Imaging using compressive sensing |
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Compressive sensing approaches the problem from a different perspective. It starts from the notion that there exists a basis function that can be transformed into through the linear operator in which very few non-zero elements are needed to represent the signal. The compressive sensing approach is then to set up the missing data problem in two phases. First, estimate the elements of the sparse basis function through,
Figure 1 shows an example of this technique applied to a 2-D missing data problem. In this case, we are trying to recover the image seen in Figure 1(a). We start from the data points seen in Figure 1(b), and recover the image seen in Figure 1(c). In this case, the image was subsampled by a factor of 8 and nearly perfect recovery was achieved.
before,random,after
Figure 1. The original image (a), the image sub-sampled by a factor of 8 (b), and the reconstructed image (c) using compressive sensing. Baraniuk et al. (2008). |
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Reconstruction using compressive sensing techniques is expensive. solvers are significantly more expensive than their counterparts, which in themselves can represent a significant cost. As a result, only certain classes of problems benefit from compressive sensing techniques. For compressive sensing to be useful, the cost of acquiring the full dataset must be significant. In addition, the signal must be highly compressible. The following two sections address both criterion.
Imaging using compressive sensing |