The wavelet decomposition for a given basis is in fact unique (for scalar wavelets, anyway). You are probably thinking of the wavelet-packet/best-basis algorithm, which is a generalization of the wavelet transform and chooses the basis which minimizes the information cost for a given signal.
I agree that 60kbps full-motion high-quality video is probably possible. Using 3-D wavelets would build in some lag time (like 128 frames or so, depending on the basis) so it wouldn't ever be "live" (but heck, what's 4 seconds?)
Actually, entropy is useful only as a guideline and not as a hard lower bound on the number of bits required. For example, I can code any image you give me using only 1 bit. The coder simply compares the input to the desired image; if it's the same, the bit is 1; otherwise its 0. Very efficient on that image, totally useless on anything else. This argument has nothing to do with entropy.
Furthermore, there is no conceivable way to define what an "average" entropy is for an image. Images take many many possible forms; the space of all possible pixel images is not infinite-dimensional but it has more dimensions than the number of particles in the universe. By using statistical tools we can try to characterize this immense variability, and that is where the use of measures like entropy come into play. But entropy says nothing about the absolute lower bound on the number of bits required to code an image.
Okay, so all of the descriptions that people have given here for wavelet compression is wrong. I've got a Ph.D. in Applied Math and do research in wavelet compression of 3-D data (not geometry data, mind you, but 3 dimensional real data, like images, but in 3 dimensions instead of 2). The basic idea behind wavelet compression is the following:
In most natural or real-world data (i.e. images, geometry data, etc.) the information at a given point in the data is very highly dependent on the data at nearby points. Thus, there is a certain amount of redundancy in the data, and this redundancy is spatially localized. The concept in transform coding is to apply some transformation (either linear or nonlinear; the wavelet transform and Fourier transforms are linear) to this data to reduce the statistical redundancy.
Even after applying the transform, you haven't saved anything in terms of the space required to store the data; all you've done is change the basis used to represent the data. Now you take the transformed data and place it into a bunch of bins, each of which is identified with an integer. At this stage, called quantization, you are modifying the information present, because the best accuracy with which you can recover the data is given by the width of the bins. At this stage, you take the sequence of integers and apply a lossless coding scheme to it to reduce the number of bits required to represent the stream of integers. The compression happens at this stage. Wavelets do a better job than blocked discrete Cosing transform (used in JPEG) at reducing the statistical redundancy of the input data; thus wavelet-based image compression compresses more efficiently than JPEG.
What Schroeder and Sweldens have done is taken an a very general, widely applicable method for constructing wavelet transforms (known as the lifting scheme, invented by Sweldens) and adapted it for representing mesh nodes and connectivity information, i.e. geometry (which incidentally could just as easily be higher dimensional data). Thus they have a wavelet transform for geometry. They achieve compression by using the EZW coding scheme, developed for coding wavelet coefficients of images and used in the JPEG2000 standard, and applying it to their geometry wavelets.
It should be very nice for low-bitrate storage and transmission of geometry, as well as successive-refinement transmission (i.e. the 3-d data gets better and better looking as more bits arrive).
Gee, if I spew out a bunch of crap opinions, interspersed with "references" (also to crap opinions!) does that mean Slashdot will call me smart? I think so! This article pretty much confirms for me that Slashdot is now completely worthless. I'll stick to nytimes.com and sciencedaily.com, thank you very much.
...would be more like 50 pages, and it would actually contain equations. You can use harmonic analysis for lots of things, and this is a pretty simple application, that arguably doesn't really rely on it to work, rather Microsoft just uses it to give a hand-wavy justification of why it works.
Slashdot Mirror
I agree that 60kbps full-motion high-quality video is probably possible. Using 3-D wavelets would build in some lag time (like 128 frames or so, depending on the basis) so it wouldn't ever be "live" (but heck, what's 4 seconds?)
Furthermore, there is no conceivable way to define what an "average" entropy is for an image. Images take many many possible forms; the space of all possible pixel images is not infinite-dimensional but it has more dimensions than the number of particles in the universe. By using statistical tools we can try to characterize this immense variability, and that is where the use of measures like entropy come into play. But entropy says nothing about the absolute lower bound on the number of bits required to code an image.
In most natural or real-world data (i.e. images, geometry data, etc.) the information at a given point in the data is very highly dependent on the data at nearby points. Thus, there is a certain amount of redundancy in the data, and this redundancy is spatially localized. The concept in transform coding is to apply some transformation (either linear or nonlinear; the wavelet transform and Fourier transforms are linear) to this data to reduce the statistical redundancy.
Even after applying the transform, you haven't saved anything in terms of the space required to store the data; all you've done is change the basis used to represent the data. Now you take the transformed data and place it into a bunch of bins, each of which is identified with an integer. At this stage, called quantization, you are modifying the information present, because the best accuracy with which you can recover the data is given by the width of the bins. At this stage, you take the sequence of integers and apply a lossless coding scheme to it to reduce the number of bits required to represent the stream of integers. The compression happens at this stage. Wavelets do a better job than blocked discrete Cosing transform (used in JPEG) at reducing the statistical redundancy of the input data; thus wavelet-based image compression compresses more efficiently than JPEG.
What Schroeder and Sweldens have done is taken an a very general, widely applicable method for constructing wavelet transforms (known as the lifting scheme, invented by Sweldens) and adapted it for representing mesh nodes and connectivity information, i.e. geometry (which incidentally could just as easily be higher dimensional data). Thus they have a wavelet transform for geometry. They achieve compression by using the EZW coding scheme, developed for coding wavelet coefficients of images and used in the JPEG2000 standard, and applying it to their geometry wavelets.
It should be very nice for low-bitrate storage and transmission of geometry, as well as successive-refinement transmission (i.e. the 3-d data gets better and better looking as more bits arrive).
Gee, if I spew out a bunch of crap opinions, interspersed with "references" (also to crap opinions!) does that mean Slashdot will call me smart? I think so! This article pretty much confirms for me that Slashdot is now completely worthless. I'll stick to nytimes.com and sciencedaily.com, thank you very much.
...would be more like 50 pages, and it would actually contain equations. You can use harmonic analysis for lots of things, and this is a pretty simple application, that arguably doesn't really rely on it to work, rather Microsoft just uses it to give a hand-wavy justification of why it works.