Universal sampling sets in subsampling and reconstructing image

Description

For this project I plan to investigate the use of universal sampling sets in subsampling and reconstructing images. I will take images which are relatively sparse in the DFT domain, and construct universal sampling sets according to their size.

For two dimensional signals, the cartesian product of two one-dimensional universal sampling sets can be used. So, for the example of a 512*512 image, we could nd any universal sampling set for length 512 = 29 signals via the results from [1] and form the cartesian product of it with itself.

Using this universal sampling set I will subsample the image and attempt to reconstruct it. The interesting part will be to see how faithfully an image can be reconstructed and to what extent noise aFFects the reconstruction. Using this method, the reconstruction algorithm amounts to inverting a speci c submatrix of the DFT matrix, and requires knowledge of the bandlimit set J .

Bibliography

[1] Brad Osgood, Aditya Siripuram, and William Wu, Discrete Sampling and Interpolation: Universal Sampling Sets for Discrete Bandlimited Spaces, IEEE Transactions on Information Theory 58 (2012), 4176-4200.

[2] Aditya Siripuram, William Wu, and Brad Osgood, Discrete Sampling and Interpolation: Orthogonal Interpolation for Discrete Bandlimited Signals (Accesed 10/28/2015), available at http://arxiv.org/abs/1411.7086.

[3] Boris Alexeev, Jameson Cahill, and Dustin G Mixon, Full Spark Frames, The Journal of Fourier Analysis and Applications 18 (2012), 1167-1194.

[4] D. Needell and J. A. Tropp, CoSaMP: Iterative Signal Recovery from Incomplete and Inaccurate Samples (Accessed 4/7/2014), available at http://arxiv.org/pdf/0803.2392v2.pdf.

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