The dyadic wavelet transform is an effective tool for processing piecewise smooth signals; however, its poor frequency resolution (its low Q-factor) limits its effectiveness for processing oscillatory signals like speech, EEG, and vibration measurements, etc. This paper develops a more flexible family of wavelet transforms for which the frequency resolution can be varied. The new wavelet transform can attain higher Q-factors (desirable for processing oscillatory signals) or the same low Q-factor of the dyadic wavelet transform. The new wavelet transform is modestly overcomplete and based on rational dilations. Like the dyadic wavelet transform, it is an easily invertible ‘constant-Q’ discrete transform implemented using iterated filter banks and can likewise be associated with a wavelet frame for L2(R). The wavelet can be made to resemble a Gabor function and can hence have good concentration in the timefrequency plane. The construction of the new wavelet transform depends on the judicious use of both the transform’s redundancy and the flexibility allowed by frequency-domain filter design. I.

The richness of natural images makes the quest for optimal representations in image processing and computer vision challenging. The latter observation has not prevented the design of image representations, which trade off between efficiency and complexity, while achieving accurate rendering of smooth regions as well as reproducing faithful contours and textures. The most recent ones, proposed in the past decade, share an hybrid heritage highlighting the multiscale and oriented nature of edges and patterns in images. This paper presents a panorama of the aforementioned literature on decompositions in multiscale, multi-orientation bases or dictionaries. They typically exhibit redundancy to improve sparsity in the transformed domain and sometimes its invariance with respect to simple geometric deformations (translation, rotation). Oriented multiscale dictionaries extend traditional wavelet processing and may offer rotation invariance. Highly redundant dictionaries require specific algorithms to simplify the search for an efficient (sparse) representation. We also discuss the extension of multiscale geometric decompositions to non-Euclidean domains such as the sphere or arbitrary meshed surfaces. The etymology of panorama suggests an overview, based on a choice of partially overlapping “pictures”.

Abstract—In this paper, we propose a new sparsity-based algorithm for automatic target detection in hyperspectral imagery (HSI). This algorithm is based on the concept that a pixel in HSI lies in a low-dimensional subspace and thus can be represented as a sparse linear combination of the training samples. The sparse representation (a sparse vector corresponding to the linear combination of a few selected training samples) of a test sample can be recovered by solving an 0-norm minimization problem. With the recent development of the compressed sensing theory, such minimization problem can be recast as a standard linear programming problem or efficiently approximated by greedy pursuit algorithms. Once the sparse vector is obtained, the class of the test sample can be determined by the characteristics of the sparse vector on reconstruction. In addition to the constraints on sparsity and reconstruction accuracy, we also exploit the fact that in HSI the neighboring pixels have a similar spectral characteristic (smoothness). In our proposed algorithm, a smoothness constraint is also imposed by forcing the vector Laplacian at each reconstructed pixel to be minimum all the time within the minimization process. The proposed sparsity-based algorithm is applied to several hyperspectral imagery to detect targets of interest. Simulation results show that our algorithm outperforms the classical hyperspectral target detection algorithms, such as the popular spectral matched filters, matched subspace detectors, adaptive subspace detectors, as well as binary classifiers such as support vector machines. Index Terms—Hyperspectral imagery, sparse recovery, sparse representation, spatial correlation, target detection. I.

Most wavelet transforms used in practice are based on integer sampling factors. Wavelet transforms based on rational sampling factors offer in principle the potential for time-scale signal representations having a finer frequency resolution. Previous work on rational wavelet transforms and filter banks includes filter design methods and frequency domain implementations. We present several specific examples of Daubechies-type filters for a discrete orthonormal rational wavelet transform (FIR filters having a maximum number of vanishing moments) obtained using Gröbner bases. We also present the design of overcomplete rational wavelet transforms (tight frames) with FIR filters obtained using polynomial matrix spectral factorization.

Human vision system has limitations in distinguishing the broad range of gray level values. Human eye can discriminate pixel intensities up to 15-30 gray levels. This restricts the qualitative analysis of radiological images. Hence quantitative analysis is preferred to reveal more information from the image. This article presents texture feature based approach for biomedical image analysis. Images acquired from Computerized Tomography (CT) scan machine have been used for the work. A novel method of texture feature extraction based on Ridgelet transform has been reported in this paper. In the first step, work involves determination of texture features from Region of Interest (ROI). Energy and entropy in partitions of Ridgelet transform images represent texture features. During the next step of work, two-class and multi-class classification has been carried out. Percentage Correct Classification for Ridgelet based energy and entropy features and comparative analysis of performance measures for different organ images have been reported. Keywords: Texture; Ridgelet Transform; Classification 1.

For radio systems there are two resources, frequency and time. Division by frequency (or time), so that each pair of communicators is allocated part of the spectrum for all of the time (or all of the spectrum for part of the time), results in Frequency (or Time) Division Multiple Access (FDMA or TDMA). In Code Division Multiple Access (CDMA), every communicator will be allocated the entire spectrum all of the time (Fig. 1). CDMA has applications in wireless cellular communication as well as navigation (e.g., GPS) systems. CDMA uses codes (here referred to as signature sequences) to identify connections and is an interference limited multiple access system. Because all users transmit on the same frequency, internal interference generated by the system is the most significant factor in determining system capacity and communication quality (e.g., quality of voice in mobile phones). In what follows, we will review frame-theoretic results associated with the problem of designing optimal, i.e. minimum interference, signature sequences in CDMA communication systems. (a) (b) Fig. 1. (a) Demonstration of code-division multiplexing (b) Transmitter structure of a CDMA system.

Besides basis expansions, frames representations play a key role in signal processing. We thus consider the problem of frame domain signal processing, which is more complex and challenging than transform domain processing. Examples of such processing abound, from overlap-add/save convolution, to frequency domain LMS, and frame magnitude reconstruction. We develop a unified view of all these situations by using a common Hilbert space view of the problem, and consider algorithms in this common framework. In addition to a synthetic view of multiple signal processing methods in frames, we derive several original results. This include a direct solution to spectral modification (which usually uses an iterative algorithm) and a unicity condition for reconstruction from frame coefficient magnitudes. Index Terms — frames, orthogonal projection, spectral modification, short-time Fourier transform.

We lay a philosophical framework for the design of overcomplete multidimensional signal decompositions based on the union of two or more orthonormal bases. By combining orthonormal bases in this way, tight (energy preserving) frames are automatically produced. The advantage of an overcomplete (tight) frame over a single orthonormal decomposition is that a signal is likely to have a more sparse representation among the overcomplete set than by using any single orthonormal basis. We discuss the question of the relationship between pairs of bases and the various criteria that can be used to measure the goodness of a particular pair of bases. A particular case considered is the dual-tree Hilbert-pair of wavelet bases. Several definitions of optimality are presented along with conjectures about the subjective characteristics of the ensembles where the optimality applies. We also consider relationships between sparseness and approximate representations.

not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution, the U.S. government or any other entity. Keywords: signal processing, image compression, independent component analysis, sparse Representation of natural signals such as sounds and images is critically important in a broad range of fields such as multimedia, data communication and storage, biomedical imaging, robotics, and computational neuroscience. Often it is crucial that the representation be efficient, i.e., the signals of interest are encoded economically. It is also desirable that the representation be robust to various types of noise. In this thesis, we advocate several ways to expand current signal encoding approaches via the framework of adaptive representations. In recent decades, the multiresolution paradigm has provided powerful mathematical and algorithmic tools to signal encoding. In spite of widely proven effectiveness, such methods ignore statistical structure of the class of signals they should represent. On the other hand, high computational costs artificially confine standard

In this paper we characterize and construct novel oversampled filter banks implementing fusion frames. A fusion frame is a sequence of orthogonal projection operators whose sum can be inverted in a numerically stable way. When properly designed, fusion frames can provide redundant encodings of signals which are optimally robust against certain types of noise and erasures. However, up to this point, few implementable constructions of such frames were known; we show how to construct them using oversampled filter banks. In this work, we first provide polyphase domain characterizations of filter bank fusion frames. We then use these characterizations to construct filter bank fusion frame versions of discrete wavelet and Gabor transforms, emphasizing those specific finite impulse response filters whose frequency responses are well-behaved.