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New Sampling Techniques Make Up For Lost Data
An unnamed reader writes: "Professors at Vanderbilt and the University of Connneticut have published a non-uniform sampling theory that could yield better quality digital signals than the standard Uniform sampling techniques pioneered by Shannon at Bell Labs.
The Vanderbilt press release and link to the published paper can be found here."
No.
As the abstract says
"The new theory, however, handles situations where the sampling is non-uniform and the signal is not band-limited."
So it isn't applicable to digital music (as this is band-limited by our hearing, and we can pick the sampling interval) but other signals that cannot be sampled well by regular sampling (either in time or in space). Examples given are seismic surveys and MRI scans. But you knew this as you'd have taken the time to read the linked article first, wouldn't you?
What's the practicality of this? Well, spiral MRIs, for example, where for mechanical reasons you don't want to have to stop-and-start the very heavy "scanner", wasting time and jarring sensitive equipment. As I said, niche applications.
As for compressing audio, there are already plenty of other psychoacoustic compression schemes -- whether non-uniform sampling is better or worse will likely depend on the application.
It was at Bell Labs ... but the guy who developed the Uniform Sampling Theorem was Nyquist, not Shannon.
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In fact, you're not limited by the Nyquist frequency when you are sampling non-uniformly, so it has some strengths in that respect. However, it has to be more to it than this for it to be news. Can anybody who understands this better than I provide any insights?
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About 7 years ago, I was involved in a research project, trying to use video teleconferencing and doctors for remote diagnosis of patients.
We found that jpeg compression of images made medical diagnosis unreliable. Hairline fractures in x-rays are exactly the kind of small details that tend to get washed away in 'lossy' compression, and the banding caused can lead to false assumptions as well.
The article suggests that this is still a lossy compression with small amounts of data loss. I know Doctors that would take that admission as a condemnation of the technique.
Hereby I donate the following algorithm to the public. It's called GNU-squat.
Step 1:
Non-uniformly sample your favorite music using just 1 bit. This will ofcourse take up at least 8 bits on your harddisk but let's not nitpick. The good part is you don't even need special hardware to sample the music, just enter if the music is loud (1) or soft (0).
Step 2:
Use the Vanderbilt mathematical routines to extrapolate the rest of the data, and presto: the complete song re-appears from just one bit of data.
Doctor to patient, after looking at the reconstructed images: "Ah there is the problem. The cause of your headaches is that you have a bunch of inch-long bony spikes sticking out of your neck, plus a bunch of holes in your skull."
example. It was not provided to show a compression mechanism in which the original image could be compressed. It was intended to show that if you sample randomly, then their algorithm can come up with a highly accurate representation of the original. The implication here is that given current capability to sample, if you apply the new technique, you can get a better image/audio recording using their technique, than you can using the current fixed sampling interval technique, making the image more vivid, or the musical recording more lifelike than current sampling provides.
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There are already many other methods for reconstructing functions from sparse, non-uniformly sampled data, so this paper doesn't solve an unsolved problem. Rather, it provides one more solution under a set of assumptions that are mathematically a bit more like those of the original sampling theorem.
Will it be useful? That's hard to tell at this point. I think it will take a lot more work to figure out whether this method is any better than existing methods on real-world problems, whether its application can be justified in real problems, and whether it leads to algorithms that are practical. It may also turn out that the method is closely related to methods already in use in other fields; for example, the kinds of function spaces they study have received some attention in neural networks, but the authors cite no papers from that work and may not be aware of it.
Along your point, there's actually a technique that uses the self similarity of images to help you compress themselves. For example, you might have seen the "Sierpinsky Triangle." You can generate this image with a few very simple recursive move/resize/draw operations.
Fractal compression uses this technique on abstract images. It aims to find a set of operations (sometimes very large) to generate any given input picture. It's very cool, and you can get more information (including example pictures) at this page.
The "state of the art" of fractal compression beats JPEG compression at some compression ratios, but looses at others. It's also interesting that a fractally-compressed image has no implicit size (ie: 640x460), so it enlarges MUCH better than simple image enlargement.
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This is not quite accurate. The original signal is not "required" to be band-limited. Rather, it is accepted that frequencies outside of your design bandwidth will not be captured. The signal can stray outside of the "defined limits", but should it do so that information will be lost. Furthermore, Fourier's math tells us that a signal that is limited in time is unlimited in frequency, and a signal that is limited in frequency is unlimited in time. This has important ramifications. The biggest - and most obvious - is that all man-made signals are limited in time and therefore unlimited in frequency. Ergo there will ALWAYS be information lost no matter what bandwidth you design for.
Now to read the rest of the article - it sounds intriguing...
I was making up missing data for lab reports twenty years ago. It filled in the gaps well enough to fool the teachers :)
News article was, as usual, totally lacking in technical details. But they did link to technical articles at the bottom of the story.
NON-UNIFORM SAMPLING AND RECONSTRUCTION IN SHIFT-INVARIANT SPACES.
I skimmed the technical article (heavy math alert), and the results seem to be along the lines that: given an irregular (and possibly noisy) sample of data, reconstruct a
function that gives smoothed (continuous, not discrete) approximation for entire data set.
There is some nice mathematics that make it suitable for such purposes. The algorithms are selected to limit number of terms and guarantee convergance, and are computationally efficient. If you think of it as fancy interpolation, you are not far off the mark from what I saw.
This is not to disparage the efforts here (it looks to be quite useful in several domains), but it is a technique for generate a smooth, continuous function to represent a set of non-uniform samples. It cannot magically find missing results not were not evident in the limited sample data.
The author
Any medical imaging technique can only be so accurate, due to either machine or physical limitations. This defines a maximal meaningful sampling rate or resolution for that imaging modality. For example, positron emission tomography (PET) has a physical resolution limit of 10mm because positrons can propagate up to 10mm from where they are generated before they decay into gamma radiation that can be detected by the machine. With this technique, a doctor can get an image with better than 10mm resolution, something that the machine by itself could never do.
BTW, sampling doesn't mean that you're guessing. The sampled data points are the actual measured values of the signal at specified points in time or space. You have to sample because there is no way that you could collect all values for the signal for all points in time or space, and there is usually a sampling rate at which point you're collecting more data than you need to accurately represent the signal.
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From what I've read, some people seem to be thinking this is some kind of "magic bullet". For example, one comment, which emanated stupidity, was titled something like, "Infinite Zooming" and the implication of the post was that it might be possible with this method to "zoom in" on an image and accurately reconstruct the image. In other words, the idea is you could zoom in on a tiny head on a photograph and accurately reconstruct all of the details.
This, my friends, is complete nonsense. You cannot zoom in on an image and accurately reconstruct further details. To imply that this is possible is to imply that you can add accurately representative data where there was none before.
As for "zooming technology" it is possible to better reconstruct a zoomed-in image, though not any more accurately. For example, when I go into MS Paint and zoom in, it simply blows up all the pixels as larger blocks. This clearly is not good. You could create some kind of algorithm to determine the "shapes" of sharp edges, as well as where gradients where, and scale those up when zooming in...for example, small a circle can be composed of four pixels -- such a technology would scale this up, not as four very large blocks, but as a circle.
But this involves assumptions about what the original pattern was representative of? Was it representative of a circle, or of four large blocks seen from a distance? So you're not really adding data, but just attempting to "zoom in" on an image "better" based on a set of good assumptions which generally work.
Such a thing could be accomplished. Indeed, it already has been accomplished -- in us. When we look at a small photograph and want to draw a poster from it, we don't draw a large, blocky, pixelated image. We are able to tell what things -- such as frecles -- are details to be scaled up in our drawing; what things are gradients -- such as a dark to light gradient going from the near to the far side of a forehead -- to be scaled up and gradiated; and what are sharp borders, to kept sharp -- such as the sides of one's face.
However, even this amazing system we have of reconstructing larger images from smaller one's cannot add detail where there is none. If a woman is freckled with tiny freckles, they won't be visible from 10 feet away; a picture taken from that distance won't show them, and if we wanted to make a portrait of her head based on that picture, we wouldn't know to add freckles.
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(Side note: It seems ironic that as storage space grows, this becomes less and less necessary.)
Compression research continues because in the domain where latency is less than one minute (that is, not FedEx), data communication throughput does not increase nearly as quickly as storage space. Sure, you have 100 GB to store uncompressed images and audio, but how are you going to transfer the information to another computer?
Will I retire or break 10K?
The reason the usual techniques are band limited is the problem of aliasing (as we all can remember from watching the wagon wheels go backwards in old Westerns). The limitation of our ears makes uniform sampling techniques feasible for digital audio, but that doesn't mean that the new theory isn't applicable to digital audio.
I have a question/theory about nonuniform sampling rates. Okay, sticking with a 44kHz sample rate, will you hear the differeces between 8, 16, and 24 bit samples? Yes, of course. It's common in digital audio to use 16 bit samples to save space, not because it's the ultimate sample size. (While it's arguable the 44kHz rate side of the equation is pretty darn good.) It's subjective and some ears don't need any "more" audio information to be happy, but I see the choice of sample size as more of a variable than the "provable" sufficient rate for 20kHz audio cutoff behing 44kHz. All I'm saying is that there is potentially audible information below 20kHz that isn't getting encoded and recreated not because of sample rate, but because of sample size. For example, if my source material didn't "need" 44kHz througout a song, could the sample rate be trimmed back in places while the sample size was increased? In the end, it's all just a stream of x samples per second, y bits deep. So if a new sampling technique allows us to reproportion (optimize) those two dimensionons in the same amount of overall space, it's possible that better audio will result. Thoughts?
There are quite some examples in math how non equidistant sampling methods can vastly improve the order of accuracy, let's think about quadratures (numerical Integration):
Integrating a function f(x) from a to b means measuring the area below the graph. So the first estimation would be to split the interval from a to b into equidistant parts and sum up the area of the rectangles below or over the graph (that would be about f(x_n)*h, where h is the width). This method is called Riemann-Sums or iterated Trapezodial-Rule.
But you could also try to plot piece-wise polynomials through these equidistant points and calculate the areas below. This would yield better (order) results; these methods are then called iterated simpsons or millne rules. But if you go higher than polynomials of 4th degree, you will get to methods that could compute negative integrals of positive functions, which does not make sense. The reason is that high order polynomials tend to "oszillate" or "run out of bonds" at the end of the intervals. Thus these "Newton-Cotes" methods of equidistant sampling points are of limited capabilites...
But if you drop the assumption that you need to take equidistant (uniform) sampling points, you will get to far better methods: With Gaussian Quadratures the sampling points are far more dense at start and end of the intervals and thus the interpolating polynomials yield far better order results!
Thus if you know what you are going to use your data for, then you can always find better sampling methods to optimize for your needs- IMO it really doesn't make sense to simply sample the voltage of the signal at equidistant time frames when trying to digitally represent sound! Where as "lossy compressions" like ogg or mp3 drop information that is less interesting, this equidistant 44kHz sampling just drops anything that does not fit into this sampling; it's kind of a "brute-force" method. And if you then compress to ogg or mp3 it's the same problem like why you should never convert mp3s to ogg... It can (and will) only get worse.
If you are interested in that quadrature methods then read "Numerical Analysis" by "Kendall E. Atkinson" Chapter 5.
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I was reading somhere (can't remember where) that although we can't hear above 20Khz, sounds that are above that range will lower in frequency when they bounce around the room and fall into some peoples hearing range.
CD's sampled at 44khz miss some of these sounds and that is what audiophiles complain about when they say digital audio sounds flat.
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The variable bit-rate in MP3 compression does not
alter the amount of time between each sample. In
terms of sampling frequency MP3, even VBR is still
uniform, uniform as in time. VBR changes how many
bits are in a sample, not the time between samples.
If you are sampling audio at 44100 Hz, then an 8000 Hz tone will only be sampled at about 5 spots in its cycle. Although the frequency information of that 8000 Hz tone is retained, the actual waveform is lost. Exactly what the reconstructed waveform will look like is up to the DAC.
Whether the human ear can hear the difference at higher sampling rates is another question, however.
Take a look at the triad of MRI images in this article. If you look at the image on the left, it appears to have been scaled up about 2-3x from the original size. If you zoom in on it, you can see that the smallest represented detail in the picture is about 3 pixels across. It looks like they just imported the MRI into Photoshop and did a Bicubic scale to 300%!
They then remove 50% of the data in the second picture, and proceed to mathematically reconstruct it in the third. In my mind, this would be a great feat, except for two things:
- More than 50% of the data was unnecessary to present the data in the first place. The original is quite obviously scaled up from its native size.
- The mathematical reconstruction introduces artifacts that were not even present in the random image, such as huge horizontal pixel smears.
Can someone point to a better demo of this set of algorithms?
Justin
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I looked through the paper quickly, and it is a survey of existing techniques. The benefits of non-uniform sampling have long been known. Current low-end graphics hardware uses non-uniform sub-sampling grids to give better anti-aliasing results.
It was shown in the 70's or early 80's by A. Ahumada that the human eye uses a non-uniform distribution of rods and cones (outside the fovea) because it can give better frequency response than a uniform grid (given the same number of cones over a given area).
In short, while this paper makes good reading, don't think that it represent a breakthrough in the field.
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... given enough training data you can make a GA to give 'guesses' into any dataset.
PhotoCD works with a differential 'error' image that was created by comparing the resampled to the original, and then that was compressed. Effect? Take a small image, blow it up by a factor of 2x, apply this itty bitty 'error' transform, and you have a nearly perfect 'fixed' image for the cost of some small change on disk space
Then there is the 'much better clarity' etc statement- there's 'inverse point transform' for getting out defects.. they used that on the Hubble Telescope. Looked pretty good for being wildly out of focus.
Everything you've mentioned is already available... the technique looks interesting but it's all data dependent
Actually, the bandwidth used by music is not limited. What humans can hear is limited. What audiophiles think they can hear is not so limited.
A low-frequency note is shaped by high-frequency components. If a difference in shape of the lower-frequency can somehow be detected, then inaudible frequences still make a difference.
Normal telephone IIRC cuts off about 3.5kHz.
Hmmm, this might be good for:
Non-skipping CD players.
De-scratching old LP records.
Reconstructing old photographs.
Looking at the 'restored' pic I see only 'horizontal' distortion, imagine how well the picture would have been restored if they would have applied their maths in *two* dimensions...
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The point is, the Nyquist rate tells you the highest spectral component that can be sampled without aliasing. But a triangular wave has frequency components higher than this threshold. These components will be lost in the sampling, and the waveform will not be preserved, although its spectrum will be -- up to the Nyquist rate.
It's essentially a POINT - it has no dimensions. When you see those little squares you actually see a poor (and fast) representation of pixels - pixels themselves are not square or non-square. Pixels won't come in various sizes, they'll still be regular 0-sized points.
Here's a good paper on why it's important to keep in mind the true nature of pixels (by Alvy Ray Smith):
A Pixel Is Not A Little Square, A Pixel Is Not A Little Square, A Pixel Is Not A Little Square! (And a Voxel is Not a Little Cube)
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I finally found a better explanation of the new sampling theory. It has to take repeated passes at the same analogue data. First pass is sampled at regular intervals, as usual. This data is analyzed, then on the second pass areas where the data changed fast are sampled at a higher rate. Repeat if needed...
This will usually give results similar to scanning at the maximum sample rate, then "compressing" by throwing out data points where the values are not changing much -- you need less RAM, but the maximum digitizer speed is the same, and you have to replay the analog data somehow. For instance, in an MRI, the multiple scans might mean holding the patient in the machine longer. That's not good, and enough RAM to hold everything isn't going to add much to the cost of the machine. Also, there is one condition where the results could be different -- if a detail such as a hairline fracture is so fine that it might be entirely missed between the points on the first coarse scan. If you scan at maximum resolution first, you won't miss that.