Geometric Transformation in Gradient Domain Possible?

I work with images in gradient domain because I sometimes need to manipulate image gradients (i.e. Laplacian filter responses) rather than intensity values.

However, I need to apply a geometric transform (distortion) to an image already in gradient domain.

Is is then meaningful to reintegrate the distorted image back to spatial domain?

Here is my earlier question related to gradient domain reconstruction with some images.

The geometric transformation needs to be independent from the pixel coordinates $x$ and $y$. assume that $G$ is your transformation matrix: $$\int\frac{d^{2} (G I)}{dx dy} = G \int\frac{d^{2} (I)}{dx dy}$$ is true only when $G$ is independent from the coordinates.

The restoration will succeed with constant (i.e. global) image rotations, scalings, and shifts.

• A "pixel" in the gradient domain is actually divergence of a vector field, so both $x$ and $y$ dimensions are taken into account. But it is now obvious that most geometric transforms are not usable directly in gradient domain. Thanks. – Libor Sep 22 '13 at 17:09
• The independence needs to be satisfied between the transformation and the pixel locations. If your transformation is not dependent on the location of the 'pixels', restoration should work. Here is an example of a transformation that is not going to succeed: rotations where the rotation angle change depending on the x or y coordinate. Rotations with angle $\theta$ where \$theta$ is a constant can be restored. – Hasan Sep 22 '13 at 20:58
• I see. I was thinking about nonlinear transformations (e.g. projection to spherical map) so I will avoid doing this with images in gradient domain. But I will keep in mind that some linear transformations are allowed. – Libor Sep 22 '13 at 21:57