Say Song Goh

A study of the rank-ambiguity issues in direction-of-arrival estimation

We first extend a theorem on linear independence of steering vectors proposed by Godara and Cantoni to include more array-sensor scenarios. We then show that an array can have pairwise linearly independent steering vectors even when all its intersensor spacings are more than /spl lambda//2 where /spl lambda/ is the wavelength of the signals. We next propose a theorem for characterizing rank-2 ambiguity, which is applicable to direction-of-arrival estimation applications where the array sensor locations are fixed and known. Subsequently, we identify a class of three-sensor arrays and a class of uniform circular arrays that have pairwise linearly independent steering vectors and are free of rank-2 ambiguity. We also show that collinearity of sensors, uniformity in intersensor spacings, the dimensions of intersensor spacings, or a combination of some or all of these may give rise to linearly dependent steering vectors. In particular, we demonstrate that for a m-sensor array, m linearly dependent steering vectors exist if the aperture is greater than [(m-1)/2]/spl lambda//2, or when at least ([(m+1)/2]+1) sensors are collinear.

Reconstruction of bandlimited signals from irregular samples

Abstract The classical theorem of Shannon enables one to reconstruct a finite-energy, bandlimited signal from a set of regularly spaced samples. Recently, Benedetto and Heller applied the theory of frames to derive a series of sampling theorems with irregularly spaced sampling sequences. In this paper, we study one of these theorems with emphasis on its implementation. To implement the theorem, sampling sequences and sampled coefficients are required. Here, general schemes to construct sampling sequences, and to evaluate sampled coefficients, are established. In addition, we provide an error analysis on the approximation of sampled coefficients. Numerical results are furnished to illustrate the theory and to study various related issues. These issues include the choice of sampling sequences and functions, the effect of truncating the sampling formula, and the influence of the irregularity of sampling sequences.

Matrix extension and biorthogonal multiwavelet construction

Abstract Suppose that P(z) and P (z) are two r × n matrices over the Laurent polynomial ring R[z], where r P(z) P (z)∗ = I r on the unit circle T . We develop an algorithm that produces two n × n matrices Q(z) and Q (z) over R[z], satisfying the identity Q(z) Q (z)∗ = I n on T such that the submatrices formed by the first r rows of Q(z) and Q (z) are P(z) and P (z) respectively. Our algorithm is used to construct compactly supported biorthogonal multiwavelets from multiresolutions generated by univariate compactly supported biorthogonal scaling functions with an arbitrary dilation parameter m ∈ Z, where m >1.

Automated Microaneurysm Segmentation and Detection using Generalized Eigenvectors

Diabetic retinopathy is a major cause of blindness and microaneurysms are the first clinically observable manifestations of diabetic retinopathy. Regular screening and timely intervention can halt or reverse the progression of this disease. This paper describes an approach that is based on the generalized eigenvectors of affinity matrix to extract microaneurysms from digital retinal images. Microaneurysms are in the low intensity regions and detection is complicated by their small sizes, the presence of retinal vessels, and their similarity to another type of retinal abnormality - haemorrhages. In order to accurately detect microaneurysms, the affinity matrix is defined to suppress larger structures such as blood vessels, haemorrhages, etc and to create uniform affinity distribution for pixels belonging to microaneurysms. The generalized eigenvector solution seeks to find the optimal segmentation for microaneurysms and provides indication to the possible locations of microaneurysms. We differentiate the true microaneurysms by studying their feature characteristics. Experiments on 70 retinal sub-images of diabetic patients indicate that we are able to achieve 93% accuracy in the detection of microaneurysms.

Joint Estimation of Time Delay and Doppler Shift for Band-Limited Signals

The standard approach for joint estimation of time delay and Doppler shift of a signal is to estimate the point at which the cross ambiguity function of the original and modified signals attains its maximum modulus. Since band-limited signals can be expressed exactly by their Shannon series, we here consider approximated signals gained by truncating their Shannon series to involve only the sampled signal values. We then estimate the time delay and Doppler shift by calculating a point at which the cross ambiguity function of the approximated signals attains its maximum modulus. This cross ambiguity function has an analytic expression which allows its evaluation at any point, and we may apply Newtons method to calculate accurately and efficiently a point where the maximum modulus is attained. In the numerical experiments we conducted, our method generally outperformed other methods for estimation of both time delay and Doppler shift.

Local discriminant time-frequency atoms for signal classification

Abstract Three methods to select discriminant time-frequency atoms from the Gabor time-frequency dictionary are proposed. The first method performs a discriminant pursuit in its selection, the second method leads to atoms with the most discriminant power, and the third method combines the first two methods. The time-frequency atoms selected by the methods extract discriminant features among different classes of signals. Experimental results on the classification of simulated data sets (triangular waveforms) and real data sets (speech signals) using the extracted features are presented.

Fourier-like frames on locally compact abelian groups

We consider a class of functions, defined on a locally compact abelian group by letting a class of modulation operators act on a countable collection of functions. We derive sufficient conditions for such a class of functions to form a Bessel sequence or a frame and for two such systems to be dual frames. Explicit constructions are obtained via various generalizations of the classical B-splines to the setting of locally compact abelian groups.

Mathematics and Computation in Imaging Science and Information Processing

The explosion of data arising from rapid advances in communication, sensing and computational power has concentrated research effort on more advanced techniques for the representation, processing, analysis and interpretation of data sets. In view of these exciting developments, the program Mathematics and Computation in Imaging Science and Information Processing was held at the Institute for Mathematical Sciences, National University of Singapore, from July to December 2003 and in August 2004 to promote and facilitate multidisciplinary research in the area. As part of the program, a series of tutorial lectures were conducted by international experts on a wide variety of topics in mathematical image, signal and information processing. This compiled volume contains survey articles by the tutorial speakers, all specialists in their respective areas. They collectively provide graduate students and researchers new to the field a unique and valuable introduction to a range of important topics at the frontiers of current research.

An algorithm for constructing multidimensional biorthogonal periodic multiwavelets

This paper deals with the problem of constructing multidimensional biorthogonal periodic multiwavelets from a given pair of biorthogonal periodic multiresolutions. Biorthogonal polyphase splines are introduced to reduce the problem to a matrix extension problem, and an algorithm for solving the matrix extension problem is derived. Sufficient conditions for collections of periodic multiwavelets to form a pair of biorthogonal Riesz bases of the entire function space are also obtained.

Uncertainty products of local periodic wavelets

This paper is on the angle–frequency localization of periodic scaling functions and wavelets. It is shown that the uncertainty products of uniformly local, uniformly regular and uniformly stable scaling functions and wavelets are uniformly bounded from above by a constant. Results for the construction of such scaling functions and wavelets are also obtained. As an illustration, scaling functions and wavelets associated with a family of generalized periodic splines are studied. This family is generated by periodic weighted convolutions, and it includes the well‐known periodic B‐splines and trigonometric B‐splines.