discrete second order derivative operator for unequally spaced data

Hi,

I need help to obtain a discrete second order derivative operator, that could be applied on a matrix. The best way, I guess, is to reshape the matrix into vector form and create a diagonal matrix with the operator, which is easy for equally spaced data in the matrix,but how can I incorporate the distances (width and heigth of the cells) for unequally spaced data?

Thanks

[Matt J]

I don't see how a diagonal matrix could ever give you a derivative operator, even with equally spaced data. A diagonal matrix can only weight each element of the operand independently, not take differences between them. Please describe the organization of your data more clearly. Does each column of your matrix contain signal samples to be differentiated? Is the spacing between these samples the same for all columns?

[David Kusnirak]

sorry, I wrote it unclear, I meant diagonal matrix with 4 off-diagonals.

example:

if ma data matrix is

M = magic(3)

and the width and height of each cell is 1, the operator looks like:

L=[ -4 1 0 0 1 0 0 0 0;
    1 -4 1 0 0 1 0 0 0;
    0 1 -4 1 0 0 1 0 0;
    0 0 1 -4 1 0 0 1 0;
    1 0 0 1 -4 1 0 0 1;
    0 1 0 0 1 -4 1 0 0;
    0 0 1 0 0 1 -4 1 0;
    0 0 0 1 0 0 1 -4 1;
    0 0 0 0 1 0 0 1 -4];

so the product of the operator (in vector form) is

dz = L * reshape(m,1,[]);

and back to matrix

dz = reshape(dz,3,3)

My question is how to implement proper geometry of the cells, for example width = 2*heigth

Answer 1

[Teja Muppirala]

Could you possibly use the GRADIENT command? This allows you to pass in unevenly spaced values for X and Y.

help gradient

[Youssef Khmou]

hi, you mean, apply the gradient twice ?

[Teja Muppirala]

I though he meant a first derivative (to second order accuracy), so just calling GRADIENT once.

[Youssef Khmou]

ok

Answer 2

[Youssef Khmou]

Discrete Laplacian :

 del2(matrix)

Example :

 [x,y]=meshgrid(-4:.1:4);
 z=exp(-x.^2-y.^2);
 dz2=del2(z);

Answer 3

[Jan]

A fast gradient with 2nd order method for unevenly distributed data: FEX: DGradient.

Answer 4

[David Kusnirak]

Thanks for hints. However I'm not as interested in the solution as in the operator itself, I need to know how does it look and work.

thanks for any help