Pdf protein image alignment via tensor product cubic splines. This example shows how to construct splines in various ways using the spline functions in curve fitting toolbox. 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. Tensor product surfaces usually, domain is rectangular. Functions used for the definition of tensor product smooths and interactions within gam model formulae. Multivariate tensor product splines introduction to multivariate tensor product 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. Surfaces must be reconstructed from sparse depth measurements that may contain outliers. A 1way tensor is a vector and a 2way tensor is a matrix.
As the name suggests, it uses arbitrary or scattered translates. Fitting values at nd grid with tensorproduct splines. Tensors also known as multidimensional arrays or nway arrays are used in a variety of applications ranging from chemometrics to network analysis. Constructing and working with stform splines introduction to the stform. A spline is a series of polynomials joined at knots. Once the surfaces are recon structed onto a uniform grid, the surfaces must be segmented into different. Using the curve fitting app or the fit function, you can fit. Curve fitting toolbox splines and matlab splines curve fitting toolbox splines. Most spline construction commands in the toolbox take advantage of this. 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 small matlab program, which generates the figure, can be downloaded. A bivariate tensor product is any weighted sum of products of a function in the first.
Using matlab to fit tensor product splines to bivariate. The mathematical equation for a tensor product spline is shown in the my question. Correspondingly, this spline is of order k in x, of order l in y, and of order m in z. Can you explain me what is the code for import a tensor. Curve fitting toolbox supports a variety of interpolation methods, including b splines, thin plate splines, and tensor product splines. 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. Spline interpolation matlab spapi mathworks benelux. When the coefficients are 2vectors or 3vectors, f is a curve in r 2 or r 3 and the. 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. 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.
We used the subdivision algorithm on closed b spline curves. Splines can be useful in scenarios where using a single approximating polynomial is impractical. Tensor product spline surfaces all spline curves studied before work well in space of any dimension. This example shows how to use the spline commands in curve fitting toolbox to fit tensor product splines to bivariate gridded data. Algorithms and data structures for truncated hierarchical. 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.
Define tensor product smooths or tensor product interactions in gam formulae description. There are several methods for multidimansional interpolation, but in your. How to take outer product of more than two matrices in one. Vectorvalued splines are also used in the approximation to gridded data, in any number of variables, using tensor product splines. In the curve fitting toolbox spline functions, vectors are column vectors. This section discusses these aspects of the chebyshev spline construction. The coefficients may be columnvectors, matrices, even ndarrays. Because it helps show the relevance of this work, please cite this software and associated papers. 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.
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. 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. However, you might be interested in seeing a detailed description of how approximation to gridded data by tensor products is actually done. You want to specify breaks, optimize knot placement, and use specialized functions for spline manipulation such as differentiation and integration. The source code of this part is located in the splines folder. Construction create splines including bform, tensor product, nurbs and other rational splines. Because the toolbox can handle splines with vector coefficients, it is easy to implement interpolation or approximation to gridded data by tensor product splines. How can i solve a multidimensional interpolation problem. Tensor product b spline surfaces are commonly used as standard modeling tool in computer aided geometric design and for numerical simulation in isogeometric analysis.
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. Finally, in all of this, one might be looking for functions of more than one variable, such as tensor product splines. 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. Continuity constraints for surfaces are more interesting than for curves our curves have form. A sparse tensor is a tensor where only a small fraction of the elements are nonzero.
Catmullrom splines can be easily generalized to any. This illustrates that cubic spline interpolation is essentially local. Use csaps to obtain the new, smoothed data points and the smoothing parameters csaps determines for the fit. Trivariate splines solve a special case of scattered data interpolation problem in the volume bounded by two concentric spheres. 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. Curve fitting toolbox documentation mathworks deutschland. Run the command by entering it in the matlab command window. However, when i write this code in matlab, it gives the following error. After the concatenation, it might be necessary to change the order of dimensional indices so that the first dimension is the newlycreated dimension. Tensorproduct spline interpolation to data on or near a. Fit a smoothing spline to bivariate data generated by the peaks function with added uniform noise.
You want other types of splines such as ppform, bform, tensor product, rational, and stform thinplate splines. This is the intent in all other cases, except that, for a dvalued mvariate function, dvectors replaces mvectors. From point clouds to tensor product b spline surfaces by lavanya sita tekumalla a masters thesis submitted to the faculty of the university of utah in partial ful. Evaluate spline function matlab fnval mathworks benelux. K krona,b returns the kronecker tensor product of matrices a and b. The structure of the data in cases a and b is such that the corresponding algorithms are particularly. Spline interpolation matlab spapi mathworks deutschland.
For example, here is a good approximation to a torus. A multivariate function form quite different from the tensor product construct is the scattered translates form, or stform for short. Matlab sparse matrix facilities are used in the spline toolbox when that is. Constructing and working with stform splines matlab. 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. See multivariate tensor product splines for a discussion of multivariate splines. How do i compute this outer product efficiently in matlab if there are more than two vectors. I am trying to construct a toy example but i cant seem to get my example to work. 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. 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. Now i want to fit a tensor product spline to this data points.
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. I need to convert this equation into a form matlab the curve fitting tool can understand and work with. You can also run the example bivariate tensor product splines. For example, a trivariate spline in bform is given. 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. Adaptive scattered data tting by extension of local. Spline interpolation matlab spapi mathworks italia. Create splines including bform, tensorproduct, nurbs, and other rational splines. You want to combine the results with other splines, e. Curve fitting toolbox provides functions for advanced spline operations, including breakknot manipulation, optimal knot placement, and datapoint weighting. One need just take its control points in corresponding space. Bform, tensor product, rational, and stform thinplate splines. 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.
Bivariate tensor product splines with values in r3 give surfaces. The functions do not evaluate the smooth they exists purely to help set up a model using tensor product based smooths. 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. All scripts are written in matlab and comments are in french. I n order n1 bar over operator indicates contracted product. Options for spline fitting in curve fitting toolbox, including curve fitting app, using the fit function, or using specialized spline functions. Product of two bspline tensor product surfaces matlab. Curve fitting toolbox splines and matlab splines matlab. A univariate spline f is specified by its nondecreasing knot sequence t and by its b spline coefficient sequence a. Continuity constraints for surfaces are more interesting than for curves. 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. How can i extract the values of data plotted in a graph which is available in pdf form. We used 2d plane as since it is enough and the simplest for computer representation. The toolbox provides polynomial spline functions in any number of variables, as tensor products of univariate splines.
The spapi function then provides a corresponding knot sequence for the i. Tensor spline approximation eric grosse computer science department stanford university stanford, california 94305 submitted by robert j. You can use csapi with scalars, vectors, matrices, and ndarrays. Curve fitting toolbox splines and matlab splines mathworks.
I just have data points in three vectors x,y,z which i loaded in the curve fitting tool. Fitting values at nd grid with tensor product splines. For tensor product bsplines, knot placement has a global effect. Multivariate splines can be obtained from univariate splines by the tensor product construct. Curve fitting toolbox provides an app and functions for fitting curves and surfaces to data. Volume data interpolation using tensor products of.
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. Here, each knorki is either a knot sequence, or a positive integer specifying the polynomial order used in the ith variable. There are two fundamental problems with surfaces in machine vision. A is a second order tensor and b is a fourth order tensor. This article introduces the tensor network b spline model for the regularized identi cation of nonlinear systems using a nonlinear autoregressive exogenous narx approach. Use vectorvalued splines to approximate gridded data in any number of variables using tensorproduct splines. Fitting values at nd grid with tensorproduct splines matlab.
Feb 06, 2015 a tensor is a multidimensional or nway array. Tensor multiplication with a vector llet a be a tensor of size i 1. In matlab you can do these operations in a vectorized way using the commands reshape, shiftdim, and permute. Volume data interpolation using tensor products of spherical and radial splines v. In this case, it is more efficient to store just the nonzeros and their indices. The same spline construction commands are used, only the form of the input differs. 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. Adaptive scattered data fitting by extension of local. How does a tensor product spline equation in matlab code. My lack of understanding of how the spline tensor product formalism is supposed to work is the issue, not the code. For this example, the grid is a 51by61 uniform grid. Approximation by tensor product splines fitting a spline to titanium test data cubic spline interpolation fitting a spline to titanium test data.
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