1. Digital Image Processing Lecture 21: Principal Components for Description
Prof. Charlene Tsai
*Chapter 11.4 of Gonzalez

2. Introduction Applicable to boundaries and regions.
First developed by Hotelling
Goal: Remove the correlation among the element of a random vector.
Aside: x is a random vector if each element xi of x is a random variable.
What is a random variable?

3. Mean and Covariance is a population of random vectors of length n
Mean vector is
Covariance matrix is defined as
Expected value

4. Example (Gonzalez, pg 677)
Considering 4 vectors x1=(0,0,0)T, x2=(1,0,0)T, x3=(1,1,0)T and x4=(1,0,1)T
Mean vector and covariance matrix are
How to interpret entries of the covariance matrix?

5. What to do with Cx?
We want to transform the vectors such that elements of the new vectors are uncorrelated.
Making the off-diagonal elements of covariance matrix 0.
Cx is real and symmetric, so there exists a set of n orthonormal eigenvectors, ei with corresponding eigenvalues in descending order.

6. Hotelling Transform (Principal-Component Analysis)
Let A be a matrix in the form
Create a new set of vectors
Largest eigenvalue
Smallest eigenvalue

7. my and Cy for Mean vector is 0 (zero vector) Covariance matrix is
Exactly what we want: 0 off-diagonal elements
Cx and Cy share the same eigenvalues and eigenvectors
Still remember matrix diagonalization?

8. Reconstruction of x from y
A is orthonormal, i.e. A-1 = AT
x can be recovered from corresponding y
Approximation can be made by using the first k eigenvectors of Cx to form Ak

9. Application: Boundary Description
e1 and e2 are the eigenvectors of the object

10. (con’d) are the points on the boundary
x=(a,b)T, where a and b are the coordinate values w.r.t. x1- and x2-axes.
The result of Hotelling (principal component) transformation is a new coordinate system:
Origin at the centroid of the
boundary points
Axes are the direction of the
eigenvectors of Cx

11. (con’d) Aligning the data with the eigenvectors, i.e. , we get
Aside: is the variance of component yi along eigenvector ei

12. Description Invariant to …
Rotation:
Aligning with the principal axes removes rotation
Scaling:
Normalization using eigenvalues (variance)
Translation:
Accounted for by centering the object about its mean