What is the difference between 1x2 and 1x3 image gradient kernel filter definition

Recently I have a debate with my colleague on image gradient operation.

Normally, the image gradient is defined as:

dI_dx(j,k) = I(j,k+1) - I(j,k) # x partial derivative of image

dI_dy(j,k) = I(j+1,k) - I(j,k) # y partial derivative of image

For x partial derivative of image, this operation can be represented by a 1x2 filter array:

[1 -1]

But there is also another definition:

dI_dx(j,k) = I(j,k+1) - I(j,k-1) => [1 0 -1] (filter array)

So my colleague asked: What is the difference between them, and why is the lat开发者_Python百科ter 1x3 filter more often used than the 1x2 filter?

We have discussed some possible reasons:

1. 1x3 sampling is more robust than 1x2

   My colleague : No, they both sample 2 pixels for each image gradient pixel, the probability that a noise occurs on the sampled pixels is the same among these filters.

2. 1x3 is smoother than 1x2

   My colleague : No, the definition of the 1x2 and 1x3 filters are not smoothed at all. The Sobel filter is the one smoothed by a gaussian...

Extended question: Does image gradient's spatial filter kernel have so called "window size"?

By the way, I and my colleague are not persuaded by the following reference webpage...

http://www.cis.rit.edu/people/faculty/rhody/EdgeDetection.htm

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One way to examine these filters in the context of digital filter design, specifically an ideal FIR differentiator. The ideal digital differentiator has an antisymmetric unit sample response h(n)=-h(-n) and hence h(0)=0. Thus [1 0 -1] is closer to the ideal differentiator than [1 -1]. Take a look at any DSP reference for more theory. In practice I always use a smoothed derivative with the Sobel operator.

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The 1x2 filter will produce samples that lie between the inputs, while the 1x3 filter output will line up with the input. However the 1x2 filter will capture finer detail.