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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."
Even better sound than what we now know as almost perfect? Great! Makes you wonder how much better it will get in the future, even when we have perfect sound....
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.
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.
"It take 9 months to bear a child, no matter how many women you assign to the job."
This is why ADCs do not just sample the incoming voltage -- they integrate over a period of time, to "boil down" the voltage over that time period to an average value, that best represents what the signal was doing during that sampling period.
Now, moving on to your point, which is to vary the sampling rate according to the characteristics of the source; this is somewhat a wasted effort, since in order to determine the source characteristics, you must perform some type of frequency analysis, or autoregression. This is intensive computation, and you would be better off spending that time doing some real compression, such as spectral quantization, or perceptual coding.
Varying the sampling rate from sample-to-sample would be the ultimate, if it were possible to gain anything from it. Unfortunately, if you vary the sampling rate at each sample, then in order to transmit the sampled stream you must transmit not only the samples, but the duration between samples as well. In the worst case you have doubled your data rate, not compressed it.
However, as you say, this could work wonders for the fidelity of the sampled signal. Instead of sampling at regular time intervals, we could build a predictive ADC that samples only when the predicted signal value becomes different from the actual by some predetermined amount. Then, send two values: the sample itself, and the duration since the last sample. This works because the DAC which converts the signal also does interpolation. It would be possible to keep the error arbitrarily small, no matter what the characteristics of the signal, up to the limits of the ADC chip itself.
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
"Why would God give us a waist if we wasn't supposed to rest our pants on it?" - Rev. Roy McDaniels
I don't see that as being true - although you may be able to create a better image of the sound through the non-uniform technique, you still have to have a highly controlled time resolution so that you at least know where your samples are. If your technology is fast enough to give you that time resolution with a low jitter, you're still better off sampling uniformly and getting rid of data later than randomly deleting data at the source.
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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