id>
              <ml:id xml:space="preserve">k</ml:id>
            </ml:apply>
            <ml:apply>
              <ml:indexer />
              <ml:id xml:space="preserve">C4</ml:id>
              <ml:apply>
                <ml:minus />
                <ml:id xml:space="preserve">levels</ml:id>
                <ml:real>1</ml:real>
              </ml:apply>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="39" />
    </region>
    <region region-id="369" left="0" top="1962" width="6000" height="6" align-x="0" align-y="1962" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="276" left="6" top="1974.75" width="176.25" height="28.5" align-x="6" align-y="1986" 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">
          <b>
            <c val="#800000">Calculate cpf for the Hypothetical </c>
          </b>
        </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="#800000">Average Detector Element</c>
          </b>
        </p>
      </text>
    </region>
    <region region-id="290" left="6" top="2010.75" width="304.5" height="42.75" align-x="6" align-y="2022" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Now a reference cumulative histogram is needed for matching. This is calculated in the same sequence of steps as before, but for the whole image.</p>
      </text>
    </region>
    <region region-id="291" left="6" top="2058.75" width="192.75" height="14.25" align-x="6" align-y="2070" 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">Calculate the histogram of the entire image:</p>
      </text>
    </region>
    <region region-id="292" left="36" top="2082.75" width="100.5" height="15.75" align-x="52.5" align-y="2094" 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">H</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">imhist</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">R</ml:id>
              <ml:id xml:space="preserve">levels</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="40" />
    </region>
    <region region-id="293" left="6" top="2106.75" width="156.75" height="14.25" align-x="6" align-y="2118" 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">and build the cumulative histogram:</p>
      </text>
    </region>
    <region region-id="294" left="36" top="2136.75" width="45.75" height="17.25" align-x="57.75" align-y="2148" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedBIUnit">
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">C</ml:id>
            <ml:real>0</ml:real>
          </ml:apply>
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">H</ml:id>
            <ml:real>0</ml:real>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="41" />
    </region>
    <region region-id="295" left="102" top="2136.75" width="84" height="17.25" align-x="123.75" align-y="2148" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">C</ml:id>
            <ml:id xml:space="preserve">p</ml:id>
          </ml:apply>
          <ml:apply>
            <ml:plus />
            <ml:apply>
              <ml:indexer />
              <ml:id xml:space="preserve">H</ml:id>
              <ml:id xml:space="preserve">p</ml:id>
            </ml:apply>
            <ml:apply>
              <ml:indexer />
              <ml:id xml:space="preserve">C</ml:id>
              <ml:apply>
                <ml:minus />
                <ml:id xml:space="preserve">p</ml:id>
                <ml:real>1</ml:real>
              </ml:apply>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="42" />
    </region>
    <region region-id="296" left="204" top="2127" width="85.5" height="36.75" align-x="233.25" align-y="2148" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">cpf</ml:id>
            <ml:id xml:space="preserve">k</ml:id>
          </ml:apply>
          <ml:apply>
            <ml:div />
            <ml:apply>
              <ml:indexer />
              <ml:id xml:space="preserve">C</ml:id>
              <ml:id xml:space="preserve">k</ml:id>
            </ml:apply>
            <ml:apply>
              <ml:indexer />
              <ml:id xml:space="preserve">C</ml:id>
              <ml:apply>
                <ml:minus />
                <ml:id xml:space="preserve">levels</ml:id>
                <ml:real>1</ml:real>
              </ml:apply>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="43" />
    </region>
    <region region-id="299" left="6" top="2172.75" width="307.5" height="28.5" align-x="6" align-y="2184" 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">For illustrative purposes, plot the reference cpf together with the individual detector cpfs:</p>
      </text>
      <indexes>
        <phrase>cpf, detector</phrase>
        <phrase>cpf, reference</phrase>
      </indexes>
    </region>
    <region region-id="298" left="6" top="2208" width="324" height="257.25" align-x="6" align-y="2208" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <plot disable-calc="false" item-idref="44" />
      <rendering item-idref="45" />
    </region>
    <region region-id="102" left="0" top="2472" width="6000" height="6" align-x="0" align-y="2472" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="300" left="6" top="2484.75" width="258" height="14.25" align-x="6" align-y="2496" 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">
          <b>
            <c val="#800000">Construct the Histogram Matching Lookup Tables</c>
          </b>
        </p>
      </text>
    </region>
    <region region-id="383" left="6" top="2502.75" width="344.25" height="128.25" align-x="6" align-y="2514" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">In the third and final step, the lookup tables for the restoration of the individual sensor data values are constructed as follows:</p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit" />
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">For each individual detector and for all possible output values (0 to 255 in this case) compare the value of the individual cpf with the reference cpf. </p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit" />
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">If the reference cpf is either equal to or the smallest value greater than the sensor cpf, retrieve the corresponding table index (the gray level of the "average" detector).<sp count="2" />This retrieval is performed by the following program:</p>
      </text>
    </region>
    <region region-id="418" left="6" top="2646.75" width="310.5" height="212.25" align-x="36" align-y="2658" 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">LUT</ml:id>
          <ml:program>
            <ml:for>
              <ml:id xml:space="preserve">q</ml:id>
              <ml:range>
                <ml:real font="0">0</ml:real>
                <ml:apply>
                  <ml:minus />
                  <ml:id xml:space="preserve">detectors</ml:id>
                  <ml:real>1</ml:real>
                </ml:apply>
              </ml:range>
              <ml:for>
                <ml:id xml:space="preserve">k</ml:id>
                <ml:range>
                  <ml:real>0</ml:real>
                  <ml:apply>
                    <ml:minus />
                    <ml:id xml:space="preserve">levels</ml:id>
                    <ml:real>1</ml:real>
                  </ml:apply>
                </ml:range>
                <ml:localDefine>
                  <ml:apply>
                    <ml:indexer />
                    <ml:id xml:space="preserve">LUT</ml:id>
                    <ml:sequence>
                      <ml:id xml:space="preserve">q</ml:id>
                      <ml:id xml:space="preserve">k</ml:id>
                    </ml:sequence>
                  </ml:apply>
                  <ml:real>-1</ml:real>
                </ml:localDefine>
              </ml:for>
            </ml:for>
            <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">levels</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">levels</ml:id>
                    <ml:real>1</ml:real>
                  </ml:apply>
                </ml:range>
                <ml:program>
                  <ml:ifThen>
                    <ml:apply>
                      <ml:mult style="default" />
                      <ml:parens>
                        <ml:apply>
                          <ml:equal />
                          <ml:apply>
                            <ml:indexer />
                            <ml:id xml:space="preserve">LUT</ml:id>
                            <ml:sequence>
                              <ml:real>0</ml:real>
                              <ml:id xml:space="preserve">i</ml:id>
                            </ml:sequence>
                          </ml:apply>
                          <ml:real>-1</ml:real>
                        </ml:apply>
                      </ml:parens>
                      <ml:parens>
                        <ml:apply>
                          <ml:lessOrEqual />
                          <ml:apply>
                            <ml:indexer />
                            <ml:id xml:space="preserve">cpf1</ml:id>
                            <ml:id xml:space="preserve">i</ml:id>
                          </ml:apply>
                          <ml:apply>
                            <ml:indexer />
                            <ml:id xml:space="preserve">cpf</ml:id>
                            <ml:id xml:space="preserve">j</ml:id>
                          </ml:apply>
                        </ml:apply>
                      </ml:parens>
                    </ml:apply>
                    <ml:localDefine>
                      <ml:apply>
                        <ml:indexer />
                        <ml:id xml:space="preserve">LUT</ml:id>
                        <ml:sequence>
                          <ml:real>0</ml:real>
                          <ml:id xml:space="preserve">i</ml:id>
                        </ml:sequence>
                      </ml:apply>
                      <ml:id xml:space="preserve">j</ml:id>
                    </ml:localDefine>
                  </ml:ifThen>
                  <ml:ifThen>
                    <ml:apply>
                      <ml:mult style="default" />
                      <ml:parens>
                        <ml:apply>
                          <ml:equal />
                          <ml:apply>
                            <ml:indexer />
                            <ml:id xml:space="preserve">LUT</ml:id>
                            <ml:sequence>
                              <ml:real>1</ml:real>
                              <ml:id xml:space="preserve">i</ml:id>
                            </ml:sequence>
                          </ml:apply>
                          <ml:real>-1</ml:real>
                        </ml:apply>
                      </ml:parens>
                      <ml:parens>
                        <ml:apply>
                          <ml:lessOrEqual />
                          <ml:apply>
                            <ml:indexer />
                            <ml:id xml:space="preserve">cpf2</ml:id>
                            <ml:id xml:space="preserve">i</ml:id>
                          </ml:apply>
                          <ml:apply>
                            <ml:indexer />
                            <ml:id xml:space="preserve">cpf</ml:id>
                            <ml:id xml:space="preserve">j</ml:id>
                          </ml:apply>
                        </ml:apply>
                      </ml:parens>
                    </ml:apply>
                    <ml:localDefine>
                      <ml:apply>
                        <ml:indexer />
                        <ml:id xml:space="preserve">LUT</ml:id>
                        <ml:sequence>
                          <ml:real>1</ml:real>
                          <ml:id xml:space="preserve">i</ml:id>
                        </ml:sequence>
                      </ml:apply>
                      <ml:id xml:space="preserve">j</ml:id>
                    </ml:localDefine>
                  </ml:ifThen>
                  <ml:ifThen>
                    <ml:apply>
                      <ml:mult style="default" />
                      <ml:parens>
                        <ml:apply>
                          <ml:equal />
                          <ml:apply>
                            <ml:indexer />
                            <ml:id xml:space="preserve">LUT</ml:id>
                            <ml:sequence>
                              <ml:real>2</ml:real>
                              <ml:id xml:space="preserve">i</ml:id>
                            </ml:sequence>
                          </ml:apply>
                          <ml:real>-1</ml:real>
                        </ml:apply>
                      </ml:parens>
                      <ml:parens>
                        <ml:apply>
                          <ml:lessOrEqual />
                          <ml:apply>
                            <ml:indexer />
                            <ml:id xml:space="preserve">cpf3</ml:id>
                            <ml:id xml:space="preserve">i</ml:id>
                          </ml:apply>
                          <ml:apply>
                            <ml:indexer />
                            <ml:id xml:space="preserve">cpf</ml:id>
                            <ml:id xml:space="preserve">j</ml:id>
                          </ml:apply>
                        </ml:apply>
                      </ml:parens>
                    </ml:apply>
                    <ml:localDefine>
                      <ml:apply>
                        <ml:indexer />
                        <ml:id xml:space="preserve">LUT</ml:id>
                        <ml:sequence>
                          <ml:real>2</ml:real>
                          <ml:id xml:space="preserve">i</ml:id>
                        </ml:sequence>
                      </ml:apply>
                      <ml:id xml:space="preserve">j</ml:id>
                    </ml:localDefine>
                  </ml:ifThen>
                  <ml:ifThen>
                    <ml:apply>
                      <ml:mult style="default" />
                      <ml:parens>
                        <ml:apply>
                          <ml:equal />
                          <ml:apply>
                            <ml:indexer />
                            <ml:id xml:space="preserve">LUT</ml:id>
                            <ml:sequence>
                              <ml:real>3</ml:real>
                              <ml:id xml:space="preserve">i</ml:id>
                            </ml:sequence>
                          </ml:apply>
                          <ml:real>-1</ml:real>
                        </ml:apply>
                      </ml:parens>
                      <ml:parens>
                        <ml:apply>
                          <ml:lessOrEqual />
                          <ml:apply>
                            <ml:indexer />
                            <ml:id xml:space="preserve">cpf4</ml:id>
                            <ml:id xml:space="preserve">i</ml:id>
                          </ml:apply>
                          <ml:apply>
                            <ml:indexer />
                            <ml:id xml:space="preserve">cpf</ml:id>
                            <ml:id xml:space="preserve">j</ml:id>
                          </ml:apply>
                        </ml:apply>
                      </ml:parens>
                    </ml:apply>
                    <ml:localDefine>
                      <ml:apply>
                        <ml:indexer />
                        <ml:id xml:space="preserve">LUT</ml:id>
                        <ml:sequence>
                          <ml:real>3</ml:real>
                          <ml:id xml:space="preserve">i</ml:id>
                        </ml:sequence>
                      </ml:apply>
                      <ml:id xml:space="preserve">j</ml:id>
                    </ml:localDefine>
                  </ml:ifThen>
                </ml:program>
              </ml:for>
            </ml:for>
            <ml:id xml:space="preserve">LUT</ml:id>
          </ml:program>
        </ml:define>
      </math>
      <rendering item-idref="46" />
    </region>
    <region region-id="328" left="6" top="2880.75" width="328.5" height="114" align-x="6" align-y="2892" 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">Notice what happens as each <b>if</b> instruction is executed; each time the average cpf is less than the individual cpf, a <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">-1</inlineAttr></f> is returned to the LUT. If, however, the average cpf is at some level greater than the individual cpf, the level, <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">j</inlineAttr></f>, is returned and remains fixed in the table. </p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit" />
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
          <b>
            <c val="#008080">Note:</c>
          </b> The construction of the lookup tables can probably be more efficiently coded as a C routine, which is easily linked into Mathcad Professional. See your <i>Mathcad User's Guide</i><i>.</i></p>
      </text>
    </region>
    <region region-id="329" left="0" top="3006" width="6000" height="6" align-x="0" align-y="3006" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="330" left="6" top="3018.75" width="164.25" height="14.25" align-x="6" align-y="3030" 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">
          <b>
            <c val="#800000">Destriping the Image</c>
          </b>
        </p>
      </text>
    </region>
    <region region-id="349" left="6" top="3036.75" width="320.25" height="28.5" align-x="6" align-y="3048" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">Now destripe the image by replacing gray levels with corresponding levels in the LUT, as follows:</p>
      </text>
    </region>
    <region region-id="339" left="36" top="3072.75" width="123.75" height="15.75" align-x="46.5" align-y="3084" 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">i</ml:id>
          <ml:range>
            <ml:sequence>
              <ml:real>0</ml:real>
              <ml:id xml:space="preserve">detectors</ml:id>
            </ml:sequence>
            <ml:id xml:space="preserve">last_row</ml:id>
          </ml:range>
        </ml:define>
      </math>
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        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">[1] Horn, B. K. P., "Destriping LANDSAT MSS Images by Histogram Modification," <i>Computer Graphics and Image Processing</i>, 10, pp. 69-83, 1979.</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">[2] Wegener, M., "Destriping Multiple Sensor Imagery by Improved Histogram Matching," <i>International Journal of Remote Sensing</i>, Vol. 11, No. 5, pp. 859-75, 1990.</p>
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