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Compressed Sensing for Error Protection in JPWL

Error correction of JPEG2000 images is handled in ITU-T Rec. T.810 “Information technology – JPEG 2000 image coding system: Wireless” which is referred to as JPWL. The main features of JPWL are header error protection, error sensitivity and unequal error protection. Although Reed-Solomon (RS) FEC codes are defined in the standard as the error correction mechanism, there may be some advantages to using compressed sensing instead.

Compressed sensing (CS) is a technique where a signal x is multiplied by an M × N (M < N) sampling matrix in order to both sample and compress it in a single operation. The signal x is recovered by finding the ℓ1 norm of the sparse version of x which could be represented by the received samples y. In other words, out of the infinite signals which could have been used to create the received y, the sparsest one is recovered as the original signal. As long as x is “sparse enough”, it can be recovered exactly using this technique. CS techniques can also be used as an error correction mechanism directly by oversampling the data (i.e. M > N) and using the ℓ1 norm to determine the sparse error signal.

Images which are compressed using CS are naturally resilient to errors. Based on this, it is possible to use CS to compress the image and simultaneously protect it from errors (and also encrypt the image). There are two things which need to be addressed for this scheme to work. First, the location of the errors must be determined. This can be done using a simple even parity scheme. Once the location of these errors (or at least most of theme) is found, the image is reconstructed as if the incorrect samples had never been transmitted.

Since the quality of the reconstructed image is dependent on the number of samples which are used to reconstruct the image, the quality of the image will decrease if there are a large number of errors. This can be avoided by increasing the number of samples encoded based on the estimated bit error rate (BER).

Compressed sensing can also be used for error correction. The sampling matrix is created so that the number of samples is more than the number of items in the signal to be transmitted in order to create redundancy. After the signal is received, the same ℓ1 minimization techniques used in CS reconstruction can be used to solve for the sparse error vector.

There are advantages and disadvantages to both approaches. Using CS as as the compression scheme makes sense in resource constrained transmitting devices because three things can be accomplished with the same operation. However, this technique will only work on the image data and not on the headers. This means that a RS or similar code must still be used on the header data. CS as error correction doesn’t have this problem. It relies on the sparsity of the errors instead of the sparsity of the signal which means that the entire signal can be protected using this method. The disadvantage to this scheme is that another protocol must be used, though this would still work to encrypt the data.

Only the second scheme allows for unequal error protection. In the first case, the more errors expected, the more samples used to represent the image. With perfect BER estimation, the exact number of samples needed can be calculated and transmitted. Using less samples will correspond to a decrease in image quality rather than a decrease in protection. In the second case, however, CS not only works but has two major advantages over RS. First, the number of levels and the strength of the levels used for UEP is only restricted by the granularity of the samples. A user can use optimization techniques to find the best possible error correction strengths for each of the sensitivity levels.

VOCAL Technologies, Ltd.
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Amherst New York 14228
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