Summary

Our purpose is to identify new vector splines based on the minimization of a regularization functional that induces some coupling between the vector components. We also investigate the application of such vector splines to the problems of (i) image registration and to (ii) the reconstruction of vector fields from incomplete data.

Introduction

A fruitful approach for defining splines in multiple dimensions is to formulate the data-fitting task (interpolation or approximation) as a minimization problem where the cost is a weighted sum of two terms. The first data term is a quadratic measure of the goodness of fit, whereas the second regularization term is a smoothness functional; typically, a semi-norm involving differential operators. In the scalar case, this naturally leads to the definition of thin-plate splines and of radial-basis functions which are very popular in engineering applications. Our present goal is to show that this type of formulation can also yield true vector solutions that are interesting alternatives to thin-plate splines for image registration and vector-field reconstruction.

Main Contribution

An important practical requirement is that the spline solution above be invariant with respect to scaling and rotation of the input data. We proved that this is achieved whenever the smoothness functional is both scale- and rotation-invariant. In the case of vector data, this leads to the identification of an augmented regularization functional that tends to induce coupling among the vector components. It is a sum of two sub-functionals: (i) Duchon's scalar semi-norm applied on the divergence field; and (ii) the same, applied to each component of the rotational field.

A natural application of such vector splines is elastic image registration. Specifically, we have proposed to solve the registration problem by minimizing a pixelwise mean-square distance measure between the target image and the warped source subject to some vector-spline regularization constraint. For computational efficiency, we express the deformation field in a B-spline basis. Our algorithm is also able to handle soft-landmark constraints, which is particularly useful when parts of the images contain little information or when its repartition is uneven.

We have also investigated a nonstandard vector-reconstruction problem with data consisting of ultrasound Doppler measurements. The main difficulty here is that the measurements are incomplete, for they do only capture the velocity component along the beam direction. The ill-posedness of the problem can be alleviated by acquiring measurements along two different directions. So far, we have implemented the first-order version of the method (the direct vector counterpart of thin-plate splines) using the same type of B-spline discretization as in the previous application. The advantage of this scheme is that it yields a well-conditioned, sparse system of equations, similar to what we have achieved previously in the scalar case. We did apply this vector-spline reconstruction method to cardiac motion-field recovery. We have validated our method using real phantom data for which the ground truth is known. More recently, we were able to reconstruct the blood flow in the carotid bifurcation of a human subject, confirming the presence of a flow asymmetry that had been predicted theoretically. We are quite excited about the prospects of this novel approach to flow imaging. The quality of the work has also been recognized internally by the EPFL which has awarded its 2006 best thesis award to Muthuvel Arigovindan.