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              <c val="#800080">Bilinear Warping</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">This function performs image warping, a geometric transformation akin to projecting an image on a sheet of rubber and then stretching the sheet. The "tiepoints" are four pairs of points in the input image and corresponding positions in the output image. Image warping is commonly used for image registration and image distortion correction.</p>
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          </f>, a grayscale image matrix</p>
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              <c val="#008080">•</c>
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          </f>, the 8 x 2 matrix containing the four pairs of tiepoints.</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 function returns a matrix containing the warped 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">Image warping is a geometric transformation that can be viewed as a process of projecting an image on a sheet of rubber and then stretching the sheet. The "tiepoints," T, are four points in the input image and the corresponding positions of the four points for the output image; they are used to compute coefficients for a bilinear warping equation that transforms the input to the output. Since the output positions for each input pixel do not necessarily correspond to integer coordinates, the output pixel values are calculated as weighted sums of the four nearest corresponding input pixels, with higher weight given to nearer pixels.</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">Bilinear warping using the following equations to transform a pixel at (x,y) to its new location (u,v):</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">The coefficients c are estimated from the four pairs of tie points, which are given as the second input of the function (stored with the row and column input coordinates in the first four rows, and the row and column output coordinates in the last four rows). </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">Since u and v calculated from the equations above are usually not integers, the actual output pixels are weighted combinations of the four nearest destination pixels. Boundaries of the image are treated as extending out with zero grayscale level.</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">Create a simple sequential matrix:</p>
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              <ml:real>21</ml:real>
              <ml:real>2</ml:real>
              <ml:real>7</ml:real>
              <ml:real>12</ml:real>
              <ml:real>17</ml:real>
              <ml:real>22</ml:real>
              <ml:real>3</ml:real>
              <ml:real>8</ml:real>
              <ml:real>13</ml:real>
              <ml:real>18</ml:real>
              <ml:real>23</ml:real>
              <ml:real>4</ml:real>
              <ml:real>9</ml:real>
              <ml:real>14</ml:real>
              <ml:real>19</ml:real>
              <ml:real>24</ml:real>
              <ml:real>5</ml:real>
              <ml:real>10</ml:real>
              <ml:real>15</ml:real>
              <ml:real>20</ml:real>
              <ml:real>25</ml:real>
              <ml:real>6</ml:real>
              <ml:real>11</ml:real>
              <ml:real>16</ml:real>
              <ml:real>21</ml:real>
              <ml:real>26</ml:real>
            </ml:matrix>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="10" />
    </region>
    <region region-id="2224" left="6" top="864.75" width="327" height="14.25" align-x="6" align-y="876" 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="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">As a first warping example, make the input and output tie points equal:</p>
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    </region>
    <region region-id="2130" left="42" top="893.25" width="62.25" height="162.75" align-x="57" align-y="978" 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">T</ml:id>
          <ml:matrix rows="8" cols="2">
            <ml:real>1</ml:real>
            <ml:real>1</ml:real>
            <ml:real>2</ml:real>
            <ml:real>2</ml:real>
            <ml:real>1</ml:real>
            <ml:real>1</ml:real>
            <ml:real>2</ml:real>
            <ml:real>2</ml:real>
            <ml:real>1</ml:real>
            <ml:real>2</ml:real>
            <ml:real>1</ml:real>
            <ml:real>2</ml:real>
            <ml:real>1</ml:real>
            <ml:real>2</ml:real>
            <ml:real>1</ml:real>
            <ml:real>2</ml:real>
          </ml:matrix>
        </ml:define>
      </math>
      <rendering item-idref="11" />
    </region>
    <region region-id="2131" left="150" top="966.75" width="91.5" height="15.75" align-x="174.75" align-y="978" 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">M1</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">warp</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">M</ml:id>
              <ml:id xml:space="preserve">T</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="12" />
    </region>
    <region region-id="2225" left="6" top="1068.75" width="330" height="28.5" align-x="6" align-y="1080" 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="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The result should be the same as the input, except at the boundaries, and neglecting roundoff errors:</p>
      </text>
    </region>
    <region region-id="2134" left="42" top="1104.75" width="189.75" height="99.75" align-x="66" align-y="1158" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:id xml:space="preserve">M1</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="5" cols="7">
              <ml:real>0</ml:real>
              <ml:real>5</ml:real>
              <ml:real>10</ml:real>
              <ml:real>15</ml:real>
              <ml:real>20</ml:real>
              <ml:real>1</ml:real>
              <ml:real>6</ml:real>
              <ml:real>11</ml:real>
              <ml:real>16</ml:real>
              <ml:real>21</ml:real>
              <ml:real>2</ml:real>
              <ml:real>7</ml:real>
              <ml:real>12</ml:real>
              <ml:real>17</ml:real>
              <ml:real>22</ml:real>
              <ml:real>3</ml:real>
              <ml:real>8</ml:real>
              <ml:real>13</ml:real>
              <ml:real>18</ml:real>
              <ml:real>23</ml:real>
              <ml:real>4</ml:real>
              <ml:real>9</ml:real>
              <ml:real>14</ml:real>
              <ml:real>19</ml:real>
              <ml:real>24</ml:real>
              <ml:real>5</ml:real>
              <ml:real>10</ml:real>
              <ml:real>15</ml:real>
              <ml:real>20</ml:real>
              <ml:real>25</ml:real>
              <ml:real>6</ml:real>
              <ml:real>11</ml:real>
              <ml:real>16</ml:real>
              <ml:real>21</ml:real>
              <ml:real>26</ml:real>
            </ml:matrix>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="13" />
    </region>
    <region region-id="1771" left="36" top="1221.75" width="219" height="50.25" align-x="233.25" align-y="1248" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:apply>
            <ml:summation />
            <ml:lambda>
              <ml:boundVars>
                <ml:id xml:space="preserve">i</ml:id>
              </ml:boundVars>
              <ml:apply>
                <ml:summation />
                <ml:lambda>
                  <ml:boundVars>
                    <ml:id xml:space="preserve">j</ml:id>
                  </ml:boundVars>
                  <ml:apply>
                    <ml:absval />
                    <ml:apply>
                      <ml:minus />
                      <ml:apply>
                        <ml:id xml:space="preserve">floor</ml:id>
                        <ml:apply>
                          <ml:plus />
                          <ml:apply>
                            <ml:indexer />
                            <ml:id xml:space="preserve">M1</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:real>0.5</ml:real>
                        </ml:apply>
                      </ml:apply>
                      <ml:apply>
                        <ml:indexer />
                        <ml:id xml:space="preserve">M</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:apply>
                </ml:lambda>
                <ml:bounds>
                  <ml:real>1</ml:real>
                  <ml:real>5</ml:real>
                </ml:bounds>
              </ml:apply>
            </ml:lambda>
            <ml:bounds>
              <ml:real>1</ml:real>
              <ml:real>3</ml:real>
            </ml:bounds>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>0</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="14" />
    </region>
    <region region-id="2302" left="0" top="1284" width="6000" height="6" align-x="0" align-y="1284" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="2226" left="6" top="1296.75" width="184.5" height="14.25" align-x="6" align-y="1308" 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="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The same transformation on a real image:</p>
      </text>
    </region>
    <region region-id="2230" left="36" top="1326.75" width="296.25" height="15.75" align-x="51.75" align-y="1338" 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">B</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">submatrix</ml:id>
            <ml:sequence>
              <ml:apply>
                <ml:id xml:space="preserve">READ_IMAGE</ml:id>
                <ml:str xml:space="preserve">bridge.bmp</ml:str>
              </ml:apply>
              <ml:real>0</ml:real>
              <ml:real>100</ml:real>
              <ml:real>0</ml:real>
              <ml:real>100</ml:real>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="15" />
    </region>
    <region region-id="2229" left="36" top="1362.75" width="85.5" height="15.75" align-x="57.75" align-y="1374" 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">B1</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">warp</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">B</ml:id>
              <ml:id xml:space="preserve">T</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="16" />
    </region>
    <region region-id="2333" left="36" top="1404" width="77.25" height="105.75" align-x="73.5" align-y="1404" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <component hide-arguments="false" clsid-buddy="8442da2d-08d7-45ad-a4a6-f28bee6ae06a" item-idref="17" disable-calc="false">
        <inputs>
          <ml:id xml:space="preserve">B</ml:id>
        </inputs>
        <outputs />
      </component>
      <rendering item-idref="18" />
    </region>
    <region region-id="2334" left="168" top="1404" width="77.25" height="105.75" align-x="205.5" align-y="1404" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <component hide-arguments="false" clsid-buddy="8442da2d-08d7-45ad-a4a6-f28bee6ae06a" item-idref="19" disable-calc="false">
        <inputs>
          <ml:id xml:space="preserve">B1</ml:id>
        </inputs>
        <outputs />
      </component>
      <rendering item-idref="20" />
    </region>
    <region region-id="2146" left="0" top="1548" width="4800" height="6" align-x="0" align-y="1548" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="2231" left="6" top="1566.75" width="294.75" height="14.25" align-x="6" align-y="1578" 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="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This warping stretches the image horizontally by a factor of two:</p>
      </text>
    </region>
    <region region-id="2234" left="42" top="1614.75" width="42.75" height="15.75" align-x="52.5" align-y="1626" 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:real>4</ml:real>
            <ml:real>7</ml:real>
          </ml:range>
        </ml:define>
      </math>
      <rendering item-idref="21" />
    </region>
    <region region-id="2232" left="102" top="1605" width="73.5" height="35.25" align-x="130.5" align-y="1626" 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">T</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">i</ml:id>
              <ml:real>1</ml:real>
            </ml:sequence>
          </ml:apply>
          <ml:apply>
            <ml:div />
            <ml:apply>
              <ml:indexer />
              <ml:id xml:space="preserve">T</ml:id>
              <ml:sequence>
                <ml:apply>
                  <ml:minus />
                  <ml:id xml:space="preserve">i</ml:id>
                  <ml:real>4</ml:real>
                </ml:apply>
                <ml:real>1</ml:real>
              </ml:sequence>
            </ml:apply>
            <ml:real>2</ml:real>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="22" />
    </region>
    <region region-id="2233" left="186" top="1614.75" width="70.5" height="17.25" align-x="214.5" align-y="1626" 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">T</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">i</ml:id>
              <ml:real>0</ml:real>
            </ml:sequence>
          </ml:apply>
          <ml:apply>
            <ml:indexer />
            <ml:id xml:space="preserve">T</ml:id>
            <ml:sequence>
              <ml:apply>
                <ml:minus />
                <ml:id xml:space="preserve">i</ml:id>
                <ml:real>4</ml:real>
              </ml:apply>
              <ml:real>0</ml:real>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="23" />
    </region>
    <region region-id="2236" left="42" top="1650.75" width="91.5" height="15.75" align-x="66.75" align-y="1662" 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">M1</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">warp</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">M</ml:id>
              <ml:id xml:space="preserve">T</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="24" />
    </region>
    <region region-id="2235" left="168" top="1650.75" width="85.5" height="15.75" align-x="189.75" align-y="1662" 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">B1</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">warp</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">B</ml:id>
              <ml:id xml:space="preserve">T</ml:id>
            </ml:sequence>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="25" />
    </region>
    <region region-id="2237" left="318" top="1577.25" width="75.75" height="162.75" align-x="332.25" align-y="1662" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:id xml:space="preserve">T</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="8" cols="2">
              <ml:real>1</ml:real>
              <ml:real>1</ml:real>
              <ml:real>2</ml:real>
              <ml:real>2</ml:real>
              <ml:real>1</ml:real>
              <ml:real>1</ml:real>
              <ml:real>2</ml:real>
              <ml:real>2</ml:real>
              <ml:real>1</ml:real>
              <ml:real>2</ml:real>
              <ml:real>1</ml:real>
              <ml:real>2</ml:real>
              <ml:real>0.5</ml:real>
              <ml:real>1</ml:real>
              <ml:real>0.5</ml:real>
              <ml:real>1</ml:real>
            </ml:matrix>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="26" />
    </region>
    <region region-id="1986" left="42" top="1686.75" width="216.75" height="99.75" align-x="66" align-y="1740" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:id xml:space="preserve">M1</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:matrix rows="5" cols="7">
              <ml:real>0</ml:real>
              <ml:real>5</ml:real>
              <ml:real>10</ml:real>
              <ml:real>15</ml:real>
              <ml:real>20</ml:real>
              <ml:real>0.5</ml:real>
              <ml:real>5.5</ml:real>
              <ml:real>10.5</ml:real>
              <ml:real>15.5</ml:real>
              <ml:real>20.5</ml:real>
              <ml:real>1</ml:real>
              <ml:real>6</ml:real>
              <ml:real>11</ml:real>
              <ml:real>16</ml:real>
              <ml:real>21</ml:real>
              <ml:real>1.5</ml:real>
              <ml:real>6.5</ml:real>
              <ml:real>11.5</ml:real>
              <ml:real>16.5</ml:real>
              <ml:real>21.5</ml:real>
              <ml:real>2</ml:real>
              <ml:real>7</ml:real>
              <ml:real>12</ml:real>
              <ml:real>17</ml:real>
              <ml:real>22</ml:real>
              <ml:real>2.5</ml:real>
              <ml:real>7.5</ml:real>
              <ml:real>12.5</ml:real>
              <ml:real>17.5</ml:real>
              <ml:real>22.5</ml:real>
              <ml:real>3</ml:real>
              <ml:real>8</ml:real>
              <ml:real>13</ml:real>
              <ml:real>18</ml:real>
              <ml:real>23</ml:real>
            </ml:matrix>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="27" />
    </region>
    <region region-id="2346" left="42" top="1800" width="77.25" height="105.75" align-x="79.5" align-y="1800" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <component hide-arguments="false" clsid-buddy="8442da2d-08d7-45ad-a4a6-f28bee6ae06a" item-idref="28" disable-calc="false">
        <inputs>
          <ml:id xml:space="preserve">B</ml:id>
        </inputs>
        <outputs />
      </component>
      <rendering item-idref="29" />
    </region>
    <region region-id="2347" left="198" top="1800" width="77.25" height="105.75" align-x="235.5" align-y="1800" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <component hide-arguments="false" clsid-buddy="8442da2d-08d7-45ad-a4a6-f28bee6ae06a" item-idref="30" disable-calc="false">
        <inputs>
          <ml:id xml:space="preserve">B1</ml:id>
        </inputs>
        <outputs />
      </component>
      <rendering item-idref="31" />
    </region>
    <region region-id="2238" left="6" top="1920.75" width="321" height="57" align-x="6" align-y="1932" 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="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The resulting image is expanded horizontally by a factor of 2 as expected. If we want to see the whole image in the output, we have to pad the matrix with zeros before warping, because the output image is always the same size as the input image, and pixels may be lost.</p>
      </text>
    </region>
    <region region-id="2251" left="0" top="1986" width="4800" height="6" align-x="0" align-y="1986" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="2239" left="6" top="2016.75" width="211.5" height="14.25" align-x="6" align-y="2028" 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="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This warping is a translation up and to the right:</p>
      </text>
    </region>
    <region region-id="2248" left="36" top="2046.75" width="42.75" height="15.75" align-x="46.5" align-y="2058" 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:real>4</ml:real>
            <ml:real>7</ml:real>
          </ml:range>
        </ml:define>
      </math>
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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">This warping example rotates clockwise by 45 degrees:</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">More general affine transformations using warping:</p>
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        <phrase>affine transformations using warping</phrase>
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