Vector Splines 
Investigators: Pouya Dehghani Tafti, Michael Unser 

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. 

A fruitful approach for defining splines in multiple dimensions is to formulate the datafitting 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 seminorm involving differential operators. In the scalar case, this naturally leads to the definition of thinplate splines and of radialbasis 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 thinplate splines for image registration and vectorfield reconstruction. 

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 rotationinvariant. 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 subfunctionals: (i) Duchon's scalar seminorm 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 meansquare distance measure between the target image and the warped source subject to some vectorspline regularization constraint. For computational efficiency, we express the deformation field in a Bspline basis. Our algorithm is also able to handle softlandmark 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 vectorreconstruction 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 illposedness of the problem can be alleviated by acquiring measurements along two different directions. So far, we have implemented the firstorder version of the method (the direct vector counterpart of thinplate splines) using the same type of Bspline discretization as in the previous application. The advantage of this scheme is that it yields a wellconditioned, sparse system of equations, similar to what we have achieved previously in the scalar case. We did apply this vectorspline reconstruction method to cardiac motionfield 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. 

Collaborations: Dr. Patrick Hunziker (Kantonsspital Basel) 

Past Investigators: Muthuvel Arigovindan 


Funding: Swiss Science Foundation under Grant 200020109415 


[2]  M. Arigovindan, M. Sühling, C. Jansen, P. Hunziker, M. Unser, "Full Flow/MotionField Recovery from PulsedWave Ultrasound Doppler Data," Proceedings of the Third IEEE International Symposium on Biomedical Imaging: From Nano to Macro (ISBI'06), Arlington VA, USA, April 69, 2006, pp. 213216.

[5]  M. Arigovindan, M. Sühling, C. Jansen, P. Hunziker, M. Unser, "Bimodal Ultrasound Motion Recovery from Incomplete Data," Proceedings of the 2004 Annual Meeting of the Swiss Society of Biomedical Engineering (SSBE'04), Zürich ZH, Swiss Confederation, September 23, 2004, poster 47.


