Banana Wavelets

 Time to fire up the wayback machine once again, and take a look at a fascinating wavelet representation apparently lost to the annuls of time. Let's take a look at the classic 'banana wavelet'.

All of this is from the paper titled 'Learning Object Representations by Clustering Banana Wavelet Responses by Peters and Kruger', which you can find here.

A banana wavelet is composed of the product of a warped gaussian and a curved wave function.

We have discussed both scale space pyramids as well as oriented steerable filters recently. So you can now view this banana wavelet through that lens. You can build this beast with different orientations, as shown below.

The mobius topology thing above might seem confusing, but if you think about how you have to compute distance between points in an orientation space, it should start to make sense.  If you are still confused, think about hue distance in a colorspace with a hue axis. Angle has the same modulo property.

So now of course you can immediately recontextualize this whole approach, and use it to construct an oriented banana wavelet scale space pyramid if you so wish.  Have fun.


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