# Gradient Domain Reconstruction - Scaling Problem

I am implementing reconstruction of image from gradient domain. This requires solving the following partial differential equation (a Poisson equation) on a 2D grid:

$$\nabla^{2}I=\mathbb{div} G$$

$\mathbb{div} G$ is divergence of a gradient field, which is simply an image with Laplacian operator applied (second-order derivative):

The original image have range of values 0 to 255 (8-bit image). The divergence image has values ranging from -341 to +318, which is expected.

However, when I solve the above equation for $I$, the result have very small scale. The values of reconstructed image range from -0.000038 to +0.000029.

I know the gradient domain stores information up to a constant. In this case however, the scale has been changed considerably.

When I stretch the reconstructed image to 0-255 interval, the correct result is obtained:

I am using Poisson solver based on the Full Multigrid Method (FMG) which works spectacularly in different application (HDR images). Here the values are stretched anyway so I never noticed the huge change of scale.

Is this scale change normal for gradient domain reconstruction or may there be

My ideas:

1. Some glitch in the numerical solution.
2. The $\mathbb{div} G$ image is padded with zeros so its size conforms with the solver - could this affect the scale?
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