Fitting values at nd grid with tensorproduct splines matlab. If a is an mbyn matrix and b is a pbyq matrix, then krona,b is an mpbynq matrix formed by taking all possible products between the elements of a and the matrix b. This example shows how to construct splines in various ways using the spline functions in curve fitting toolbox. In this case, it is more efficient to store just the nonzeros and their indices. Most spline construction commands in the toolbox take advantage of this. This is the intent in all other cases, except that, for a dvalued mvariate function, dvectors replaces mvectors. Introduction since curve fitting toolbox can handle splines with vector coefficients, it is easy to implement interpolation or approximation to gridded data by tensor product splines. Constructing and working with stform splines introduction to the stform. In this case the spapi function calls the aptknt function to determine a workable, but not necessarily optimal, knot sequence for the given sites x. Curve fitting toolbox documentation mathworks deutschland. All scripts are written in matlab and comments are in french.
The ppform of such a bivariate spline comprises, analogously, a cell array of break sequences, a multidimensional coefficient array, a vector of number pieces, and a vector of polynomial orders. After the concatenation, it might be necessary to change the order of dimensional indices so that the first dimension is the newlycreated dimension. This example shows how to use the spline commands in curve fitting toolbox to fit tensor product splines to bivariate gridded data. You want other types of splines such as ppform, bform, tensor product, rational, and stform thinplate splines. A 1way tensor is a vector and a 2way tensor is a matrix. This illustrates that cubic spline interpolation is essentially local. Curve fitting toolbox provides an app and functions for fitting curves and surfaces to data. Tensor product surfaces usually, domain is rectangular. The fit function only allows scalarvalued splines you want other types of splines such as ppform, bform, tensor product, rational, and stform thinplate splines. The spapi function then provides a corresponding knot sequence for the i. The structure of the data in cases a and b is such that the corresponding algorithms are particularly. These multivariate splines come in both standard forms, the bform and the ppform, and their construction and use parallels entirely that of the univariate splines discussed in previous.
Product of two bspline tensor product surfaces matlab. From point clouds to tensor product bspline surfaces by lavanya sita tekumalla a masters thesis submitted to the faculty of the university of utah in partial ful. The mathematical equation for a tensor product spline is shown in the my question. For example, here is a good approximation to a torus. You want to combine the results with other splines, e. Here, each knorki is either a knot sequence, or a positive integer specifying the polynomial order used in the ith variable.
We used 2d plane as since it is enough and the simplest for computer representation. Catmullrom splines can be easily generalized to any. Splines can be useful in scenarios where using a single approximating polynomial is impractical. Constructing and working with stform splines matlab. Trivariate splines solve a special case of scattered data interpolation problem in the volume bounded by two concentric spheres. I need to convert this equation into a form matlab the curve fitting tool can understand and work with.
Spline interpolation matlab spapi mathworks italia. A univariate spline f is specified by its nondecreasing knot sequence t and by its b spline coefficient sequence a. There are several methods for multidimansional interpolation, but in your. For example, a trivariate spline in bform is given. Evaluate spline function matlab fnval mathworks benelux. A bivariate tensor product is any weighted sum of products of a function in the first. We have designed a license that enables researchers to freely use the toolbox for their work and others to evaluate the toolbox at no cost. Fitting values at nd grid with tensorproduct splines. Functions used for the definition of tensor product smooths and interactions within gam model formulae.
Spline interpolation matlab spapi mathworks deutschland. Tensor product b spline surfaces are commonly used as standard modeling tool in computer aided geometric design and for numerical simulation in isogeometric analysis. Here, each knorki is either a knot sequence, or else is a positive integer specifying the polynomial order to be used in the ith variable, thus leaving it to spapi to provide a corresponding knot sequence for the ith variable. Construction create splines including bform, tensor product, nurbs and other rational splines. If a is a true 3d array, something like a rand4,10,3 and assuming that b stays as a 2d array, then each a. The functions do not evaluate the smooth they exists purely to help set up a model using tensor product based smooths. A sparse tensor is a tensor where only a small fraction of the elements are nonzero. Can you explain me what is the code for import a tensor. I just have data points in three vectors x,y,z which i loaded in the curve fitting tool. Multivariate splines can be obtained from univariate splines by the tensor product construct. You can also run the example bivariate tensor product splines. Tensor product spline surfaces all spline curves studied before work well in space of any dimension.
How do i compute this outer product efficiently in matlab if there are more than two vectors. This article introduces the tensor network b spline model for the regularized identi cation of nonlinear systems using a nonlinear autoregressive exogenous narx approach. How can i extract the values of data plotted in a graph which is available in pdf form. This section discusses these aspects of the chebyshev spline construction.
Run the command by entering it in the matlab command window. I n order n1 bar over operator indicates contracted product. Finally, in all of this, one might be looking for functions of more than one variable, such as tensor product splines. Bivariate tensor product splines with values in r3 give surfaces. The source code of this part is located in the splines folder. A multivariate function form quite different from the tensor product construct is the scattered translates form, or stform for short. In the curve fitting toolbox spline functions, vectors are column vectors. Vectorvalued splines are also used in the approximation to gridded data, in any number of variables, using tensor product splines. See multivariate tensor product splines for a discussion of multivariate splines. Note that the hierarchical b spline model may be applied in combination with uniform and nonuniform re nement, di erent degrees and smoothness, and related. The essential idea is that contraction of a tensor with a vector is equivalent to matrix multiplication of that vector with an unfolded version of the tensor.
Curve fitting toolbox splines and matlab splines matlab. Curve fitting toolbox spline functions contain versions of the essential matlab programs of the bspline package extended to handle also vectorvalued splines as described in a practical guide to splines, applied math. There are two fundamental problems with surfaces in machine vision. Similarly, the ppform of a tensor product spline is specified by break sequences in each of the variables and, for each hyperrectangle thereby specified, a coefficient array. Nov 20, 2008 the twodimensional tensor product spline model with noknot cubic spline latency function and linear spline with one knot at the median for exposure gave a statistically better fit relative to a model that fitted a noknot cubic spline weight function for latency with a global linear function for exposure lr 61. Fitting values at nd grid with tensor product splines. Tensor network theory is used to alleviate the curse of dimensionality of multivariate b splines by representing the highdimensional weight tensor as a lowrank approximation. Volume data interpolation using tensor products of. Curve fitting toolbox splines and matlab splines mathworks. A is a second order tensor and b is a fourth order tensor. Curve fitting toolbox provides functions for advanced spline operations, including breakknot manipulation, optimal knot placement, and datapoint weighting.
How can i solve a multidimensional interpolation problem. Algorithms exist for leastsquares tensor product splineapproximation to data a on a rectangular mesh, b on a family of parallel lines and c that is generally scattered. It is possible to retain the interpolating cubic spline in a form suitable for subsequent evaluation, or for calculating its derivatives, or for other manipulations. Once the surfaces are recon structed onto a uniform grid, the surfaces must be segmented into different. Curve fitting toolbox supports a variety of interpolation methods, including b splines, thin plate splines, and tensor product splines. Adaptive scattered data fitting by extension of local.
When the coefficients are 2vectors or 3vectors, f is a curve in r 2 or r 3 and the. A spline is a series of polynomials joined at knots. Use vectorvalued splines to approximate gridded data in any number of variables using tensorproduct splines. Algorithms and data structures for truncated hierarchical. Multivariate tensor product splines introduction to multivariate tensor product splines. Tensor spline approximation eric grosse computer science department stanford university stanford, california 94305 submitted by robert j. The small matlab program, which generates the figure, can be downloaded. You want to specify breaks, optimize knot placement, and use specialized functions for spline manipulation such as differentiation and integration. Options for spline fitting in curve fitting toolbox, including curve fitting app, using the fit function, or using specialized spline functions. Bform, tensor product, rational, and stform thinplate splines. You can use csapi with scalars, vectors, matrices, and ndarrays. Continuity constraints for surfaces are more interesting than for curves.
Now i want to fit a tensor product spline to this data points. Because the toolbox can handle splines with vector coefficients, it is easy to implement interpolation or approximation to gridded data by tensor product splines, as the following illustration is meant to show. Using matlab to fit tensor product splines to bivariate. How does a tensor product spline equation in matlab code. Because it helps show the relevance of this work, please cite this software and associated papers. Because the toolbox can handle splines with vector coefficients, it is easy to implement interpolation or approximation to gridded data by tensor product splines. The coefficients may be columnvectors, matrices, even ndarrays.
Volume data interpolation using tensor products of spherical and radial splines v. Correspondingly, this spline is of order k in x, of order l in y, and of order m in z. Define tensor product smooths or tensor product interactions in gam formulae description. For this example, the grid is a 51by61 uniform grid. Since curve fitting toolbox can handle splines with vector coefficients, it is easy to implement interpolation or approximation to gridded data by tensor product splines. Interpolation you can construct a cubic spline interpolant that matches the cosine function at the following sites x, using the csapi command. Tensor multiplication with a vector llet a be a tensor of size i 1. Continuity constraints for surfaces are more interesting than for curves our curves have form. Adaptive scattered data tting by extension of local. Tensors also known as multidimensional arrays or nway arrays are used in a variety of applications ranging from chemometrics to network analysis. We used the subdivision algorithm on closed b spline curves. Surfaces must be reconstructed from sparse depth measurements that may contain outliers. The toolbox provides polynomial spline functions in any number of variables, as tensor products of univariate splines. How to take outer product of more than two matrices in one.
For tensor product bsplines, knot placement has a global effect. However, when considering tensor product grids, there is no possibility of a localized mesh refinement without propagation of the refinement outside the region of interest. As the name suggests, it uses arbitrary or scattered translates. My lack of understanding of how the spline tensor product formalism is supposed to work is the issue, not the code. I am trying to construct a toy example but i cant seem to get my example to work. Spline interpolation matlab spapi mathworks benelux. Using the curve fitting app or the fit function, you can fit.
One need just take its control points in corresponding space. Fit a smoothing spline to bivariate data generated by the peaks function with added uniform noise. The class of tensor product cubic splines is considered in an optimization process for twodimensional polyacrylamide gel electrophoresis 2dpage image alignment problems arising in proteomics. The same spline construction commands are used, only the form of the input differs. However, when i write this code in matlab, it gives the following error.
Pdf protein image alignment via tensor product cubic splines. Matlab sparse matrix facilities are used in the spline toolbox when that is. Feb 06, 2015 a tensor is a multidimensional or nway array. Approximation by tensor product splines fitting a spline to titanium test data cubic spline interpolation fitting a spline to titanium test data. Curve fitting toolbox splines and matlab splines curve fitting toolbox splines. Here, each knorki is either a knot sequence, or else is a positive integer specifying the polynomial order to be used in the ith variable, thus leaving it to spapi to. Gouraud shading on a rectangle this gives a bilinear interpolate of the rectangles vertex values. In contrast,direct algorithms for tensor product splineinterpolation are available only for data type a.
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