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Image registration is a widely used task of image analysis with applications in many fields. Its classical formulation and current improvements are given in the spatial domain. In this paper a regularization term based on fractional order derivatives is formulated. This term is defined and implemented in the frequency domain by translating the energy functional into the frequency domain and obtaining the Euler-Lagrange equations which minimize it. The new regularization term leads to a simple formulation and design, being applicable to higher dimensions by using the corresponding multidimensional Fourier transform. The proposed regularization term allows for a real gradual transition from a diffusion registration to a curvature registration which is best suited to some applications and it is not possible in the spatial domain. Results with 3D actual images show the validity of this approach.

Image registration is the process of finding out the global and/or local correspondence between two images, template

Recently, new approaches have arisen for regularization terms in nonparametric image registration, for example, minimal curvature [

This paper is structured as follows: we start out with the variational framework for the registration problem, proposing the hybrid regularization term. In the following section, the energy functional is adapted to handle discrete

The

The nonparametric registration can be approached in terms of the variational calculus, by defining the joint energy functional to be minimized:

The distance measure

The regularization term

Because digital datasets are handled, which are typically encoded by uniformly distributed picture elements, the finite difference method is the natural approach to approximate (

This paper proposes a novel regularization term for

Taking into account that the Fourier transform of a Kronecker's delta shifted to

The resulting frequency operator

The regularization term (

According to the variational calculus, a necessary condition for a minimizer

For the Gâteaux derivative of

For the Gâteaux derivative of

To solve the Euler-Lagrange equations (

As shown in (

The frequency point of view allows to understand the internal forces, with the restrictions imposed on the displacement field by the regularizer, as a low pass filtering. In (

Spectra of

The practical implementation takes into account that datasets are discrete and then the spatial variable

In this section, the proposed regularization term is tested on a experiment involving three-dimensional (

3D view of two cylinders of an engine block: (a) reference dataset and (b) template dataset.

Figure

Parameters and numerical results for the registration of 3D industrial images. Each row corresponds, respectively, to registered datasets shown in Figure

Regularizer | PSNR | CR | |||
---|---|---|---|---|---|

Diffusion ( | 50 | 39.43 dB | 99.73% | ||

Curvature ( | 65 | 41.55 dB | 99.85% | ||

Fractional ( | 40 | 41.64 dB | 99.86% |

Registration results of two cylinders of an engine block: (a) diffusion (

Registration of industrial images using the proposed fractional regularization term. First column: views of the reference dataset. Second column: views of the template dataset. Third column: views of the registered template (

Registration of industrial images using the proposed fractional regularization term. First column: slices of the reference dataset. Second column: slices of the template dataset. Third column: slices of the registered template (

One point to take into account is about the boundary conditions considered, since spatial domain-based schemes impose Von Neumann boundary conditions, and the frequency domain-based scheme imposes periodic boundary conditions (actually, due to the use of the

In this paper, a fractional regularization term is proposed for approaching the variational image registration problem. The joint energy functional

The proposed regularization term implemented in the frequency domain allows for a gradual transition between diffusion registration and curvature registration, thus providing better registration results in terms of both similarity of the images and smoothness of the transformation, and in a lower number of iterations of the algorithm. The use of the frequency domain (especially if the

The Gâteaux derivative of the proposed energy term is given by

This work is partially supported by