Complete Communications Engineering

There are a number of cases in imaging where the act of acquiring the image itself can be very difficult and resource intensive. One obvious example of this is in the medical field with MRI imaging. A typical MRI test could take up to two hours to perform, during which time the patient must stay entirely still. This can be very difficult for small children, people with claustrophobia, etc. Another obvious example is with the Gigapixel Project, which is a project to take extremely detailed images of cityscapes and landmarks (~ 4 billion pixels) for preservation. One of these images takes approximately 24 Gb of disk space and must be captured with very specialized (and expensive) analog equipment and then converted to digital for storage.

In most digital image capture technology, an image is first captured as a raw data file and then compressed in order to save storage space, transmission time or both. This means that a huge amount of data (sometimes more than 90% of the data) is obtained and then almost immediately thrown away. Compressed sensing (CS) offers us an alternative to this by allowing the number of samples required to capture an image to be reduced at the cost of more total samples required to represent that same image.

The theory of compressed sensing states that if a signal x ∈N has a sparse representation in some domain (such as DCT or wavelet), it can be captured by taking y = Φx where y ∈M, Φ is a known matrix M × N and M &lt N. It can then be approximately recovered using ℓ1 minimization of the sparse representation of the image. This means that instead of capturing the entire image and then compressing it, the compressed samples are directly captured by the imaging device.

The trade off to this is that it takes roughly five times as many samples as it would take wavelet coefficients (or DCT or whatever sparsifying transform is used) in order to reconstruct the original image with the same quality. For example, a 1024 x 1024 image could be compressed to 5% (52,000 wavelet coefficients) or 25% (262,000 samples) by CS to achieve a given image quality. However, in systems where the actual capture of an image can be difficult, there could be a great advantage to using this system. If an MRI system takes 1 hour to achieve a given quality, the same quality can be achieved in 15 minutes with compressed sensing.

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