Moment Invariants

Functions > Image Processing > Feature Extraction > Moment Invariants

Moment Invariants

• moment_invariant(M)—Returns a vector containing the seven typical moment invariants of M.

The invariant moments (defined by Hu in 1962) are combinations of normalized spatial moments up to the third order. They remain unchanged under translation, rotation, and scaling of the image. Invariant moments are very useful in pattern recognition problems, such as optical character recognition (OCR).

Invariant moments are defined as follows:

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  <math resultRef="5">
    <apply xmlns="http://schemas.mathsoft.com/math50">
      <equal />
      <apply>
        <indexer />
        <id xml:space="preserve">n</id>
        <sequence>
          <id xml:space="preserve">p</id>
          <id xml:space="preserve">q</id>
        </sequence>
      </apply>
      <apply>
        <div />
        <apply>
          <indexer />
          <id xml:space="preserve">μ</id>
          <sequence>
            <id xml:space="preserve">p</id>
            <id xml:space="preserve">q</id>
          </sequence>
        </apply>
        <apply>
          <pow />
          <parens>
            <apply>
              <indexer />
              <id xml:space="preserve">μ</id>
              <sequence>
                <real>0</real>
                <real>0</real>
              </sequence>
            </apply>
          </parens>
          <id xml:space="preserve">γ</id>
        </apply>
      </apply>
    </apply>
  </math>
</region>

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where p and q are the orders of the normalized moment in the horizontal and vertical directions, and μp, q is the spatial central moment of order (p, q).

The central moment is defined as:

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  <math resultRef="9">
    <apply xmlns="http://schemas.mathsoft.com/math50">
      <equal />
      <apply>
        <indexer />
        <id xml:space="preserve">γ</id>
        <sequence>
          <id xml:space="preserve">p</id>
          <id xml:space="preserve">q</id>
        </sequence>
      </apply>
      <apply>
        <summation />
        <lambda>
          <boundVars>
            <id xml:space="preserve">i</id>
          </boundVars>
          <apply>
            <summation />
            <lambda>
              <boundVars>
                <id xml:space="preserve">j</id>
              </boundVars>
              <parens>
                <apply>
                  <mult />
                  <apply>
                    <mult />
                    <apply>
                      <pow />
                      <parens>
                        <apply>
                          <minus />
                          <id xml:space="preserve">j</id>
                          <apply>
                            <indexer />
                            <id xml:space="preserve">c</id>
                            <id xml:space="preserve">j</id>
                          </apply>
                        </apply>
                      </parens>
                      <id xml:space="preserve">p</id>
                    </apply>
                    <apply>
                      <pow />
                      <parens>
                        <apply>
                          <minus />
                          <id xml:space="preserve">i</id>
                          <apply>
                            <indexer />
                            <id xml:space="preserve">c</id>
                            <id xml:space="preserve">i</id>
                          </apply>
                        </apply>
                      </parens>
                      <id xml:space="preserve">q</id>
                    </apply>
                  </apply>
                  <apply>
                    <indexer />
                    <id xml:space="preserve">M</id>
                    <sequence>
                      <id xml:space="preserve">i</id>
                      <id xml:space="preserve">j</id>
                    </sequence>
                  </apply>
                </apply>
              </parens>
            </lambda>
            <lowerBound>
              <real>0</real>
            </lowerBound>
            <upperBound>
              <apply>
                <minus />
                <id xml:space="preserve">ncol</id>
                <real>1</real>
              </apply>
            </upperBound>
          </apply>
        </lambda>
        <lowerBound>
          <real>0</real>
        </lowerBound>
        <upperBound>
          <apply>
            <minus />
            <id xml:space="preserve">nrow</id>
            <real>1</real>
          </apply>
        </upperBound>
      </apply>
    </apply>
  </math>
</region>

where nrow and ncol are the number of rows and columns of the input image M, and (cj, ci) is the center of mass of the image.

Arguments

• M is an image matrix.