How to calculate laplacian form gradient?

Question

I am dealing with shadow removal. First I have to calculate gradient from the image in both directions. g(x,y) = -2*f(x,y) + f(x+1,y) + f(x,y+1). Afterwards, I do some calculation on the gradient and modify it a bit.

The problem comes when I have to calculate Laplacian (second derivative) from the modified gradient.

I know that the Laplacian filter has a matrix: [0,1,0: 1,-4,1; 0,1,0]. But the questions is how to calculate the Laplacian from the gradient, if the cells are already modified?

Calculation of gradient:

[Gx,Gy] = imgradientxy(img_G,'intermediate');
greenGradient = Gx + Gy;

Thanks!

Answer 1

When calculating gradient using imgradientxy(I,'intermediate'):

GX(j, i)  = I(j, i+1) - I(j, i)
GY(j, i)  = I(j+1, i) - I(j, i)

And the Laplacian:

L(j, i)   = I(j, i-1) + I(j, i+1) + I(j-1, i) + I(j+1, i) - 4*I(j, i)

Now if we calculate gradient of GX and GY:

GGX(j, i) = GX(j, i) - GX(j, i-1)
          = I(j, i+1) - I(j, i) - I(j, i) + I(j, i-1)
          = I(j, i-1) + I(j, i+1) - 2*I(j, i)
GGY(j, i) = I(j-1, i) + I(j+1, i) - 2*I(j, i)

So

L(j, i) = GGX(j, i) + GGY(j, i)

Note that there is on pixel offset between methods used to find gradient of I and gradient of GX and GY.

I =   im2double(imread('coins.png'));
[GX, GY] = imgradientxy(I,'intermediate');
L =   imfilter(I, [0 1 0; 1 -4 1; 0 1 0], 'replicate');
GGX = imfilter(GX, [0 0 0; -1 1 0; 0 0 0], 'replicate');
GGY = imfilter(GY, [0 -1 0; 0 1 0; 0 0 0], 'replicate');
L2 =  GGX+GGY;
E =   (L2-L).^2;