Showing papers on "S transform published in 1994"

Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform.

Abstract: Two definitions of a fractional Fourier transform have been proposed previously. One is based on the propagation of a wave field through a graded-index medium, and the other is based on rotating a function's Wigner distribution. It is shown that both definitions are equivalent. An important result of this equivalency is that the Wigner distribution of a wave field rotates as the wave field propagates through a quadratic graded-index medium. The relation with ray-optics phase space is discussed.

Wavelet-based post-processing of low bit rate transform coded images

Abstract: We propose a novel method based on wavelet thresholding for enhancement of decompressed transform coded images. Transform coding at low bit rates typically introduces artifacts associated with the basis functions of the transform. In particular, the method works remarkably well in "deblocking" of DCT compressed images. The method is nonlinear, computationally efficient, and spatially adaptive and has the distinct feature that it removes artifacts yet retain sharp features in the images. An important implication of this result is that images coded using the JPEG standard can efficiently be postprocessed to give significantly improved visual quality in the images. The algorithm can use a conventional JPEG encoder and decoder for which VLSI chips are available. >

Application of wavelet transform signal processor to ultrasound

Abstract: The wavelet transform is applied to the analysis of ultrasonic waves for improved signal detection and analysis. In instances where the mother wavelet is well defined, the wavelet transform has relative insensitivity to noise and does not need windowing. Peak detection of ultrasonic pulses using the wavelet transform is described and results show good detection even when large white noise was added. The use of the wavelet transform to extract the frequency dispersion relation of the Lamb wave velocity is also described. The two-dimensional wavelet transform allows for both time and frequency analysis, thus making it particularly suitable for dispersion studies. Experimental and numerical results show the superior performance of the wavelet transform signal processor

Linear and parabolic [tau]-p transforms revisited

Abstract: New derivations for the conventional linear and parabolic [tau]-p transforms in the classic continuous function domain provide useful insight into the discrete [tau]-p transformations. For the filtering of unwanted waves such as multiples, the derivation of the [tau]-p transform should define the inverse transform first, and then compute the forward transform. The forward transform usually requires a p-direction deconvolution to improve the resolution in that direction. It aids the wave filtering by improving the separation of events in the [tau]-p domain. The p-direction deconvolution is required for both the linear and curvilinear [tau]-p transformations for aperture-limited data. It essentially compensates for the finite length of the array. For the parabolic [tau]-p transform, the deconvolution is required even if the input data have an infinite aperture. For sampled data, the derived [tau]-p transform formulas are identical to the DRT equations obtained by other researchers. Numerical examples are presented to demonstrate event focusing in [tau]-p space after deconvolution.

An efficient algorithm to compute the complete set of discrete Gabor coefficients

Abstract: The discrete Gabor (1946) transform algorithm is introduced that provides an efficient method of calculating the complete set of discrete Gabor coefficients of a finite-duration discrete signal from finite summations and to reconstruct the original signal exactly from the computed expansion coefficients. The similarity of the formulas between the discrete Gabor transform and the discrete Fourier transform enables one to employ the FFT algorithms in the computation. The discrete 1-D Gabor transform algorithm can be extended to 2-D as well. >

Analytic properties of the Hartley transform and their implications

Abstract: The Hartley transform is an integral transform closely related to the Fourier transform. It has some advantages over the Fourier transform in the analysis of real signals as it avoids the use of complex arithmetic. However, the Hartley transform has other applications in signal and image reconstruction related to traditional phase retrieval problems. These can he understood by examining the analytic properties of the Hartley transform in the complex plane. In this paper, the analytic continuation of the Hartley transform into the complex plane is derived and its properties discussed. It is shown that for signals or images of finite extent, the Hartley transform is analytic in the entire finite complex plane, and this is used to derive properties of its complex zeros. Hilbert transform-type relationships for the Hartley transform, related to causal and analytic-signals, are also derived. The analytic properties derived are used to study the problem of image reconstruction from the Hartley transform intensity. Uniqueness and reconstruction algorithms for one- and two-dimensional problems are discussed, and examples are presented. Generation of image moments from the Hartley transform intensity is also described. >

Short-time Fourier transform and wavelet transform with Fourier-domain processing

Abstract: We discuss the semicontinuous short-time Fourier transform (STFT) and the semicontinual wavelet transform (WT) with Fourier-domain processing, which is suitable for optical implementation. We also systematically analyze the selection of the window functions, especially those based on the biorthogonality and the orthogonality constraints for perfect signal reconstruction. We show that one of the best substitutions for the Gaussian function in the Fourier domain is a squared sinusoid function that can form a biorthogonal window function in the time domain. The merit of a biorthogonal window is that it could simplify the inverse STFT and the inverse WT. A couple of optical architectures based on Fourier-domain processing for the STFT and the WT, by which real-time signal processing can be realized, are proposed.

Gabor’s signal expansion and the Zak transform

Abstract: Gabor’s expansion of a signal into a discrete set of shifted and modulated versions of an elementary signal is introduced, and its relation to sampling of the sliding-window spectrum is shown. It is shown how Gabor’s expansion coefficients can be found as samples of the sliding-window spectrum, in which the window function is related to the elementary signal in such a way that the set of shifted and modulated elementary signals is biorthonormal to the corresponding set of window functions. The Zak transform is introduced, and its intimate relationship to Gabor’s signal expansion is demonstrated. It is shown how the Zak transform can be helpful in determining the window function that corresponds to a given elementary signal and how it can be used to find Gabor’s expansion coefficients. The continuous-time and the discrete-time cases are considered, and, by sampling the continuous frequency variable that still occurs in the discrete-time case, the discrete Zak transform and the discrete Gabor transform are introduced. It is shown how the discrete transforms enable us to determine Gabor’s expansion coefficients by a fast computer algorithm, which is analogous to the well-known fast Fourier-transform algorithm.

Efficient systolic architecture for the one-dimensional wavelet transform

Abstract: We present in this paper an architectural design for a wavelet transform chip for use in real-time one-dimensional signal processing applications. Based on the observation that further levels of the wavelet transform require only as much computation as the first level, our architecture requires only one row of processing elements to compute the complete transform. This is compared to previous designs requiring one row of processing elements per level of the transform. Our architecture provides the output of the transform in two forms, one with all levels multiplexed on one line (useful for transmission or compression) and the other as individual levels on separate lines synchronized in time to facilitate real-time analysis. We consider the usefulness of this architecture for real-time analysis of audio signals (typically 40kHz sampling rate) and discuss the design and implementation benefits of the computational simplicity of the presented architecture.

Quadrature shape detection using the flow integration transform

Abstract: A transform, the Flow Integration Transform (FIT), is applied for determining the presence of a preconceived shape in a gray scale image. The FIT represents a form of Quadrature Shape Detection. The expected contour serves as a filter for detecting potential targets as periodic signals. The FIT performs a line integral of two vectors: (1) the flow, a vector equal to the gradient of the image's intensity but rotated by 90 degrees, and (2) the local tangent to the path of integration. The path of integration follows the expected contour. The integration is performed starting at each point in the image, producing a two-dimensional transform whose pixel value corresponds to the relative presence of the contour at that location in the input image. The transform exhibits the feature that information widely dispersed in the image becomes concentrated in a local area of the transform.

Sign Haar Transform

Abstract: A non-linear transform, called "Sign Haar Transform" has been introduced. The transform is unique and converts binary/ternary vectors into ternary spectral domain. Recursive definitions and Fast Transforms for the calculation of Sign Haar Transform have been developed. The new transform is extremely computationally effective both in terms of memory requirements and processing time. >

Two-dimensional optical wavelet transform in space domain and its performance analysis.

Abstract: A new architecture of an optical wavelet transform system with a lenslet array is proposed, and its optical performance and optical limits are analyzed.

Enhancement of block transform coded images using residual spectra adaptive postfiltering

Abstract: Image block transform techniques usually introduce several types of spatial periodic distortion which are mostly noticeable at low bit rates. One way to reduce these artifacts to obtain an acceptable visual quality level is to postfilter the decoded images using nonlinear space-variant adaptive filters derived from the structural relationships and residual spectral information provided by the discrete-time Fourier transform (DTFT) of block transforms such as the discrete cosine transform (DCT) and the lapped orthogonal transform (LOT). A method for analyzing and filtering the DCT blocking noise and the LOT ringing noise for moderate and highly compressed images is described and several test cases are presented. A generalized Fourier analysis of the block transform distortion as seen in the frequency domain is discussed in conjunction with an outline of a separable adaptive postfiltering algorithm for decoded image enhancement. >

Geometric invariance in space-variant vision systems: the exponential chirp transform

Abstract: Outlines a method to derive geometric invariance kernels which may be applied to a space-variant sensor architecture. The basic idea as to transform a kernel with desired symmetry properties (e.g. the Fourier kernel) in the domain to the range of the transform. By combining this transformed kernel with the Jacobian of the transformation, the authors obtain a new integral transform, in the range, which has similar properties to the original transform. The authors illustrate this idea with a variant of the Mellin-Fourier transform, applied to an image which has been transformed by a log-polar mapping. The kernel obtained, which the authors call an "exponential chirp" has properties (unlike the Mellin-Fourier transform) which are both consistent with the spatial nature of human vision and can be applied directly in the space-variant image plane. The authors outline applications to visual template matching and auto-correlation, and show a one-dimensional example of a generalization of cepstral auto-correlation using this method.

Optical wavelet transforms from computer-generated holography

Abstract: An optical implementation of a wavelet transform is presented. Optical Haar wavelets are created by the use of computer-generated holography. Two different holographic techniques are explored: (1) interferogram and (2) detour-phase. A discrete representation of a continuous wavelet transform is obtained by the optical correlation of an image with a Haar mother wavelet. Experimental results are compared with their digital simulations.

Comparison of transform coding techniques for two-dimensional arbitrarily shaped images

Abstract: Envisioned advanced multimedia video services include arbitrarily shaped (AS) image segments as well as regular rectangular images. Image segments of the TV weather report produced by the chromo-key technique [1] and image segments produced by video analysis and image segmentation [2–4] are typical examples of AS image segments. This paper explores efficient intraframe transform coding techniques for general two-dimensional (2D) AS image segments, treating the traditional rectangular images as a special case. In particular, we focus on the transform coding of the partially defined image blocks along the boundary of the AS image segments. We recognize two different approaches — thebrute force transform coding approach and theshape-adaptive transform coding approach. The former fills the uncovered area with the optimal redundant data such that the resulting transform spectrum is compact. A simple but efficient mirror image extension technique is proposed. Once augmented into full image blocks, these boundary blocks can be processed by traditional block-based transform techniques like the popular discrete cosine transform (DCT). In the second approach, we change either the transform basis or the coefficient calculation process adaptively based on the shape of the AS image segment. We propose an efficientshape-projected problem formulation to reduce the dimension of the problem. Existing coding algorithms, such as the orthogonal transform by Gilge [5] and the iterative coding by Kaup and Aach [6], can be interpreted intuitively. We also propose a new adaptive transform based on the same principle as that used in deriving the DCT from the optimal Karhunen-Loeve transform (KLT). We analyze the tradeoff relationship between compression performance, computational complexity, and codec complexity for different coding schemes. Simulation results show that complicated algorithms (e.g., iterative, adaptive) can improve the quality by 5–10 dB at some computational or hardware cost. Alternatively, the simple mirror image extension technique improves the quality by 3–4 dB without any overheads. The contributions of this paper lie in efficient problem formulations, new transform coding techniques, and numerical tradeoff analyses.

Inexpensive Gabor decompositions

Abstract: This is a paper on the discrete Gabor transform. We discuss the calculation of dual and tight Gabor atoms, for Gabor atoms which satisfy certain support restrictions related to the relevant time- and frequency lattice constants. These conditions imply that the Gabor frame operator is just a pointwise multiplication operator and therefore computing the inverse or the square root of the inverse frame operator are computationally inexpensive.

Texture analysis using a generalised wavelet transform

Abstract: A novel frequency domain approach to the analysis of texture is presented. It consists of two main components: texture synthesis and segmentation. The synthesis method models the relation between pairs of texture patches by an affine transform. Segmentation is accomplished by detecting the texture boundary using a gradient operator. This scheme is implemented in the Fourier domain with varying scales using a generalised wavelet transform, the multiresolution Fourier transform (MFT). The technique has the potential to deal effectively with textures having varying amounts of structural information. >

Use of the wavelet transform in optimal receiver design.

Abstract: Wavelet transforms as they apply to optimal receiver design are studied. We start with an overview of the Karhunen–Loeve transform and explore the relationship between wavelet bases and the Karhunen–Loeve transform. We show that the dyadic wavelet basic can constitute the eigenfunction basis of a random process. With the help of this foundation, the design of an optimal receiver by the use of a the wavelet expansion is described. The relationship of this receiver to the wavelet-transform-based adaptive filter is also established.

Wavelet transform adaptive filtering

Abstract: An LMS adaptive filtering algorithm is presented utilizing wavelet transforms. Its performance is compared to DCT and Walsh-Hadamard transform-based adaptive filtering. The experimental analysis is performed in the case of the system identification of an unknown system or filter for stationary input signals. The results show some improvement in the weight modelling of the filter with comparable convergence rates. A new performance criteria, the diagonality factor, is introduced in order to show the specific effect of the wavelet transform on a signal. A Mean Average Difference is also utilized to compare the weight modelling performance of the various transform-based LMS adaptive filterings studied in this paper.

Skeletonisation using an extended Euclidean distance transform

Abstract: A standard method to perform skeletonisation is to use a distance transform. Unfortunately such an approach has the drawback that only the Symmetric axis transform can be computed and not the more practical smoothed local symmetries or the more general symmetry set. Using singularity theory we introduce an extended distance transform which may be used to capture more of the symmetries of a shape. We describe the relationship of this extended distance transform to the skeletal shape descriptors themselves and other geometric phenomema related to the boundary of the curve. We then show how the extended distance transform can be used to derive skeletal descriptions of an object.

Performance analysis of wavelet transform-based adaptive filtering

Abstract: The wavelet transform has been introduced for quite some time now. We examine the performance of wavelet transform-based adaptive filters. The application of two typical wavelets (D4 and Haar wavelets) are studied extensively as indicators of wavelet transform-based adaptive filter performance in general. Experimental results for a system identification application show an improvement in signal modelling and satisfactory convergence speed for a variety of equalisation conditions. We also compare the wavelet transform-based adaptive filters of Lee and Un (1986) to other transform-based adaptive filters such as discrete-cosine transform (DCT), discrete sine transform (DST), discrete Fourier transform (DFT), Walsh Hadamard transform (WHT) and discrete Hartley transform (DHT). The results of this comparison are mixed. We find that the performance depends on many factors but are consistent with the findings of Lee and Un. The results are also largely dependent on the properties of wavelets which have been found to be well suited for analyzing non-stationary signals. Finally, we present computational complexity considerations between these various transform-domain adaptive filters. >

Gabor transform in texture analysis

Abstract: Gabor transform has recently been exploited to do texture analysis, including texture edge detection, texture segmentation/discrimination, and texture synthesis. For most of the applications using Gabor transform, people convolve the given texture image with a set of Gabor filters with some user specified parameters. Although the mathematical formulation of applications involve the Fourier transform, few have investigated mathematical properties of the relationship between Gabor filters and their Fourier transform. This paper mainly studies mathematical properties of real Gabor filters and their corresponding Fourier transform. The goal is to select a set of `interesting' Gabor filters, or say, a set of parameters for Gabor filters to do texture analysis. We demonstrate, by means of 3-D graphical displays, that a Gabor filter or its corresponding Fourier transform may have a single peak or double peaks according to different parameters. Experiments for texture discrimination are given to demonstrate the applications of Gabor transform.© (1994) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Digital image coding using Legendre transform

Abstract: Image coding can be implemented through DPCM, transform, hybrid, or segmentation coding techniques. Some transform coding techniques, such as cosine and Hadamard, have been exhaustively analyzed and evaluated, while others, such as Legendre, have not. This paper introduces the use of Legendre transform in image coding. The transform matrix for different block sizes is calculated, the fast algorithm is derived, and the performance is evaluated through both mean square error and subjective quality. The results obtained have indicated that the system performance is comparable with that of optimum KLT and cosine transforms; moreover, it is simpler in implementation.

Computation and conditioning of the finite-discrete Gabor transform

Abstract: We present a new approach to analysing the continuous Gabor transform and two critically sampled discretizations of it: the periodic finite-discrete and non-periodic finite-discrete versions. In particular, we distinguish between the analysis and synthesis forms of the transform, and introduce an intermediate operation that decomposes both transforms to collections of independent Toeplitz operators. In the continuous and the periodic finite-discrete case this decomposition allows us to show that for appropriate window functions the analysis and synthesis transforms are inverses of each other. In the nonperiodic finite-discrete case this relation no longer holds, but we are still able to use the decomposition and results on Toeplitz matrices to show that both transforms and their inverses are computable in O(PlogP) operations (after a setup cost of O(Plog/sup 2/P)) for P discrete samples. Moreover the decomposition also facilitates analysis of the conditioning of finite versions of the transform. >

Wavelet based detectors

Abstract: Many processes can be described as time-scale in that signals arising from these processes exhibit constant time-bandwidth products. This feature is exploitable by wavelet transform techniques. A wavelet transform based detector is detailed and its performance evaluated relative to the short time Fourier transform. >

Optimal zone coding using the slant transform

Abstract: Discrete orthogonal transforms (DOTs) are widely used in digital signal processing, image coding and compression, systems theory, communication, and control. A special representative of the class of DOTs with nonsinusoidal basis functions is the slant transform, which is distinguished by the presence of a slanted vector with linearly decreasing components in its basis. The slant transform of fourth and eighth orders was introduced in 1971 by Enomoto and Shibata especially for efficient representation of the video signal in line sections with smooth variation of brightness. It has been used for television image coding. Pratt, Chen, and Welch generalized the slant transform to vectors of any dimension N = 2{sup n} and two-dimensional arrays, and derived posterior estimates of reconstruction error with zonal image compression (the zones were chosen by trial and error) for various transforms. These estimates show that, for the same N and the same compression ratio {tau}, the slant transform is inferior to the Karhunen - Loeve transform and superior to Walsh and Fourier transforms. In this paper, we derive prior estimates of the reconstruction error for the slant transform in zone coding and suggest an optimal technique for zone selection.