Comment by peter_d_sherman | Hacker News Reader

Comment by peter_d_sherman | Hacker News Reader

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peter_d_sherman

Hi derf_!

Those are some very interesting comments!

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.

[0] See: typical sets. https://en.wikipedia.org/wiki/Typical_set

> (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.)

See https://ieeexplore.ieee.org/abstract/document/660992 for the connection between fractal image coding and wavelets + vector quantization.

An excellent link!

Thank you very much!

Let's just transcribe the name of this paper here on HN (should links be broken in the future), for posterity:

"A Wavelet-Based Analysis of Fractal Image Compression" by Geoffrey M. Davis, 1997, IEEE Transactions On Image Processing Vol. 7 Issue 2.

Incidentally, I found a non-paywalled version:

https://www.geoffdavis.net/papers/ieee.pdf

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?

That's what I search for...

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