Near Best Tree Approximation

Tree approximation is a form of nonlinear wavelet approximation that appears naturally in applications such as image compression and entropy encoding. The distinction between tree approximation and the more familiar n-term wavelet approximation is that the wavelets appearing in the approximant are required to align themselves in a certain connected tree structure. This makes their positions easy to encode. Previous work [4,6] has established upper bounds for the error of tree approximation for certain (Besov) classes of functions. This paper, in contrast, studies tree approximation of individual functions with the aim of characterizing those functions with a prescribed approximation error. We accomplish this in the case that the approximation error is measured in L ₂, or in the case p2, in the Besov spaces B _p ⁰(L _p ), which are close to (but not the same as) L _p . Our characterization of functions with a prescribed approximation order in these cases is given in terms of a certain maximal function applied to the wavelet coefficients.     

Recent advances in terahertz imaging

We review recent progress in the field of terahertz “T-ray” imaging. This relatively new imaging technique, based on terahertz time-domain spectroscopy, has the potential to be the first portable far-infrared imaging spectrometer. We give several examples which illustrate the possible applications of this technology, using both the amplitude and phase information contained in the THz waveforms. We describe the latest results in tomographic imaging, in which waveforms reflected from an object can be used to form a three-dimensional representation. Advanced signal processing tools are exploited for the purposes of extracting tomographic results, including spectroscopic information about each reflecting layer of a sample. We also describe the application of optical near-field techniques to the THz imaging system. Substantial improvements in the spatial resolution are demonstrated. Received: 29 January 1999 / Published online: 7 April 1999     

Random Projections of Smooth Manifolds

We propose a new approach for nonadaptive dimensionality reduction of manifold-modeled data, demonstrating that a small number of random linear projections can preserve key information about a manifold-modeled signal. We center our analysis on the effect of a random linear projection operator Φ:ℝ ^N →ℝ ^M , M<N, on a smooth well-conditioned K-dimensional submanifold ℳ⊂ℝ ^N . As our main theoretical contribution, we establish a sufficient number M of random projections to guarantee that, with high probability, all pairwise Euclidean and geodesic distances between points on ℳ are well preserved under the mapping Φ. Our results bear strong resemblance to the emerging theory of Compressed Sensing (CS), in which sparse signals can be recovered from small numbers of random linear measurements. As in CS, the random measurements we propose can be used to recover the original data in ℝ ^N . Moreover, like the fundamental bound in CS, our requisite M is linear in the “information level” K and logarithmic in the ambient dimension N; we also identify a logarithmic dependence on the volume and conditioning of the manifold. In addition to recovering faithful approximations to manifold-modeled signals, however, the random projections we propose can also be used to discern key properties about the manifold. We discuss connections and contrasts with existing techniques in manifold learning, a setting where dimensionality reducing mappings are typically nonlinear and constructed adaptively from a set of sampled training data. This research was supported by ONR grants N00014-06-1-0769 and N00014-06-1-0829; AFOSR grant FA9550-04-0148; DARPA grants N66001-06-1-2011 and N00014-06-1-0610; NSF grants CCF-0431150, CNS-0435425, CNS-0520280, and DMS-0603606; and the Texas Instruments Leadership University Program. Web: dsp.rice.edu/cs.     

Democracy in action: Quantization, saturation, and compressive sensing

Recent theoretical developments in the area of compressive sensing (CS) have the potential to significantly extend the capabilities of digital data acquisition systems such as analog-to-digital converters and digital imagers in certain applications. To date, most of the CS literature has been devoted to studying the recovery of sparse signals from a small number of linear measurements. In this paper, we study more practical CS systems where the measurements are quantized to a finite number of bits; in such systems some of the measurements typically saturate, causing significant nonlinearity and potentially unbounded errors. We develop two general approaches to sparse signal recovery in the face of saturation error. The first approach merely rejects saturated measurements; the second approach factors them into a conventional CS recovery algorithm via convex consistency constraints. To prove that both approaches are capable of stable signal recovery, we exploit the heretofore relatively unexplored property that many CS measurement systems are democratic, in that each measurement carries roughly the same amount of information about the signal being acquired. A series of computational experiments indicate that the signal acquisition error is minimized when a significant fraction of the CS measurements is allowed to saturate (10–30% in our experiments). This challenges the conventional wisdom of both conventional sampling and CS.     

A Simple Proof of the Restricted Isometry Property for Random Matrices

We give a simple technique for verifying the Restricted Isometry Property (as introduced by Candès and Tao) for random matrices that underlies Compressed Sensing. Our approach has two main ingredients: (i) concentration inequalities for random inner products that have recently provided algorithmically simple proofs of the Johnson–Lindenstrauss lemma; and (ii) covering numbers for finite-dimensional balls in Euclidean space. This leads to an elementary proof of the Restricted Isometry Property and brings out connections between Compressed Sensing and the Johnson–Lindenstrauss lemma. As a result, we obtain simple and direct proofs of Kashin’s theorems on widths of finite balls in Euclidean space (and their improvements due to Gluskin) and proofs of the existence of optimal Compressed Sensing measurement matrices. In the process, we also prove that these measurements have a certain universality with respect to the sparsity-inducing basis.      

Terahertz reflection imaging using Kirchhoff migration

We describe a new imaging method that uses single-cycle pulses of terahertz (THz) radiation. This technique emulates data-collection and image-processing procedures developed for geophysical prospecting and is made possible by the availability of fiber-coupled THz receiver antennas. We use a simple migration procedure to solve the inverse problem; this permits us to reconstruct the location and shape of targets. These results demonstrate the feasibility of the THz system as a test-bed for the exploration of new seismic processing methods involving complex model systems.     

Micellar electrokinetic chromatographic and capillary zone electrophoretic methods for screening urinary biomarkers of human disorders: a critical review of the state-of-the-art

Human urine plays a central role in clinical diagnostic being one of the most-frequently used body fluid for detection of biological markers. Samples from patients with different diseases display patterns of biomarkers that differ significantly from those obtained from healthy subjects. The availability of fast, reproducible, and easy-to-apply analytical techniques that would allow identification of a large number of these analytes is thus highly desiderable since they may provide detailed information about the progression of a pathological process. From among the variety of methods so far applied for the determination of urinary metabolites, capillary electrophoresis, both in the capillary zone electrophoresis (CZE) and micellar electrokinetic chromatography (MEKC) modes, represents a robust and reliable analytical tool widely used in this area. The aim of the present article is to focus the interest of the reader on recent applications of MEKC and CZE in the field of urinary biomarkers and to discuss advantages and/or limitations of each mode.     

Analysis of the sinusitis nasal lavage fluid proteome using capillary liquid chromatography interfaced to electrospray ionization-quadrupole time of flight- tandem mass spectrometry

The nasal lavage fluids (NLFs) from four subjects with acute sinusitis were analyzed to investigate the amount of proteins expressed in this pathology at the beginning of the event (day 1) and after 6 days of treatment with antibiotics and a nasal steroid spray. The protein identification was performed with capillary liquid chromatography-electrospray-quadrupole time of flight-(LC-ESI-Q-TOF)-mass spectrometry. The samples collected on the first day contained high-abundant plasma proteins, such as albumin and immunoglobulins, glandular serous cell proteins (lysozyme, lactoferrin, and polymeric immunoglobulin receptor), epithelial keratins, and inflammatory cell proteins (myeloperoxidase, IL-16, and IL-17E). After six days of therapy, the complexity of the proteome was reduced to plasma proteins and lysozyme with no inflammatory markers. The presence of hemoglobin, however, suggested that significant squamous metaplasia with breaches in the epithelial barrier, or nasal steroid-related bleeding, had occurred. The proteomic approach presented here allowed us to identify, in the high complexity of acute sinusitis nasal secretions, the proteins that respond to a pharmacological treatment and that could be suitable as markers of this pathology.     

Terahertz imaging with compressed sensing and phase retrieval

We describe a novel, high-speed pulsed terahertz (THz) Fourier imaging system based on compressed sensing (CS), a new signal processing theory, which allows image reconstruction with fewer samples than traditionally required. Using CS, we successfully reconstruct a 64 x 64 image of an object with pixel size 1.4 mm using a randomly chosen subset of the 4096 pixels, which defines the image in the Fourier plane, and observe improved reconstruction quality when we apply phase correction. For our chosen image, only about 12% of the pixels are required for reassembling the image. In combination with phase retrieval, our system has the capability to reconstruct images with only a small subset of Fourier amplitude measurements and thus has potential application in THz imaging with cw sources.