Re: reconstruct image from its gradient field

From: Uwe Schmitt (schmitt_at_num.uni-sb.de)
Date: 08/04/04 

Date: Wed, 4 Aug 2004 10:52:21 +0200

>
> Innersmile wrote:
>
> > Hi there,
> >
> > Given a gradient field grad(I) = (I_x, I_y) of an
> > image I(x,y), and some boundary conditions or constraints
> > like average intensity, can we reconstruct the unknown
> > image I(x,y)?

Just an idea, I'm not an image-processing expert:

Assume your image is denoted by x and has size N x N:
If the gradients are exakt, you get for one horizontal
line: x_(i,1) + sum_j^n g_(i,j) = x_(i,n)

If you choose n=N you connect left and
right border of the image.
(It is just the discrete version of the main theorem
of differentical calculus).

But if your gradient is given as integer values,
you should add (unknown) weights for each gradient
in the sum above:

   x_(i,0) + sum_j^N a_(i,j) g_(i,j) = x_(i,N) for all i
   x_(0,j) + sum_i^N a_(i,j) g_(i,j) = x_(N,j) for all j

If you transport the boundary values from the left side to
the right side you achieve an underdetermined system for
the coefficients a_(i,j): A * a = z

Which you can solve by: (AA^T) u = z, a := A^T u

Maybe the system has bad condition (you will see this
as highly oscillating solutions u), you shoud try
tikhonov-phillips regularization:

     (A A^T + lambda I) u = z; a:= A^T u

Maybe this helps,

Greetings, Uwe.

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