Chapter 5: Computing scale-space derivatives

Chapter 5 in Scale-Space Theory in Computer Vision shows how discrete derivative approximations can be defined so that scale-space properties hold exactly also in the discrete domain. A family of kernels is derived which constitute discrete analogues to the continuous Gaussian derivatives, and possesses an algebraic structure similar to that possessed by the derivatives of the traditional scale-space representation in the continuous domain.

The representation has theoretical advantages compared to other discretizations of scale-space theory in the sense that operators which commute before discretization commute after discretization. Some computational implications of this are that derivative approximations can be computed directly from smoothed data (without any need for repeating the smoothing operation), and this will give exactly the same result as convolution with the corresponding derivative approximation kernel. Moreover, a number of normalization conditions are automatically satisfied.