C. Sinan Güntürk

Sigma delta quantization for compressed sensing

Recent results make it clear that the compressed sensing paradigm can be used effectively for dimension reduction. On the other hand, the literature on quantization of compressed sensing measurements is relatively sparse, and mainly focuses on pulse-code-modulation (PCM) type schemes where each measurement is quantized independently using a uniform quantizer, say, of step size ¿. The robust recovery result of Cande¿s et al. and Donoho guarantees that in this case, under certain generic conditions on the measurement matrix such as the restricted isometry property, ¿1 recovery yields an approximation of the original sparse signal with an accuracy of O(¿). In this paper, we propose sigma-delta quantization as a more effective alternative to PCM in the compressed sensing setting. We show that if we use an rth order sigma-delta scheme to quantize m compressed sensing measurements of a k-sparse signal in ¿N, the reconstruction accuracy can be improved by a factor of (m/k)(r-1/2)¿ for any 0 < ¿ < 1 if m ¿r k(log N)1/(1-¿) (with high probability on the measurement matrix). This is achieved by employing an alternative recovery method via rth-order Sobolev dual frames.

VQ-adaptive block transform coding of images

Two new design techniques for adaptive orthogonal block transforms based on vector quantization (VQ) codebooks are presented. Both techniques start from reference vectors that are adapted to the characteristics of the signal to be coded, while using different methods to create orthogonal bases. The resulting transforms represent a signal coding tool that stands between a pure VQ scheme on one extreme and signal-independent, fixed block transformation-like discrete cosine transform (DCT) on the other. The proposed technique has superior compaction performance as compared to DCT both in the rendition of details of the image and in the peak signal-to-noise ratio (PSNR) figures.

Distributed Noise-Shaping Quantization: I. Beta Duals of Finite Frames and Near-Optimal Quantization of Random Measurements

This paper introduces a new algorithm for the so-called “analysis problem” in quantization of finite frame representations that provides a near-optimal solution in the case of random measurements. The main contributions include the development of a general quantization framework called distributed noise shaping and, in particular, beta duals of frames, as well as the performance analysis of beta duals in both deterministic and probabilistic settings. It is shown that for random frames, using beta duals results in near-optimally accurate reconstructions with respect to both the frame redundancy and the number of levels at which the frame coefficients are quantized. More specifically, for any frame E of m vectors in

Sobolev duals of random frames

\mathbb {R}^k

Distributed Noise-Shaping Quantization: II. Classical Frames

Rk except possibly from a subset of Gaussian measure exponentially small in m and for any number

Sobolev Duals for Random Frames and Sigma-Delta Quantization of Compressed Sensing Measurements

L \ge 2