Showing papers on "Hadamard transform published in 2007"

Hadamard matrices and their applications

Abstract: In Hadamard Matrices and Their Applications, K. J. Horadam provides the first unified account of cocyclic Hadamard matrices and their applications in signal and data processing. This original work is based on the development of an algebraic link between Hadamard matrices and the cohomology of finite groups that was discovered fifteen years ago. The book translates physical applications into terms a pure mathematician will appreciate, and theoretical structures into ones an applied mathematician, computer scientist, or communications engineer can adapt and use. The first half of the book explains the state of our knowledge of Hadamard matrices and two important generalizations: matrices with group entries and multidimensional Hadamard arrays. It focuses on their applications in engineering and computer science, as signal transforms, spreading sequences, error-correcting codes, and cryptographic primitives. The book's second half presents the new results in cocyclic Hadamard matrices and their applications. Full expression of this theory has been realized only recently, in the Five-fold Constellation. This identifies cocyclic generalized Hadamard matrices with particular "stars" in four other areas of mathematics and engineering: group cohomology, incidence structures, combinatorics, and signal correlation. Pointing the way to possible new developments in a field ripe for further research, this book formulates and discusses ninety open questions.

Multiplexing for Optimal Lighting

Abstract: Imaging of objects under variable lighting directions is an important and frequent practice in computer vision, machine vision, and image-based rendering. Methods for such imaging have traditionally used only a single light source per acquired image. They may result in images that are too dark and noisy, e.g., due to the need to avoid saturation of highlights. We introduce an approach that can significantly improve the quality of such images, in which multiple light sources illuminate the object simultaneously from different directions. These illumination-multiplexed frames are then computationally demultiplexed. The approach is useful for imaging dim objects, as well as objects having a specular reflection component. We give the optimal scheme by which lighting should be multiplexed to obtain the highest quality output, for signal-independent noise. The scheme is based on Hadamard codes. The consequences of imperfections such as stray light, saturation, and noisy illumination sources are then studied. In addition, the paper analyzes the implications of shot noise, which is signal-dependent, to Hadamard multiplexing. The approach facilitates practical lighting setups having high directional resolution. This is shown by a setup we devise, which is flexible, scalable, and programmable. We used it to demonstrate the benefit of multiplexing in experiments.

Mutually unbiased bases and Hadamard matrices of order six

Abstract: We report on a search for mutually unbiased bases (MUBs) in six dimensions. We find only triplets of MUBs, and thus do not come close to the theoretical upper bound 7. However, we point out that the natural habitat for sets of MUBs is the set of all complex Hadamard matrices of the given order, and we introduce a natural notion of distance between bases in Hilbert space. This allows us to draw a detailed map of where in the landscape the MUB triplets are situated. We use available tools, such as the theory of the discrete Fourier transform, to organize our results. Finally, we present some evidence for the conjecture that there exists a four dimensional family of complex Hadamard matrices of order 6. If this conjecture is true the landscape in which one may search for MUBs is much larger than previously thought.

Pseudo-Paley graphs and skew Hadamard difference sets from presemifields

Abstract: Let (K, + ,*) be an odd order presemifield with commutative multiplication. We show that the set of nonzero squares of (K, *) is a skew Hadamard difference set or a Paley type partial difference set in (K, +) according as q is congruent to 3 modulo 4 or q is congruent to 1 modulo 4. Applying this result to the Coulter---Matthews presemifield and the Ding---Yuan variation of it, we recover a recent construction of skew Hadamard difference sets by Ding and Yuan [7]. On the other hand, applying this result to the known presemifields with commutative multiplication and having order q congruent to 1 modulo 4, we construct several families of pseudo-Paley graphs. We compute the p-ranks of these pseudo-Paley graphs when q = 34, 36, 38, 310, 54, and 74. The p-rank results indicate that these graphs seem to be new. Along the way, we also disprove a conjecture of Rene Peeters [17, p. 47] which says that the Paley graphs of nonprime order are uniquely determined by their parameters and the minimality of their relevant p-ranks.

Some hadamard’s inequalities for co-ordinated convex functions in a rectangle from the plane

Abstract: A monotonic nondecreasing mapping connected with Hadamard type inequalities in two variables is given and some Hadamard type inequalities for Lipschizian mapping in two variables are established.

Z4-linear Hadamard and extended perfect codes

Abstract: If $N=2^k > 8$ then there exist exactly $[(k-1)/2]$ pairwise nonequivalent $Z_4$-linear Hadamard $(N,2N,N/2)$-codes and $[(k+1)/2]$ pairwise nonequivalent $Z_4$-linear extended perfect $(N,2^N/2N,4)$-codes A recurrent construction of $Z_4$-linear Hadamard codes is given

On Hadamard integral inequalities involving two log-preinvex functions.

Abstract: Preinvex functions, Hermite-Hadamard integral inequalities, log-prinvex func- tions. Abstract: In this paper, we establish some new Hermite-Hadamard type integral inequali- ties involving two log-preinvex functions. Note that log-preinvex functions are nonconvex functions and include the log-convex functions as special cases. As special cases, we obtain the well known results for the convex functions.

What is Q-extension?

Abstract: In this paper we present a developed software in the area of Coding Theory. Using it, codes with given properties can be classified. A part of this software can be used also for investigations (isomorphisms, automorphism groups) of other discrete structures-combinatorial designs, Hadamard matrices, bipartite graphs etc.

Optimality Conditions for Nonsmooth Multiobjective Optimization Using Hadamard Directional Derivatives

Abstract: We establish both necessary and sufficient optimality conditions for weak efficiency and firm efficiency by using Hadamard directional derivatives and scalarizing the multiobjective problem under consideration via signed distances. For the first-order conditions, the data of the problem need not even be continuous; for the second-order conditions, we assume only that the first-order derivatives of the data are calm. We include examples showing the advantages of our results over some recent papers in the literature.

Improved image watermarking scheme using fast Hadamard and discrete wavelet transforms

Abstract: We propose a robust image watermarking scheme by applying the fast Hadamard transform (FHT) to small blocks computed from the four discrete wavelet transform (DWT) subbands. Different transforms have different properties that can effectively match various aspects of the signal's frequencies. Our approach consists of four main steps: (1) we decomposed the original image into four subbands, (2) the four subbands are divided into blocks; (3) FHT is applied to each block; and (4) the singular-value decomposition (SVD) is applied to the watermark image prior to distributing the singular values over the DC components of the transformed blocks. The proposed technique improves the data embedding system effectively, the watermark imperceptibility, and its resistance to a wide range of intentional attacks. The experimental results demonstrate the improved performance of the proposed method in comparison with existing techniques in terms of the watermark imperceptibility and the robustness against attacks.

On $Z_{2^k}$ -Dual Binary Codes

Abstract: A new generalization of the Gray map is introduced. The new generalization Phi:Z2 kn rarr Z2 2k-1n is connected with the known generalized Gray map phi in the following way: if we take two dual linear Z2 k-codes and construct binary codes from them using the generalizations phi and Phi of the Gray map, then the weight enumerators of the binary codes obtained will satisfy the MacWilliams identity. The classes of Z2 k-linear Hadamard codes and co-Z2 k-linear extended 1-perfect codes are described, where co-Z2 k-linearity means that the code can be obtained from a linear Z2 k-code with the help of the new generalized Gray map

A Note on Low-Correlation Zone Signal Sets

Abstract: In this correspondence, we present a connection between designing low-correlation zone (LCZ) sequences and the results of correlation of sequences with subfield decompositions presented in a recent book by the first two authors. This results in LCZ signal sets with huge sizes over three different alphabetic sets: finite field of size, integer residue ring modulo , and the subset in the complex field which consists of powers of a primitive th root of unity. We show a connection between these sequence designs and ldquocompletely noncyclicrdquo Hadamard matrices and a construction for those sequences. We also provide some open problems along this direction.

On symmetric signatures in holographic algorithms

Abstract: In holographic algorithms, symmetric signatures have been particularly useful.We give a complete characterization of these symmetric signatures over all bases of size 1. These improve previous results [4] where only symmetric signatures over the Hadamard basis (special basis of size 1) were obtained. In particular, we give a complete list of Boolean symmetric signatures over bases of size 1. It is an open problem whether signatures over bases of higher dimensions are strictly more powerful. The recent result by Valiant [18] seems to suggest that bases of size 2 might be indeed more powerful than bases of size 1. This result is with regard to a restrictive counting version of #SAT called #Pl-Rtw-Mon-3CNF. It is known that the problem is #P-hard, and its mod 2 version is ⊕P-hard. Yet its mod 7 version is solvable in polynomial time by holographic algorithms. This was accomplished by a suitable symmetric signature over a basis of size 2 [18]. We show that the same unexpected holographic algorithm can be realized over a basis of size 1. Furthermore we prove that 7 is the only modulus for which such an "accidental algorithm" exists.

Using Reed–Muller ${\hbox{RM}}\,(1, m)$ Codes Over Channels With Synchronization and Substitution Errors

Abstract: We analyze the performance of a Reed-Muller RM(1,m) code over a channel that, in addition to substitution errors, permits either the repetition of a single bit or the deletion of a single bit; the latter feature is used to model synchronization errors. We first analyze the run-length structure of this code. We enumerate all pairs of codewords that can result in the same sequence after the deletion of a single bit, and propose a simple way to prune the code by dropping one information bit such that the resulting linear subcode has good post-deletion and post-repetition minimum distance. A bounded distance decoding algorithm is provided for the use of this pruned code over the channel. This algorithm has the same order of complexity as the usual fast Hadamard transform based decoder for the RM(1,m) code

Determining the nonlinearity of a new family of APN functions

Abstract: We compute the Walsh spectrum and hence the nonlinearity of a new family of quadratic multi-term APN functions. We show that the distribution of values in the Walsh spectrum of these functions is the same as the Gold function.

Power Modeling and Efficient FPGA Implementation of FHT for Signal Processing

Abstract: Fast Hadamard transform (FHT) belongs to the family of discrete orthogonal transforms and is used widely in image and signal processing applications. In this paper, a parameterizable and scalable architecture for FHT with time and area complexities of O(2(W+1)) and O(2N2), respectively, has been proposed, where W and N are the word and vector lengths. A novel algorithmic transformation for the FHT based on sparse matrix factorization and distributed arithmetic (DA) principles has been presented. The architecture has been parallelized and pipelined in order to achieve high throughput rates. Efficient and optimized field-programmable gate array implementation of the proposed architecture that yield excellent performance metrics has been analyzed in detail. Additionally, a functional level power analysis and modeling methodology has been proposed to characterize the various power and energy metrics of the cores in terms of system parameters and design variables. The mathematical models that have been derived provide quick presilicon estimate of power and energy measures, allowing intelligent tradeoffs when incorporating the developed cores as subblocks in hardware-based image and video processing systems

Method of reducing computations in intra-prediction and mode decision processes in a digital video encoder

Abstract: A method of improving the computation speed of the sum of absolute transformed distances (SATD) for different intra-prediction modes is described. Determining the SATD quicker provides the benefits of better coding performance without suffering the drawbacks of longer computation times. The method of reducing intra-prediction and mode decision processes in a video encoder, implements Hadamard transforms with improvements. Hadamard transforms are performed on an original block and predicted blocks and calculations are only performed where coefficients are non-zero thus skipping the coefficients that are zero. Using such an approach, the calculations required for the Vertical Prediction, Horizontal Prediction and DC Prediction are reduced significantly. Thus, the best intra-prediction mode is able to be determined very efficiently.

Optimal multiplexed sensing: bounds, conditions and a graph theory link.

Abstract: Measuring an array of variables is central to many systems, including imagers (array of pixels), spectrometers (array of spectral bands) and lighting systems. Each of the measurements, however, is prone to noise and potential sensor saturation. It is recognized by a growing number of methods that such problems can be reduced by multiplexing the measured variables. In each measurement, multiple variables (radiation channels) are mixed (multiplexed) by a code. Then, after data acquisition, the variables are decoupled computationally in post processing. Potential benefits of the use of multiplexing include increased signal-to-noise ratio and accommodation of scene dynamic range. However, existing multiplexing schemes, including Hadamard-based codes, are inhibited by fundamental limits set by sensor saturation and Poisson distributed photon noise, which is scene dependent. There is thus a need to find optimal codes that best increase the signal to noise ratio, while accounting for these effects. Hence, this paper deals with the pursuit of such optimal measurements that avoid saturation and account for the signal dependency of noise. The paper derives lower bounds on the mean square error of demultiplexed variables. This is useful for assessing the optimality of numerically-searched multiplexing codes, thus expediting the numerical search. Furthermore, the paper states the necessary conditions for attaining the lower bounds by a general code. We show that graph theory can be harnessed for finding such ideal codes, by the use of strongly regular graphs.

Multi-Layer Broadcasting over a Block Fading MIMO Channel

Abstract: This paper introduces extensions for the broadcast approach for a multi-input multi-output (MIMO) block fading channel, with receiver only channel state information (CSI). Previous works have not been able to fully characterize the fundamental MIMO broadcasting upper bound. As it seems that analytical solution for this problem is quite difficult to achieve, we consider here sub-optimal schemes, for which achievable rates may be computed. In particular, finite level coding over a MIMO channel instead of continuous layering is analyzed, the expressions derived for the decoding probability regions allow numerical computation of finite level coding upper bounds. Noticing that the gains of two level coding over a MIMO channel are rather small, we consider sub-optimal techniques, which are more straightforward to implement. Among these techniques is the multiple-access channel (MAC) approach with single level coded streams, which is similar in concept to V-BLAST. Closed form expressions for probabilities of decoding regions here are derived, allowing numerical evaluation. We further consider multi-access permutation codes (MAPC). A Hadamard transform is compared with a suggested diagonal permutation code, which are shown to have similar performance, while diagonal permutation has lower implementation complexity. For all approaches, we derive information theoretic upper bounds of achievable rates.

An Image Compression Scheme Using AC Predictions

Abstract: An improved image compression scheme using AC coefficient prediction of Hadamard transform is presented in this paper. In the prediction phase, information of the nearest neighbor dc values of Hadamard transform blocks is utilized to predict the center block AC coefficients. Linear programming based optimization technique is used to predict the AC coefficients. Results indicate that this method is superior in terms of improving peak signal to noise ratio (PSNR) and minimizing blocking artifacts .

Some Results on Matchgates and Holographic Algorithms.

Abstract: We establish a 1-1 correspondence between Valiant's character theory of matchgate/matchcircuit [14] and his signature theory of planar-matchgate/matchgrid [16], thus unifying the two theories in expressibility. In [3], we had established a complete characterization of general matchgates, in terms of a set of useful Grassmann-Plucker identities. With this correspondence, we give a corresponding set of identities which completely characterizes planar-matchgates and their signatures. Applying this characterization we prove some negative results for holographic algorithms. On the positive side, we also give a polynomial time algorithm for a simultaneous node-edge deletion problem, using holographic algorithms. Finally we give characterizations of symmetric signatures realizable in the Hadamard basis.

On Bloch-Type Functions with Hadamard Gaps

Abstract: We give some sufficient and necessary conditions for an analytic function f on the unit ball B with Hadamard gaps, that is, for f(z)=∑k=1∞Pnk(z) (the homogeneous polynomial expansion of f) satisfying nk

Walsh-Hadamard Transform for Facial Feature Extraction in Face Recognition

Abstract: This Paper proposes a new facial feature extraction approach, Wash-Hadamard Transform (WHT). This approach is based on correlation between local pixels of the face image. Its primary advantage is the simplicity of its computation. The paper compares the proposed approach, WHT, which was traditionally used in data compression with two other known approaches: the Principal Component Analysis (PCA) and the Discrete Cosine Transform (DCT) using the face database of Olivetti Research Laboratory (ORL). In spite of its simple computation, the proposed algorithm (WHT) gave very close results to those obtained by the PCA and DCT. This paper initiates the research into WHT and the family of frequency transforms and examines their suitability for feature extraction in face recognition applications.

Construction of a new code for spectral amplitude coding in optical code-division multiple-access systems

Abstract: An enhanced code structure for spectral amplitude coding in optical code division multiple access systems based on double weight (DW) code families is proposed. Enhanced double weight (EDW) codes possess ideal cross-correlation properties such as maximum cross correlation of 1 and a weight that can be any odd number greater than 1. It has been observed through theoretical analysis and experimental simulation that EDW codes perform significantly better than Hadamard and modified frequency-hopping (MFH) codes. In this study, point-to-point transmission with three EDW-encoded channels was tested at the bit rate of 10 Gbit/s per channel.

The relative isoperimetric inequality in Cartan-Hadamard 3-manifolds

Abstract: Abstract We prove that a region D outside a convex set C with smooth boundary in a three-dimensional Cartan-Hadamard manifold M satisfies the relative isoperimetric inequality area , with equality if and only if D is isometric to a Euclidean half ball. We also prove similar sharp inequalities when the sectional curvature of M is bounded above by a negative constant.

On Some New Inequalities of Hermite-Hadamard-Fejer Type Involving Convex Functions

Abstract: In this paper, we establish some inequalities of Hermite-Hadamard- Fejer type for m-convex functions and s-convex functions.

A Simple Element Inverse Jacket Transform Coding

Abstract: Jacket transforms are a class of transforms that are simple to calculate, easily inverted, and size-flexible. Previously reported jacket transforms were generalizations of the well-known Walsh-Hadamard transform (WHT) and the center-weighted Hadamard transform (CWHT). In this letter, we present a new class of jacket transform not derived from either the WHT or the CWHT. This class of transform can be applied to any even length vector, is applicable to finite fields, and is useful for constructing error control codes

SIFME: A Single Iteration Fractional-Pel Motion Estimation Algorithm and Architecture for HDTV Sized H.264 Video Coding

Abstract: This paper presents a set of fast algorithm and VLSI architecture for HDTV-sized H.264 fractional motion estimation. To solve the long computational latency in HD-sized application, we propose to use the single iteration algorithm with only six search points. This single iteration method halves the cycle count of two iteration methods in previous approaches. Moreover, we propose to use 4×4 Hadamard instead of 8×8 Hadamard as cost function for H.264 high profiles without significant video quality loss. By these techniques, the resulted architecture can save 20% of area and provide over 40% of throughput improvement than the previous work, and is able to support HDTV applications.