我谈Sobel算子与高斯一阶微分的关系

现在算力提升了，最常用的一阶差分边缘检测算子已经不是Sobel算子了，而是高斯一阶微分。

高斯一阶微分

顾名思义，高斯函数的一阶导数。
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Derivative of Gaussian 1D

一维直接扩展到二维。

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禹晶、肖创柏、廖庆敏《数字图像处理（面向新工科的电工电子信息基础课程系列教材）》P238

MATLAB中的实现。

    t = (-width:width);
    gau = exp(-(t.*t)/(2*ssq))/(2*pi*ssq);     % the gaussian 1D filter

    % Find the directional derivative of 2D Gaussian (along X-axis)
    % Since the result is symmetric along X, we can get the derivative along
    % Y-axis simply by transposing the result for X direction.
    [x,y]=meshgrid(-width:width,-width:width);
    dgau2D=-x.*exp(-(x.*x+y.*y)/(2*ssq))/(pi*ssq);

    % Convolve the filters with the image in each direction
    % The canny edge detector first requires convolution with
    % 2D Gaussian, and then with the derivative of a Gaussian.
    % Since Gaussian filter is separable, for smoothing, we can use
    % two 1D convolutions in order to achieve the effect of convolving
    % with 2D Gaussian.  We convolve along rows and then columns.

    %smooth the image out
    aSmooth=imfilter(a,gau,'conv','replicate');   % run the filter across rows
    aSmooth=imfilter(aSmooth,gau','conv','replicate'); % and then across columns

    %apply directional derivatives
    ax = imfilter(aSmooth, dgau2D, 'conv','replicate');
    ay = imfilter(aSmooth, dgau2D', 'conv','replicate');

    mag = sqrt((ax.*ax) + (ay.*ay));
    magmax = max(mag(:));
    if magmax>0
        mag = mag / magmax;   % normalize
    end