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Comment by dognotdog | Hacker News Reader

When seen as a [1 3 3 1] (the 4th row of Pascal's Triangle) kernel, it is more easily revealed as a discrete Gaussian kernel, which is for example used in scale-space representations in image processing.

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Whoa. Your one sentence completely demystified this article for me.

The Gaussian function is actually the distribution function of the normal distribution, and the normal distribution’s discrete equivalent is the binomial distribution whose coefficients (binomial coefficients) are the rows of Pascal's triangle.

Right. And the advantages of the Gaussian kernel with respect to image processing:

1. It has no negative coefficients (hence the output will always be a valid color value.)

2. It is circular symmetric, yet separable (X and Y may be filtered independently) -- a property which simplifies the implementation considerably.

The same applies to {0.05, 0.25, 0.4, 0.25, 0.05} mentioned later: it's proportional to 1 5 8 5 1, which is two rows down from 1 3 3 1.

The row is 1 4 6 4 1, but these are the closest 5-cent approximations to the 1/16th that add to 1.

Thanks for the correction, and additional information.

The rows after 1 3 3 1 are 1 4 6 4 1 and 1 5 10 10 5 1. Where does 1 5 8 5 1 come from?

From me being bad at arithmetic. Sorry.

Doing arithmetic in public is dangerous.

This makes heaps of sense, they say "wow it looks so much like a Gaussian kernel" and my first thought was "is Gaussian not the go-to kernel for image processing?"

no, gaussian is a bad reconstruction kernel.

The argument is that, in practice, it performs better for resampling images than the more correct sinc kernel from sampling and reconstruction theory.

Well, you can't actually use sinc in practice because it has an infinite support. But something like windowed sinc will work better than gaussian.

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