Assume that both the template and reference images are defined on the same domain . Because reference and template images are obtained from different distances, angles, times, sensors, and sometimes even by different individuals, a displacement field may occur between the reference and template images. One of the major goals of this paper is to compute the deformation field in a systematic way.
A deformation field is a vector image that maps reference image pixel coordinates to the coordinates of the corresponding template image pixels. Consider the deformations of the form
where denotes a displacement field. Because both the template and reference images are almost surely ‘the same’ images, we can write
In this paper, we exploit 3-D Haar wavelets. When expanding the displacement field in terms of a wavelet decomposition, it is necessary to take into account the wavelet parameters (coefficients) α, which yields that
The major goal of this paper is to compute the displacement field in a systematic way. We express each of , , in terms of wavelet coefficients. Therefore, computing these wavelet coefficients will be enough to obtain the displacement field .
A multiresolution analysis of , is a partially ordered set of closed linear subspaces
, for every , ;
, for every , ;
There exists such that is a Riesz basis for .
Here , , , , . The function is called the scaling function of the multiresolution analysis. Detailed information about multiresolution analysis and wavelets might be seen in . Let be the open unit cube. For any , we introduce the function space : the space of piecewise constant functions on a uniform grid with mesh size . These grids are uniformly spaced in each of the three coordinate directions, but possibly with a different mesh size in the different directions. The volume of these cells is denoted by . The functions in are all constant in each cell defined by
A basis may be generated from a scaling function Φ. In order to deal with 3-D deformation field, to each component of the displacement field, a multiresolution decomposition is applied. At scale m, i.e., in the space , the displacement field will be parameterized by the vector
Elements (scaling functions) of a basis are 3-D functions that are translated across the cubical grid . These functions are a tensor product of the 1-D scaling and wavelet functions as