The author talks about JPEG, and says the principle is surprisingly similar, but the connection between the two is actually pretty direct. I will link a previous comment of mine from another discussion of SVD image compression to explain: https://news.ycombinator.com/item?id=34732922
To go from JPEG to lossy WebP (really, VP8 intra frames), the main differences are
(a) using smaller (4x4) transform blocks instead of 8x8,
(b) noticing that if you take just the first coefficient in each block, it looks like a lower resolution of the original image, and applying another 4x4 transform to those coefficients,
(c) adding more sophisticated prediction between blocks (e.g., subtract the pixel on the right edge of the previous block from all the pixels in the same row of the current block before transforming them, or something similar to predict in other directions),
(d) filtering along block edges to reduce blocking artifacts (many JPEG decoders do this, but WebP/VP8 make it a mandatory part of the standard), and
(e) using arithmetic coding instead of Huffman coding.
There are of course other minor differences in the details.
But the main mechanism that underlies both is a transform that can be derived from the SVD if one assumes a simple statistical model of natural image data.
I'd be interested (and this is purely "academic" curiousity) IF there was:
Something mathematical -- which would underly all image compression algorithms; that is, if all image compression algorithms could be shown ("proven" I guess -- I am not a professional Mathematician, only armchair, so bear with me!) to belong to a specific class of generalized compression algorithms or (ideally!) a single "parent" (for lack of a better word) image compression algorithm -- that would give rise to all other image compression algorithms, no matter how diverse or not mathematically related they would all seem...
The use of Matrices (which model systems of linear equations, which in turn, enumerate/list/make clear/table -- relationships between factors in those equations) -- would seems to be a good starting point for this Mathematical inquiry, should it be made...
Maybe I should phrase this a little bit better...
Maybe it's something like (as a future goal for the Mathematicians of the world), "for every image compression algorithm X, show/prove that X is related to <parent Mathematical principle involving matrices>"...
Or something like that...
Sort of like a grand unifying theory of image compression...
Anyway, your comments are highly interesting.
It seems like you might be part of the way there for something like a rigorously mathematically unified (and proven!) future understanding of image compression!
I don't think that it's truly universal, but quite a few image compression schemes have a similar structure if you stand back and squint:
1. Use a reversible linear transform to transform the image into some other space. (It'll still require the same amount of data to represent)
2. Throw away a bunch of the values (set them to zero, or round them to low precision, etc)
3. Now that there are some repetitive zeroes or whatever, a conventional data compression algorithm (like LZW, Huffman, or interval/arithmetic) can make it smaller
Even simple things like resizing an image smaller can be fit into this framework: step 1 is a simple blur, step 2 sets a bunch of pixels to zero, step 3 discards them.
(I don't think fractal compression can be fit into this framework, but I could be wrong. That's the only exception I can think of offhand.)
Broadly, the entire space of (lossy, lossless) compression falls under the purview of information theory, which tells us that almost all [0] encoding schemes with sufficient codebook entropy can achieve the desired capacity / compression subject to constraints on distortion. Unfortunately, this isn't directly useful since the proof is nonconstructive, but it suggests (to me, at least) that there is an infinitude of compression schemes that don't have to have any common given structure.
The special thing that image compression gets to rely on is that human vision is inherently lossy, so we can crush color spaces, or throw away tiny details, or even clone regions wholesale (JBIG2, anyone?). And unless you're looking very closely, you probably won't notice a thing.
One can go very deep in making universal statements about compression, or images, probably deeper than you realize, but it will never be so simple and wrapped up with a pure-mathematical bow as you are implying.
Correction: I am not implying, I am searching for.
There is a difference, a huge difference, between implying and searching for.
I search for things, in the above case, a mathematical (or algorithmic) theory or theories which could link disparate compression algorithms to a common ancestor or universal pattern -- because if one exists, and if it could be found -- it would greatly enhance humanity's understanding of all of the subcases that emanate from that universal pattern.
That is, it would greatly enhance humanity's understanding of all compression algorithms, and why specifically, WHY specifically -- they work...
That would be useful knowledge to have.
Especially since Cable TV, Video Streaming, The Internet and many other technologies -- all usually use different forms of compression...
The modern world is based on such things as: Electricity, Radio, Signals, Computers, Algorithms, Digital Data -- and compression usually plays a large associated role in those technologies...
So I search for a unified understanding of compression, in the above case, specifially that related to Images, but more generally with respect to any form of compression, that is:
What are the mathematical roots of Compression; what is the mathematical origin of Compression?
