Geometric Transformation in Gradient Domain Possible?
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.