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A very common filter in signal / image processing is the Laplacian of Gaussian filter, and its approximation, the difference of gaussians filter.

The DoG filter is the difference of 2 Gaussian PDFs that have slightly different standard deviations.

My question is: what would a filter that uses the difference of 2 Gaussian CDFs (instead of PDFs) look like and what properties would it have?

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  • $\begingroup$ hm, honestly, you seem capable enough, why don't you try? $\endgroup$
    – Marcus Müller
    Commented Aug 25, 2016 at 19:06
  • $\begingroup$ From a quick search I've just never seen anyone using it before, so just curious if it would have certain interesting/unexpected properties or if it's used in some niche applications, etc. $\endgroup$
    – sambajetson
    Commented Aug 25, 2016 at 19:51

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You need to ask yourself why do we use the difference of Gaussians from the first place? The reason is because the difference will give us a measurement for the change in value around the point we apply it to as a function of the variances of the Gaussians. If we have a big change the difference between the Gaussians it means that we have some frequencies that were low passed by the Gaussain with the bigger variance. If the change is low it means that there i no information lost. So we can think of it as a band-pass filter.

A simplified reason why it acts like that is because of the shape of the Gaussian PDF as can be seen in the following figure.

Taken from http://www.olympusmicro.com/primer/java/digitalimaging/processing/diffgaussians/

If we would use the CDF of the Gaussains and not the PDF it will give us a positive difference on one side and a negative difference on the other side.

enter image description here

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  • $\begingroup$ Visually, it would seem like the difference of CDFs would appear as a heartbeat signal. E.g. if your difference is: large SD Gaussian - small SD Gaussian (orange - blue curve above), you get a positive spike on the left and a negative spike on the right, appearing as a heartbeat pulse. So maybe could be some naive matched filter for pulses with heartbeat-like structures? $\endgroup$
    – sambajetson
    Commented Aug 25, 2016 at 22:14
  • $\begingroup$ May be. Sounds like an interesting idea. $\endgroup$
    – Amitay Nachmani
    Commented Aug 26, 2016 at 3:57

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