Derivative computation by multiscale filters

摘要

It is a common problem to compute the derivative of a signal in image processing and computer vision. So far, in most of the computational methods, the nth order derivative of a noisy signal is obtained by filtering the signal by a nth Order Derivative Filter (NODF), which is the nth order derivative of a smooth filter. In order to do this, the given smooth filter has to be an Analytic Smooth Function Derivable up to nth order (ASFDN).In this paper, a new methodology for the NODF design is presented. It allows us to design directly the nth order derivative filter without using the ASFDN. The importance of this new design method is that we can find a number of NODFs satisfying certain desired optimization criteria but their corresponding ASFDN may not exist.We propose a new set of the NODFs constructed by multiscale filters. It is shown that a NODF can be designed as the weighted sum of a number of functions, with the same kernel but different scales.We compare our filter with some well-known filters. It has been considered for a long time in the computer vision community that DoG is a second derivative filter only in the sense that it is a good approximation of LoG when its scale ratio is equal to 0.625. But, we prove that DoG with any scale ratio is itself a second derivative filter. In addition, using the criteria proposed by Sarkar and Boyer, it is shown that the best scale ratio of the DoG for edge detection is 0.176 rather than 0.625. Grimson and Pavalidis proposed a second derivative filter which is the difference of a δ function and a smooth function. Shen and Castan proposed a filter which is the difference of a δ function and an exponential function. It is shown that these filters are particular filters with scale ratio 0, which can be designed by our method. We proposed a better second derivative filter DoE, which is the difference of two exponential functions with the scale ratio 0.3. Extension for two dimensional partial derivative filter is also presented.