Why Convolutional Networks Learn Oriented Bandpass Filters: A Hypothesis

Richard P. Wildes

Keywords: convolutional networks, computer vision, oriented bandpass filters, linear systems theory

TL;DR: This paper offers a hypothesis for why convolutional networks learn oriented bandpass filters when applied to image understanding.

Abstract:

It has been repeatedly observed that convolutional architectures when applied to
image understanding tasks learn oriented bandpass filters. A standard explanation
of this result is that these filters reflect the structure of the images that they have
been exposed to during training: Natural images typically are locally composed
of oriented contours at various scales and oriented bandpass filters are matched
to such structure. The present paper offers an alternative explanation based not
on the structure of images, but rather on the structure of convolutional architectures.
In particular, complex exponentials are the eigenfunctions of convolution.
These eigenfunctions are defined globally; however, convolutional architectures
operate locally. To enforce locality, one can apply a windowing function to the
eigenfunctions, which leads to oriented bandpass filters as the natural operators
to be learned with convolutional architectures. From a representational point of
view, these filters allow for a local systematic way to characterize and operate on
an image or other signal.