Affine Transformation

[Avinash Raj]

Affine Transformation Matrix is said to be formed by initializing it using a learned projection matrix from a conventional algorithm like Eigenfaces or Fisherfaces; then the singular value decomposition T=UAV' is used where T is the transformation matrix.

Could somebody explain how the decomposition is obtained and what it is?

 

[Godfrey]

This should get you started:
http://en.wikipedia.org/wiki/Affine_transformation

References there will point to more useful data.

 

[Jan Brittenson]

Matrix decomposition is pretty basic linear algebra. It's the solution to Ax = b, for a given A and b. There are many ways to solve this (see http://en.wikipedia.org/wiki/Matrix_decomposition ) but generally the first method taught in any linalg course is LU decomposition because it builds directly on Gaussian Elimination. (During elimination you collect various properties, which become the LU solution.) I strongly suggest an introductory textbook to linear algebra. Any textbook. They'll all cover it.

The MIT Open Course Ware has a *great* video lecture series on linear algebra as well. But reading a book might be quicker depending on prior exposure and how quickly you pick it up. http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/

 

[Avinash Raj]

Thank You

I did learn about the SVD of a matrix. An IEEE paper (Face Verification With Balanced Thresholds) that I read few days back says "the right orthogonal matrix of SVD of a transformation matrix does not affect the similarity measure if based on Euclidean distance." I drew blanks in my attempts to understand how it is so and wikipedia wasnt a help at all. Could you tell me why the measure is invariant to the right orthogonal matrix?

Note - The right unitary matrix becomes orthogonal as only real matrices are considered in the problem.

 

[Jan Brittenson]

I don't think it's possible to answer that without understanding the measurement and the underlying basis. My use of linalg is mainly for physics, so measurement to me means a projection that reduces the dimensionality of the basis. For example, a point expressed as two angles and a radius can be projected to the x axis (orthonormal basis) in an x,y,z basis. This would be a measurement of the point's x property. (It could also be projected to say the radius basis to measure the radius, I just wanted to illustrate that it can cross between basis, although maybe technically this is a transform.) Not sure if this is how they use the term, but they're telling you something about the measurement or the basis(es).

 

[Jan Brittenson]

And Euclidean is obvious. That's just a qualifier to keep people from wandering off into differentiable geometries even if totally inapplicable.

By the way, if I were you I'd ask in a more technical forum, like http://www.physicsforums.com/ in the Mathematics section.