Multidimensional polynomial transform algorithm formultidimensional discrete W transform

The multidimensional (MD) polynomial transform is used to convert the MD W transform (MDDWT) into a series of one-dimensional (1-D) W transforms (DWTs). Thus, a new polynomial transform algorithm for the MDDWT is obtained. The algorithm needs no operations on complex data. The number of multiplications for computing an r-dimensional DWT is only 1 times that of the commonly used row-column method. The number of additions is also reduced considerably     

The fast discrete Radon transform. I. Theory

An inversion scheme for reconstruction of images from projections based on the slope-intercept form of the discrete Radon transform is presented. A seminal algorithm for the forward and the inverse transforms proposed by G. Beylkin (1987) demonstrated poor dispersion characteristics for steep slopes and could not invert transforms based on nonlinear slope variations. By formulating the computation of a discrete computation of the continuous Radon transform formula, the authors explicitly derive fast generalized inversion methods that overcome the original shortcomings. The generalized forward (FRT) and inverse (IFRT) algorithms proposed are fast, eliminate interpolation calculations, and convert directly between a raster scan grid and a rectangular/polar grid in one step     

Efficient blind image restoration using discrete periodic Radon transform

Restoring an image from its convolution with an unknown blur function is a well-known ill-posed problem in image processing. Many approaches have been proposed to solve the problem and they have shown to have good performance in identifying the blur function and restoring the original image. However, in actual implementation, various problems incurred due to the large data size and long computational time of these approaches are undesirable even with the current computing machines. In this paper, an efficient algorithm is proposed for blind image restoration based on the discrete periodic Radon transform (DPRT). With DPRT, the original two-dimensional blind image restoration problem is converted into one-dimensional ones, which greatly reduces the memory size and computational time required. Experimental results show that the resulting approach is faster in almost an order of magnitude as compared with the traditional approach, while the quality of the restored image is similar.     

The discrete polynomial-phase transform

The discrete polynomial-phase transform (DPT) is a new tool for analyzing constant-amplitude polynomial-phase signals. The main properties of the DPT are its ability to identify the degree of the phase polynomial and to estimate its coefficients. The transform is robust to deviations from the ideal signal model, such as slowly-varying amplitude, additive noise and nonpolynomial phase. The authors define the DPT, derive its basic properties, and use it to develop computationally efficient estimation and detection algorithms. A statistical accuracy analysis of the estimated parameters is also presented     

The discrete Hilbert transform

A Hilbert transformation procedure for discrete data has been developed. This transform could be useful in a variety of applications such as the analysis of sampled data systems and the simulation of filters.     

The fractional discrete cosine transform

The extension of the Fourier transform operator to a fractional power has received much attention in signal theory and is finding attractive applications. The paper introduces and develops the fractional discrete cosine transform (DCT) on the same lines, discussing multiplicity and computational aspects. Similarities and differences with respect to the fractional Fourier transform are pointed out     

The undersampled discrete Gabor transform

Conventional studies on discrete Gabor transforms have generally been confined to the cases of critical sampling and oversampling in which the Gabor families span the whole signal space. In this paper, we investigate undersampled discrete Gabor transforms. For an undersampled Gabor triple (g,a,b), i.e. a·b>N, we show that the associated generalized dual Gabor window (GDGW) function is the same as the one associated with the oversampled (g,N/b,N/a), except for the constant factor (ab/N). Computations of undersampled Gabor transforms are made possible. By applying the methods (algorithms) developed in oversampled settings, the undersampled GDGW is determined. Then, we are able to obtain the best approximation of a signal x by linear combinations of vectors in the Gabor family     

The discrete rotational Fourier transform

We define a discrete version of the angular Fourier transform and present the properties of the transform that show it to be a rotation in time-frequency space, this new transform is a generalization of the DFT. Efficient algorithms for its computation can then be based on the FFT and the eigenstructure of the DFT     

The discrete fractional Fourier transform

We propose and consolidate a definition of the discrete fractional Fourier transform that generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform generalizes the continuous ordinary Fourier transform. This definition is based on a particular set of eigenvectors of the DFT matrix, which constitutes the discrete counterpart of the set of Hermite-Gaussian functions. The definition is exactly unitary, index additive, and reduces to the DFT for unit order. The fact that this definition satisfies all the desirable properties expected of the discrete fractional Fourier transform supports our confidence that it will be accepted as the definitive definition of this transform     

Wavelet localization of the Radon transform

The authors develop an algorithm which significantly reduces radiation exposure in X-ray tomography, when a local region of the body is to be imaged. The algorithm uses the properties of wavelets to essentially localize the Radon transform. This algorithm differs from previous algorithms for doing local tomography because it recovers an approximation to the original image, not the image module the nullspace of the local tomography operator, or the Lambda transform of the image. This is possible because the authors do not truly invert the interior Radon transform, but rather sample the Radon transform sparsely away from the local region of interest. Much attention in the field has been directed towards localized tomography. The authors believe that this technique represents a significant contribution towards this effort     

A generalized discrete transform

A class of discrete transforms for signal processing is defined. Elementary properties of this set of transforms are compiled and presented.     

Mixed-radix discrete cosine transform

Presents two new fast discrete cosine transform computation algorithms: a radix-3 and a radix-6 algorithm. These two new algorithms are superior to the conventional radix-3 algorithm as they (i) require less computational complexity in terms of the number of multiplications per point, (ii) provide a wider choice of the sequence length for which the DCT can be realized and, (iii) support the prime factor-decomposed computation algorithm to realize the 2^m3ⁿ-point DCT. Furthermore, a mixed-radix algorithm is also proposed such that an optimal performance can be achieved by applying the proposed radix-3 and radix-6 and the well-developed radix-2 decomposition techniques in a proper sequence     

The discrete Laguerre transform: derivation and applications

The discrete Laguerre transform (DLT) belongs to the family of unitary transforms known as Gauss-Jacobi transforms. Using classical methodology, the DLT is derived from the orthonormal set of Laguerre functions. By examining the basis vectors of the transform matrix, the types of signals that can be best represented by the DLT are determined. Simulation results are used to compare the DLT's effectiveness in representing such signals to that of other available transforms in applications such as data compression and transform-domain adaptive filters     

Combined polynomial transform and radix-q algorithm for MD discreteW transform

The type-II r-dimensional discrete W transform (rD-DWT-II) with size q^l1×q^l2 ×···×^lr q where q is an odd prime number, is converted into a series of one-dimensional (1-D) reduced DWT-IIs by using the multidimensional polynomial transform and an index permutation. Then, a radix-q algorithm and a cyclic convolution algorithm are presented for the computation of the 1-D reduced DWT-IIs. The new fast algorithm substantially reduces the overall computational complexity compared with the row-column method. Especially, the number of multiplications required by the proposed algorithm for computing an rD-DWT-II is only 1/r times that needed by the commonly used row-column method     

DGT与DCT在图像编码中的性能比较

介绍了二维实值离散Gabor变换（RDGT）的快速算法,并着重探讨了二维实值离散Gabor变换与二维离散余弦变换在图像编码中的性能及差异。     

A modified generalized discrete transform

A modified version of the generalized discrete transform described earlier is now developed. The transform matrix of this modified version has a number of zeros as its elements, and consequently its matrix factors are more sparse. This results in fewer arithmetic operations and corresponding savings in computer time, when information is processed.     

Three-dimensional discrete wavelet transform architectures

The three-dimensional (3-D) discrete wavelet transform (DWT) suits compression applications well, allowing for better compression on 3-D data as compared with two-dimensional (2-D) methods. This paper describes two architectures for the 3-D DWT, called the 3DW-I and the 3DW-II. The first architecture (3DW-I) is based on folding, whereas the 3DW-II architecture is block-based. Potential applications for these architectures include high definition television (HDTV) and medical data compression, such as magnetic resonance imaging (MRI). The 3DW-I architecture is an implementation of the 3-D DWT similar to folded 1-D and 2-D designs. It allows even distribution of the processing load onto 3 sets of filters, with each set performing the calculations for one dimension. The control for this design is very simple, since the data are operated on in a row-column-slice fashion. Due to pipelining, all filters are utilized 100% of the time, except for the start up and wind-down times. The 3DW-II architecture uses block inputs to reduce the requirement of on-chip memory. It has a central control unit to select which coefficients to pass on to the lowpass and highpass filters. The memory on the chip will be small compared with the input size since it depends solely on the filter sizes. The 3DW-I and 3DW-II architectures are compared according to memory requirements, number of clock cycles, and processing of frames per second. The two architectures described are the first 3-D DWT architectures     

一种基于Radon变换的微表情识别算法

微表情是人们处在一些与平时生活环境不同的高强度环境下试图控制和掩饰的情感表现,也是一种不曾意识到的瞬时脸部表情,持续时间短,强度弱。为了提高其准确率,提出了基于Radon变换的微表情识别算法。首先,对数据库中的视频序列进行灰度归一化、尺寸归一化和二维主成分分析法(Two-dimensional Principal Component Analysis,2DPCA)降维预处理,使用光流法对降维后图像提取运动特征;然后使用Radon变换算法对光流图像进行处理,得到对应微表情的特征值和特征图像;最后使用支持向量机进行微表情分类识别。实验结果表明,使用Radon变换后得到的微表情特征图像得到了较好的识别效果,在微表情数据集CASME和CASMEⅡ上识别率分别为81. 48%和82. 17%,通过与选取的其他方法对比说明了该方法具有更好的识别性能。     

Recursive computation of discrete Legendre polynomial coefficients

Discrete Legendre polynomials play an important role in image processing, signal processing, and control. This paper presents two new recursive methods to compute Legendre polynomials' coefficients. One method computes the coefficients of the (m + 1)th order term using the coefficient of themth order term of the same degree plynomial. The other method computes thenth order coefficient of a higher degree polynomial using themth order coefficient of a lower degree polynomial.