matlab 散点拟合面

function [zgrid,xgrid,ygrid] = gridfit(x,y,z,xnodes,ynodes,varargin) % gridfit: estimates a surface on a 2d grid, based on scattered data % Replicates are allowed. All methods extrapolate to the grid % boundaries. Gridfit uses a modified ridge estimator to % generate the surface, where the bias is toward smoothness. % % Gridfit is not an interpolant. Its goal is a smooth surface % that approximates your data, but allows you to control the % amount of smoothing. % % usage #1: zgrid = gridfit(x,y,z,xnodes,ynodes); % usage #2: [zgrid,xgrid,ygrid] = gridfit(x,y,z,xnodes,ynodes); % usage #3: zgrid = gridfit(x,y,z,xnodes,ynodes,prop,val,prop,val,...); % % Arguments: (input) % x,y,z - vectors of equal lengths, containing arbitrary scattered data % The only constraint on x and y is they cannot ALL fall on a % single line in the x-y plane. Replicate points will be treated % in a least squares sense. % % ANY points containing a NaN are ignored in the estimation % % xnodes - vector defining the nodes in the grid in the independent % variable (x). xnodes need not be equally spaced. xnodes % must completely span the data. If they do not, then the % 'extend' property is applied, adjusting the first and last % nodes to be extended as necessary. See below for a complete % description of the 'extend' property. % % If xnodes is a scalar integer, then it specifies the number % of equally spaced nodes between the min and max of the data. % % ynodes - vector defining the nodes in the grid in the independent % variable (y). ynodes need not be equally spaced. % % If ynodes is a scalar integer, then it specifies the number % of equally spaced nodes between the min and max of the data. % % Also see the extend property. % % Additional arguments follow in the form of property/value pairs. % Valid properties are: % 'smoothness', 'interp', 'regularizer', 'solver', 'maxiter' % 'extend', 'tilesize', 'overlap' % % Any UNAMBIGUOUS shortening (even down to a single letter) is % valid for property names. All properties have default values, % chosen (I hope) to give a reasonable result out of the box. % % 'smoothness' - scalar - determines the eventual smoothness of the % estimated surface. A larger value here means the surface % will be smoother. Smoothness must be a non-negative real % number. % % Note: the problem is normalized in advance so that a % smoothness of 1 MAY generate reasonable results. If you % find the result is too smooth, then use a smaller value % for this parameter. Likewise, bumpy surfaces suggest use % of a larger value. (Sometimes, use of an iterative solver % with too small a limit on the maximum number of iterations % will result in non-convergence.) % % DEFAULT: 1 % % % 'interp' - character, denotes the interpolation scheme used % to interpolate the data. % % DEFAULT: 'triangle' % % 'bilinear' - use bilinear interpolation within the grid % (also known as tensor product linear interpolation) % % 'triangle' - split each cell in the grid into a triangle, % then linear interpolation inside each triangle % % 'nearest' - nearest neighbor interpolation. This will % rarely be a good choice, but I included it % as an option for completeness. % % % 'regularizer' - character flag, denotes the regularization % paradignm to be used. There are currently three options. % % DEFAULT: 'gradient' % % 'diffusion' or 'laplacian' - uses a finite difference % approximation to the Laplacian operator (i.e, del^2). % % We can think of the surface as a plate, wherein the % bending rigidity of the plate is specified by the user % as a number relative to the importance of fidelity to % the data. A stiffer plate will result in a smoother % surface overall, but fit the data less well. I've % modeled a simple plate using the Laplacian, del^2. (A % projected enhancement is to do a better job with the % plate equations.) % % We can also view the regularizer as a diffusion problem, % where the relative thermal conductivity is supplied. % Here interpolation is seen as a problem of finding the % steady temperature profile in an object, given a set of % points held at a fixed temperature. Extrapolation will % be linear. Both paradigms are appropriate for a Laplacian % regularizer. % % 'gradient' - attempts to ensure the gradient is as smooth % as possible everywhere. Its subtly different from the % 'diffusion' option, in that here the directional % derivatives are biased to be smooth across cell % boundaries in the grid. % % The gradient option uncouples the terms in the Laplacian. % Think of it as two coupled PDEs instead of one PDE. Why % are they different at all? The terms in the Laplacian % can balance each other. % % 'springs' - uses a spring model connecting nodes to each % other, as well as connecting data points to the nodes % in the grid. This choice will cause any extrapolation % to be as constant as possible. % % Here the smoothing parameter is the relative stiffness % of the springs connecting the nodes to each other compared % to the stiffness of a spting connecting the lattice to % each data point. Since all springs have a rest length % (length at which the spring has zero potential energy) % of zero, any extrapolation will be minimized. % % Note: I don't terribly like the 'springs' strategy. % It tends to drag the surface towards the mean of all % the data. Its been left in only because the paradigm % interests me. % % % 'solver' - character flag - denotes the solver used for the % resulting linear system. Different solvers will have % different solution times depending upon the specific % problem to be solved. Up to a certain size grid, the % direct \ solver will often be speedy, until memory % swaps causes problems. % % What solver should you use? Problems with a significant % amount of extrapolation should avoid lsqr. \ may be % best numerically for small smoothnesss parameters and % high extents of extrapolation. % % Large numbers of points will slow down the direct % \, but when applied to the normal equations, \ can be % quite fast. Since the equations generated by these % methods will tend to be well conditioned, the normal % equations are not a bad choice of method to use. Beware % when a small smoothing parameter is used, since this will % make the equations less well conditioned. % % DEFAULT: 'normal' % % '\' - uses matlab's backslash operator to solve the sparse % system. 'backslash' is an alternate name. % % 'symmlq' - uses matlab's iterative symmlq solver % % 'lsqr' - uses matlab's iterative lsqr solver % % 'normal' - uses \ to solve the normal equations. % % % 'maxiter' - only applies to iterative solvers - defines the % maximum number of iterations for an iterative solver % % DEFAULT: min(10000,length(xnodes)*length(ynodes)) % % % 'extend' - character flag - controls whether the first and last % nodes i