Piecewise Polynomial (pp) for Bicubic Interpolation

Hi, The piecewise polynomial (pp) for a cubic or spline interpolant is extremely easy to obtain by the formula:

pp = interp1(x,Y,method,'pp') where the piecewise polynomial gets stored in 'pp'.

But when we go for interpolation in 2 dimensions using interp2 function, there is no way I can get the 'pp' form of the bicubic interpolant. I need the bicubic interpolation function in Image Processing for implementing few Digital Image Correlation algorithms, as I need the continuous gray function for the 8-bit Tiff Images (usual resolution in the range of 300*300) to further get the derivatives of the function (which is not possible if I don't have the 'pp').

Please help me coz I am bugged quite bad right now, and my project has been freezed just because of this shortcoming :(

Any help/hint even from spline toolbox is acceptable too. Thanks in advance.....

Answer 1

[Andrew Newell]

MATLAB hasn't implemented multidimensional polynomials, and there doesn't seem to be anything in the File Exchange. However, if what you really need is derivatives, why not use gradient?

[sudhir sharma]

Well that's really sad if it's not implemented (built in) in Matlab. I think I have to go through the spline toolbox for some help. I can't use the gradient as I don't have my function continuous. It's a gray Image and so the gray vale (or intensity)function is discrete matrix.

Rather that's what the whole catch is about, and since Matlab can do the interpolation in two dimensions, then I am pretty sure there must be a function representing 'pp' which is being used to generate interpolated points.

[Andrew Newell]

The function GRADIENT is designed to work on a discrete set of data using finite difference formulae. It's not really so different from using splines. Instead of fitting the data to cubic polynomials, it approximates neighboring points by Taylor expansions and then combines them to estimate the derivatives.