Is there any reverse operation to (image) gradient?

Question

Suppose I have calculated gradient (of a grayscale image).

Gradient is a difference between neighboring pixels in X and Y directions.

Can I calculate image back, having this gradient information?

Can I filter gradient data somehow so that reverse operation give some reasonable results?

Answer 1

Gradient is the difference between the color of two neighboring pixels. To get back the image you need one piece of information: the initial color of the boundary pixels.

Like in math, a derivation can be reversed by integration as far as a constant term is involved. Or, if you have d = a - b you can get back a only if you also know b.

Without the boundary values you can still recover the image but not at the same saturation & contrast. There will be a constant term missing from the entire image.

Example

Consider an image which has only 3 pixels: 42, 142, 100. The gradient will be 0, 100, -42 (computing it as the difference between the current pixel and the previous one). To get back the initial image we assume that the first pixel had been 0 and then do addition instead of subtraction: the recovered image will be 0, 100, (100-42). As you see, adding 42 to all of them will give us back the initial image.