A rigid transformation is a mapping from R n to R n that is the composition of
Question:
A rigid transformation is a mapping from Rn to Rn that is the composition of a translation and a rotation. Mathematically, we can express a rigid transformation Φ orthogonal transformation and a vector.
We are given a set of pairs of pointsand wish to find a rigid transformation that best matches them. We can write the problem as
where In is the n x n identity matrix.
The problem arises in image processing, to provide ways to deform an image (represented as a set of two-dimensional points) based on the manual selection of a few points and their transformed counterparts.
1. Assume that R is fixed in problem (5.2). Express an optimal r as a function of R.
2.
3. Show that the problem can be written as
for appropriate matrices X,Y, which you will determine.
4. Show that the problem can be further written as
for an appropriate n x n matrix Z, which you will determine.
5. Show that R = VUT is optimal, where Z = USVT is the SVD of Z.
6. Show the result you used in the previous question: assume Z is diagonal, and show that R = In is optimal for the problem above.
7. How woud you apply this technique to make Mona Lisa smile more?
Step by Step Answer:
Optimization Models
ISBN: 9781107050877
1st Edition
Authors: Giuseppe C. Calafiore, Laurent El Ghaoui