Pier Luigi Dragotti

Sampling Moments and Reconstructing Signals of Finite Rate of Innovation: Shannon Meets Strang–Fix

Consider the problem of sampling signals which are not bandlimited, but still have a finite number of degrees of freedom per unit of time, such as, for example, nonuniform splines or piecewise polynomials, and call the number of degrees of freedom per unit of time the rate of innovation. Classical sampling theory does not enable a perfect reconstruction of such signals since they are not bandlimited. Recently, it was shown that, by using an adequate sampling kernel and a sampling rate greater or equal to the rate of innovation, it is possible to reconstruct such signals uniquely . These sampling schemes, however, use kernels with infinite support, and this leads to complex and potentially unstable reconstruction algorithms. In this paper, we show that many signals with a finite rate of innovation can be sampled and perfectly reconstructed using physically realizable kernels of compact support and a local reconstruction algorithm. The class of kernels that we can use is very rich and includes functions satisfying Strang-Fix conditions, exponential splines and functions with rational Fourier transform. This last class of kernels is quite general and includes, for instance, any linear electric circuit. We, thus, show with an example how to estimate a signal of finite rate of innovation at the output of an RC circuit. The case of noisy measurements is also analyzed, and we present a novel algorithm that reduces the effect of noise by oversampling

Sparse Sampling of Signal Innovations

Sparse sampling of continuous-time sparse signals is addressed. In particular, it is shown that sampling at the rate of innovation is possible, in some sense applying Occams razor to the sampling of sparse signals. The noisy case is analyzed and solved, proposing methods reaching the optimal performance given by the Cramer-Rao bounds. Finally, a number of applications have been discussed where sparsity can be taken advantage of. The comprehensive coverage given in this article should lead to further research in sparse sampling, as well as new applications. One main application to use the theory presented in this article is ultra-wide band (UWB) communications.

Directionlets: anisotropic multidirectional representation with separable filtering

In spite of the success of the standard wavelet transform (WT) in image processing in recent years, the efficiency of its representation is limited by the spatial isotropy of its basis functions built in the horizontal and vertical directions. One-dimensional (1-D) discontinuities in images (edges and contours) that are very important elements in visual perception, intersect too many wavelet basis functions and lead to a nonsparse representation. To efficiently capture these anisotropic geometrical structures characterized by many more than the horizontal and vertical directions, a more complex multidirectional (M-DIR) and anisotropic transform is required. We present a new lattice-based perfect reconstruction and critically sampled anisotropic M-DIR WT. The transform retains the separable filtering and subsampling and the simplicity of computations and filter design from the standard two-dimensional WT, unlike in the case of some other directional transform constructions (e.g., curvelets, contourlets, or edgelets). The corresponding anisotropic basis functions (directionlets) have directional vanishing moments along any two directions with rational slopes. Furthermore, we show that this novel transform provides an efficient tool for nonlinear approximation of images, achieving the approximation power O(N/sup -1.55/), which, while slower than the optimal rate O(N/sup -2/), is much better than O(N/sup -1/) achieved with wavelets, but at similar complexity.

The Distributed Karhunen–Loève Transform

The Karhunen-Loeve transform (KLT) is a key element of many signal processing and communication tasks. Many recent applications involve distributed signal processing, where it is not generally possible to apply the KLT to the entire signal; rather, the KLT must be approximated in a distributed fashion. This paper investigates such distributed approaches to the KLT, where several distributed terminals observe disjoint subsets of a random vector. We introduce several versions of the distributed KLT. First, a local KLT is introduced, which is the optimal solution for a given terminal, assuming all else is fixed. This local KLT is different and in general improves upon the marginal KLT which simply ignores other terminals. Both optimal approximation and compression using this local KLT are derived. Two important special cases are studied in detail, namely, the partial observation KLT which has access to a subset of variables, but aims at reconstructing them all, and the conditional KLT which has access to side information at the decoder. We focus on the jointly Gaussian case, with known correlation structure, and on approximation and compression problems. Then, the distributed KLT is addressed by considering local KLTs in turn at the various terminals, leading to an iterative algorithm which is locally convergent, sometimes reaching a global optimum, depending on the overall correlation structure. For compression, it is shown that the classical distributed source coding techniques admit a natural transform coding interpretation, the transform being the distributed KLT. Examples throughout illustrate the performance of the proposed distributed KLT. This distributed transform has potential applications in sensor networks, distributed image databases, hyper-spectral imagery, and data fusion

Filter bank frame expansions with erasures

We study frames for robust transmission over the Internet. In our previous work, we used quantized finite-dimensional frames to achieve resilience to packet losses; here, we allow the input to be a sequence in l/sub 2/(Z) and focus on a filter-bank implementation of the system. We present results in parallel, R/sup N/ or C/sup N/ versus l/sub 2/(Z), and show that uniform tight frames, as well as newly introduced strongly uniform tight frames, provide the best performance.

Rate-distortion optimized tree-structured compression algorithms for piecewise polynomial images

This paper presents novel coding algorithms based on tree-structured segmentation, which achieve the correct asymptotic rate-distortion (R-D) behavior for a simple class of signals, known as piecewise polynomials, by using an R-D based prune and join scheme. For the one-dimensional case, our scheme is based on binary-tree segmentation of the signal. This scheme approximates the signal segments using polynomial models and utilizes an R-D optimal bit allocation strategy among the different signal segments. The scheme further encodes similar neighbors jointly to achieve the correct exponentially decaying R-D behavior (D(R)/spl sim/c/sub 0/2/sup -c1R/), thus improving over classic wavelet schemes. We also prove that the computational complexity of the scheme is of O(NlogN). We then show the extension of this scheme to the two-dimensional case using a quadtree. This quadtree-coding scheme also achieves an exponentially decaying R-D behavior, for the polygonal image model composed of a white polygon-shaped object against a uniform black background, with low computational cost of O(NlogN). Again, the key is an R-D optimized prune and join strategy. Finally, we conclude with numerical results, which show that the proposed quadtree-coding scheme outperforms JPEG2000 by about 1 dB for real images, like cameraman, at low rates of around 0.15 bpp.

Wavelet footprints: theory, algorithms, and applications

Wavelet-based algorithms have been successful in different signal processing tasks. The wavelet transform is a powerful tool because it manages to represent both transient and stationary behaviors of a signal with few transform coefficients. Discontinuities often carry relevant signal information, and therefore, they represent a critical part to analyze. We study the dependency across scales of the wavelet coefficients generated by discontinuities. We start by showing that any piecewise smooth signal can be expressed as a sum of a piecewise polynomial signal and a uniformly smooth residual (Theorem 1). We then introduce the notion of footprints, which are scale space vectors that model discontinuities in piecewise polynomial signals exactly. We show that footprints form an overcomplete dictionary and develop efficient and robust algorithms to find the exact representation of a piecewise polynomial function in terms of footprints. This also leads to efficient approximation of piecewise smooth functions. Finally, we focus on applications and show that algorithms based on footprints outperform standard wavelet methods in different applications such as denoising, compression, and (nonblind) deconvolution. In the case of compression, we also prove that at high rates, footprint-based algorithms attain optimal performance (Theorem 3).

The distributed Karhunen-Loeve transform

The Karhunen-Loeve transform is a key element of many signal processing tasks, including classification and compression. In this paper, we consider distributed signal processing scenarios with limited communication between correlated sources, and we investigate a distributed Karhunen-Loeve transform (KLT). In particular, a partial (where only a subset of sources are observed) and a conditional KLT (where some sources act as side information) are posed and solved in a rate-distortion sense. The partial KLT leads to an original bit allocation problem, while the conditional KLT leads to a Wyner-Ziv solution which is separable at the sources. These two cases can be seen as extreme cases of a distributed KLT.

Sensing reality and communicating bits: a dangerous liaison

To illustrate the conceptual issues related to sampling, source representation/coding and communication in sensor networks, we review the underlying theory and discuss specific examples. We show how the structure of the distributed sensing and communication problem dictates new processing architectures. The key challenge lies in the discretization of space, time and amplitude, since most of the advanced signal processing systems operate in discrete domain.

Exact Feature Extraction Using Finite Rate of Innovation Principles With an Application to Image Super-Resolution

The accurate registration of multiview images is of central importance in many advanced image processing applications. Image super-resolution, for example, is a typical application where the quality of the super-resolved image is degrading as registration errors increase. Popular registration methods are often based on features extracted from the acquired images. The accuracy of the registration is in this case directly related to the number of extracted features and to the precision at which the features are located: images are best registered when many features are found with a good precision. However, in low-resolution images, only a few features can be extracted and often with a poor precision. By taking a sampling perspective, we propose in this paper new methods for extracting features in low-resolution images in order to develop efficient registration techniques. We consider, in particular, the sampling theory of signals with finite rate of innovation and show that some features of interest for registration can be retrieved perfectly in this framework, thus allowing an exact registration. We also demonstrate through simulations that the sampling model which enables the use of finite rate of innovation principles is well suited for modeling the acquisition of images by a camera. Simulations of image registration and image super-resolution of artificially sampled images are first presented, analyzed and compared to traditional techniques. We finally present favorable experimental results of super-resolution of real images acquired by a digital camera available on the market.