Gagan Mirchandani

A wreath product group approach to signal and image processing .I. Multiresolution analysis

We propose the use of spectral analysis on certain noncommutative finite groups in digital signal processing and, in particular, image processing. We pay significant attention to groups constructed as wreath products of cyclic groups. Within this large class of groups, our approach recovers the discrete Fourier transform (DFT), the Haar wavelet transform, various multichannel pyramid filter banks, and other aspects of multiresolution analysis as special cases of a more general phenomenon. In addition, the group structure provides a rich algebraic structure that can be exploited for the analysis and manipulation of signals. Our approach relies on a synthesis of ideas found in the early work of Holmes (1987, 1990), Karpovsky and Trachtenberg (1985), and others on noncommutative filtering, as well as Diaconiss (1989) spectral analysis approach to understanding data.

A wreath product group approach to signal and image processing .II. Convolution, correlation, and applications

For pt.I see ibid., vol.48, no.1, p.102-32 (2000). This paper continues the investigation of the use of spectral analysis on certain noncommutative finite groups-wreath product groups-in digital signal processing. We describe the generalization of discrete cyclic convolution in convolution over these groups and show how it reduces to multiplication in the spectral domain. Finite group-based convolution is defined in both the spatial and spectral domains and its properties established. We pay particular attention to wreath product cyclic groups and further describe convolution properties from a geometric view point in terms of operations with specific signals and filters. Group-based correlation is defined in a natural way, and its properties follow from those of convolution (the detection of similarity of perceptually similar signals) and an application of correlation (the detection of similarity of group-transformed signals). Several examples using images are included to demonstrate the ideas pictorially.

A frequency-domain method for generation of discrete-time analytic signals

We consider a common frequency-domain procedure hilbert for generating discrete-time analytic signals and show how it fails for a specific class of signals. A new frequency-domain technique ehilbert is formulated that solves the defect. Moreover, the new technique is applicable to all discrete-time real signals of even length. It is implemented by the introduction of one additional zero of the continuous spectrum of the analytic signal hilbert at a negative frequency. Both frequency-domain methods generate equal length discrete-time analytic signals. The new analytic signal preserves the original signal (real part) and also the zeros of the discrete spectrum hilbert in the negative frequencies. The greater attenuation at the negative frequencies affects the degree of aliasing of the analytic signal. It is measured by applying the analytic signal to an orthogonal wavelet transform and determining the improved transform shiftability

Two-dimensional wreath product group-based image processing

A theoretical foundation to the notion of 2D transform and 2D signal processing is given, focusing on 2D group-based transforms, of which the 2D Haar and 2D Fourier transforms are particular instances. Conditions for separability of these transforms are established. The theory is applied to certain groups that are wreath products of cyclic groups to give separable and inseparable 2D wreath product transforms and their filter bank implementations.

Wreath products for image processing

We present a wreath product approach for matched filtering to detect rotated copies of a template in an image. We view the image as a homogeneous space for a wreath product, a noncommutative symmetry group. The corresponding Fourier analysis has a natural multiresolution structure and accompanying efficient algorithm which we explain and illustrate with an example. The associated matched filter is a new example of the use of a noncommutative convolution for image processing. Numerical experiments are described in which this noncommutative approach outperforms standard Fourier-based methods.

Efficient implementation of neural nets using an optimal relationship between number of patterns, input dimension and hidden nodes

Some key issues in the design of neural nets for pattern classification are topology and associated training samples required to obtain adequate performance with test samples. Currently, there does not exist an analytical framework within which to formulate the design of multilayer perceptrons. A theorem that relates input dimension, number of hidden nodes, and number of separable regions is given. The results of application to some experiments reported in the literature and to new experiments are analyzed.<<ETX>>

Discrete-time analytic signals with improved shiftability

We consider a common procedure (Marple, S.L., Jr., IEEE Trans. Sig. Process., vol.47, no.9, p.2600-3, 1999) for generating analytic signals and show how it fails for specific discrete-time real signals. A new frequency domain technique is formulated that solves the defect. Both methods have the same redundancy. The new analytic signal preserves the original signal (real part) and also the zeros of its discrete spectrum in the negative frequencies. The superiority of the new method is in the introduction of one additional zero of the continuous spectrum of the original signal at a negative frequency and a corresponding reduction in shiftability.

Experiments in partitioning and scheduling signal processing algorithms for parallel processing

Implementing signal-processing algorithms on multiple programmable signal processors has the potential for providing increased throughput. The capability can be utilized for implementing more complex code, extending the frequency range of application of programmable signal processors, or substituting less costly processors for expensive fast processors. Results obtained in applying heuristic algorithms to the scheduling of code of seven practical applications in signal and image processing are reported.<<ETX>>

A novel method for generating complex half-band filters

We propose a simple, novel and efficient method for generating complex half-band FIR filters, which we use in the generation of discrete-time analytic (DTA) signals. These filters have properties of linear phase and real-time implementation while the DTA signals generated are orthogonal and invertible (the original real signal is recoverable). The filter design, in contrast to some other methods, is easily scalable and stable. The new method is evaluated for performance (i.e. aliasing) by comparing its shiftability property with that of other transforms. Using a total variation measure for determining function variation, we see that its shiftability either matches or exceeds that of other methods. Furthermore, this design method lends itself to an enhancement, thereby allowing additional improvement in shiftability. We prove an important theoretical aspect of the new method: the amplitude spectrum of the length N filter converges almost everywhere to the ideal complex half-band amplitude spectrum as N /spl rarr/ +/spl infin/, thereby assuring shiftability.

Analytic Functions, Singularities and Edges: A New Formalism

Methods for detecting edges, be they multidimensional or multiresolution, ultimately reduce to finding extremal points, first derivatives or zeros of second derivatives. However, problems such as missing edges, weak edges due to thresholding, derivatives not existing and false edge generation, are some of the consequences. We adopt a new formalism: edges are singularities of the mathematically smoothest function possible - the complex analytic function. We embed a real image into the real part of an analytic function. After solving the conjugate harmonic problem, edges in discrete images are identified from the imaginary part. The analytic function model is inherently two-dimensional and an invariant measure. Comparisons are made with other standard edge detection methods. We outline issues that need to be considered for establishing analytic functions for edge detection