Modified Zak transform for the quincunx-type Gabor lattice

Gabor's (1946) signal expansion and the Gabor transform are formulated on a non-orthogonal time-frequency lattice instead of on the traditional rectangular lattice. The reason for doing so is that a non-orthogonal sampling geometry might be better adapted to the form of the window functions (in the time-frequency domain) than an orthogonal one. Oversampling in the Gabor scheme, which is required to have mathematically more attractive properties for the analysis window, then leads to better results in combination with less oversampling. The new procedure presented in this paper is based on considering the non-orthogonal lattice as a sub-lattice of a denser orthogonal lattice that is oversampled by a rational factor. In doing so, Gabor's signal expansion on a non-orthogonal lattice can be related to the expansion on an orthogonal lattice (restricting ourselves, of course, to only those sampling points that are part of the non-orthogonal sub-lattice), and all the techniques that have been derived for rectangular sampling can be used, albeit in a slightly modified form.

Gabor's expansion and the Zak transform for continuous-time and discrete-time signals

Advanced Topics in Shannon Sampling and Interpolation Theory is the second volume of a textbook on signal analysis solely devoted to the topic of sampling and restoration of continuous time signals and images. Sampling and reconstruction are fundamental problems in any field that deals with real-time signals or images, including communication engineering, image processing, seismology, speech recognition, and digital signal processing. This second volume includes contributions from leading researchers in the field on such topics as Gabor's signal expansion, sampling in optical image formation, linear prediction theory, polar and spiral sampling theory, interpolation from non-uniform samples, an extension of Papoulis' generalized sampling expansion to higher dimensions, and applications of sampling theory to optics and to time-frequency representations.

Image interpolation based on median-type filters is investigated. We study this problem for two decimation schemes that are most commonly used in practice: (1) quincunx lattice and (2) rectangular lattice. For each decimation lattice, we propose a number of interpolation structures that are based on some median or finite impulse response (FIR)-median hybrid filters. Extensive simulations show that the proposed median-based interpolators outperform significantly as compared with linear schemes. To further verify the effectiveness of these medianbased interpolators, we compare them with multilayer neural networks designed through a training process, and still show a preference for the median-based interpolators. As an application, they are used in an interpolative Joint Photographic Experts Group (JPEG) coding scheme to suppress blocking artifacts at very low bit rates.

A coprime antenna array consists of two or more sparse subarrays featuring enhanced degrees of freedom (DOF) and reduced mutual coupling. This paper introduces a new class of planar coprime arrays, based on the theory of ideal lattices. In quadratic number fields, a splitting prime p can be decomposed into the product of two distinct prime ideals, which give rise to the two sparse subarrays. Their virtual difference coarray enjoys a quadratic gain in DOF, thanks to the generalized Chinese remainder theorem (CRT). To enlarge the contiguous aperture of the coarray, we present hole-free symmetric CRT arrays with simple closed-form expressions. The ring of Gaussian integers and the ring of Eisenstein integers are considered as examples to demonstrate the procedure of designing coprime arrays. With Eisenstein integers, our design yields a difference coarray that is a subset of the hexagonal lattice, offering a significant gain in DOF over the rectangular lattice, given the same physical areas. Maximization of CRT arrays in the aspect of essentialness and the superior performance in the context of angle estimation will be presented in the companion paper (Part II).

Recent development in vision and image understanding related study reveals that a signal decomposition before processing may provide enormous useful information about the signal. Various signal decomposition models, such as the Gabor and wavelet transforms have been proposed. While the Gabor signal expansion creates a fixed resolution space-frequency signal representation, the wavelet transform provides a multi-resolution signal space-scale decomposition. Digital implementation of these transforms are computationally intensive both because of the nature of the coordinate-doubling of the transforms and due to the large quantity of convolution/correlation operations to be performed. Optics with its inherent parallel processing capability has been applied to many useful linear signal and image transformations for feature analysis and extraction. This paper is intended to study the suitability of using optical processing techniques for the signal Gabor and wavelet analysis. Gabor and wavelet transforms of both one- and two-dimensional signals and images are discussed. System parameters and limitation are analyzed. Preliminary experimental results are presented.

Recent development in vision and image understanding related study reveals that a signal decomposition before processing may provide an enormous amount of useful information aboutthe signal. Various signal decomposition models such as the Gabor and wavelet expansions have been proposed. While the Gabor signal expansion creates a fixed resolution space-frequency signal representation, the wavelet transform provides a multiresolution signal space-scale decomposition. Digital implementation of these expansions are computationally intensive both because of the nature of the coordinate-doubling of the transforms and because ofthe large quantity of convolution/correlation operations to be performed. Optics with its inherent parallel-processing capability has been applied to many useful linear signal and image transformations for feature analysis and extraction. We studied the suitability of using optical processing techniques for the signal Gabor and wavelet analysis. Gabor and wavelet expansions of both 1- and 2-D signals and images are discussed. System parameters and limitations are analyzed. Preliminary experimental results are presented.

An Innovative Approach to Multidimensional Signals and Systems Theory for Image and Video Processing In this volume, Eric Dubois further develops the theory of multi-D signal processing wherein input and output are vector-value signals. With this framework, he introduces the reader to crucial concepts in signal processing such as continuous- and discrete-domain signals and systems, discrete-domain periodic signals, sampling and reconstruction, light and color, random field models, image representation and more. While most treatments use normalized representations for non-rectangular sampling, this approach obscures much of the geometrical and scale information of the signal. In contrast, Dr. Dubois uses actual units of space-time and frequency. Basis-independent representations appear as much as possible, and the basis is introduced where needed to perform calculations or implementations. Thus, lattice theory is developed from the beginning and rectangular sampling is treated as a special case. This is especially significant in the treatment of color and color image processing and for discrete transform representations based on symmetry groups, including fast computational algorithms. Other features include: An entire chapter on lattices, giving the reader a thorough grounding in the use of lattices in signal processing Extensive treatment of lattices as used to describe discrete-domain signals and signal periodicities Chapters on sampling and reconstruction, random field models, symmetry invariant signals and systems and multidimensional Fourier transformation properties Supplemented throughout with MATLAB examples and accompanying downloadable source code Graduate and doctoral students as well as senior undergraduates and professionals working in signal processing or video/image processing and imaging will appreciate this fresh approach to multidimensional signals and systems theory, both as a thorough introduction to the subject and as inspiration for future research.

A simple, practical method for generating an hexagonal sampling lattice is described for an E-O scanning sensor. Matched filtering techniques are developed for hexagonally sampled data. An efficient approximate matched filter is described which uses integer weights and takes advantage of the structure of the data. Relative to this filter, the number of processing operations increases for a rectangular sampling lattice, by a factor of six for an ideal matched filter and by 25 percent for an approximate filter with integer weights.© (1991) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

An established method for precise determination of optical absorption is the so called laser calorimetry. According to ISO 115511 laser calorimetry is preferred to other photothermal test methods, because of its capability to deliver absolute calibration. Many optical materials have low heat conductivity, which can affect the calibration significantly. The timeand spatial dependent temperature profile in a sample of materials with low heat conductivity requires accurate temperature measurement strategies to determine material-independent and absolutely calibrated absorption values. For thin cylindrical samples, ISO 11551 provides a strategy to compensate heat conductivity effects. The optimal temperature sensor position, where accordingly calibrated measurement results2 can be obtained, is simply based on the symmetric sample geometry. For thick geometries an additional temperature distribution along propagation direction of the heating beam must be considered. The current version of ISO 11551 does not provide a sophisticated solution for this problem, because the heating scheme of a sample is usually unknown. Therefore, a reliable calibration procedure can only be applied to samples of well-known absorption properties of surfaces and bulk material. Utilizing such kind of specifically prepared reference samples in combination with Finite Element Method (FEM) calculations, a general measurement and data evaluation concept based on laser calorimetry is presented, that allows deriving absolutely calibrated absorption measurement results for rectangular sample geometries.

We show experimentally as well as theoretically that patterned magnetic tunnel junctions can be characterized using the current-in-plane tunneling (CIPT) method, and the key parameters, the resistance-area product (RA) and the tunnel magnetoresistance (TMR), can be determined. The CIPT method relies on four-point probe measurements performed with a range of different probe pitches and was originally developed for infinite samples. Using the method of images, we derive a modified CIPT model, which compensates for the insulating boundaries of a finite rectangular sample geometry. We measure on square tunnel junction pads with varying sizes and analyze the measured data using both the original and the modified CIPT model. Thus, we determine in which sample size range the modified CIPT model is needed to ensure validity of the extracted sample parameters, RA and TMR. In addition, measurements as a function of position on a square tunnel junction pad are used to investigate the sensitivity of the measurement results to probe misalignment.