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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 latter 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

Community
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Bossliaw
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2 Answers2

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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.

jeff7
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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.

Mark Ransom
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