6.8 Inference using compressive measurements

While the compressive sensing (CS) literature has focused almost exclusively on problems in signal reconstruction/approximation , this is frequently not necessary. For instance, in many signal processing applications(including computer vision, digital communications and radar systems), signals are acquired only for the purpose of making a detection orclassification decision. Tasks such as detection do not require a reconstruction of the signal, but only require estimates of therelevant sufficient statistics for the problem at hand.

As a simple example, suppose a surveillance system (based on compressive imaging) observes the motion of a person across a static background. The relevant information to be extracted from the data acquired by this system would be, for example, the identity of the person, or the location of this person with respect to a predefined frame of coordinates. There are two ways of doing this:

• Reconstruct the full data using standard sparse recovery techniques and apply standard computer vision/inference algorithms on the reconstructed images.
• Develop an inference test which operates directly on the compressive measurements, without ever reconstructing the full images.

A crucial property that enables the design of compressive inference algorithms is the information scalability property of compressive measurements. This property arises from the following two observations:

• For certain signal models, the action of a random linear function on the set of signals of interest preserves enough information to perform inference tasks on the observed measurements.
• The number of random measurements required to perform the inference task usually depends on the nature of the inference task. Informally, we observe that more sophisticated tasks require more measurements.

We examine three possible inference problems for which algorithms that directly operate on the compressive measurements can be developed: detection (determining the presence or absence of an information-bearing signal), classification (assigning the observed signal to one of two (or more) signal classes), and parameter estimation (calculating a function of the observed signal).

Detection

In detection one simply wishes to answer the question: is a (known) signal present in the observations?To solve this problem, it suffices to estimate a relevant sufficient statistic . Based on a concentration of measure inequality, it is possible to show that such sufficient statistics for a detection problem can be accurately estimated from random projections, where the quality of this estimate depends on the signal to noise ratio (SNR)  [link] . We make no assumptions on the signal of interest $s$ , and hence we can build systems capable of detecting $s$ even when it is not known in advance. Thus, we can use random projections fordimensionality-reduction in the detection setting without knowing the relevant structure.