DERIVATIVE COMPUTATION BY MULTISCALE FILTERS

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 m ethods, the nth order derivative of a noisy signal is obtained by filt ering 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 giv en smooth filter has to be an Analytic Smooth Function Derivable up to nth order (ASFDN). In this paper, a new methodology for the NODF desi gn is presented. It allows us to design directly the nth order derivat ive filter without using the ASFDN. The importance of this new design method is that we can find a number of NODFs satisfying certain desire d optimization criteria but their corresponding ASFDN may not exist. W e 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 ou r filter with some well-known filters. It has been considered for a lo ng time in the computer vision community that DoG is a second derivati ve filter only in the sense that it is a good approximation of LoG whe n its scale ratio is equal to 0.625. But, we prove that DoG with any s cale ratio is itself a second derivative filter. In addition, using th e criteria proposed by Sarkar and Boyer, it is shown that the best sca le ratio of the DoG for edge detection is 0.176 rather than 0.625. Gri mson and Pavalidis proposed a second derivative filter which is the di fference of a delta function and a smooth function. Shen and Castan pr oposed a filter which is the difference of a delta function and an exp onential function. It is shown that these filters are particular filte rs with scale ratio 0, which can be designed by our method. We propose d a better second derivative filter DoE, which is the difference of tw o exponential functions with the scale ratio 0.3. Extension for two di mensional partial derivative filter is also presented. (C) 1998 Elsevi er Science B.V.