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."

Comments

Re:So....

[pmc]

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?

Nyquist, not Shannon

[s20451]

It was at Bell Labs ... but the guy who developed the Uniform Sampling Theorem was Nyquist, not Shannon.

Some folks seem to be missing the point on the MRI

[fatboy-fitz]

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.

Fractal compression is pure math

[Leeji]

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.

Time vs. Frequency

[PingXao]

Classical techniques also require that the original signal be "band limited" - a technical term meaning that the signal must stay within certain, defined limits.

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...

Re:Time vs. Frequency

[madsatod]

You're right about the Fourier-stuff.
But I think you misunderstood the "band limited" thing.
When you sample you have to the filter out frequencies above the Nyquist-freq., if you want to avoid aliasing-problems.
Aliasing comes from the mirroring of the spectrum around n*Fsample. So if you don't want your original signal to get distorted when sampling, one have to use an anti-aliasing filter, that "band-limits" the signal to below Fsample/2.
Does this new technique mean, you can skip anti-aliasing filters?

Re:Some useful niche applications

[hrieke]

Accually the width value of 80 for CGA display goes back to the punchcard days, not as you state in trying to keep memory and bandwidth costs down.

And, I'm still trying to figure out by what you mean by non-square pixels. Are you trying to say the physical size on the screen, or how they are stored in memory on the graphics adaptor?

If these guys have the ability to return useful data from non reporting areas I can see a whole range of non niche applications - and real word applications where data recovery would be useful.

Both are right:

[volpe]

From Engineering Fundamentals

The sampling theorem is considered to have been articulated by Nyquist in 1928 and mathematically proven by Shannon in 1949. Some books use the term "Nyquist Sampling Theorem", and others use "Shannon Sampling Theorem". They are in fact the same sampling theorem.

Re:Brain scans?

[perky]

sampling is not guessing. The Nyquist sampling theorem shows that a signal that is sampled at twice the frequency of the highest frequency component in the signal can be reconstructed perfectly. With music this doesn't matter though, because humans have bandlimited hearing, so all we have to do is sample at twice the maximum frequency we can hear.

Re:Varying audio sample rates

[SpinyNorman]

Nope - the sampling accuracy and quantization is only going to affect the accuracy to which you can reconstruct the component frequencies. Whether or not your sampling is capturing given frequency components is a matter of the sampling rate (or more generally - as is applicable here in the case on non-uniform sampling - the minimum inter-sample delays). Higher sampling rate will only gain you higher frequency components; the lower frequency components are already going to be there unless you deliberately chose to lose them via a high pass filter.

Regarding 16 bit vs 24 bit "samples", note that there's a difference between sampling accuracy and the number of bits to store your quantized samples. The two are only the same if you're using linear quantization and thus, for example, storing your 24-bit accuracy sample "itself" (i.e. linearly quantized into 2**24 discrete steps). Linear quantization is rather wasteful as the human hearing system does not have equal discrimination at all volume levels, so you might want to quantize more roughly at higher volume levels something like this:

(0) (1) (2) .. (10 11) (12 13) ... (20 21 22) (23 24 25) etc

So you could sample at 24 bits to capture additional detail at low volume and yet non-linearly quantize to store your samples in 16 bits wihtout losing that detail.

Not like Variable bit-rate

[rhh]

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.

Re:Varying audio sample rates

[perky]

Realistically, you start getting diminishing returns at around (IIRC) 1/4 the sampling rate.

That's not true. The whole point is that if a signal is sampled at frequency f, then it can be reconstructed perfectly if its bandwidth is less than f/2. Go learn the maths instead of making vague statements that you think must be right intuitively, but which you actually don't know about.

Re:Some useful niche applications

[ncc74656]

And, I'm still trying to figure out by what you mean by non-square pixels. Are you trying to say the physical size on the screen, or how they are stored in memory on the graphics adaptor?

The pixels that make up a CGA image aren't square...they were drawn on a 640x200 grid. The pixels on a VGA display at most resolutions are square (1280x1024 is the most common exception)...for instance, (1024/4)/(768/3)=1. With CGA, (640/4)/(200/3)=2.4, which means it's stretched vertically.

A Pixel Is Not A Little Square

[Peter Lake]

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)