Mark Lammers

Sobolev Duals for Random Frames and ΣΔ Quantization of Compressed Sensing Measurements

Quantization of compressed sensing measurements is typically justified by the robust recovery results of Candès, Romberg and Tao, and of Donoho. These results guarantee that if a uniform quantizer of step size δ is used to quantize m measurements y=Φx of a k-sparse signal x∈ℝN, where Φ satisfies the restricted isometry property, then the approximate recovery x# via ℓ1-minimization is within O(δ) of x. The simplest and commonly assumed approach is to quantize each measurement independently. In this paper, we show that if instead an rth-order ΣΔ (Sigma–Delta) quantization scheme with the same output alphabet is used to quantize y, then there is an alternative recovery method via Sobolev dual frames which guarantees a reduced approximation error that is of the order δ(k/m)(r−1/2)α for any 0<α<1, if m≳r,αk(logN)1/(1−α). The result holds with high probability on the initial draw of the measurement matrix Φ from the Gaussian distribution, and uniformly for all k-sparse signals x whose magnitudes are suitably bounded away from zero on their support.

Alternative dual frames for digital-to-analog conversion in sigma-delta quantization

We design alternative dual frames for linearly reconstructing signals from sigma–delta (ΣΔ) quantized finite frame coefficients. In the setting of sampling expansions for bandlimited functions, it is known that a stable rth order sigma–delta quantizer produces approximations where the approximation error is at most of order 1 / λr, and λ > 1 is the oversampling ratio. We show that the counterpart of this result is not true for several families of redundant finite frames for

Sigma delta quantization for compressed sensing

\mathbb{R}^d

BRACKET PRODUCTS FOR WEYL-HEISENBERG FRAMES

when the canonical dual frame is used in linear reconstruction. As a remedy, we construct alternative dual frame sequences which enable an rth order sigma–delta quantizer to achieve approximation error of order 1/Nr for certain sequences of frames where N is the frame size. We also present several numerical examples regarding the constructions.

On quantization of finite frame expansions: sigma-delta schemes of arbitrary order

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.

Duality principles, localization of frames, and gabor theory

We provide a detailed development of theL 1function-valued inner product onL 2(ℝ) known as the bracket product. In addition to some of the more basic properties, we show that this inner product has a Bessel’s inequality, a Riesz Representation Theorem, and a Gram—Schmidt process. We then apply this to Weyl—Heisenberg frames to show that there exist “compressed” versions of the frame operator, the frame transform and the preframe operator. Finally, we introduce the notion of an a-frame and show that there is an equivalence between the frames of translates for this function-valued inner product and Weyl—Heisenberg frames.

The finite Balian-Low conjecture

In this note we will show that the so called Sobolev dual is the minimizer over all linear reconstructions using dual frames for stable rth order ΣΔ quantization schemes under the so called White Noise Hypothesis (WNH) design criteria. We compute some Sobolev duals for common frames and apply them to audio clips to test their performance against canonical duals and another alternate dual corresponding to the well known Blackman filter.

Sobolev duals of random frames

The theory of localized frames is a recently introduced concept with broad implications to frame theory in general, as well as to the special cases of Gabor and wavelet frames. Using the new notion of a R-dual sequence associated with a Bessel sequence, we derive several duality principles concerning localization in abstract frame theory. As applications of our results we prove a duality principle of localization of Gabor systems in the spirit of the Ron-Shen duality principle, and obtain a Janssen representation for general frame operators.

The Finite Fractional Zak Transform

We present a conjecture for a finite version of the celebrated Balian-Low Theorem for Gabor systems in L²(R). We proceed to prove a special case of the conjecture for C⁹.

Duality Principles in Frame Theory

Sobolev dual frames have recently been proposed as optimal alternative reconstruction operators that are specifically tailored for Sigma-Delta (¿¿) quantization of frame coefficients. While the canonical dual frame of a given analysis (sampling) frame is optimal for the white-noise type quantization error of Pulse Code Modulation (PCM), the Sobolev dual offers significant reduction of the reconstruction error for the colored-noise of ¿¿ quantization. However, initial quantitative results concerning the use of Sobolev dual frames required certain regularity assumptions on the given analysis frame in order to deduce improvements of performance on reconstruction that are similar to those achieved in the standard setting of bandlimited functions. In this paper, we show that these regularity assumptions can be lifted for (Gaussian) random frames with high probability on the choice of the analysis frame. Our results are immediately applicable in the traditional oversampled (coarse) quantization scenario, but also extend to compressive sampling of sparse signals.