I am working on extremely large, symmetric matrices of counts, and attempting to identify patterns/shapes within them. Wavelets are a popular tool in image processing, and have some nice statistical properties when applies to discrete random variable.
The problem I face is this: 2D wavelet transforms on the diagonal contain redundant values, making the statistical interpretation of their coefficients difficult.
HOWEVER, i noticed that the lower triangular portion of my dataset (i.e. the dataset without any redundant values) has a natural analogue as a Möbius strip!
The steps taken to remove the redundant entries in my matrix are illustrated in the image below:
- Remove the upper right half of the matrix.
- Slice the image diagonally through the middle.
- Twist and flip the lower right portion and "glue" it to the upper left portion so that the dashed black lines are aligned and pointing in the same direction.
- Give the square a half twist and "glue" the yellow sides so that the arrows align in the same direction.
If you're incredulous that these steps are justified, see this video about redundant ordered pairs.
This is all well an good, but is there a natural extension of wavelets (or ANY orthogonal transformation) onto a non-orientable surface? If so, how does one interpret a multi-resolution analysis?
My intuition is that horizontal detail wavelets will carry over, but the non-orientable property will complicate vertical and diagonal detail wavelets.
I am interested in sources that discuss the topic. Web searches haven't turned up anything. I have emailed the topology faculty at my school, and have yet to hear back.