Image Processing Reference
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where L a is the total affine arc length of the considered curve, ∧ represents the cross product
between two vectors and, ||·|| denotes the Euclidean norm.
2.2.2 Contours alignment using geometrical affine parameters estimation
We consider two closed curves O 1 and O 2 which define the same shape. O 1 and O 2 are said to be
related by an affine transformation if and only if:
where B is a translation vector, A is a linear transformation, l 0 is the shift value, f and h are the
affine reparametrization of two contours having the same affine shape. In the Fourier space
we get:
where U k ( h ) and U k ( f ) are, respectively, the Fourier coefficients of f and h .
So estimation of affine motion can be resumed to the estimation of its three parameters: the
affine matrix A , the shift value l 0 , and finally the scale factor α if we consider a normalization
under translation.
Estimation of the scale factor α
The scale factor can be estimated using the following formula:
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