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              <c val="#800080">Moment Invariants</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 computes seven invariant spatial moments for an image M. Invariant moments are features of an image that are unchanged under translation, rotation, and scaling of the image. These moments are very useful in pattern-recognition problems, such as optical character recognition (OCR). </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 1-by-7 column vector of invariant moments. In this function we calculate the seven typical invariant moments (up to the third order) developed by Hu in 1962.</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">This function computes the seven typical invariant spatial moments defined by Hu in 1962. These moment features are invariant under translation, rotation, and scaling of the input image and are combinations of normalized spatial moments up to the third order. The normalized spatial moments, which are invariant under translation and scaling of the input image, are defined as follows:</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">where p and q are the orders of the normalized moment in the horizontal and vertical directions, and <region region-id="2416" left="176.25" top="519" width="24.75" height="18" align-x="187.5" align-y="531" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""><math optimize="false" disable-calc="false"><ml:apply><ml:indexer /><ml:id xml:space="preserve">μ</ml:id><ml:sequence><ml:id xml:space="preserve">p</ml:id><ml:id xml:space="preserve">q</ml:id></ml:sequence></ml:apply></math><rendering item-idref="7" /></region><f family="Symbol" charset="2" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><sp /></inlineAttr></f>is the spatial central moment of order (p,q).<f family="Symbol" charset="2" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><sp /></inlineAttr></f><f family="Symbol" charset="2" size="12"><inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"><sp /></inlineAttr></f></p>
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            <rendering item-idref="10" />
          </region>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false">) is the center of mass</inlineAttr>
          </f>
          <f family="Times New Roman" charset="0" size="12">
            <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> of the image.</inlineAttr>
          </f>
        </p>
      </text>
    </region>
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      <pageBreak />
    </region>
    <region region-id="2533" left="6" top="708.75" width="324" height="28.5" align-x="6" align-y="720" 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">Next we describe some examples of the use of these invariant moments.</p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Let's first generate a simple image, M.<sp count="2" /></p>
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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>
            <ml:real>29</ml:real>
          </ml:range>
        </ml:define>
      </math>
      <rendering item-idref="11" />
    </region>
    <region region-id="2177" left="120" top="756.75" width="50.25" height="15.75" align-x="132" align-y="768" 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">j</ml:id>
          <ml:range>
            <ml:real>0</ml:real>
            <ml:real>39</ml:real>
          </ml:range>
        </ml:define>
      </math>
      <rendering item-idref="12" />
    </region>
    <region region-id="2176" left="210" top="756.75" width="75.75" height="17.25" align-x="240" align-y="768" 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">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:plus />
            <ml:apply>
              <ml:mult style="default" />
              <ml:id xml:space="preserve">i</ml:id>
              <ml:real>40</ml:real>
            </ml:apply>
            <ml:id xml:space="preserve">j</ml:id>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="13" />
    </region>
    <region region-id="2568" left="42" top="780" width="39" height="59.25" align-x="60.75" align-y="780" 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="14" disable-calc="false">
        <inputs>
          <ml:id xml:space="preserve">M</ml:id>
        </inputs>
        <outputs />
      </component>
      <rendering item-idref="15" />
    </region>
    <region region-id="2540" left="6" top="882.75" width="295.5" height="14.25" align-x="6" align-y="894" 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">Calculate the moments up to the second order:</p>
      </text>
    </region>
    <region region-id="2542" left="36" top="906.75" width="84.75" height="15.75" align-x="67.5" align-y="918" 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">nrow</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">rows</ml:id>
            <ml:id xml:space="preserve">M</ml:id>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="16" />
    </region>
    <region region-id="2239" left="138" top="906.75" width="75.75" height="15.75" align-x="165" align-y="918" 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">ncol</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">cols</ml:id>
            <ml:id xml:space="preserve">M</ml:id>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="17" />
    </region>
    <region region-id="2541" left="6" top="930.75" width="186.75" height="14.25" align-x="6" align-y="942" 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">Compute the center of mass of the image:</p>
      </text>
    </region>
    <region region-id="2543" left="6" top="960" width="151.5" height="102.75" align-x="33" align-y="1014" 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">mnc</ml:id>
          <ml:apply>
            <ml:div />
            <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:parens>
                      <ml:apply>
                        <ml:mult style="default" />
                        <ml:id xml:space="preserve">j</ml:id>
                        <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:parens>
                  </ml:lambda>
                  <ml:bounds>
                    <ml:real>0</ml:real>
                    <ml:apply>
                      <ml:minus />
                      <ml:id xml:space="preserve">ncol</ml:id>
                      <ml:real>1</ml:real>
                    </ml:apply>
                  </ml:bounds>
                </ml:apply>
              </ml:lambda>
              <ml:bounds>
                <ml:real>0</ml:real>
                <ml:apply>
                  <ml:minus />
                  <ml:id xml:space="preserve">nrow</ml:id>
                  <ml:real>1</ml:real>
                </ml:apply>
              </ml:bounds>
            </ml:apply>
            <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: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:lambda>
                  <ml:bounds>
                    <ml:real>0</ml:real>
                    <ml:apply>
                      <ml:minus />
                      <ml:id xml:space="preserve">ncol</ml:id>
                      <ml:real>1</ml:real>
                    </ml:apply>
                  </ml:bounds>
                </ml:apply>
              </ml:lambda>
              <ml:bounds>
                <ml:real>0</ml:real>
                <ml:apply>
                  <ml:minus />
                  <ml:id xml:space="preserve">nrow</ml:id>
                  <ml:real>1</ml:real>
                </ml:apply>
              </ml:bounds>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="18" />
    </region>
    <region region-id="2544" left="198" top="960" width="148.5" height="102.75" align-x="223.5" align-y="1014" 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">mnr</ml:id>
          <ml:apply>
            <ml:div />
            <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:parens>
                      <ml:apply>
                        <ml:mult style="default" />
                        <ml:id xml:space="preserve">i</ml:id>
                        <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:parens>
                  </ml:lambda>
                  <ml:bounds>
                    <ml:real>0</ml:real>
                    <ml:apply>
                      <ml:minus />
                      <ml:id xml:space="preserve">ncol</ml:id>
                      <ml:real>1</ml:real>
                    </ml:apply>
                  </ml:bounds>
                </ml:apply>
              </ml:lambda>
              <ml:bounds>
                <ml:real>0</ml:real>
                <ml:apply>
                  <ml:minus />
                  <ml:id xml:space="preserve">nrow</ml:id>
                  <ml:real>1</ml:real>
                </ml:apply>
              </ml:bounds>
            </ml:apply>
            <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: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:lambda>
                  <ml:bounds>
                    <ml:real>0</ml:real>
                    <ml:apply>
                      <ml:minus />
                      <ml:id xml:space="preserve">ncol</ml:id>
                      <ml:real>1</ml:real>
                    </ml:apply>
                  </ml:bounds>
                </ml:apply>
              </ml:lambda>
              <ml:bounds>
                <ml:real>0</ml:real>
                <ml:apply>
                  <ml:minus />
                  <ml:id xml:space="preserve">nrow</ml:id>
                  <ml:real>1</ml:real>
                </ml:apply>
              </ml:bounds>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="19" />
    </region>
    <region region-id="2545" left="6" top="1080.75" width="231" height="14.25" align-x="6" align-y="1092" 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">Compute the central moments up to the third order:</p>
      </text>
    </region>
    <region region-id="2257" left="36" top="1104.75" width="46.5" height="15.75" align-x="50.25" align-y="1116" 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">p</ml:id>
          <ml:range>
            <ml:real>0</ml:real>
            <ml:real>3</ml:real>
          </ml:range>
        </ml:define>
      </math>
      <rendering item-idref="20" />
    </region>
    <region region-id="2258" left="108" top="1104.75" width="46.5" height="15.75" align-x="122.25" align-y="1116" 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">q</ml:id>
          <ml:range>
            <ml:real>0</ml:real>
            <ml:real>3</ml:real>
          </ml:range>
        </ml:define>
      </math>
      <rendering item-idref="21" />
    </region>
    <region region-id="2437" left="36" top="1137.75" width="249.75" height="50.25" align-x="68.25" align-y="1164" 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">μ</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">p</ml:id>
              <ml:id xml:space="preserve">q</ml:id>
            </ml:sequence>
          </ml:apply>
          <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:parens>
                    <ml:apply>
                      <ml:mult style="default" />
                      <ml:apply>
                        <ml:mult style="default" />
                        <ml:apply>
                          <ml:pow />
                          <ml:parens>
                            <ml:apply>
                              <ml:minus />
                              <ml:id xml:space="preserve">j</ml:id>
                              <ml:id xml:space="preserve">mnc</ml:id>
                            </ml:apply>
                          </ml:parens>
                          <ml:id xml:space="preserve">p</ml:id>
                        </ml:apply>
                        <ml:apply>
                          <ml:pow />
                          <ml:parens>
                            <ml:apply>
                              <ml:minus />
                              <ml:id xml:space="preserve">i</ml:id>
                              <ml:id xml:space="preserve">mnr</ml:id>
                            </ml:apply>
                          </ml:parens>
                          <ml:id xml:space="preserve">q</ml:id>
                        </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:parens>
                </ml:lambda>
                <ml:bounds>
                  <ml:real>0</ml:real>
                  <ml:apply>
                    <ml:minus />
                    <ml:id xml:space="preserve">ncol</ml:id>
                    <ml:real>1</ml:real>
                  </ml:apply>
                </ml:bounds>
              </ml:apply>
            </ml:lambda>
            <ml:bounds>
              <ml:real>0</ml:real>
              <ml:apply>
                <ml:minus />
                <ml:id xml:space="preserve">nrow</ml:id>
                <ml:real>1</ml:real>
              </ml:apply>
            </ml:bounds>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="22" />
    </region>
    <region region-id="2439" left="0" top="1194" width="6000" height="6" align-x="0" align-y="1194" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
      <pageBreak />
    </region>
    <region region-id="2546" left="6" top="1206.75" width="192.75" height="14.25" align-x="6" align-y="1218" 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 normalized moments are computed as:</p>
      </text>
    </region>
    <region region-id="2549" left="36" top="1232.25" width="111.75" height="59.25" align-x="69" align-y="1254" 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">η</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">p</ml:id>
              <ml:id xml:space="preserve">q</ml:id>
            </ml:sequence>
          </ml:apply>
          <ml:apply>
            <ml:div />
            <ml:apply>
              <ml:indexer />
              <ml:id xml:space="preserve">μ</ml:id>
              <ml:sequence>
                <ml:id xml:space="preserve">p</ml:id>
                <ml:id xml:space="preserve">q</ml:id>
              </ml:sequence>
            </ml:apply>
            <ml:apply>
              <ml:pow />
              <ml:parens>
                <ml:apply>
                  <ml:indexer />
                  <ml:id xml:space="preserve">μ</ml:id>
                  <ml:sequence>
                    <ml:real>0</ml:real>
                    <ml:real>0</ml:real>
                  </ml:sequence>
                </ml:apply>
              </ml:parens>
              <ml:apply>
                <ml:div />
                <ml:apply>
                  <ml:plus />
                  <ml:apply>
                    <ml:plus />
                    <ml:id xml:space="preserve">p</ml:id>
                    <ml:id xml:space="preserve">q</ml:id>
                  </ml:apply>
                  <ml:real>2</ml:real>
                </ml:apply>
                <ml:real>2</ml:real>
              </ml:apply>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="23" />
    </region>
    <region region-id="2547" left="6" top="1302.75" width="204.75" height="14.25" align-x="6" align-y="1314" 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">and the first moment invariant is calculated as:</p>
      </text>
    </region>
    <region region-id="2552" left="36" top="1332" width="94.5" height="18" align-x="57" align-y="1344" 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">φ1</ml:id>
          <ml:apply>
            <ml:plus />
            <ml:apply>
              <ml:indexer />
              <ml:id xml:space="preserve">η</ml:id>
              <ml:sequence>
                <ml:real>2</ml:real>
                <ml:real>0</ml:real>
              </ml:sequence>
            </ml:apply>
            <ml:apply>
              <ml:indexer />
              <ml:id xml:space="preserve">η</ml:id>
              <ml:sequence>
                <ml:real>0</ml:real>
                <ml:real>2</ml:real>
              </ml:sequence>
            </ml:apply>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="24" />
    </region>
    <region region-id="2553" left="162" top="1327.5" width="100.5" height="21" align-x="182.25" align-y="1344" 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">φ1</ml:id>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>0.00025456097010781529</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="25" />
    </region>
    <region region-id="2548" left="6" top="1356.75" width="316.5" height="14.25" align-x="6" align-y="1368" 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">Now we call the <b>moment_invariant</b> function and compare the result:</p>
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    </region>
    <region region-id="2446" left="36" top="1380.75" width="125.25" height="15.75" align-x="51" align-y="1392" 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">P</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">moment_invariant</ml:id>
            <ml:id xml:space="preserve">M</ml:id>
          </ml:apply>
        </ml:define>
      </math>
      <rendering item-idref="26" />
    </region>
    <region region-id="2254" left="192" top="1380" width="88.5" height="18" align-x="258.75" align-y="1392" 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:absval />
            <ml:apply>
              <ml:minus />
              <ml:apply>
                <ml:indexer />
                <ml:id xml:space="preserve">P</ml:id>
                <ml:sequence>
                  <ml:real>0</ml:real>
                  <ml:real>0</ml:real>
                </ml:sequence>
              </ml:apply>
              <ml:id xml:space="preserve">φ1</ml:id>
            </ml:apply>
          </ml:apply>
          <result xmlns="http://schemas.mathsoft.com/math20">
            <ml:real>2.7105054312137611E-19</ml:real>
          </result>
        </ml:eval>
      </math>
      <rendering item-idref="27" />
    </region>
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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 see that the first moment invariant exactly matches the definition.</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 let's use another image. </p>
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      <math optimize="false" disable-calc="false">
        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">M</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">READ_IMAGE</ml:id>
            <ml:str xml:space="preserve">flower.bmp</ml:str>
          </ml:apply>
        </ml:define>
      </math>
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    <region region-id="2610" left="6" top="1488.75" width="319.5" height="42.75" align-x="6" align-y="1500" 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">Perform three types of geometric transforms on the image - translation, rotation, and scaling - and show that the invariant moments are unchanged.</p>
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      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">M1</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">translate</ml:id>
            <ml:sequence>
              <ml:id xml:space="preserve">M</ml:id>
              <ml:real>-10</ml:real>
              <ml:real>-5</ml:real>
              <ml:real>0</ml:real>
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          </ml:apply>
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      <math optimize="false" disable-calc="false">
        <ml:define>
          <ml:id xml:space="preserve">M2</ml:id>
          <ml:apply>
            <ml:id xml:space="preserve">zoom</ml:id>
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              <ml:id xml:space="preserve">M</ml:id>
              <ml:real>0.8</ml:real>
              <ml:real>0.8</ml:real>
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        </ml:define>
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            <ml:id xml:space="preserve">rotate</ml:id>
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              <ml:id xml:space="preserve">M</ml:id>
              <ml:real>45</ml:real>
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      <pageBreak />
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        <outputs />
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      <component hide-arguments="false" clsid-buddy="8442da2d-08d7-45ad-a4a6-f28bee6ae06a" item-idref="38" disable-calc="false">
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        <outputs />
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    <region region-id="2321" left="42" top="1902" width="129" height="16.5" align-x="60.75" align-y="1914" 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="WarnRedefinedBIFunction">
          <ml:id xml:space="preserve">Φ</ml:id>
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            <ml:id xml:space="preserve">moment_invariant</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:id xml:space="preserve">Φ1</ml:id>
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            <ml:id xml:space="preserve">moment_invariant</ml:id>
            <ml:id xml:space="preserve">M1</ml:id>
          </ml:apply>
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    <region region-id="2323" left="42" top="1950" width="140.25" height="16.5" align-x="66" align-y="1962" 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">Φ2</ml:id>
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            <ml:id xml:space="preserve">moment_invariant</ml:id>
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          </ml:apply>
        </ml:define>
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    <region region-id="2324" left="42" top="1974" width="140.25" height="16.5" align-x="66" align-y="1986" 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">Φ3</ml:id>
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            <ml:id xml:space="preserve">moment_invariant</ml:id>
            <ml:id xml:space="preserve">M3</ml:id>
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    <region region-id="2325" left="42" top="2011.5" width="441.75" height="21" align-x="60" align-y="2028" 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">Φ</ml:id>
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            <ml:matrix rows="1" cols="7">
              <ml:real>0.0018140717263038784</ml:real>
              <ml:real>1.3075391093693361E-06</ml:real>
              <ml:real>1.2112634350069453E-09</ml:real>
              <ml:real>4.6092501093300703E-10</ml:real>
              <ml:real>3.0620029021354592E-19</ml:real>
              <ml:real>2.9923503236778981E-13</ml:real>
              <ml:real>-1.5765025015146539E-19</ml:real>
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          <ml:id xml:space="preserve">Φ1</ml:id>
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              <ml:real>0.0018140717263038784</ml:real>
              <ml:real>1.3075391093693361E-06</ml:real>
              <ml:real>1.2112634350069453E-09</ml:real>
              <ml:real>4.6092501093300703E-10</ml:real>
              <ml:real>3.0620029021354592E-19</ml:real>
              <ml:real>2.9923503236778981E-13</ml:real>
              <ml:real>-1.5765025015146539E-19</ml:real>
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        <ml:eval>
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            <ml:matrix rows="1" cols="7">
              <ml:real>0.0018149647689488071</ml:real>
              <ml:real>1.3071051928238221E-06</ml:real>
              <ml:real>1.2127522787511866E-09</ml:real>
              <ml:real>4.62004986839144E-10</ml:real>
              <ml:real>3.0739606829528969E-19</ml:real>
              <ml:real>2.9985441374522055E-13</ml:real>
              <ml:real>-1.5843736942664827E-19</ml:real>
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      <math optimize="false" disable-calc="false">
        <ml:eval>
          <ml:id xml:space="preserve">Φ3</ml:id>
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            <ml:matrix rows="1" cols="7">
              <ml:real>0.0018215721127572053</ml:real>
              <ml:real>1.3606806414531088E-06</ml:real>
              <ml:real>1.2275810788873474E-09</ml:real>
              <ml:real>4.6333748843105925E-10</ml:real>
              <ml:real>3.1008313647910334E-19</ml:real>
              <ml:real>3.2792051079433497E-13</ml:real>
              <ml:real>-1.6110864460783295E-19</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">The above results are in reasonable agreement with the invariants computed from the original image. The small errors are due to zero-padding and pixel interpolation that take place in the </p>
        <p style="Normal" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">transformations.</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">In the next step, we combine all these three transformations together</p>
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">and compare the moment invariants to the original ones:</p>
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      <math optimize="false" disable-calc="false">
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          <ml:id xml:space="preserve">M1</ml:id>
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              <ml:id xml:space="preserve">M</ml:id>
              <ml:real>-60</ml:real>
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          <ml:id xml:space="preserve">M2</ml:id>
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            <ml:id xml:space="preserve">zoom</ml:id>
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              <ml:id xml:space="preserve">M1</ml:id>
              <ml:real>0.75</ml:real>
              <ml:real>0.75</ml:real>
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          <ml:id xml:space="preserve">M3</ml:id>
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            <ml:id xml:space="preserve">translate</ml:id>
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              <ml:id xml:space="preserve">M2</ml:id>
              <ml:real>-10</ml:real>
              <ml:real>10</ml:real>
              <ml:real>0</ml:real>
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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">Here is the final image:</p>
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          <ml:id xml:space="preserve">Φ1</ml:id>
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          <ml:id xml:space="preserve">Φ2</ml:id>
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        <ml:define warning="WarnRedefinedUDScalar">
          <ml:id xml:space="preserve">Φ3</ml:id>
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              <ml:real>0.0018140717263038784</ml:real>
              <ml:real>1.3075391093693361E-06</ml:real>
              <ml:real>1.2112634350069453E-09</ml:real>
              <ml:real>4.6092501093300703E-10</ml:real>
              <ml:real>3.0620029021354592E-19</ml:real>
              <ml:real>2.9923503236778981E-13</ml:real>
              <ml:real>-1.5765025015146539E-19</ml:real>
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      <math optimize="false" disable-calc="false">
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              <ml:real>2.975791941904326E-13</ml:real>
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              <ml:real>2.9760597179154935E-13</ml:real>
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        <ml:eval>
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              <ml:real>4.6379198855120306E-10</ml:real>
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              <ml:real>2.9760597179153486E-13</ml:real>
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      <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 above results are also in reasonable agreement with the invariants </p>
        <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">computed from the original image.</p>
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