Code covered by the BSD License

# Regular Control Point Interpolation Matrix with Boundary Conditions

by

### Matt J (view profile)

06 Jan 2010 (Updated )

Creates Toeplitz-like matrices representing interpolation operations with edge conditions.

Example2D
```function Example2D
%Two dimensional cubic B-Spline fitting example using interpMatrix()
%in conjunction with KronProd().

%%Data

s = @(t) cos(2*pi*t).*exp(-abs(2*t))+ 1;  %signal to fit

cubicBspline = @(t) (t>-1 & t<1).*(2/3 - t.^2 +abs(t).^3/2) +...
(abs(t)>=1 & abs(t)<2).*((2-abs(t)).^3/6);

tCoarse=linspace(-1.2, 1.2,9);     %Coarse sample locations on t-axis
dtCoarse=tCoarse(2)-tCoarse(1);
tFine=linspace(-1.2, 1.2,81);      %Fine sample locations on t-axis
dtFine=tFine(2)-tFine(1);

SampRatio=round(dtCoarse/dtFine); %Sampling ratio

%sample the signal
sCoarse1D=s(tCoarse(:));  sCoarse1D=sCoarse1D;
sFine1D=s(tFine(:));      sFine1D=sFine1D;

sCoarse=sCoarse1D*sCoarse1D.';
sFine=sFine1D*sFine1D.';

figure; subplot(121)
surf(tFine,tFine,sFine);

title 'Signal Samples'

%%Engine

kernel=cubicBspline(-2:1/SampRatio:2 );
nCtrlPts=length(tCoarse);

%create interpolation system matrices

Basis1DFine=interpMatrix(kernel, 'max', nCtrlPts, SampRatio, 'mirror');
BasisFine=KronProd({Basis1DFine}, [1 1]);

Basis1DCoarse=Basis1DFine(1:SampRatio:end,:);
BasisCoarse=KronProd({Basis1DCoarse}, [1 1]);

%Alternatively, we could have used KronProd class cellfun() method:
%
%   BasisCoarse=cellfun(@(A) A(1:SampRatio:end,:) ,BasisFine);

%%Do the fit!!!
sFit = BasisFine*(BasisCoarse\sCoarse);

subplot(122)
surf(tFine,tFine,sFit);
title 'Cubic B-Spline Reconstructions'

```