`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 ∧, 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, ∧, u i } , where i = 1, ∧, m and usually m << n. The averaged, or
1    2        n
mean, object u = { u 1, u 2, ∧, u n } of this group is defined as follows:
m
1
 ul .
k
u l = ---
m
k=1
m∵m
Covariance matrix C = |cij| is a square symmetric matrix                         :
n

i             j
( ul  ? ul ) ? ( ul ? ul ) .
c ij =
l=1
Eigen objects basis e i = { e i, ei, ∧, e i } , i = 1, ∧ ,           m1 ? m of the input objects group
1   2      n
may be calculated using the following relation:
m
1
 vk ? ( ul ? ul ) ,
i                    i      k
= ---------
-
el
λi
k=1
i    i       i
i
where λi and                = { v 1 , v2 , ∧, 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:
 el ? ( u l ? u l ) .
i
wi =
l=1
Using these coefficients, we may calculate projection u = { u 1, u 2 ∧, 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
 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```

Last-modified: 2007-05-08 (残) 17:02:00 (4489d)