Thin plate splines tps are a spline based technique for data interpolation and smoothing. We now turn to a least squares method based on thinplate splines. Thin plate spline thin plate spline tps is a technique for interpolating and smoothing surface over scattered data 1. Automatic lung lobe segmentation using particles, thin plate. Duchon 11for thin plate spline interpolation can beimproved byh12. I am using thin plate spline to interpolation rainfall data. Thinplate splines have the edge in terms of mse over the cubic regression spline, but are much more costly to set up the basis function for. Introduction to tps thin plate spline, or tps for short, is an interpolation method that finds a minimally bended smooth surface that passes through all given points. The equation of an infinite thin plate under the influence of point loads and mid. Weimer and warren subdivision schemes for thin plate splines 1. The purpose of this chapter is to present an introduction to thinplate spline interpolation and indicate how it can be a useful tool in medical imaging applications. Weillustrate that h matrix techniques can successfully be employed to solve very large thin plate spline interpolation problems.
There are no free parameters that need manual tuning. An important diagnostic associated with thin plate smoothing spline analyses is the signal of the fitted spline, as estimated by the trace of the influence matrix. Thin plate spline interpolation problem geonet, the esri. Thin plate spline interpolation and deformation github. This can be useful in its own right, shows how the mathematics work, and allows easy visualization. Duchon 11 for thin plate spline interpolation can be improved by. The thinplate spline was visualized as a technique which provided an optimal solution to the problem of deforming a flat piece of sheet metal at a finite number of locations wm04. Thinplate spline example image processing development. The thinplate spline method uses the tpaps function. They are an important special case of a polyharmonic spline. Thinplate splines 4 comparing kriging and spline interpolation 5 references d g rossiter cu empirical interpolation. There are quite a few techniques out there that provide a smooth interpolation between a set of control points. Some algorithms for thin plate spline interpolation to functions of two. However, both models have almost identical results in terms of deviance explained and pattern of residuals.
To solve this problem, this paper proposes an improved skewness balancing filtering algorithm based on thin plate spline tps interpolation. This occurred with both transformed and untransformed data. The purpose of this chapter is to present an introduction to thin plate spline interpolation and indicate how it can be a useful tool in medical imaging applications. These techniques have a wide range of applications in computer graphics and computer vision. The smoothers are constructed by a simple transformation and truncation of the basis that arises from the solution of the thin plate spline smoothing problem and are optimal in the sense that the truncation is designed to result in the minimum. Washko6 1 channing laboratory, brigham and womens hospital, boston, ma 2 laboratory of mathematics in imaging, brigham and womens hospital. A very classical example of such a local subdivision pro. Thinplate spline interpolation is an effective interpolation method under these conditions, and has been widely used in the context of deforming one image into registration with another. Apr 16, 2014 thin plate spline interpolation and deformation. Error indicators and adaptive refinement of finite element.
Creating highresolution digital elevation model using. An example of such scattered data is visualized in fig. Subdivision schemes for thin plate splines rice computer science. Automatic lung lobe segmentation using particles, thin plate splines, and maximum a posteriori estimation james c. In 3we then consider twodimensional surface interpolation. Thin plate splines tps are a splinebased technique for data interpolation and smoothing. Robust point matching rpm is a common extension and shortly known as the tpsrpm algorithm. Automatic lung lobe segmentation using particles, thin. The goal of scattered data interpolation techniques is to construct a typically smooth function from a set of unorganized samples. Smooth data interpolation ainterpolating spline bsmoothing spline figure. The spline surface represents a thin metal sheet that is constrained not to move at the sample points x i. The linear and nearest neighbor methods are fast, but the resulting curves are not very smooth.
Initial application of the procedure splina to fit a second order thin plate smoothing spline to the data set yielded exact interpolation. In this paper, the idea of thin plate spline interpolation will be carried over from the. The thin plate spline tps is a commonly used basis function for represen ting co ordinate mappings from r 2 to. This is referred to as the eulerlagrange differential equation. First, we adopt a semiautomatic algorithm based on ziplock. Approximatethinplatesplinemappings cornell university. As the smoothing parameter varies from 0 to 1, the smoothing spline varies, from the leastsquares approximation to the data by a linear polynomial when p is 0, to the thin plate spline interpolant to the data when p is 1. In chapter 4 the procedure of the dem creation is presented. At last, they are used in a typical registration framework where a few matching points are provided. I have the shape of a face with together with the reconstruction of that face and i want to model the corresponding image of the initial shape. Thin plate regression splines wood 2003 journal of.
Pdf an improved skewness balancing filtering algorithm. A thinplate spline is welldefined on any dimension. Error estimates for thin plate spline approximation in the disc. I discuss the production of low rank smoothers for d.
My algorithm seems to work correctly when the bounding box of the initial scattered data has an aspect ratio close to 1. The thin plate spline is the 2d generalization of the cubic spline. Warping using thin plate splines file exchange matlab central. But for thin plate spline it is needed to modify the mathematics so that it can take into account both the dimensions in single mathematical interpretation. Thin plate spline interpolation is an effective interpolation method under these conditions, and has been widely used in the context of deforming one image into registration with another. Interpolation of rainfall data with thin plate smoothing. It was reasoned that this was due to short range correlation in the. The thin plate spline tps is an interpolation approach that has been developed. Before introducing the thin plate spline warping algorithm, we will quickly go through the radial basis function interpolation, which is the general form of the thin plate spline interpolation problem. I thought it was the name of a regularization technique for bsplines surface fitting i. To develop the spherical tps, we utilize the idea of an elastic shell that is attributed with the bending energy and the. Most filtering algorithms suffer from complex parameter settings or threshold adjusting. Tps is 2d interpolation which represents a simple thin metal sheet and the sheet is formed in such a. A thinplate spline interpolates over a nonuniformly sampled input space, which in this case is a training set of image thumbnails and associated illumination chromaticities.
The construction is based on choosing a function that minimizes an integral that represents the bending energy of a surface. Based mostly on approximation methods for thin plate spline mappings and principal warps by gianluca donato and serge belongie, 2002. As an example, let us construct the function fx for which the arclength integral is a. Approximation methods for thin plate spline mappings and. Approximate thin plate spline mappings cornell vision pages. Id be interested to see whether the effects of the two splines are really that different given the uncertainty in their estimation at end points of a variable, esp if the data is patchy. The origins of thin plate splines in 2d appears to be 1,2. Pdf image morphing with snake model and thin plate. A disadvantage of thin plate spline interpolation to n values of a function of two variables is. For example, he showed that a gain of h12 is possible if f. Thin plate splines thin plate splines were introduced to geometric design by duchon 6 in 1976 as the minimizer of the functional uuf mapsir2 f2 2f2 uv f 2 vv dudv 1 with respect to some additional interpolation constraints. The smoothers are constructed by a simple transformation and truncation of the basis that arises. The smoothers are constructed by a simple transformation and truncation of the basis that arises from the solution of the thin plate spline smoothing problem and are optimal in the sense that the truncation is designed to result in the.
The thin plate spline tps is an interpolation approach that has been developed to investigate a frequently occurring problem in geosciences. Generally speaking, the computation cost of initially solving the inverse matrix for the thin plate spline is on3 where n is the number of points used in. I am not sure i understand what thinplatesplines are i thought it was the name of a regularization technique for bsplines surface fitting i. Creating highresolution digital elevation model using thin. Thin plate splines 4 comparing kriging and spline interpolation 5 references d g rossiter cu empirical interpolation. Warping using thin plate splines file exchange matlab. There are quite a few techniques out there that provide a. Thus, a thin plate spline can be defined as the function f. The thin plate spline method uses the tpaps function. Use of a cubic spline for x, as above, results in quite a different shape for the fitted function compared to the default thin plate spline for that part of the space where x values are sparse. To reduce the size of the training set, incremental k medians are applied. Opencv suitable implementation for thin plate spline warping.
I am not sure i understand what thin plate splines are. But reading papers, i understand they are a new type of splines i. In this paper, we carry over the concept of the thin plate spline from the plane to the sphere. Examples illustrating the behavior of the interpolation function are given.
As i understand it tps is just a special case of radial basis function interpolation. The problem is that i get negative values for rainfall which is impossible is that normal. Key features of thin plate smoothing spline analyses are their robustness and operational simplicity. However, scaling one of the data points coordinates changes the interpolation result. Illumination estimation via thinplate spline interpolation.
Ultrasonic elastogram generation by 2d thin plate smoothing. The type of interpolant to use depends on the characteristics of the data being fit, the required smoothness of the curve, speed considerations, postfit analysis requirements, and so on. They were introduced to geometric design by duchon. Outline 1 spatial prediction 2 1d splines 1d natural splines 1d smoothing splines 3 2d. Standard thin plate splines require the solution of a dense linear system of equations whose size increases with the number of. Opencv suitable implementation for thin plate spline. Thinplate spline thinplate spline tps is a technique for interpolating and smoothing surface over scattered data 1. Compute thin plate spline interpolation and deformation on a 3d surface.
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