December 2011 Archives

This is interesting, and sure to make some people angry: A published patent from Feb. 2008 for a "System and Method for Compressive Sampling for Multimedia Coding." Here is the abstract:

An apparatus comprising a decorrelator, a compressive sampler coupled to the decorrelator, and an encoder coupled to the compressive sampler, wherein the compressive sampler is configured to receive sparse data and compress the sparse data using compressive sampling. Also included is a network component comprising at least one processor configured to implement a method comprising decorrelating sparse data or data including sparse data, compressing the sparse data using compressive sampling, and encoding the data. Also included is a method comprising receiving a data stream comprising a sparse data portion, compressing the sparse data portion using compressive sampling, and compressing the remaining data portion without using compressive sampling.
I like this graphic from the patent: CSpatent.png Seems to me, that would cover all possible devices for compression. Data not sparse? Ok, compress it with other patented method, and now all your codecs are belong to us. K THX.

The first author, Jun Tian, apparently has many many other patent applications. Here is one for a "Communication System with Compressive Sensing". I cannot tell whether either have been granted. (What does "published" mean?)

What is "exact recovery"?

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Since compressive sampling (CS) is all about signal acquisition, its faithful recoverability of sensed signals is of paramount importance. Thus, "exact recovery" is a good thing to measure when it comes time for experimental work. However, we find in the CS literature at least three definitions of "exact recovery":

  1. When the signal support is recovered with no false alarms, and no missed detections;
  2. When the normalized squared model error is less than \(\epsilon^2\).
  3. When the largest magnitude difference in the model error is less than \(\epsilon\).
In a digital world of limited precision, the second and third definitions give trouble if we define \(\epsilon = 0\). Thus, this value is usually made a little larger. In Maleki and Donoho, e.g., they set \(\epsilon^2 = 0.0001\) in the second criterion. Others set it to \(\epsilon^2 = 0.000001\). Sometimes I find work that does not mention what value was used, or even the criterion for "exact recovery"! So what does this value mean? And how should it be set?

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