Implementation of a new class of FFT algorithms on transputer computational structures

Lin, Chung and Han introduced in 2014 the LCH-basis in order to derive FFT-based multipoint evaluation and interpolation algorithms with respect to this polynomial basis. Considering an affine space of n = 2j points, their algorithms require O(n · log2 n) operations in F2r. The LCH-basis has then been extended over finite fields of characteristic p by Lin et al. in 2016 and an n-point evaluation algorithm has been derived for n = pj with complexity O(n · logp n · p). However, the problem of interpolating polynomials represented in such a basis over FPr has not been addressed.

This paper presents a novel approach to efficiently compute the 4-, 8-, and 16-point variants of the Discrete Fourier Transform (DFT). Our solution requires as many (non-trivial) multiplications and additions of real numbers as fast algorithms of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{FFT}$</tex> family. However, we saved operations in a completely different way. Instead of factorizing the complex matrix that describes the transform, we consider separately the real and imaginary parts of the DFT matrix. Both parts are rank-deficient matrices, which contain pairs of linearly dependent rows and columns, but are structured and symmetric. We factorize each of them into a product of sparse matrices by applying shear transforms to rows and columns, similarly as in two-side diagonalization. The obtainable factorizations describe data flow graphs in which non-trivial multiplications are well grouped, as they occur in the midst of sequences of lifting steps related to trivial multiplications by powers of 2, which can be implemented by bit-shifting. Such schemes could be used to implement the DFT in hardware, as they offer alternative possibilities of modularization, pipelining, and folding of digital circuits.

Integer DCTs have a wide range of applications in lossless coding, especially in image compression. An integer-to-integer DCT of radix-2-length n is a nonlinear, left-invertible mapping, which acts on /spl Zopf//sup n/ and approximates the classical discrete cosine transform (DCT) of length n. All known integer-to-integer DCT-algorithms of length 8 are based on factorizations of the cosine matrix C/sub 8//sup II/ into a product of sparse matrices and work with lifting steps and rounding off. For fast implementation one replaces floating point numbers by appropriate dyadic rationals. Both rounding and approximation leads to truncation errors. In this paper, we consider an integer-to-integer transform for (2/spl times/2) rotation matrices and give estimates of the truncation errors for arbitrary approximating dyadic rationals. Further, using two known integer-to-integer DCT-algorithms, we show examplarily how to estimate the worst-case truncation error of lifting based integer-to-integer algorithms in fixed-point arithmetic, whose factorizations are based on (2/spl times/2) rotation matrices.

The tensor product is a fundamental mathematical concept with applications spanning linear algebra, graph theory, quantum computing, and representation theory. In graph theory, the tensor product provides a framework for analyzing structural relationships, particularly through the study of complete graphs, which yield complex networks from simple structures. Closely related is the Kronecker product of matrices, an essential tool for investigating tensor products via adjacency matrices. The Kronecker product preserves key algebraic properties, including linearity, distributivity, and associativity, and has played a central role in matrix analysis, systems theory, and signal processing. This work presents the definitions and core properties of the tensor product, supported by illustrative examples with complete graphs, and explores the Kronecker product along with its fundamental properties and theorems. By combining theoretical foundations with applications, the study offers both conceptual insights and practical perspectives on these algebraic constructions.

Binary CRCs are very effective for error detection, but their software implementation is not very efficient. Thus, many binary nonCRC codes (which are not as strong as CRCs, but can be more efficiently implemented in software) are proposed as alternatives to CRCs. The nonCRC codes include WSC, CXOR, one's-complement checksum, Fletcher checksum, and block-parity code. We present a general algorithm for constructing a family of binary error-detection codes. This family is large because it contains all these nonCRC codes, CRCs, perfect codes, as well as other linear and nonlinear codes. In addition to unifying these apparently disparate codes, our algorithm also generates some nonCRC codes that have minimum distance 4 (like CRCs) and efficient software implementation.

Invariants and the study of the map preserving a certain invariant play vital roles in the study of the theoretical mathematics. The preserver problems are the researches on linear operators that preserve certain invariants between matrix sets. Based on the result of linear $k$-power preservers on general matrix spaces, in terms of the advantages of matrix tensor products which is not limited by the size of matrices as well as the immense actual background, the study of the structure of the linear $k$-power preservers on tensor products of matrices is essential, which is coped with in this paper. That is to characterize a linear map $f:M_{m_{1}\cdots m_{l}}\rightarrow M_{m_{1}\cdots m_{l}}$ satisfying $f(X_{1}\otimes \cdots \otimes X_{l})^{k}=f\left( (X_{1}\otimes \cdots \otimes X_{l})^{k}\right) $ for all $X_{1}\otimes \cdots \otimes X_{l}\in M_{m_{1}\cdots m_{l}}$.

Considering multidimensional structure of the multi-exposure images, a new Tensor product and Tensor-singular value decomposition based Multi-Exposure image Fusion (TT-MEF) method is proposed. The main innovation of this work is to explore a new feature representation of multi-exposure images in the new tensor domain and design the fusion strategy on this basis. Specifically, the luminance and the chrominance channels are fused separately to maintain color consistency. For the luminance fusion, the luminance channel of multi-exposure images is divided into two parts, that is, de-mean term and mean term. The de-mean term is represented as a tensor to extract the feature. Then, the tensor product and tensor-singular value decomposition (T-SVD) are used to design a tensor feature extractor. Furthermore, a fusion strategy of the de-mean term is presented according to the visual saliency model, and a fusion strategy of the mean term is defined by the local and the global visual weights to control counterpoise between the local and global luminance. For the chrominance fusion, a new fusion strategy is also designed by the tensor product and T-SVD, similar to the luminance fusion. Finally, the fused image is obtained by combining the luminance and chrominance fusion. Experimental results show that the proposed TT-MEF method generally outperforms the existing state-of-the-art in terms of subjective visual quality and objective evaluation.

Importance of DES: NBS, ANSI and ISO (in study) have DES as standards.The available devices or programs have some tedious properties for an extensive use: ? hardware is expensive or slow, and limited, ? software is slow.We describe methods for obtaining efficient hardware and software implementations for the DES, i.e.

In Defense of the Demand Pattern, a Software Approach

Neural fields have revolutionized the area of 3D reconstruction and novel view synthesis of rigid scenes. A key challenge in making such methods applicable to articulated objects, such as the human body, is to model the deformation of 3D locations between the rest pose (a canonical space) and the deformed space. We propose a new articulation module for neural fields, Fast-SNARF, which finds accurate correspondences between canonical space and posed space via iterative root finding. Fast-SNARF is a drop-in replacement in functionality to our previous work, SNARF, while significantly improving its computational efficiency. We contribute several algorithmic and implementation improvements over SNARF, yielding a speed-up of 150×. These improvements include voxel-based correspondence search, pre-computing the linear blend skinning function, and an efficient software implementation with CUDA kernels. Fast-SNARF enables efficient and simultaneous optimization of shape and skinning weights given deformed observations without correspondences (e.g. 3D meshes). Because learning of deformation maps is a crucial component in many 3D human avatar methods and since Fast-SNARF provides a computationally efficient solution, we believe that this work represents a significant step towards the practical creation of 3D virtual humans.

Cognitive radio (CR) is considered an enabler of wireless technologies in safety-critical applications such as connected mobility and industrial automation, due to its high reliability and resource efficiency. Here, accurate and fast estimation of the quality of the radio channel is crucial to CR’s adaptive capability. In this paper, we propose a channel quality estimation architecture that devises a novel quality metric based solely on physical layer (PHY) measurements. The proposed metric utilizes a decision tree of PHY sub-metrics, without decoding the packets at higher layers. We validate the proposed architecture using a software defined radio (SDR) experimental setup, and investigate real-time capability and hardware implementation complexity. Our measurements show a high correlation between the proposed metric and the application layer quality of service (QoS). Our measurements also show that by using efficient software implementation, complexity is substantially reduced, which enables real-time functionality.

The mathematical description of the algorithm presented in a previous paper and a more efficient software implementation is given.

An efficient method for special least squares solution of the complex matrix equation [formula omitted

Many Markovian stochastic structured modeling formalisms like Petri nets, automata networks and process algebra represent the infinitesimal generator of the underlying Markov chain as a descriptor instead of a traditional sparse matrix. A descriptor is a compact and structured storage based on a sum of tensor (Kronecker) products of small matrices that can be handled by many algorithms allowing affordable stationary and transient solutions even for very large Markovian models. One of the most efficient algorithms used to compute iterative solutions of descriptors is the Shuffle algorithm which is used to perform the multiplication by a probability vector. In this paper we propose an alternative algorithm called Split, since it offers a flexible solution between the pure sparse matrix approach and the Shuffle algorithm using a hybrid solution. The Split algorithm puts the Shuffle approach in perspective by presenting a faster execution time for many cases and at least the same efficiency for the worst cases. The Split algorithm is applied to solve two SAN models based on real problems showing the practical contribution of this paper.

Fourier transform algorithms are described using tensor (Kronecker) products and an associated class of permutations. Algebraic properties of tensor products and the related permutations are used to derive variants of the Cooley-Tukey fast Fourier transform algorithm. These algorithms can be implemented by translating tensor products and permutations to programming constructs. An implementation can be matched to a specific computer architecture by selecting the appropriate variant. This methodology is carried out for the Cray X-MP and the AT&T DSP32.