Sobel Operator - Alternative Operators

Alternative Operators

The Sobel operator, while reducing artifacts associated with a pure central differences operator, does not have perfect rotational symmetry. Scharr looked into optimizing this property. Filter kernels up to size 5 x 5 have been presented there, but the most frequently used one is:

Scharr operators result from an optimization minimizing weighted mean squared angular error in Fourier domain. This optimization is done under the condition that resulting filters are numerically consistent. Therefore they really are derivative kernels rather than merely keeping symmetry constraints.

A similar optimization strategy and resulting filters were also presented by Farid and Simoncelli. They also investigate higher-order derivative schemes. In contrast to the work of Scharr, these filters are not enforced to be numerically consistent.

The problem of derivative filter design has been revisited e.g. by Kroon.

Orientation-optimal derivative kernels drastically reduce systematic estimation errors in optical flow estimation. Larger schemes with even higher accuracy and optimized filter families for extended optical flow estimation have been presented in subsequent work by Scharr. Second order derivative filter sets have been investigated for transparent motion estimation. It has been observed that the larger the resulting kernels are, the better they approximate Derivative of Gaussian filters.