An affine shape constraint for geometric active contours - Emerging Trends in Image Processing, Computer Vision, and Pattern Recognition - page 424

where and are 2 × 2 matrix formed by Fourier coeicients
on some fixed index k and U * is the U complex conjugate.

Computation of the shift value l 0

Let's consider and

(13)

where k 1 and k 2 are two fixed index and arg( U ) is the complex argument of U .

Computation of the matrix A 's parameters
Since scale factor α and shift value l 0 have been previously estimated (Equations 12 and 13 ) , we
are going to estimate affine matrix A parameters. Lets O 1 , O 2 be two curves and U f , U h , respect-
ively, their Fourier coefficients. In Fourier space we have (Equation 14 )

(14)

Using matrix, we have the following formula:

(15)

So we have to estimate a 1 , a 2 , a 3 , and a 4 . This system can be represented as following: