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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。

一维直接扩展到二维。

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 outaSmooth=imfilter(a,gau,'conv','replicate');   % run the filter across rowsaSmooth=imfilter(aSmooth,gau','conv','replicate'); % and then across columns%apply directional derivativesax = imfilter(aSmooth, dgau2D, 'conv','replicate');ay = imfilter(aSmooth, dgau2D', 'conv','replicate');mag = sqrt((ax.*ax) + (ay.*ay));magmax = max(mag(:));if magmax>0mag = mag / magmax;   % normalizeend

Sobel

Sobel算子可以分解为一步高斯平滑和一步中心差分。所以，它可以看做高斯一阶微分的整数近似形式。

Sobel算子的数学推导是四个方向的相加，模板推导出来后，我们可以从上面的角度解释。