Reconstruct a scalar field from its gradient

Hi,

I have a cartesian grid over the rectangle [0,N]x[0,M]. In each point of the grid, I know the gradient of a certain scalar field F; that is, I know the derivative of F with respect to x, Fx, and the derivative of F with respect to y, Fy.

The question then is: what is the best way to reconstruct the scalar field F?

Of course, the easiest thing is to integrate by lines: Integrate along a straigth line keeping y fixed (for example) and then integrate along a bunch of perpendicular straigth lines keeping x fixed. But this is quite inefficient and, since this is a numerical problem, the directionality could play (undesirably) an important role (although theoretically, of course, that shouldn't be the case if the gradient truly comes from a scalar field)

So, briefly put: how can I do this in Matlab? Is there a function or subroutine to which I can feed a gradient so that it will return to me the scalar field?

Thank you

[Jan]

Do you have a reason to claim that "this is quite inefficient"? Are you aware that you cannot reconstruct a scalar field from the gradients, because you will miss the constant factor? Do you have any further information about the field like symmetry, smoothness, discontinuities?

[Diego Soler Polo]

" Are you aware that you cannot reconstruct a scalar field from the gradients, because you will miss the constant factor?" Yes, of course, but that is unimportant. Just fix some value, like F(0,0)=0.

"Do you have a reason to claim that "this is quite inefficient"?" My reason is that I have done it and I think it is too slow. The second part of the integration consists in integrating along several hundreds lines, one after each other. It is doable, of course, but I was thinking that there must be something better and optimized.

"Do you have any further information about the field like symmetry, smoothness, discontinuities?"

No symmetry. The reconstruction is purely numerical, so it wouldn't make sense to apply the concept "smooth" here, but the physical concept associated to this field I want to reconstruct, a free energy map, can be thougth in this context to be continously differentiable.

Thanks

[Jan]

Could you post the code, which is too slow currently?

Answer 1

[John D'Errico]

Just download my intgrad tools from the FEX. They are designed to recover a scalar field, given gradient information on a regular gridded domain. Of course they allow you to provide the constant of integration, or just assume it is zero.