一种基于紧支撑连续可微小波的图象边缘检测算法

针对一般的梯度算子及Laphcian算子在边缘检测中对噪声敏感的问题，用一种二维不可分离的平滑函数来平滑图象，以降低噪声对边缘检出的影响，即提出一类紧支撑连续可微的函数，在2-范数下与高斯函数在有效支撑区间逼近，得到一变形平滑函数。简要证明了变形平滑函数的导数满足容许条件为一维连续小波函数。并给出该小波函数时域和频域的主要性质。将一维变形平滑函数推广至二维，用多尺度的该二维平滑函数对图象进行平滑．用模极大值法和过零点法检测图象的边缘，给出两种算法实现的关键之处。仿真结果表明。模极大值法能有效地从小噪声图象中检测细节边缘，在抗噪声方面优于一般的梯度算子；过零点方法从噪声图象中检出边缘。该算子与LOG算子检测效果相当，从侧面验证了LOG算子的鲁棒性。

Algorithm for Image Edge Detection Based on Compactly Supported and Continuously Differentiable Wavelet

As the common gradient operator and Laplacian operator are sensitive in edge detection of a noised image, we smooth the image with a kind of inseparable 2D function to reduce influence of noise, i.e. first, present a kind of compactly supported and continuously differentiable function; second, achieve a deformed function through approximating Gauss function in its effective supported interval with 2-'s norm; then, prove that the derivatives of deformed function are one-dimensional wavelets which satisfy admissibility condition, and also discuss the main character of wavelets in time and frequency domain. Extend the deformed function to 2-d and smoothed image, given the key realization process of 2-d continuous wavelets transform in maximum and zero-crossing arithmetic of image edge detection. The imitator of test makes known that, the maximum law arithmetic effectively examines the particular edge in the image, and is superior to the ordinary gradient operator; the zero-crossing arithmetic examines out the great fringe from the noised image, the effect of this operator detecting edges can match LOG operator, verifying that LOG operator is robust for edge detecting through the aspect