Compressed sensing is a signal processing technique. It is used to acquire and then reconstruct a signal by finding solutions within under-determined linear systems. The theory and applications are based on the principle that, with optimization, a signal’s sparsity can be exploited to recover it using fewer samples than other techniques.

Within the compressed sensing theory and applications, there are two conditions where a signal recovery is possible.

1. The signal must be sparse in some domain, which is defined as a matrix where most of the elements are zero. Dense matrixes do not benefit from the application of compressed sensing.

2. The incoherence applied through isometric properties must be sufficient for the spare signals within the evaluated matrix.

Compressed sensing does not require prior knowledge of the signal or assumptions to be made to initiate the reconstruction process. A simple series of measurements, from an engineering standpoint, is all that is required to begin the reconstruction process.

### How Compressed Sensing Is Able to Work

Compressed sensing is able to take advantage of the redundancies that can be found in an “interesting signal.” That means the signal must contain something more than pure noise. Although the process is most effective when the sparsity rates where the elements can be defined as zero, it is still possible for this theory and its applications to work when a majority of the coefficients are close to zero.

The process begins most often when a weighted linear combination of samples, referred to as a compressive measurement, is taken in a basis that is different from the basis where the sparsity of the signal is known. The comparative process allows for small measurements to be discovered, which usually contain useful information for signal reconstruction. As the domain is analyzed, it becomes possible to convert the image back into the intended domain.

This is possible even though the compressive measurements will be smaller than the number of pixels or other types of information that are found within the signal.

For the compressed sensing theory and applications to work properly, the sparsity constraint must sometimes be enforced. This process is started when one must solve for an under-determined or undetermined system of linear equations. Enforcement allows for the number of non-zero components to be minimized, allowing for the solution to be found.

### How the Compressed Sensing Theory is Applied Today

The most common place where someone will encounter the compressed sensing theory and its applications is within the camera sensor of a mobile phone. The technology allows for a reduction in image acquisition by up to a factor of 15. Bell Laps is even using single-pixel cameras that takes a still image by using repetitive snapshots of random apertures chosen from a grid.

Holographic technologies are also based on compressed sensing theories. It is used to improve image reconstruction because it can increase the number of voxels permitted during the volume-rendering stage of the image. The applications can also be used to retrieve images from under-sampled measurements in millimeter-wave and optical holographic technologies.

Additional technologies which take advantage of the compressed sensing theory and applications are facial recognition, magnetic resonance imaging, radio astronomy through aperture synthesis, and network tomography.

### Does the Nyquist-Shannon Sampling Theorem Apply?

One of the earliest breakthroughs in the field of signal processing was the Nyquist-Shannon sampling theorem. To summarize the theorem, if the highest frequency of a signal is less than 50% of the sampling rate, then the signal can be perfectly reconstructed by using sinc interpolation. With prior knowledge regarding the constraints of signal frequencies, fewer samples are then required to reconstruct the signal.

Compressed sensing utilizes an evolution of the theorem for its application. With given knowledge about the sparsity of a signal, even fewer samples are required to complete a reconstruction than what Nyquist-Shannon sampling theorem requires. This is what provides the foundation for the compressed sensing theory and its applications.

Although it seems like compressed sensing might violate the ideas behind the Nyquist-Shannon sampling theorem, the two actually use different determining factors. The Nyquist-Shannon sampling theorem focuses on its highest frequency. Compressed sensing focuses on the sparsity of the signal. The former creates a reconstruction based on sufficient instead of necessary conditions.

The latter creates a reconstruction based on necessary instead of sufficient conditions.

By using the principles here, engineers can reconstruct numerous signals to retain the information it contains. That is why it has become such an integral component of modern technologies.