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              <c val="#800080">Image Distortions</c>
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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">Commercial PC imaging programs often provide "effects" that include geometric distortions of the image. To see how these might be generated by some simple mathematics, we take an image of a dog and subject it to some transformations. Here's the original undistorted image:</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 the first example, the image is wrapped around the surface of a cylinder and then viewed looking perpendicular to the axis of the cylinder; this fattens up the central portion of the image and reduces the left and right edges. The sine function would map the horizontal coordinate of a point on a vertical half-cylinder onto the axial plane, so, to make the cylindrical image, the inverse sine can be used to find the pre-image on the cylinder of each column of pixels on the axial plane. The column coordinates of these pre-images are not usually integers, so the columns are averaged with the two nearest integer coordinates. It's been assumed that the image has an even number of columns.</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 cylinder image <i>seems</i> wider, but if you line up images C and A vertically you'll see that they have the same number of pixels both vertically and horizontally.</p>
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        <phrase>sphere - image distortion</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">Next we try wrapping the image on a hemisphere. This transformation is more complicated. To keep things simple we extract a square subimage. </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">Now we have to range not just over columns but over every pixel of our distorted image, so the calculation takes a while. Here's one way of carrying out the arithmetic. </p>
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      <math optimize="false" disable-calc="false">
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          <ml:id xml:space="preserve">D</ml:id>
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            <ml:id xml:space="preserve">cols</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">For convenience we store the image coordinates in the matrix <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">I</inlineAttr></f> as complex numbers, centering the image at the origin. Then we can carry out transformations on the coordinates just by applying complex functions to <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">I.</inlineAttr></f></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 mapping we have chosen is the inverse of </p>
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    <region region-id="407" left="6" top="1464.75" width="309.75" height="28.5" align-x="6" align-y="1476" 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">To form<f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false"> Q, we scale and map the coordinates by f and t</inlineAttr></f>hen translate the origin back to the lower left corner.</p>
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                  <ml:real>2</ml:real>
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              <ml:parens>
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    <region region-id="409" left="0" top="1548" width="6000" height="6" align-x="0" align-y="1548" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
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    <region region-id="410" left="6" top="1560.75" width="330" height="85.5" align-x="6" align-y="1572" 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 we build the distorted image as follows: We treat the mapped coordinates in <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">Q</inlineAttr></f> as coordinates in the original image. They won't usually be integers, so we round them to the nearest integer. If these coordinates fall outside the original image, we set the new pixel to black, otherwise we set it to the pixel value in the original image. The function <f family="Times New Roman" charset="0"><b><inlineAttr line-through="false">rd</inlineAttr></b></f> rounds and <f family="Times New Roman" charset="0"><inlineAttr font-weight="normal" line-through="false">T</inlineAttr></f> checks whether the mapped coordinates are in the original image. </p>
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        <ml:define>
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          <ml:apply>
            <ml:id xml:space="preserve">floor</ml:id>
            <ml:apply>
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              <ml:real>.5</ml:real>
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                      <ml:apply>
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                        <ml:sequence>
                          <ml:id xml:space="preserve">m</ml:id>
                          <ml:id xml:space="preserve">n</ml:id>
                        </ml:sequence>
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                  <ml:apply>
                    <ml:id xml:space="preserve">rd</ml:id>
                    <ml:apply>
                      <ml:id xml:space="preserve">Im</ml:id>
                      <ml:apply>
                        <ml:indexer />
                        <ml:id xml:space="preserve">Q</ml:id>
                        <ml:sequence>
                          <ml:id xml:space="preserve">m</ml:id>
                          <ml:id xml:space="preserve">n</ml:id>
                        </ml:sequence>
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                    </ml:apply>
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              <ml:real>0</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">Here is the result:</p>
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        <outputs />
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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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">
          <b>
            <c val="#800000">Swirl</c>
          </b>
        </p>
      </text>
      <indexes>
        <phrase>swirl - image distortion</phrase>
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    </region>
    <region region-id="443" left="6" top="1974.75" width="318" height="57" 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="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Another popular effect is a rotation that spirals out from the center of the image. We can generate this effect by using a mapping that rotates the complex coordinates by an amount proportional to the square of their distance from the origin:</p>
      </text>
    </region>
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      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDFunction">
          <ml:function>
            <ml:id xml:space="preserve">f</ml:id>
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            </ml:boundVars>
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            <ml:apply>
              <ml:id xml:space="preserve">exp</ml:id>
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                <ml:apply>
                  <ml:mult style="default" />
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                    <ml:apply>
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                  <ml:real>2</ml:real>
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                <ml:apply>
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                  <ml:real>2</ml:real>
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                <ml:apply>
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                <ml:id xml:space="preserve">rd</ml:id>
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                  <ml:apply>
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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">Finally, we illustrate another radial transform, one that imitates ripples on the surface of a pond.</p>
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