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            <c val="#008080">IMAGE ENHANCEMENT</c>
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              <c val="#800080">Mapping Image Intensities</c>
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        <phrase>mapping image intensities</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 document illustrates some techniques for controlling the intensity distribution in a grayscale image. It's possible to distribute intensities according to common formulas, such as with the <b>equalize</b> function, which spreads intensities uniformly. For other applications, it may also be desirable to create a new function to remap image intensities to produce a given-intensity histogram. Several classes of intensity maps are demonstrated, along with the most efficient ways to apply them using the <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">Image Processing Pack</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">We start with the famous "camera" image, shot on the campus of MIT in Cambridge, Mass.</p>
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            <c val="#800000">Equalizing an Image with a Narrow Range</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 first experiment is to compress all intensity values in an image linearly by a factor of <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">N</inlineAttr></f>, centered on 128 and using the <b>floor</b> function to produce an image with just a few midrange gray 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">Now we display the image histograms, dividing the compressed histogram by 10 to fit the same vertical scale.</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 distribution in the original image changes rapidly at low values (the large area of black) and midrange (the gray sky). In the compressed-range image, all values are concentrated in a narrow band in the middle. There are only</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 we equalize the all-gray version of the image.</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 equalized image <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">J</inlineAttr></f> is still quantized into only 8 levels, but the range has been expanded and the image is quite legible. We can plot a cumulative histogram as shown in the section <b><u><link href="./Equalize.xmcd" title="Links to section" popup="false">Equalization</link></u></b>:</p>
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          <b>
            <c val="#800000">Using Power Functions to Remap Image Intensities</c>
          </b>
        </p>
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        <phrase>power functions</phrase>
        <phrase>remap image intensities</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">As a second experiment, we rescale the intensities so that almost all the values are either very low or very high. Some image-processing software allows you to draw the desired intensity mapping with the mouse. In <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">Mathcad</inlineAttr></f> we can draw our maps with simple functions. The advantage of this is that our function can include one or more parameters that will give us a whole family of maps for the price of one. </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 map we want looks something like this:</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 get a map with a variable parameter, a couple of power functions can be glued together; the exponent is the variable parameter. For convenience, the function is defined so that it maps [0, 1] into [0, 1], and then scales appropriately when applied to an image.</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">Experiment with this map<b><sp /></b><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">f by changing n. Using n = 1 g</inlineAttr></f>ives the identity map (the image is unchanged). Higher powers push the values further toward the high and low ends. What happens when n is a fraction like 1/10?</p>
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    <region region-id="474" left="18" top="2502" width="214.5" height="133.5" align-x="18" align-y="2502" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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    <region region-id="409" left="6" top="2646.75" width="319.5" height="28.5" align-x="6" align-y="2658" 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">Applying the map to the camera image, scaling down to [0, 1] and then scaling back up yields:</p>
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    <region region-id="484" left="6" top="3000.75" width="316.5" height="71.25" align-x="6" align-y="3012" 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">Experiment by changing <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">n; try large integers as well as some fractions. Notice that a value like n = 0.4</inlineAttr></f> brings out detail in the coat that is invisible in the original image. Very small values like <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">.02 produce an image similar to our narrow-band image C. Large values like 20</inlineAttr></f><b><sp /></b>yield images that resemble a binarized picture. </p>
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        <outputs />
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      <pageBreak />
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    <region region-id="417" left="6" top="3336.75" width="242.25" height="14.25" align-x="6" align-y="3348" 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">
          <b>
            <c val="#800000">Piecewise Linear Remapping</c>
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        <phrase>piecewise linear remapping</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">Another useful class of maps expands the grayscale in one particular range. The prototype might look something like this:</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 the values around<b><sp /></b><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">t</inlineAttr></f> are stretched and the others are correspondingly compressed. A restriction is imposed on the stretched section: it must be positioned so that the compressed sections have the same slope; this makes the arithmetic a bit more complicated. Here it's easier to work directly with the gray values in the range [0, 255]. We use a range of 10 on either side of<b><sp /></b><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">t and expand this to 40 on either side. If p is the height of the graph at x = t</inlineAttr></f>, the condition of equal slopes is </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 <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">Mathcad's symbolic processor to solve for p, </inlineAttr></f>which after simplifying yields</p>
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        <ml:define>
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            <ml:apply>
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              <ml:apply>
                <ml:div />
                <ml:real>35</ml:real>
                <ml:real>47</ml:real>
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              <ml:id xml:space="preserve">t</ml:id>
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            <ml:apply>
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              <ml:real>1530</ml:real>
              <ml:real>47</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">The slope of the flatter segments is</p>
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      <math optimize="false" disable-calc="false">
        <ml:define>
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              <ml:real>255</ml:real>
              <ml:real>81</ml:real>
            </ml:apply>
            <ml:apply>
              <ml:minus />
              <ml:real>255</ml:real>
              <ml:real>21</ml:real>
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    <region region-id="431" left="0" top="3852" width="6000" height="6" align-x="0" align-y="3852" 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">The mapping function, <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">g</inlineAttr></f>, is:</p>
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      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedBIUnit">
          <ml:function>
            <ml:id xml:space="preserve">g</ml:id>
            <ml:boundVars>
              <ml:id xml:space="preserve">x</ml:id>
              <ml:id xml:space="preserve">t</ml:id>
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            <ml:id xml:space="preserve">if</ml:id>
            <ml:sequence>
              <ml:apply>
                <ml:lessOrEqual />
                <ml:apply>
                  <ml:absval />
                  <ml:apply>
                    <ml:minus />
                    <ml:id xml:space="preserve">x</ml:id>
                    <ml:id xml:space="preserve">t</ml:id>
                  </ml:apply>
                </ml:apply>
                <ml:real>10</ml:real>
              </ml:apply>
              <ml:apply>
                <ml:plus />
                <ml:apply>
                  <ml:id xml:space="preserve">p</ml:id>
                  <ml:id xml:space="preserve">t</ml:id>
                </ml:apply>
                <ml:apply>
                  <ml:mult style="default" />
                  <ml:real>4</ml:real>
                  <ml:parens>
                    <ml:apply>
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                      <ml:id xml:space="preserve">x</ml:id>
                      <ml:id xml:space="preserve">t</ml:id>
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                  </ml:parens>
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              </ml:apply>
              <ml:apply>
                <ml:id xml:space="preserve">if</ml:id>
                <ml:sequence>
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                    <ml:id xml:space="preserve">x</ml:id>
                    <ml:apply>
                      <ml:minus />
                      <ml:id xml:space="preserve">t</ml:id>
                      <ml:real>10</ml:real>
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                  <ml:apply>
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                    <ml:id xml:space="preserve">slp</ml:id>
                    <ml:id xml:space="preserve">x</ml:id>
                  </ml:apply>
                  <ml:apply>
                    <ml:minus />
                    <ml:real>255</ml:real>
                    <ml:apply>
                      <ml:mult style="default" />
                      <ml:id xml:space="preserve">slp</ml:id>
                      <ml:parens>
                        <ml:apply>
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                          <ml:id xml:space="preserve">x</ml:id>
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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">Try changing<b><sp /></b><f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">t </inlineAttr></f>to numbers between 10 and 245 and see how the mapping function adjusts.</p>
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      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">t</ml:id>
          <ml:real>110</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="51" />
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      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">x</ml:id>
          <ml:range>
            <ml:real>0</ml:real>
            <ml:real>255</ml:real>
          </ml:range>
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      </math>
      <rendering item-idref="52" />
    </region>
    <region region-id="477" left="18" top="3972" width="254.25" height="133.5" align-x="18" align-y="3972" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <plot disable-calc="false" item-idref="53" />
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    </region>
    <region region-id="442" left="6" top="4110.75" width="319.5" height="57" align-x="6" align-y="4122" 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 experiment with expanding the scale in different ranges of the camera image by changing the definition of <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">t below. For example, notice that t = 20 brings out the detail in the coat and glove, while t = 182 stretches the val</inlineAttr></f>ues in the taller building in the background.</p>
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      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">t</ml:id>
          <ml:real>20</ml:real>
        </ml:define>
      </math>
      <rendering item-idref="55" />
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      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:apply>
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        </ml:define>
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      <rendering item-idref="56" />
    </region>
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      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">expand</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">levelmap</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">Q</ml:id>
              <ml:id xml:space="preserve">vec</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
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      <component hide-arguments="false" clsid-buddy="8442da2d-08d7-45ad-a4a6-f28bee6ae06a" item-idref="58" disable-calc="false">
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        <outputs />
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