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Concept# Thin plate spline

Summary

Thin plate splines (TPS) are a spline-based technique for data interpolation and smoothing. They were introduced to geometric design by Duchon. They are an important special case of a polyharmonic spline. Robust Point Matching (RPM) is a common extension and shortly known as the TPS-RPM algorithm.
Physical analogy
The name thin plate spline refers to a physical analogy involving the bending of a thin sheet of metal. Just as the metal has rigidity, the TPS fit resists bending also, implying a penalty involving the smoothness of the fitted surface. In the physical setting, the deflection is in the z direction, orthogonal to the plane. In order to apply this idea to the problem of coordinate transformation, one interprets the lifting of the plate as a displacement of the x or y coordinates within the plane. In 2D cases, given a set of K corresponding points, the TPS warp is de

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1998We address the problem of reconstructing scalar and vector functions from non-uniform data. The reconstruction problem is formulated as a minimization problem where the cost is a weighted sum of two terms. The first data term is the quadratic measure of goodness of fit, whereas the second regularization term is a smoothness functional. We concentrate on the case where the later is a semi-norm involving differential operators. We are interested in a solution that is invariant with respect to scaling and rotation of the input data. We first show that this is achieved whenever the smoothness functional is both scale- and rotation-invariant. In the first part of the thesis, we address the scalar problem. An elegant solution having the above mentioned invariant properties is provided by Duchon's method of thin-plate splines. Unfortunately, the solution involves radial basis functions that are poorly conditioned and becomes impractical when the number of samples is large. We propose a computationally efficient alternative where the minimization is carried out within the space of uniform B-splines. We show how the B-spline coefficients of the solution can be obtained by solving a well-conditioned, sparse linear system of equations. By taking advantage of the refinable nature of B-splines, we devise a fast multiresolution-multigrid algorithm. We demonstrate the effectiveness of this method in the context of image processing. Next, we consider the reconstruction of vector functions from projected samples, meaning that the input data do not contain the full vector values, but only some directional components. We first define the rotational invariance and the scale invariance of a vector smoothness functional, and then characterize the complete family of such functionals. We show that such a functional is composed of a weighted sum of two sub-functionals: (i) Duchon's scalar semi-norm applied on the divergence field; (ii) and the same applied to each component of the rotational field. This forms a three-parameter family, where the first two are the Duchon's order of the above sub-functionals, and the third is their relative weight. Our family is general enough to include all vector spline formulations that have been proposed so far. We provide the analytical solution for this minimization problem and show that the solution can be expressed as a weighted sum of vector basis functions, which we call the generalized vector splines. We construct the linear system of equations that yields the required weights. As in the scalar case, we also provide an alternative B-spline solution for this problem, and propose a fast multigrid algorithm. Finally, we apply our vector field reconstruction method to cardiac motion field recovery from ultrasound pulsed wave Doppler data, and demonstrate its clinical potential.

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