Add demo showing steering of intermediate representation

[billbrod]

@ThomasYerxa , add details here on what you've done, why it didn't quite work, what else needs to be done, etc.

[ThomasYerxa]

The goal of this experiment is to demonstrate that the steerable is equivariant w.r.t. rotations. Specifically, a rotation in the input domain should correspond to steering the coefficients by the same amount in the pyramid domain:

• Pyramid(Rotate(Image, x)) = Steer(Pyramid(Image), x)
• The 'steer' operation above is not exactly the 'steer' operation in the pyramid sense, which simulates pyramid responses to a single image at different orientations. This steer simulates the pyramids response to a shifted image.

• A simple demonstration attempt is as follows:

  1.  Initialize a pyramid with 2 orientations (horizontal and vertical)
  2. Take an input image and rotate it by 90 degrees
  3. Push the original and rotated versions through the pyramid
  4. Show the coefficients are the same but have swapped orientation bands

  • I have tried doing a verison of this in the 'steerpyr_equivariance' branch in the examples/03_Steerable_Pyramid.ipynb notebook. Rather than rotate by 90 degrees I tried rotating by 45 degrees (and using a pyramid with more orientation bands including an oblique at 45 degrees).
  • another detail: I apply a circular mask to the image before rotating (this seemed the right thing to do, allows for rotating at intermediate angles without introducing quite as many artifacts)
  • Its possible I've made an implementation mistake in checking for the correspondence between bands, but I didn't achieve a satisfactory result. the code in this notebook is just a starting point for this type of demonstration.
  • another possibility is that the deviations are just due to the interpolation that needs to happen when rotating an image by an angle that isn't a multiple of 90 degrees.