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# discrete second order derivative operator for unequally spaced data

Asked by David Kusnirak on 14 Feb 2013

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 on 14 Feb 2013

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 on 15 Feb 2013

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

## Products

Answer by Teja Muppirala on 15 Feb 2013

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

```help gradient
```

Youssef KHMOU on 15 Feb 2013

hi, you mean, apply the gradient twice ?

Teja Muppirala on 15 Feb 2013

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

Answer by Youssef KHMOU on 15 Feb 2013

Discrete Laplacian :

` del2(matrix)`

Example :

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

Answer by Jan Simon on 15 Feb 2013

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