多尺度微分滤波器的构造与应用

DESIGN AND APPLICATION OF MULTI-SCALE DIFFERENTIAL FILTERS

微分运算广泛应用在图像处理与计算机视觉中．目前，最普遍的方法是将带有噪声的信号与某一N阶微分滤波器（NODF）进行卷积来得到信号的N阶微分．构造NODF的传统方法是先找到一个N次可微的平滑滤波器（ASFDN），其N阶微分就是NODF．本文阐述了N阶微分滤波器（NODF）构造的理论基础，给出了一滤波器成为NODF的充分必要条件．根据这一理论，我们给出了两种新的构造NODF的系统方法．这种新方法的重要性在于所构造的NODF的对应的ASFDN不一定具有解析形式，因此我们可以找到新的NODF，它满足所要求的优化准则但其ASFDN可以不存在．我们指出一些广泛使用的滤波器是我们所构造的滤波器的特例．最后我们给出了一些应用．

Differential filter is commonly used in computing the derivative of a sig-nal in image processing and computer vision. The most common approach is to con-volute the noisy signal with an nth order differential filter(NODF). However, it re-quires the given filter to be an analytic smooth function derivable up to nth order (ASFDN). This paper elaborates the theoretical foundation on NODF design.Firstly the necessary and sufficient conditions for a kind of filter to be an NODF are given. Then two systematic approaches for NODF design are proposed. The impor-tance of this new design method lies in that a number of NODFs satisfied certain de-sired optimization criteria can be found but their corresponding ASFDNs may not exit. It shows that some well known filters are the special cases of the filters de-signed by proposed method. Finally experiments are given to verify the method.