Showing papers on "Hadamard transform published in 2009"

Monotone vector fields and the proximal point algorithm on Hadamard manifolds

Abstract: The maximal monotonicity notion in Banach spaces is extended to Riemannian manifolds of nonpositive sectional curvature, Hadamard manifolds, and proved to be equivalent to the upper semicontinuity. We consider the problem of finding a singularity of a multivalued vector field in a Hadamard manifold and present a general proximal point method to solve that problem, which extends the known proximal point algorithm in Euclidean spaces. We prove that the sequence generated by our method is well defined and converges to a singularity of a maximal monotone vector field, whenever it exists. Applications in minimization problems with constraints, minimax problems and variational inequality problems, within the framework of Hadamard manifolds, are presented.

Constructing mutually unbiased bases in dimension six

Abstract: The density matrix of a qudit may be reconstructed with optimal efficiency if the expectation values of a specific set of observables are known. In dimension six, the required observables only exist if it is possible to identify six mutually unbiased complex (6 × 6) Hadamard matrices. Prescribing a first Hadamard matrix, we construct all others mutually unbiased to it, using algebraic computations performed by a computer program. We repeat this calculation many times, sampling all known complex Hadamard matrices, and we never find more than two that are mutually unbiased. This result adds considerable support to the conjecture that no seven mutually unbiased bases exist in dimension six.

A Hybrid Approach Based on PSO and Hadamard Difference Sets for the Synthesis of Square Thinned Arrays

Abstract: A hybrid approach for the synthesis of planar thinned antenna arrays is presented. The proposed solution exploits and combines the most attractive features of a particle swarm algorithm and those of a combinatorial method based on the noncyclic difference sets of Hadamard type. Numerical experiments validate the proposed solution, showing improvements with respect to previous results.

Variable density compressed image sampling

Abstract: Compressed sensing (CS) provides an efficient way to acquire and reconstruct natural images from a reduced number of linear projection measurements at sub-Nyquist sampling rates. A key to the success of CS is the design of the measurement ensemble. This paper addresses the design of a novel variable density sampling strategy, where the “a priori” information about the statistical distributions that natural images exhibit in the wavelet domain is exploited. Compared to the current sampling schemes for compressed image sampling, the proposed variable density sampling has the following advantages: 1) The number of necessary measurements for image reconstruction is reduced; 2) The proposed sampling approach can be applied to several transform domains leading to simple implementations. In particular, the proposed method is applied to the compressed sampling in the 2D ordered discrete Hadamard transform (DHT) domain for spatial domain imaging. Furthermore, to evaluate the incoherence of different sampling schemes, a new metric that incorporates the “a priori” information is also introduced. Extensive simulations show the effectiveness of the proposed sampling methods.

On The Hadamard's Inequality for Log-Convex Functions on the Coordinates

Abstract: Inequalities of the Hadamard and Jensen types for coordinated log-convex functions defined in a rectangle from the plane and other related results are given

Proximal point methods for quasiconvex and convex functions with Bregman distances on Hadamard manifolds

Abstract: This paper generalizes the proximal point method using Bregman distances to solve convex and quasiconvex optimization problems on noncompact Hadamard manifolds. We will proved that the sequence generated by our method is well defined and converges to an optimal solution of the problem. Also, we obtain the same convergence properties for the classical proximal method, applied to a class of quasiconvex problems. Finally, we give some examples of Bregman distances in non-Euclidean spaces.

An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, the unitary group and the Pauli group

Abstract: The construction of unitary operator bases in a finite-dimensional Hilbert space is reviewed through a nonstandard approach combining angular momentum theory and representation theory of SU(2). A single formula for the bases is obtained from a polar decomposition of SU(2) and is analyzed in terms of cyclic groups, quadratic Fourier transforms, Hadamard matrices and generalized Gauss sums. Weyl pairs, generalized Pauli operators and their application to the unitary group and the Pauli group naturally arise in this approach.

An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, unitary group and Pauli group

Abstract: The construction of unitary operator bases in a finite-dimensional Hilbert space is reviewed through a nonstandard approach combinining angular momentum theory and representation theory of SU(2). A single formula for the bases is obtained from a polar decomposition of SU(2) and analysed in terms of cyclic groups, quadratic Fourier transforms, Hadamard matrices and generalized Gauss sums. Weyl pairs, generalized Pauli operators and their application to the unitary group and the Pauli group naturally arise in this approach.

Rajan Transform and its Uses in Pattern Recognition

Abstract: This transform was introduced in the year 1997 by Rajan [2], [4] on the lines of Hadamard Transform. This paper presents, in addition to its formulation, the algebraic properties of the transform and its uses in pattern recognition. Rajan Transform (RT) is a coding morphism by which a number sequence (integer, rational, real or complex) of length equal to any power of two is transformed into a highly correlated number sequence of the same length. It is a homomorphism that maps a set consisting of a number sequence, its graphical inverse and their cyclic and dyadic permutations, to a set consisting of a unique number sequence ensuring the invariance property under such permutations. This invariance property is also true for the permutation class of the dual sequence of the number sequence under consideration. A number sequence and its dual are like an object and its mirror image. For example, the four point sequences x(n) = 3, 1, 3, 3 and y(n) = 2, 4, 2, 2 are duals to each other. Observe that the sum of each sequence is 10 and one sequence could be obtained from the other by subtracting the elements of the other sequence from 5, which is half of its sum. Since RT of a number sequence is an organized number sequence with a high degree of correlation, it is suitable for effective data compression. This paper describes in detail the techniques of using RT for pattern recognition purposes. For example, pattern recognition operations like extracting lines, curves, isolated points and points of intersection of lines from a digital gray or colour image could be carried out using RT based fast algorithms.

Conjugate Symmetric Sequency-Ordered Complex Hadamard Transform

Abstract: A new transform known as conjugate symmetric sequency-ordered complex Hadamard transform (CS-SCHT) is presented in this paper. The transform matrix of this transform possesses sequency ordering and the spectrum obtained by the CS-SCHT is conjugate symmetric. Some of its important properties are discussed and analyzed. Sequency defined in the CS-SCHT is interpreted as compared to frequency in the discrete Fourier transform. The exponential form of the CS-SCHT is derived, and the proof of the dyadic shift invariant property of the CS-SCHT is also given. The fast and efficient algorithm to compute the CS-SCHT is developed using the sparse matrix factorization method and its computational load is examined as compared to that of the SCHT. The applications of the CS-SCHT in spectrum estimation and image compression are discussed. The simulation results reveal that the CS-SCHT is promising to be employed in such applications.

Inequalities of Hermite-Hadamard's Type for Functions Whose Derivatives Absolute Values are Quasi-Convex

Abstract: In this paper, some inequalities of Hermite-Hadamard type for functions whose derivatives absolute values are quasi-convex, are given. Some error estimates for the midpoint formula are also obtained.

All phase biorthogonal transform and its application in JPEG-like image compression

Abstract: This paper proposes new concepts of the all phase biorthogonal transform (APBT) and the dual biorthogonal basis vectors. In the light of all phase digital filtering theory, three kinds of all phase biorthogonal transforms based on the Walsh transform (WT), the discrete cosine transform (DCT) and the inverse discrete cosine transform (IDCT) are proposed. The matrices of APBT based on WT, DCT and IDCT are deduced, which can be used in image compression instead of the conventional DCT. Compared with DCT-based JPEG (DCT-JPEG) image compression algorithm at the same bit rates, the PSNR and visual quality of the reconstructed images using these transforms are approximate to DCT, outgoing DCT-JPEG at low bit rates especially. But the advantage is that the quantization table is simplified and the transform coefficients can be quantized uniformly. Therefore, the computing time becomes shorter and the hardware implementation easier.

Optical full Hadamard matrix multiplexing and noise effects.

Abstract: Hadamard multiplexing provides a considerable SNR boost over additive random noise but Poisson noise such as photon noise reduces the boost. We develop the theory for full H-matrix Hadamard transform imaging under additive and Poisson noise effects. We show that H-matrix encoding results in no effect on average on the noise level due to Poisson noise sources while preferentially reducing additive noise. We use this result to explain the wavelength-dependent varying SNR boost in a Hadamard hyperspectral imager and argue that such a preferential boost is useful when the main noise source is indeterminant or varying.

Quasiperiodic spectra and orthogonality for iterated function system measures

Abstract: We extend classical basis constructions from Fourier analysis to attractors for affine iterated function systems (IFSs). This is of interest since these attractors have fractal features, e.g., measures with fractal scaling dimension. Moreover, the spectrum is then typically quasi-periodic, but non-periodic, i.e., the spectrum is a “small perturbation” of a lattice. Due to earlier research on IFSs, there are known results on certain classes of spectral duality-pairs, also called spectral pairs or spectral measures. It is known that some duality pairs are associated with complex Hadamard matrices. However, not all IFSs X admit spectral duality. When X is given, we identify geometric conditions on X for the existence of a Fourier spectrum, serving as the second part in a spectral pair. We show how these spectral pairs compose, and we characterize the decompositions in terms of atoms. The decompositions refer to tensor product factorizations for associated complex Hadamard matrices.

Digitally-multiplexed nanoelectrospray ionization atmospheric pressure drift tube ion mobility spectrometry.

Abstract: One of the shortcomings of atmospheric pressure drift tube ion mobility spectrometry (DTIMS) is its intrinsically low duty cycle (∼0.04−1%) caused by the rapid pulsing of the ion gate (25−400 μs) followed by a comparatively long drift time (25−100 ms), which translates into a loss of sensitivity. Multiplexing approaches via Hadamard and Fourier-type gating techniques have been reported for increasing the sensitivity of DTIMS. Here, we report an extended multiplexing approach which encompasses arbitrary binary ion injection waveforms with variable duty cycles ranging from 0.5 to 50%. In this approach, ion mobility spectra can be collected using conventional signal averaging, arbitrary, standard Hadamard and/or “extended” Hadamard operation modes. Initial results indicate signal-to-noise gains ranging from 2−7-fold for both arbitrary and “extended” Hadamard sequences. Standard Hadamard transform IMS provided increased sensitivity, with gains ranging from 9−12-fold, however, mobility spectra suffered from de...

Hermite-Hadamard-type Inequalitites for Generalized Convex Functions

Abstract: The aim of the present paper is to extend the classical Hermite-Hadamard inequality to the case when the convexity notion is induced by a Chebyshev system.

Representations of quantum permutation algebras

Abstract: We develop a combinatorial approach to the quantum permutation algebras, as Hopf images of representations of type $\pi:A_s(n)\to B(H)$. We discuss several general problems, including the commutativity and cocommutativity ones, the existence of tensor product or free wreath product decompositions, and the Tannakian aspects of the construction. The main motivation comes from the quantum invariants of the complex Hadamard matrices: we show here that, under suitable regularity assumptions, the computations can be performed up to $n=6$.

An Active Pixel CMOS separable transform image sensor

Abstract: This paper presents a 128×128 charge-mode CMOS imaging sensor that computes separable transforms directly on the focal plane. The pixel is a unique extension of the widely reported Active Pixel Sensor (APS) cell. By capacitively coupling across an array of such cells onto switched capacitor circuits, computation of any unitary 2-D transform that is separable into inner and outer products is possible. This includes the Walsh, Hadamard and Haar basis functions. This scheme offers several advantages including multiresolution imaging, inherent de-noising, compressive sampling and lower integration voltage and faster readout. The chip was implemented on a 0:5µm CMOS process and measures 9mm2 in MOSIS' submicron design rules.

Robust decoupling control design for twin rotor system using Hadamard weights

Abstract: This paper considers cross-coupling affects in a twin rotor system that leads to degraded performance during precise helicopter maneuvering. This cross-coupling can be suppressed implicitly either by declaring it as disturbance or explicitly by introducing decoupling techniques. The standard H ∞ controller synthesized by loop-shaping design procedure (LSDP) offers robustness at the cost of performance to overcome cross-coupling. However, Hadamard weights are used to decouple the system dynamics to give desired performance as well. This idea has been successfully proved by simulations and verified through implementing it on a twin rotor system.

Some results on skew Hadamard difference sets

Abstract: In this paper, we present two constructions of divisible difference sets based on skew Hadamard difference sets. A special class of Hadamard difference sets, which can be derived from a skew Hadamard difference set and a Paley type regular partial difference set respectively in two groups of orders v 1 and v 2 with |v 1 ? v 2| = 2, is contained in these constructions. Some result on inequivalence of skew Hadamard difference sets is also given in the paper. As a consequence of Delsarte's theorem, the dual set of skew Hadamard difference set is also a skew Hadamard difference set in an abelian group. We show that there are seven pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 35 or 37, and also at least four pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 39. Furthermore, the skew Hadamard difference sets deduced by Ree-Tits slice symplectic spreads are the dual sets of each other when q ? 311.

Transform-Exempted Calculation of Sum of Absolute Hadamard Transformed Differences

Abstract: Sum of absolute Hadamard transformed differences (SATD) is an important distortion metric applied in the latest video coding standard H.264/AVC, which is an alternative to the sum of absolute differences (SAD) to improve coding efficiency. However, the SATD requires more computation load due to the Hadamard transform involved. Inspired by a geometric interpretation of the simplest two-point SATD calculation, an efficient way to compute the SATD is proposed, which considers the joint effect of the Hadamard transform and the following SAD processing in the SATD calculation and thus enables the computation of SATD without performing the Hadamard transform separately. We further extend the two-point transform- exempted SATD (TE-SATD) computation scheme to four-point and 4 times 4 block SATD calculation. With the same coding performance, the proposed TE-SATD-based fast algorithms can save 38% and 17% operations in computing a 4 times 4 block SATD compared with the conventional SATD calculation and the fast Hadamard transform-based calculation, respectively.

Flexible Representation of Quantum Images and Its Computational Complexity Analysis

Abstract: Flexible Representation of Quantum Images (FRQI) is proposed to provide a representation of images that enables efficient preparation and image processing operators on quantum computers FRQI captures colors and their corresponding positions in an image into a quantum state The preparation with polynomial simple operations for FRQI turning a quantum computer from the initialized state to the FRQI state is done by using Hadamard and controlled rotation gates The Quantum Image Compression (QIC) algorithm reduces simple operations used in the preparation process based on the minimization of Boolean equations are from same color groups of positions Quantum image processing operators on FRQI based unitary transforms dealing with only colors, colors at some positions and the combination by the quantum Fourier transform of both colors and positions are addressed The experiments for the storage and retrieval of images using FRQI are implemented in Matlab The compression ratio of QIC among same color groups ranges from 667% to 3162% on the Lena image

A two-parameter family of complex Hadamard matrices of order 6 induced by hypocycloids

Abstract: We construct a 2-parameter family of complex Hadamard matrices of order 6 by a natural block construction. We combine this family with an earlier result of Zauner to derive a 2-parameter family of triplets of mutually unbiased bases (MUBs) in ℂ 6 . This invalidates some numerical evidence given by Brierley and Weigert and sheds new light on the problem of determining the maximal number of MUBs in ℂ 6 .

Method of processing a video signal

Abstract: A method of processing a video signal is disclosed. The present invention includes obtaining a DC (discrete cosine) transform coefficient for a current macroblock and partition information of a DC (direct current) component block from a bitstream, obtaining transform size information of each partition of the DC component block based on the partition information of the DC component block, performing an inverse DC transform or a Hadamard transform based on the transform size information of the partition of the DC component block, performing inverse quantization on a result value from the transform and an AC (alternating current) component, and reconstructing a residual block by performing an inverse DC transform on a result value from the inverse quantization, wherein the residual block includes a block indicating a pixel value difference between an original picture and a predicted picture.