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 unique ordered pairs.
Now, 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.
BACKGROUND
I am including this supplement to explain my data in more detail for those that are curious.
I am dealing with matrices that contain information about the proximity among the various loci along a strand of DNA. The laboratory method is known as Hi-C.
Briefly, the DNA (which is bundled up in the nucleus) is "cross-linked" or stuck together. This ensured that fragments of DNA in later steps were in fact close to one another in 3d, euclidean space.
The DNA is then shredded. I small molecule of biotin (think of it as a magnet) is added to the ends of the DNA fragments, and those tip "ligate" or click together.
(source: sciencenews.org)
The DNA is then shattered again. Now we have A LOT of DNA pieces, BUT the pieces of primary interested (the so-called "chimeric" DNA composed of two pieces from distant parts of the DNA string) have a charged molecule on them! So scientists can separate them will a magnetic charge.
After completing the lab process, each tip of the chimeric DNA molecules is sequenced and mapped onto the genome. Because each piece comes from the same chromosome, one can represent this as a symmetric matrix, where each row/column represents a unique locus on the chromosome, and the count in the cell formed by the intersection of the row/column represents the number of times those two loci were observed as a piece of chimeric DNA.
Once completed, binning at any desired resolution presents coarser/finer matrices. Naturally, counts near the diagonal are expected to be higher, since those cells represent loci that are near to each other in terms of arc-length, and thus their 3D euclidean distance is limited by the triangle theorem.
I hope this is helpful.
First of all, let's try to make sense of your question. (You will need a number of differential geometry concepts which you are probably unfamiliar with. Any textbook on differential or Riemannian geometry will cover these.)
An orthogonal transformation is a concept from linear algebra. In order to define this concept you need two things: a real vector space $V$ and a positive definite bilinear form (aka an inner product) $\langle\cdot, \cdot\rangle$ on $V$. (In the case when $V={\mathbb R}^n$, we usually take this bilinear for to be the dot product $\langle u, v\rangle= u\cdot v$.) Then a linear transformation $A$ of $V$ is called orthogonal if it preserves this bilinear form: $$ \langle u, v\rangle = \langle Au, Av\rangle $$ for all vectors $u, v\in V$.
On the other hand, a surface (oriented or not) is a topological concept; such a surface does not have linear structure (and a bilinear form). So, on the face of it, you question is meaningless. (Akin to asking to define the concept of derivative on the set of frogs in a swamp.)
Nevertheless, one can make sense of the question using Riemannian geometry:
Take a surface $S$ (oriented or not is irrelevant) equipped with a Riemannian metric $g$, which is an inner product on tangent planes to $S$ (for the record, the inner product needs to vary smoothly from point to point). Then a diffeomorphism (a smooth invertible map whose inverse is also smooth) $$ f: (S,g)\to (S,g) $$ is called a Riemannian isometry if $f$ preserves the Riemannian metric $g$: For every $p\in S$ and and two vectors $u, v$ in the tangent plane $T_pS$, we require $$ g(u,v)= g(df(u), df(v)). $$ In other words, the derivative $df: T_pS\to T_{f(p)}S$ of $f$ is orthogonal on every tangent plane $T_p$ to $S$.
For the surfaces you seem to be interested in (e.g. the Klein bottle), one can make all this a bit more concrete. Namely, the Klein bottle $K$ admits a Riemannian metric $g$ which is locally Euclidean; you can obtain such a metric by realizing $K$ as the identification surface of a rectangle $P$ in the Euclidean plane $E^2$ in the usual manner. The Euclidean Riemannian metric on $E^2$ descends to $K$. Then a Riemannian isometry $f: (K,g)\to (K,g)$ for this metric in local coordinates (coming, say, from the rectangle $P$) can be locally written as the restriction of a Euclidean isometry $E^2\to E^2$. The latter is a composition of an orthogonal transformation with a translation.
The set of self-isometries $(S,g)\to (S,g)$ forms a group $G$ (one can compose them and invert them); this group has a natural topology. In the case of a Klein bottle with the locally Euclidean metric $g$ as above, this group is 1-dimensional. (There is a 1-parameter family of self-isometries.) I can tell you more about this group if you are interested.