Assume that both the template \mathbf{T}(\mathbf{x}) and reference \mathbf{R}(\mathbf{x}) images are defined on the same domain \mathrm{\Omega}\subset {\mathbb{R}}^{3}. 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
{\varphi}_{u}(\mathbf{x}):=\mathbf{x}+u(\mathbf{x}),\phantom{\rule{1em}{0ex}}\mathbf{x}=({x}_{1},{x}_{2},{x}_{3})\in \mathrm{\Omega},u(\mathbf{x})=({u}_{1}(\mathbf{x}),{u}_{2}(\mathbf{x}),{u}_{3}(\mathbf{x})),
(1)
where u(\mathbf{x}) denotes a displacement field. Because both the template \mathbf{T}(\mathbf{x}) and reference \mathbf{R}(\mathbf{x}) images are almost surely ‘the same’ images, we can write
\mathbf{R}({x}_{1},{x}_{2},{x}_{3})\approx \mathbf{T}(({x}_{1},{x}_{2},{x}_{3})+u({x}_{1},{x}_{2},{x}_{3}))=\mathbf{T}({x}_{1}+{u}_{1}(\mathbf{x}),{x}_{2}+{u}_{2}(\mathbf{x}),{x}_{3}+{u}_{3}(\mathbf{x})).
(2)
In this paper, we exploit 3D Haar wavelets. When expanding the displacement field u(\mathbf{x}) in terms of a wavelet decomposition, it is necessary to take into account the wavelet parameters (coefficients) α, which yields that
{\varphi}_{u}(\mathbf{x})=\left[\begin{array}{c}{x}_{1}+{u}_{1}({x}_{1},{x}_{2},{x}_{3},\alpha )\\ {x}_{2}+{u}_{2}({x}_{1},{x}_{2},{x}_{3},\alpha )\\ {x}_{3}+{u}_{3}({x}_{1},{x}_{2},{x}_{3},\alpha )\end{array}\right].
The major goal of this paper is to compute the displacement field u(\mathbf{x}) in a systematic way. We express each of {u}_{i}, i=1,2,3, in terms of wavelet coefficients. Therefore, computing these wavelet coefficients will be enough to obtain the displacement field u(\mathbf{x}).
A multiresolution analysis of {L}^{2}(\mathrm{\Omega}), \mathrm{\Omega}={\mathbb{R}}^{3} is a partially ordered set of closed linear subspaces
{\{{V}_{\mathbf{n}}\subset {L}^{2}(\mathrm{\Omega})\}}_{\mathbf{n}\in {\mathbb{Z}}^{3}}
with properties:

(1)
{\bigcap}_{\mathbf{n}}{V}_{\mathbf{n}}=\{0\}; {\bigcup}_{\mathbf{n}}{V}_{\mathbf{n}}{\subset}_{\mathrm{dense}}{L}^{2}(\mathrm{\Omega});

(2)
f(\mathbf{x})\in {V}_{\mathbf{n}}\u27faf({2}^{\mathbf{m}}\mathbf{x})\in {V}_{\mathbf{n}+\mathbf{m}}, for every \mathbf{n}\in {\mathbb{Z}}^{3}, \mathbf{m}\in E;

(3)
f(\mathbf{x})\in {V}_{\mathbf{n}}\u27faf(\mathbf{x}{2}^{\mathbf{k}})\in {V}_{\mathbf{n}}, for every \mathbf{k}\in {\mathbb{Z}}^{3}, \mathbf{n}\in E;

(4)
There exists \mathrm{\Phi}\in {V}_{\mathbf{0}} such that {\{\mathrm{\Phi}(\mathbf{x}\mathbf{k})\}}_{\mathbf{k}\in {\mathbb{Z}}^{3}} is a Riesz basis for {V}_{\mathbf{0}}.
Here \mathbf{n}=({n}_{1},{n}_{2},{n}_{3}), {2}^{\mathbf{n}}=({2}^{{n}_{1}},{2}^{{n}_{2}},{2}^{{n}_{3}}), \mathbf{0}=(0,0,0)\in {\mathbb{N}}^{3}, \mathbf{x}=({x}_{1},{x}_{2},{x}_{3})\in {\mathbb{R}}^{3}, {2}^{n}=({2}^{{n}_{1}}{x}_{1},{2}^{{n}_{1}}{x}_{2},{2}^{{n}_{1}}{x}_{3}). The function \mathrm{\Phi}(\mathbf{x}) is called the scaling function of the multiresolution analysis. Detailed information about multiresolution analysis and wavelets might be seen in [6]. Let \mathrm{\Omega}={(0,1)}^{3}\subset {\mathbb{R}}^{3} be the open unit cube. For any \mathbf{n}\in {\mathbb{Z}}^{3}, we introduce the function space {V}_{\mathbf{n}}: the space of piecewise constant functions on a uniform grid with mesh size h=({2}^{{n}_{1}},{2}^{{n}_{2}},{2}^{{n}_{3}}). 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 {h}^{3}={2}^{({n}_{1}+{n}_{2}+{n}_{3})}. The functions in {V}_{\mathbf{n}} are all constant in each cell defined by
{\mathrm{\Omega}}_{\mathbf{n},\mathbf{k}}:=[{k}_{1}{2}^{{n}_{1}},({k}_{1}+1){2}^{{n}_{1}}]\times [{k}_{2}{2}^{{n}_{2}},({k}_{2}+1){2}^{{n}_{2}}]\times [{k}_{3}{2}^{{n}_{3}},({k}_{3}+1){2}^{{n}_{3}}].
(3)
A basis {V}_{m} may be generated from a scaling function Φ. In order to deal with 3D deformation field, to each component of the displacement field, a multiresolution decomposition is applied. At scale m, i.e., in the space {V}_{m}, the displacement field {u}^{m}=({u}_{1}^{m},{u}_{2}^{m},{u}_{3}^{m}) will be parameterized by the vector
{\alpha}^{m}:=[{\alpha}_{{x}_{1},i,j,k}^{m},{\alpha}_{{x}_{2},i,j,k}^{m},{\alpha}_{{x}_{3},i,j,k}^{m}]
as
{u}^{m}({x}_{1},{x}_{2},{x}_{3},\alpha )=\left[\begin{array}{c}{u}_{1}^{m}({x}_{1},{x}_{2},{x}_{3},\alpha )\\ {u}_{2}^{m}({x}_{1},{x}_{2},{x}_{3},\alpha )\\ {u}_{3}^{m}({x}_{1},{x}_{2},{x}_{3},\alpha )\end{array}\right]=\left[\begin{array}{c}{\sum}_{i,j,k}{\alpha}_{{x}_{1},i,j,k}^{m}{\mathrm{\Phi}}_{i,j,k}^{m}({x}_{1},{x}_{2},{x}_{3},\alpha )\\ {\sum}_{i,j,k}{\alpha}_{{x}_{2},i,j,k}^{m}{\mathrm{\Phi}}_{i,j,k}^{m}({x}_{1},{x}_{2},{x}_{3},\alpha )\\ {\sum}_{i,j,k}{\alpha}_{{x}_{3},i,j,k}^{m}{\mathrm{\Phi}}_{i,j,k}^{m}({x}_{1},{x}_{2},{x}_{3},\alpha )\end{array}\right].
(4)
Elements (scaling functions) of a basis {V}_{m} are 3D functions that are translated across the cubical grid {\mathrm{\Omega}}_{\mathbf{n},\mathbf{k}}. These functions are a tensor product of the 1D scaling and wavelet functions as
\begin{array}{c}{\mathrm{\Phi}}^{1}=\varphi ({2}^{j}{x}_{1}{k}_{1})\varphi ({2}^{j}{x}_{2}{k}_{2})\varphi ({2}^{j}{x}_{3}{k}_{3}),\hfill \\ {\mathrm{\Phi}}^{2}=\varphi ({2}^{j}{x}_{1}{k}_{1})\varphi ({2}^{j}{x}_{2}{k}_{2})\psi ({2}^{j}{x}_{3}{k}_{3}),\hfill \\ {\mathrm{\Phi}}^{3}=\varphi ({2}^{j}{x}_{1}{k}_{1})\psi ({2}^{j}{x}_{2}{k}_{2})\varphi ({2}^{j}{x}_{3}{k}_{3}),\hfill \\ {\mathrm{\Phi}}^{4}=\varphi ({2}^{j}{x}_{1}{k}_{1})\psi ({2}^{j}{x}_{2}{k}_{2})\psi ({2}^{j}{x}_{3}{k}_{3}),\hfill \\ {\mathrm{\Phi}}^{5}=\psi ({2}^{j}{x}_{1}{k}_{1})\varphi ({2}^{j}{x}_{2}{k}_{2})\varphi ({2}^{j}{x}_{3}{k}_{3}),\hfill \\ {\mathrm{\Phi}}^{6}=\psi ({2}^{j}{x}_{1}{k}_{1})\varphi ({2}^{j}{x}_{2}{k}_{2})\psi ({2}^{j}{x}_{3}{k}_{3}),\hfill \\ {\mathrm{\Phi}}^{7}=\psi ({2}^{j}{x}_{1}{k}_{1})\psi ({2}^{j}{x}_{2}{k}_{2})\varphi ({2}^{j}{x}_{3}{k}_{3}),\hfill \\ {\mathrm{\Phi}}^{8}=\psi ({2}^{j}{x}_{1}{k}_{1})\psi ({2}^{j}{x}_{2}{k}_{2})\psi ({2}^{j}{x}_{3}{k}_{3}).\hfill \end{array}