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          <b>
            <c val="#008080">IMAGE MANIPULATION</c>
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            <b>
              <c val="#800080">One- and Two-Dimensional Histograms</c>
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        <phrase>histograms, 1- and 2-D</phrase>
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          <b>
            <c val="#800000">Function Definitions and Syntax</c>
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              <c val="#800000">imhist(M, n)</c>
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              <c val="#800000">imhist2(M, n)</c>
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              <c val="#800000">hist2d(M, N, n)</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">These functions show you the intensity levels in an image, divided into<f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"> n</inlineAttr></f> bins. Histograms shows concentrations of gray levels or colors in an image. They can be used to determine the type of image, or to find an appropriate threshold for filtering or equalization. Two-dimensional histograms show the degree of similarity (in intensity or texture) between two images. </p>
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          <b>
            <c val="#800000">Arguments</c>
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              <b>M</b>
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          </f>, the image matrix, or M and N, two image matrices of the same size, and</p>
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              <c val="#008080">• </c>
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              <inlineAttr line-through="false">n</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">The one-dimensional histogram functions return a vector, <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">n</inlineAttr></f> elements long, containing the number of intensity levels in each bin. For <f family="Times New Roman" charset="0"><b><inlineAttr font-style="normal" line-through="false">imhist</inlineAttr></b></f><inlineAttr font-style="normal">,</inlineAttr> the n bins are spaced between intensity levels of 0 and 255, and for <f family="Times New Roman" charset="0"><b><inlineAttr font-style="normal" line-through="false">imhist2</inlineAttr></b></f> they are spaced in the actual range of intensities in the matrix. The two-dimensional histogram returns an <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">n</inlineAttr></f><f family="Symbol" charset="2">´</f><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">n</inlineAttr></f><b><sp /></b>matrix, which contains an entry<b><sp /></b><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">i,j,</inlineAttr></f> the number of pixels in corresponding locations which are in bin <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">i in image M and bin j in image N.</inlineAttr></f></p>
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          <b>
            <c val="#800000">Function Details</c>
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          <b>
            <inlineAttr font-style="normal">
              <c val="#800000">imhist(M, n)</c>
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        <phrase>imhist 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">This function generates a histogram for the elements o<inlineAttr font-weight="normal" line-through="false">f M by dividing the intensity range </inlineAttr><inlineAttr font-weight="normal" line-through="false">into n</inlineAttr> bins. A histogram is a vector of <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">n elements, or bins, where each bin contains the number of pixels which have intensity levels in that bin. So, for example, if n = 256, t</inlineAttr></f>here will be one bin for each possible intensity level in an 8-bit grayscale image. If there are fewer than 256 bins, then the intensity levels are divided into bins in the following way:</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">If there are more than 256 bins, the bin sizes are smaller than 1.</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 use a small matrix as an example to see the relationship between values and the appearance of the histogram.</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 only bin which should have more than one count is 150. Try setting <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">n</inlineAttr></f> to be a small number, such as 5, to see the difference.</p>
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      <plot disable-calc="false" item-idref="11" />
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    </region>
    <region region-id="551" left="6" top="912.75" width="187.5" height="14.25" align-x="6" align-y="924" 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">Remember that bin labeling begins with 0.</p>
      </text>
    </region>
    <region region-id="370" left="0" top="936" width="6000" height="6" align-x="0" align-y="936" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="371" left="6" top="948.75" width="324" height="42.75" align-x="6" align-y="960" 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 an image with a lot of white space, like the test pattern used in several examples of other functions, there is a large spike at the top of the histogram.</p>
      </text>
    </region>
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      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">M</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">READ_IMAGE</ml:id>
            <ml:str xml:space="preserve">pattern.bmp</ml:str>
          </ml:apply>
        </ml:define>
      </math>
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        <outputs />
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      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <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">M</ml:id>
              <ml:real>256</ml:real>
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        </ml:define>
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      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">k</ml:id>
          <ml:range>
            <ml:real>0</ml:real>
            <ml:apply>
              <ml:minus />
              <ml:apply>
                <ml:id xml:space="preserve">rows</ml:id>
                <ml:id xml:space="preserve">H</ml:id>
              </ml:apply>
              <ml:real>1</ml:real>
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      <plot disable-calc="false" item-idref="18" />
      <rendering item-idref="19" />
    </region>
    <region region-id="378" left="6" top="1254.75" width="318.75" height="28.5" align-x="6" align-y="1266" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The white background in this image shows up as the large spike on the right, along with several other distinct gray tones.</p>
      </text>
    </region>
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      <pageBreak />
    </region>
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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">The function <b>imhist2</b> may be used to bin data with values outside the range from 0 to 255. As an example, consider the matrix:</p>
      </text>
      <indexes>
        <phrase>imhist2 function</phrase>
      </indexes>
    </region>
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        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">M</ml:id>
          <ml:matrix rows="4" cols="4">
            <ml:real>15</ml:real>
            <ml:real>21.7</ml:real>
            <ml:real>195</ml:real>
            <ml:real>300</ml:real>
            <ml:real>0.5</ml:real>
            <ml:real>9</ml:real>
            <ml:real>0.3</ml:real>
            <ml:real>77</ml:real>
            <ml:real>451</ml:real>
            <ml:real>25</ml:real>
            <ml:real>28</ml:real>
            <ml:real>200</ml:real>
            <ml:real>88</ml:real>
            <ml:real>671.35</ml:real>
            <ml:real>50</ml:real>
            <ml:real>5</ml:real>
          </ml:matrix>
        </ml:define>
      </math>
      <rendering item-idref="20" />
    </region>
    <region region-id="508" left="6" top="1422.75" width="210.75" height="14.25" 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="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">We can compute an n-bin histogram by calling:</p>
      </text>
    </region>
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      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">n</ml:id>
          <ml:real>5</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="21" />
    </region>
    <region region-id="562" left="96" top="1446.75" width="90" height="15.75" align-x="112.5" align-y="1458" 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">H</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">imhist2</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">M</ml:id>
              <ml:id xml:space="preserve">n</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
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    <region region-id="494" left="6" top="1470.75" width="179.25" height="14.25" align-x="6" align-y="1482" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The endpoints of the <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">n</inlineAttr></f> bins are given by</p>
      </text>
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      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">i</ml:id>
          <ml:range>
            <ml:real>0</ml:real>
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      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">a</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">min</ml:id>
            <ml:id xml:space="preserve">M</ml:id>
          </ml:apply>
        </ml:define>
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      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">b</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">max</ml:id>
            <ml:id xml:space="preserve">M</ml:id>
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        </ml:define>
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      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">bins</ml:id>
            <ml:id xml:space="preserve">i</ml:id>
          </ml:apply>
          <ml:apply>
            <ml:plus />
            <ml:id xml:space="preserve">a</ml:id>
            <ml:apply>
              <ml:mult style="default" />
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                <ml:apply>
                  <ml:div />
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                    <ml:minus />
                    <ml:id xml:space="preserve">b</ml:id>
                    <ml:id xml:space="preserve">a</ml:id>
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                  <ml:id xml:space="preserve">n</ml:id>
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      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:id xml:space="preserve">bins</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="6" cols="1">
              <ml:real>0.3</ml:real>
              <ml:real>134.51000000000002</ml:real>
              <ml:real>268.72</ml:real>
              <ml:real>402.93</ml:real>
              <ml:real>537.14</ml:real>
              <ml:real>671.35</ml:real>
            </ml:matrix>
          </result>
        </ml:eval>
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      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:id xml:space="preserve">H</ml:id>
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            <ml:matrix rows="5" cols="1">
              <ml:real>11</ml:real>
              <ml:real>2</ml:real>
              <ml:real>1</ml:real>
              <ml:real>1</ml:real>
              <ml:real>1</ml:real>
            </ml:matrix>
          </result>
        </ml:eval>
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      <pageBreak />
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    <region region-id="568" left="6" top="1716.75" width="83.25" height="14.25" align-x="6" align-y="1728" 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">Function Details</c>
          </b>
        </p>
      </text>
    </region>
    <region region-id="566" left="114" top="1716.75" width="80.25" height="14.25" align-x="114" align-y="1728" 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>
            <inlineAttr font-style="normal">
              <c val="#800000">hist2d(M, N, n)</c>
            </inlineAttr>
          </b>
        </p>
      </text>
      <indexes>
        <phrase>hist2d  function</phrase>
      </indexes>
    </region>
    <region region-id="570" left="6" top="1734.75" width="333" height="128.25" align-x="6" align-y="1746" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <text use-page-width="false" push-down="false" lock-width="true">
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">This function generates a two-dimensional histogram of two images<b><sp /></b><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">M</inlineAttr></f><b><sp /></b>and<b><sp /></b><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">N</inlineAttr></f>, creating<b><sp /></b><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">n bins in each dimension. M and N mu</inlineAttr></f>st be the same size. If the two images have similar numbers of pixels at similar intensities, the output matrix from<b><sp /></b><b>hist2d</b> has a strong diagonal.</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">A two-dimensional histogram starts by calculating the one-dimensional histogram for each input image. Like the <inlineAttr line-through="false"><f family="Times New Roman" charset="0"><b>imhist</b></f></inlineAttr> function, <b><inlineAttr line-through="false"><f family="Times New Roman" charset="0">hist2d</f></inlineAttr></b> divides the range 0 to 255 into n bins, so we can use the <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">vectorize</inlineAttr></b></f> operator with the bin_number function we defined above to calculate the bin numbers:</p>
      </text>
      <indexes>
        <phrase>vectorize operator</phrase>
      </indexes>
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      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">n</ml:id>
          <ml:real>5</ml:real>
        </ml:define>
      </math>
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      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:id xml:space="preserve">P</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="3" cols="3">
              <ml:real>30</ml:real>
              <ml:real>40</ml:real>
              <ml:real>120</ml:real>
              <ml:real>150</ml:real>
              <ml:real>150</ml:real>
              <ml:real>150</ml:real>
              <ml:real>200</ml:real>
              <ml:real>0</ml:real>
              <ml:real>240</ml:real>
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          </result>
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          <ml:id xml:space="preserve">Q</ml:id>
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            <ml:real>150</ml:real>
            <ml:real>150</ml:real>
            <ml:real>20</ml:real>
            <ml:real>0</ml:real>
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            <ml:real>20</ml:real>
            <ml:real>0</ml:real>
            <ml:real>255</ml:real>
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        </ml:define>
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          <ml:id xml:space="preserve">Pbins</ml:id>
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              <ml:id xml:space="preserve">bin_label</ml:id>
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                <ml:id xml:space="preserve">P</ml:id>
                <ml:id xml:space="preserve">n</ml:id>
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              <ml:id xml:space="preserve">bin_label</ml:id>
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              <ml:real>0</ml:real>
              <ml:real>2</ml:real>
              <ml:real>2</ml:real>
              <ml:real>2</ml:real>
              <ml:real>2</ml:real>
              <ml:real>3</ml:real>
              <ml:real>0</ml:real>
              <ml:real>4</ml:real>
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      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:id xml:space="preserve">Qbins</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="3" cols="3">
              <ml:real>2</ml:real>
              <ml:real>2</ml:real>
              <ml:real>2</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>4</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>4</ml:real>
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          </result>
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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 two-dimensional histogram is an n <f family="Symbol" charset="2">´ </f>n matrix, with a count at column c and row r whenever there is an element of Pbins with bin number c, and the corresponding element of Qbins has count r. For instance, the first elements of Pbins and Qbins are 0 and 2, so they generate one count at column 0 and row 2. We can calculate this by first generating a matrix of zeros and then adding the counts:</p>
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              <ml:id xml:space="preserve">k</ml:id>
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          <ml:apply>
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            <ml:sequence>
              <ml:apply>
                <ml:indexer />
                <ml:id xml:space="preserve">Qbins</ml:id>
                <ml:sequence>
                  <ml:id xml:space="preserve">i</ml:id>
                  <ml:id xml:space="preserve">j</ml:id>
                </ml:sequence>
              </ml:apply>
              <ml:apply>
                <ml:indexer />
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              <ml:id xml:space="preserve">hist</ml:id>
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                <ml:apply>
                  <ml:indexer />
                  <ml:id xml:space="preserve">Qbins</ml:id>
                  <ml:sequence>
                    <ml:id xml:space="preserve">i</ml:id>
                    <ml:id xml:space="preserve">j</ml:id>
                  </ml:sequence>
                </ml:apply>
                <ml:apply>
                  <ml:indexer />
                  <ml:id xml:space="preserve">Pbins</ml:id>
                  <ml:sequence>
                    <ml:id xml:space="preserve">i</ml:id>
                    <ml:id xml:space="preserve">j</ml:id>
                  </ml:sequence>
                </ml:apply>
              </ml:sequence>
            </ml:apply>
            <ml:real>1</ml:real>
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        </ml:define>
      </math>
      <rendering item-idref="41" />
    </region>
    <region region-id="630" left="0" top="2202" width="6000" height="6" align-x="0" align-y="2202" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
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    <region region-id="581" left="6" top="2214.75" width="327" height="14.25" align-x="6" align-y="2226" 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">Compare with the output of the built-in function:</p>
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      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">H</ml:id>
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            <ml:id xml:space="preserve">hist2d</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">P</ml:id>
              <ml:id xml:space="preserve">Q</ml:id>
              <ml:id xml:space="preserve">n</ml:id>
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      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:id xml:space="preserve">hist</ml:id>
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              <ml:real>0</ml:real>
              <ml:real>2</ml:real>
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              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
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              <ml:real>0</ml:real>
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              <ml:real>1</ml:real>
              <ml:real>1</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
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          </result>
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              <ml:real>0</ml:real>
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              <ml:real>2</ml:real>
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              <ml:real>0</ml:real>
              <ml:real>0</ml:real>
              <ml:real>1</ml:real>
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    <region region-id="461" left="6" top="2382.75" width="295.5" height="14.25" align-x="6" align-y="2394" 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">For a more concrete example, consider the following color image:</p>
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          <ml:id xml:space="preserve">test</ml:id>
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            <ml:id xml:space="preserve">READRGB</ml:id>
            <ml:str xml:space="preserve">smallwar.gif</ml:str>
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    <region region-id="464" left="6" top="2592.75" width="290.25" height="28.5" align-x="6" align-y="2604" 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">Extracting the green and blue components separately allows a comparison of the two color components.</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">Here is the two-dimensional histogram shown as a surface plot and as an image of intensities on a log scale. The strong diagonal in this two-dimensional histogram means that most pixels in corresponding locations had similar intensities. This is not unreasonable in two color components from the same photograph. The diagonal would be less prominent in two images with very different textures.</p>
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                  <ml:real>10</ml:real>
                  <ml:real>-5</ml:real>
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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 zooming in on the image Hlog a couple of times for a better view of the</p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">histogram.</p>
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