3-D discrete analytical ridgelet transform.

In this paper, we propose an implementation of the 3-D Ridgelet transform: the 3-D discrete analytical Ridgelet transform (3-D DART). This transform uses the Fourier strategy for the computation of the associated 3-D discrete Radon transform. The innovative step is the definition of a discrete 3-D transform with the discrete analytical geometry theory by the construction of 3-D discrete analytical lines in the Fourier domain. We propose two types of 3-D discrete lines: 3-D discrete radial lines going through the origin defined from their orthogonal projections and 3-D planes covered with 2-D discrete line segments. These discrete analytical lines have a parameter called arithmetical thickness, allowing us to define a 3-D DART adapted to a specific application. Indeed, the 3-D DART representation is not orthogonal, It is associated with a flexible redundancy factor. The 3-D DART has a very simple forward/inverse algorithm that provides an exact reconstruction without any iterative method. In order to illustrate the potentiality of this new discrete transform, we apply the 3-D DART and its extension to the Local-DART (with smooth windowing) to the denoising of 3-D image and color video. These experimental results show that the simple thresholding of the 3-D DART coefficients is efficient.     

One- and two-dimensional minimum and nonminimum phase retrieval bysolving linear systems of equations

The discrete phase retrieval problem is to reconstruct a discrete time signal whose support is known and compact from the magnitude of its discrete Fourier transform. We formulate the problem as a linear system of equations; our methods do not require polynomial rooting, tracking zero curves of algebraic functions, or any sort of iteration like previous methods. Our solutions obviate the stagnation problems associated with iterative algorithms, and our solutions are computationally simpler and more stable than alternative noniterative algorithms. Furthermore, our methods can explicitly accommodate noisy Fourier magnitude information through the use of total least squares type techniques. We assume either of the following two types of a priori knowledge of the signal: (1) a band of known values (which may be zeros) or (2) some known values of a subminimum phase signal (whose zeros lie inside a disk of radius greater than unity). We illustrate our methods with nonminimum-phase one-dimensional (1-D) and two-dimensional (2-D) signals     

Efficient architectures for 1-D and 2-D lifting-based wavelet transforms

The lifting scheme reduces the computational complexity of the discrete wavelet transform (DWT) by factoring the wavelet filters into cascades of simple lifting steps that process the input samples in pairs. We propose four compact and efficient hardware architectures for implementing lifting-based DWTs, namely, one-dimensional (1-D) and two-dimensional (2-D) versions of what we call recursive and dual scan architectures. The 1-D recursive architecture exploits interdependencies among the wavelet coefficients by interleaving, on alternate clock cycles using the same datapath hardware, the calculation of higher order coefficients along with that of the first-stage coefficients. The resulting hardware utilization exceeds 90% in the typical case of a five-stage 1-D DWT operating on 1024 samples. The 1-D dual scan architecture achieves 100% datapath hardware utilization by processing two independent data streams together using shared functional blocks. The recursive and dual scan architectures can be readily extended to the 2-D case. The 2-D recursive architecture is roughly 25% faster than conventional implementations, and it requires a buffer that stores only a few rows of the data array instead of a fixed fraction (typically 25% or more) of the entire array. The 2-D dual scan architecture processes the column and row transforms simultaneously, and the memory buffer size is comparable to existing architectures.     

二维正交子波变换的VLSI并行计算

本文提出一个二维离散正交子波变换的ＶＬＳＩ并行结构，该结构将二维输入信号分解成不重叠的若干行组，从而使每组中的所有行被并行处理，而不同组的行的处理、不同级上的计算，以至不同信号的计算可以在此结构上流水线地进行。     

Fast Parallel Approach for 2-D DHT-Based Real-Valued Discrete Gabor Transform

Two-dimensional fast Gabor transform algorithms are useful for real-time applications due to the high computational complexity of the traditional 2-D complex-valued discrete Gabor transform (CDGT). This paper presents two block time-recursive algorithms for 2-D DHT-based real-valued discrete Gabor transform (RDGT) and its inverse transform and develops a fast parallel approach for the implementation of the two algorithms. The computational complexity of the proposed parallel approach is analyzed and compared with that of the existing 2-D CDGT algorithms. The results indicate that the proposed parallel approach is attractive for real time image processing.      

2-D DCT systolic array implementation

The algorithm and architecture of a 2-D systolic array processor for the DCT (discrete cosine transform) are proposed. It is based on the relationship between DCT and cosine DFT and sine DFT. Two systolic architectures of 1-D DCT data and control flow computation are discussed. By use of the main feature of the two systolic 1-D arrays for DCT, a full 2-D systolic DCT array is presented.<>     

2-D and 1-D multipaired transforms: frequency-time type wavelets

A concept of multipaired unitary transforms is introduced. These kinds of transforms reveal the mathematical structure of Fourier transforms and can be considered intermediate unitary transforms when transferring processed data from the original real space of signals to the complex or frequency space of their images. Considering paired transforms, we analyze simultaneously the splitting of the multidimensional Fourier transform as well as the presentation of the processed multidimensional signal in the form of the short one-dimensional (1-D) “signals”, that determine such splitting. The main properties of the orthogonal system of paired functions are described, and the matrix decompositions of the Fourier and Hadamard transforms via the paired transforms are given. The multiplicative complexity of the two-dimensional (2-D) 2^r×2^r-point discrete Fourier transform by the paired transforms is 4^r/2(r-7/3)+8/3-12 (r>3), which shows the maximum splitting of the 5-D Fourier transform into the number of the short 1-D Fourier transforms. The 2-D paired transforms are not separable and represent themselves as frequency-time type wavelets for which two parameters are united: frequency and time. The decomposition of the signal is performed in a way that is different from the traditional Haar system of functions     

Rigid-motion-invariant classification of 3-D textures

This paper studies the problem of 3-D rigid-motion-invariant texture discrimination for discrete 3-D textures that are spatially homogeneous by modeling them as stationary Gaussian random fields. The latter property and our formulation of a 3-D rigid motion of a texture reduce the problem to the study of 3-D rotations of discrete textures. We formally develop the concept of 3-D texture rotations in the 3-D digital domain. We use this novel concept to define a "distance" between 3-D textures that remains invariant under all 3-D rigid motions of the texture. This concept of "distance" can be used for a monoscale or a multiscale 3-D rigid-motion-invariant testing of the statistical similarity of the 3-D textures. To compute the "distance" between any two rotations R(1) and R(2) of two given 3-D textures, we use the Kullback-Leibler divergence between 3-D Gaussian Markov random fields fitted to the rotated texture data. Then, the 3-D rigid-motion-invariant texture distance is the integral average, with respect to the Haar measure of the group SO(3), of all of these divergences when rotations R(1) and R(2) vary throughout SO(3). We also present an algorithm enabling the computation of the proposed 3-D rigid-motion-invariant texture distance as well as rules for 3-D rigid-motion-invariant texture discrimination/classification and experimental results demonstrating the capabilities of the proposed 3-D rigid-motion texture discrimination rules when applied in a multiscale setting, even on very general 3-D texture models.     

H∞ filtering of 2-D discrete systems

This paper deals with H_∞ filtering of two-dimensional (2-D) linear discrete systems described by a 2-D local state-space (LSS) Fornasini-Marchesini (1978) second model. Several versions of the bounded real lemma of the 2-D discrete systems are established. The 2-D bounded real lemma allows us to solve the finite horizon and infinite horizon H_∞ filtering problems using a Riccati difference equation or a Riccati inequality approach. Further a solution to the infinite horizon H_∞ filtering problem based on a linear matrix inequality (LMI) approach is developed. Our results extend existing work for one-dimensional (1-D) systems to the 2-D case and give a state-space solution to the bounded realness of 2-D discrete systems as well as 2-D H_∞ filtering for the first time. Numerical examples are given to demonstrate the Riccati difference equation approach to the 2-D finite horizon H_∞ filtering problem and the LMI approach to the 2-D infinite horizon H_∞ filtering problem     

Multirate-based fast parallel algorithms for 2-D DHT-based real-valued discrete Gabor transform

Novel algorithms for the multirate and fast parallel implementation of the 2-D discrete Hartley transform (DHT)-based real-valued discrete Gabor transform (RDGT) and its inverse transform are presented in this paper. A 2-D multirate-based analysis convolver bank is designed for the 2-D RDGT, and a 2-D multirate-based synthesis convolver bank is designed for the 2-D inverse RDGT. The parallel channels in each of the two convolver banks have a unified structure and can apply the 2-D fast DHT algorithm to speed up their computations. The computational complexity of each parallel channel is low and is independent of the Gabor oversampling rate. All the 2-D RDGT coefficients of an image are computed in parallel during the analysis process and can be reconstructed in parallel during the synthesis process. The computational complexity and time of the proposed parallel algorithms are analyzed and compared with those of the existing fastest algorithms for 2-D discrete Gabor transforms. The results indicate that the proposed algorithms are the fastest, which make them attractive for real-time image processing.     

Reduction in Sampling-Point Numbers for 2-D Discrete Fourier Transform Used in Harmonic Balance Method

This paper discusses how to reduce the numbers of sampling points to obtain the 2-D discrete Fourier transform used in harmonic balance method. A method of embedding a 2-D Fourier transform into a 1-D one has already been proposed. This paper proposes a method of reducing the numbers of sampling points in a 1-D Fourier transform, by using bandpass sampling. A 2-D Fourier transform for harmonic balance method can be embedded into a 1-D one with a reduced number of sampling points by combining the previously proposed and the present methods. The extension of these methods to higher dimensional cases is also briefly discussed.      

Block classification for an adaptive 1-D/2-D DCT video coding

A classification scheme for an adaptive one- or two-dimensional discrete cosine transform (1-D/2-D DCT) technique is described and demonstrated to be a more appropriate strategy than the conventional 2-D DCT for coding motion compensated prediction error images. Two block-based classification methods are introduced and their accuracy in predicting the correct transform type discussed. The accuracy is assessed with a classification measure designed to ascertain the effectiveness of energy compaction when the predicted transform class is applied; vis-a-vis horizontally, vertically or two-dimensionally transformed blocks. Energy compaction is a useful property not only for efficient entropy coding but also for enhancing the resilience of the transform coder to quantisation noise. Improvements against the homogeneous 2-D DCT system both in terms of the peak signal to noise ratio and subjective assessments are achieved. Observable ringing artifacts along edges, which are usual in conventional transform coding, are reduced     

ECG data compression using cut and align beats approach and 2-D transforms

A new electrocardiogram (ECG) data compression method is presented which employs a two dimensional (2-D) transform. This 2-D transform method utilizes the fact that ECG signals generally show two types of redundancies--between adjacent heartbeats and between adjacent samples. A heartbeat data sequence is cut and beat-aligned to form a 2-D data array. Any 2-D compression method can then be applied. Transform coding using the 2-D discrete cosine transform (DCT) [2-D DCT] is employed here as an example. Using selections from the MIT-BIH arrhythmia and Medtronic databases, results are presented that illustrate substantial improvement in compression ratio over one-dimensional methods for comparable percent root-mean-square difference (PRD).     

On stationarizability for nonstationary 2-D random fields usingdiscrete wavelet transforms

The emphasis in this article is on the study of nonstationary two-dimensional (2-D) random fields with wide-sense stationary increments, wide-sense stationary jumps, and 2-D fractional Brownian motion (fBm) fields. The effort made in this work is to develop a realizable method of stationarization provided for nonstationary 2-D random fields. We also present the correlation functions of the discrete wavelet transform relating to 2-D fBm fields that will decay hyperbolically fast.     

二维DCT算法及其精简的VLSI设计

采用了快速算法,并通过矩阵的变化,得到了一维离散余弦变换(Discrete Cosine Transform,DCT)的一种快速实现,并由此提出一种精简的超大规模集成电路(Very-large-scale integration,VLSI)设计架构.使用了一维DCT的复用技术,带符号数的乘法器设计等技术,实现了二维DCT算法的精简的VLSI设计.实验结果表明,所设计的二维DCT设计有效,并能够获得非常精简的电路设计.     

Radix-2 × 2 × 2 algorithm for the 3-D discrete Hartleytransform

The discrete Hartley transform (DHT) has proved to be a valuable tool in digital signal/image processing and communications and has also attracted research interests in many multidimensional applications. Although many fast algorithms have been developed for the calculation of one- and two-dimensional (1-D and 2-D) DHT, the development of multidimensional algorithms in three and more dimensions is still unexplored and has not been given similar attention; hence, the multidimensional Hartley transform is usually calculated through the row-column approach. However, proper multidimensional algorithms can be more efficient than the row-column method and need to be developed. Therefore, it is the aim of this paper to introduce the concept and derivation of the three-dimensional (3-D) radix-2 × 2 × 2 algorithm for fast calculation of the 3-D discrete Hartley transform. The proposed algorithm is based on the principles of the divide-and-conquer approach applied directly in 3-D. It has a simple butterfly structure and has been found to offer significant savings in arithmetic operations compared with the row-column approach based on similar algorithms     

Computationally efficient 2-D DOA estimation for uniform rectangular arrays

A computationally efficient two-dimensional (2-D) direction-of-arrival (DOA) estimation method for uniform rectangular arrays is presented. A preprocessing transformation matrix is first introduced, which transforms both the complex-valued covariance matrix and the complex-valued search vector into real-valued ones. Then the 2-D DOA estimation problem is decoupled into two successive real-valued one-dimensional (1-D) DOA estimation problems with real-valued computations only. All these measures lead to significantly reduced computational complexity for the proposed method.     

3-D numerical mode-matching (NMM) method for resistivitywell-logging tools

A three-dimensional (3-D) numerical mode-matching (NMM) method is presented for Poisson's equation in general inhomogeneons media. It reduces the original 3-D problem into a series of two-dimensional (2-D) eigenvalue problems plus a one-dimensional (1-D) layered medium problem, which can be modeled efficiently by a recursion procedure. The algorithm is tested for several 3-D inhomogeneous media and an excellent agreement between the NMM and analytical solutions is obtained for all test cases. We demonstrate some typical applications of the 3-D NMM algorithm in resistivity well logging, including invasion zones of noncircular shape, vertical and horizontal fractures, and horizontal wells. The solution procedure proposed is directly applicable to wave propagation in 3-D inhomogeneous media     

Scalability of 2-D wavelet transform algorithms: analytical and experimental results on MPPs

This paper studies the scalability of two-dimensional (2-D) discrete wavelet transform (DWT) algorithms on massively parallel processors (MPPs). The principal operation in the 2-D DWT is the filtering operation used to implement the filter banks of the 2-D subband decomposition. This filtering operation can be implemented as a convolution in the time domain or as a multiplication in the frequency domain. We demonstrate that there exist combinations of machine size, image size, and wavelet kernel size for which the time-domain algorithms outperform the frequency domain algorithms and vice-versa. We therefore demonstrate that a hybrid approach that combines time- and frequency-domain approaches can yield linear scalability for a broad range of problem and machine sizes. Furthermore, we show the effect of processor speed versus communication overhead and the use of separable versus nonseparable wavelets on the crossover points between the algorithm approaches.