function = 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: = 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.
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