opencv/sec5.ObjectRecognition

quotation@opencv-1.0.0/docs/opencvman_old.pdf

5 Object Recognition †

Eigen Objects †

This section describes functions that operate on eigen objects.

     Let us define an object u = { u 1, u 2 ¡Ä, u n } as a vector in the n-dimensional space. For
     example, u can be an image and its components ul are the image pixel values. In this
     case n is equal to the number of pixels in the image. Then, consider a group of input
     objects u i = { ui, u i, ¡Ä, u i } , where i = 1, ¡Ä, m and usually m << n. The averaged, or
                              1    2        n
     mean, object u = { u 1, u 2, ¡Ä, u n } of this group is defined as follows:
                   m
            1
                 ­ô ul .
                         k
     u l = ---
           m
               k=1
                                                                                 m¡ßm
     Covariance matrix C = |cij| is a square symmetric matrix                         :
                  n
                ­ô
                          i             j
                      ( ul  ? ul ) ? ( ul ? ul ) .
     c ij =
              l=1
     Eigen objects basis e i = { e i, ei, ¡Ä, e i } , i = 1, ¡Ä ,           m1 ? m of the input objects group
                                                1   2      n
     may be calculated using the following relation:
                      m
               1
                     ­ô vk ? ( ul ? ul ) ,
       i                    i      k
         = ---------
                   -
     el
               ¦Ëi
                     k=1
                                        i    i       i
                               i
     where ¦Ëi and                = { v 1 , v2 , ¡Ä, v m } are eigenvalues and the corresponding eigenvectors
                             v
     of matrix C.
                                                            5-1
                                                                                          Any input object ui as well as any other object u may be decomposed in the eigen
          objectsnm1-D sub-space. Decomposition coefficients of the object u are:
                 ­ô el ? ( u l ? u l ) .
                     i
          wi =
                l=1
          Using these coefficients, we may calculate projection u = { u 1, u 2 ¡Ä, u n } of the object u
                                                                   ?     ??       ?
          to the eigen objects sub-space, or, in other words, restore the object u in that sub-space:
                 m1
                 ­ô wk e l + u l .
                        k
          ul =
           ?
                k=1
          For examples of use of the functions and relevant data types see Image Recognition
          Reference Chapter.

Embedded Hidden Markov Models †

          This section describes functions for using Embedded Hidden Markov Models (HMM)
          in face recognition task. See Reference for HMM Structures.
                                                 5-2