Dyadic Decomposition Research Articles

A novel hardware-efficient spatial orientation tree-based image compression algorithm and its field programmable gate array implementation

Set partitioning in hierarchical trees (SPIHT) has become a popular research topic for more than a decade now. This is because it is simple, besides having compression efficiency close to the state-of-the-art JPEG2000 standard. The main drawback of SPIHT is that it uses three lists to store addresses of coefficients during its operation. These lists are dynamic and in worst cases need to store more number of addresses than total coefficients. In this work, a novel hardware-efficient spatial orientation tree-based algorithm is proposed and its hardware implementation is carried out. The wavelet transformed image is partitioned into 2 $\times$ 2 blocks. Each node of spatial orientation tree (SOT) represent a block of coefficients, rather than a single coefficient. Two small state-tables are used in this algorithm to store the status of each block. In addition to this, two extremely small lists are used to store the node addresses of a single SOT tree. To store the state-tables and lists for 512 $\times$ 512 image, only 0.88 % of the memory needed by SPIHT is required for five levels of dyadic decomposition. The peak-signal-to-noise ratio (PSNR) gain of 0.1 dB to 0.3 dB at low bit rates (below 1 bpp) and 0.6 dB to 1.2 dB at high bit rates (above 1.75 bpp) in comparison to SPIHT is observed for test image Lena. A field programmable gate array (FPGA) implementation targeted for Xilinx Zynq Z-7020 is presented in the paper. The proposed architecture saves 90 % of the FPGA area used by SPIHT. The hardware efficiency of the proposed architecture is better in comparison to different existing architectures.

Inviscid limit for the damped Boussinesq equation

In this paper, we consider the limit behavior of solutions to the Cauchy problem for damped Boussinesq equation in the regime of small viscosity in Rn. It is shown that the strong solutions of damped Boussinesq equation converge to a strong solution of Boussinesq equation in any time interval [0,T]. The key idea is that we only take into account the effect of dispersion and overlook the effect of dissipation in the damped Boussinesq equation, which is a kind of new view. We analyze the dispersion by the techniques of high-low frequency decomposition and Littlewood-Paley dyadic decomposition, stationary phase estimate and some properties of Bessel function.

A Universal Coding Scheme for Remote Generation of Continuous Random Variables

Alice selects an arbitrary pdf $f$ and uses a stochastic encoder to generate a prefix-free codeword $M$ , which is sent to Bob so that he can generate a single instance of the random variable $X\sim f$ . We describe a universal coding scheme for this setup which works for any $f$ , and establish an upper bound on its expected codeword length when the pdf $f$ is bounded, orthogonally concave (which includes quasiconcave pdf), and has a finite first absolute moment. A dyadic decomposition scheme is used to express the pdf as a mixture of uniform pdfs over hypercubes. Alice randomly selects a hypercube according to its weight, encodes its position and size into $M$ , and sends it to Bob who generates $X$ uniformly over the hypercube. Compared to previous results on channel simulation, our coding scheme applies to any continuous distribution and does not require two-way communication or shared randomness. Applying our coding scheme to classical simulation of quantum entanglement, we obtain a tighter bound on the average codeword length than previously known.

Sharp bounds for Boltzmann and Landau collision operators

The aim of the work is to provide a stable method to get sharp bounds for Boltzmann and Landau operators in weighted Sobolev spaces and in anisotropic spaces. All the sharp bounds are given for the original Boltzmann and Landau operators. The sharpness means the lower and upper bounds for the operators are consistent with the behavior of the linearized operators. Moreover, we make clear the difference between the bounds for the original operators and those for the linearized ones. According to the Bobylev's formula, we introduce two types of dyadic decompositions performed in both phase and frequency spaces to make full use of the interaction and the cancellation. It allows us to see clearly which part of the operator behaves like a Laplace type operator and which part is dominated by the anisotropic structure. It is the key point to get the sharp bounds in weighted Sobolev spaces and in anisotropic spaces. Based on the geometric structure of the elastic collision, we make a geometric decomposition to capture the anisotropic structure of the collision operator. More precisely, we make it explicit that the fractional Laplace-Beltrami operator really exists in the structure of the collision operator. It enables us to derive the sharp bounds in anisotropic spaces and then complete the entropy dissipation estimates. The structures mentioned above are so stable that we can apply them to the rescaled Boltzmann collision operator in the process of the grazing collisions limit. Then we get the sharp bounds for the Landau collision operator by passing to the limit. We remark that our analysis used here will shed light on the investigation of the asymptotics from Boltzmann equation to Landau equation.

The boundedness of generalized Bessel-Riesz operators on generalized Morrey spaces

The purpose of this paper is to prove the boundedness of generalized Bessel-Riesz operators on generalized Morrey spaces. The kernel of the operators contain some parameters, one of which is related to Bessel decay. As usual, we use the usual dyadic decomposition, Hölder’s inequality, a Hedberg-type inequality for the operators, and the boundedness of Hardy-Littlewood maximal operator in the proofs. In addition, we also exploit the relationship between the parameters of the kernel and of the space. We obtain that the norm of the operators is dominated by the norm of the kernels

Distributed Simulation of Continuous Random Variables

We establish the first known upper bound on the exact and Wyner’s common information of $n$ continuous random variables in terms of the dual total correlation between them (which is a generalization of mutual information). In particular, we show that when the pdf of the random variables is log-concave, there is a constant gap of $n^{2}\log e+9n\log n$ between this upper bound and the dual total correlation lower bound that does not depend on the distribution. The upper bound is obtained using a computationally efficient dyadic decomposition scheme for constructing a discrete common randomness variable $W$ from which the $n$ random variables can be simulated in a distributed manner. We then bound the entropy of $W$ using a new measure, which we refer to as the erosion entropy.

Mapping properties for operator-valued pseudodifferential operators on toroidal Besov spaces

In this paper, we consider pseudodifferential operators on the torus with operator-valued symbols and prove continuity properties on vector-valued toroidal Besov spaces, without assumptions on the underlying Banach spaces. The symbols are of limited smoothness with respect to $x$ and satisfy a finite number of estimates on the discrete derivatives. The proof of the main result is based on a description of the operator as a convolution operator with a kernel representation which is related to the dyadic decomposition appearing in the definition of the Besov space.

Investigation of the scaling characteristics of LANDSAT temperature and vegetation data: a wavelet-based approach.

An investigation of the scaling characteristics of vegetation and temperature data derived from LANDSAT data was undertaken for a heterogeneous area in Tamil Nadu, India. A wavelet-based multiresolution technique decomposed the data into large-scale mean vegetation and temperature fields and fluctuations in horizontal, diagonal, and vertical directions at hierarchical spatial resolutions. In this approach, the wavelet coefficients were used to investigate whether the normalized difference vegetation index (NDVI) and land surface temperature (LST) fields exhibited self-similar scaling behaviour. In this study, l-moments were used instead of conventional simple moments to understand scaling behaviour. Using the first six moments of the wavelet coefficients through five levels of dyadic decomposition, the NDVI data were shown to be statistically self-similar, with a slope of approximately -0.45 in each of the horizontal, vertical, and diagonal directions of the image, over scales ranging from 30 to 960m. The temperature data were also shown to exhibit self-similarity with slopes ranging from -0.25 in the diagonal direction to -0.20 in the vertical direction over the same scales. These findings can help develop appropriate up- and down-scaling schemes of remotely sensed NDVI and LST data for various hydrologic and environmental modelling applications. A sensitivity analysis was also undertaken to understand the effect of mother wavelets on the scaling characteristics of LST and NDVI images.

Quantitative weighted estimates for rough homogeneous singular integrals

We consider homogeneous singular kernels, whose angular part is bounded, but need not have any continuity. For the norm of the corresponding singular integral operators on the weighted space L 2(w), we obtain a bound that is quadratic in A 2 constant $${\left[ w \right]_{{A_2}}}$$ . We do not know if this is sharp, but it is the best known quantitative result for this class of operators. The proof relies on a classical decomposition of these operators into smooth pieces, for which we use a quantitative elaboration of Lacey's dyadic decomposition of Dini-continuous operators: the dependence of constants on the Dini norm of the kernels is crucial to control the summability of the series expansion of the rough operator. We conclude with applications and conjectures related to weighted bounds for powers of the Beurling transform.

Separation of cyclic and starlike hierarchical dominance in evolutionary matrix games.

We study antisymmetric components of matrices characterizing pair interactions in multistrategy evolutionary games. Based on the dyadic decomposition of matrices we distinguish cyclic and starlike hierarchical dominance in the appropriate components. In the symmetric matrix games the strengths of these elementary components are determined. The general features and intrinsic symmetries of these interactions are represented by directed graphs. It is found that the variation of a single matrix component modifies simultaneously the strengths of two starlike hierarchical basis games and many other independent rock-paper-scissors type cyclic basis games. The application of the related concepts is illustrated by discussing the three-strategy voluntary prisoner's dilemma.

Smoothing for the Zakharov and Klein--Gordon--Schrödinger Systems on Euclidean Spaces

This paper studies the existence and regularity properties of solutions to the Zakharov and Klein--Gordon--Schrodinger systems at low regularity levels. The main result is that the nonlinear part of the solution flow falls in a smoother space than the initial data. This relies on a new bilinear $X^{s,b}$ estimate, which is proved using delicate dyadic and angular decompositions of the frequency domain. Such smoothing estimates have a number of implications for the long-term dynamics of the system. In this work, we give a simplified proof of the existence of global attractors for the Klein--Gordon--Schrodinger flow in the energy space for dimensions d = 2,3. Second, we use smoothing in conjunction with a high-low decomposition to show global well-posedness of the Klein--Gordon--Schrodinger evolution on $\mathbb{R}^4$ below the energy space for sufficiently small initial data.

$L^p$ boundedness for maximal functions associated with multi-linear pseudo-differential operators

In this paper, we establish the $L^p$ estimates for the maximal functions associated with the multilinear pseudo-differential operators. Our main result is Theorem 1.2. There are several major different ingredients and extra difficulties in our proof from those in Grafakos, Honzik and Seeger [ 15 ] and Honzik [ 22 ] for maximal functions generated by multipliers. First, in order to eliminate the variable $x$ in the symbols, we adapt a non-smooth modification of the smooth localization method developed by Muscalu in [ 26 , 30 ]. Then, by applying the inhomogeneous Littlewood-Paley dyadic decomposition and a discretization procedure, we can reduce the proof of Theorem 1.2 into proving the localized estimates for localized maximal functions generated by discrete paraproducts. The non-smooth cut-off functions in the localization procedure will be essential in establishing localized estimates. Finally, by proving a key localized square function estimate (Lemma 4.3) and applying the good-$\lambda$ inequality, we can derive the desired localized estimates.

A mathematical algorithm for quantification of CT image noise

Quantification of computed tomography (CT) noise helps in determination of radiation dosage requirements for adequate image quality. Clinical methods used include calculation of the standard deviation (SD) of a selected region of interest (ROI). In industry, wavelet decomposition has been used for image compression while removing high-frequency noise. We evaluated a cohort of 74 consecutive patients referred for coronary artery calcium scoring and quantitated noise within a 16×16 ROI in the ascending aorta using the traditional SD method and also using a two-dimensional dyadic wavelet decomposition method. Clinically, noise has been shown to be proportional to patient weight and also body mass index (BMI), which is a derived value from height and weight. Noise for both methods was plotted against patient parameters of height, weight, waist circumference and calculated BMI. A regression line was calculated and coefficient of determination (CoD) calculated for each. The CoD was better for height, weight, and waist circumference using the wavelet method as compared to the traditional SD method. The wavelet method of quantification of image noise may be an improved method as compared to the SD method. This method could help further refine an imaging system's determination of radiation dosage requirements to obtain a satisfactory quality image.

Lower Bounds on Blowing-Up Solutions of the Three-Dimensional Navier--Stokes Equations in $\dot H^{3/2}$, $\dot H^{5/2}$, and $\dot B^{5/2}_{2,1}$

If $u$ is a smooth solution of the Navier--Stokes equations on ${\mathbb R}^3$ with first blowup time $T$, we prove lower bounds for $u$ in the Sobolev spaces $\dot H^{3/2}$, $\dot H^{5/2}$, and the Besov space $\dot B^{5/2}_{2,1}$, with optimal rates of blowup: we prove the strong lower bounds $\|u(t)\|_{\dot H^{3/2}}\ge c(T-t)^{-1/2}$ and $\|u(t)\|_{\dot B^{5/2}_{2,1}}\ge c(T-t)^{-1}$; in $\dot H^{5/2}$ we obtain $\limsup_{t\to T^-}(T-t)\|u(t)\|_{\dot H^{5/2}}\ge c$, a weaker result. The proofs involve new inequalities for the nonlinear term in Sobolev and Besov spaces, both of which are obtained using a dyadic decomposition of $u$.

Global regularity of logarithmically supercritical 3-D LAMHD-alpha system with zero diffusion

We study the three-dimensional Lagrangian-averaged magnetohydrodynamics-alpha system with zero diffusion. Despite the dissipation strength at the logarithmically supercritical level, using dyadic decomposition techniques we show that given initial data sufficiently smooth, the solution pair remains smooth for all time. This settles the global regularity case suggested by the authors in [27].

Global existence and asymptotic behavior of solution for the sixth order Boussinesq equation with damped term

We investigate the small-data Cauchy problem for the sixth order Boussinesq equation with damped term. The decay estimates of solutions to the corresponding linear equation are given by using the dyadic decomposition and some properties of Bessel functions. Then, we prove the global existence and asymptotic of the small amplitude solution in the time-weighted Sobolev space by the contraction mapping principle.

Multiscale Reverse-Time-Migration-Type Imaging Using the Dyadic Parabolic Decomposition of Phase Space

We develop a representation of reverse-time migration (RTM) in terms of Fourier integral operators, the canonical relations of which are graphs. Through the dyadic parabolic decomposition of phase space, we obtain the solution of the wave equation with a boundary source and homogeneous initial conditions using wave packets. On this basis, we develop a numerical procedure for the reverse-time continuation from the boundary of scattering data and for RTM. The algorithms are derived from those we recently developed for the discrete approximate evaluation of the action of Fourier integral operators and inherit their conceptual and numerical properties.

Periodic Impulsive Fault Feature Extraction of Rotating Machinery Using Dual-Tree Rational Dilation Complex Wavelet Transform

Fault diagnosis of rotating machinery is very important to guarantee the safety of manufacturing. Periodic impulsive fault features commonly appear in vibration measurements when local defects occur in the key components like rolling bearings and gearboxes. To extract the periodic impulses embedded in strong background noise, wavelet transform (WT) is suitable and has been widely used in analyzing these nonstationary signals. However, a few limitations like shift-variance and fixed frequency partition manner of the dyadic WT would weaken its effectiveness in engineering application. Compared with dyadic WT, the dual-tree rational dilation complex wavelet transform (DT-RADWT) enjoys attractive properties of better shift-invariance, flexible time-frequency (TF) partition manner, and tunable oscillatory nature of the bases. In this article, an impulsive fault features extraction technique based on the DT-RADWT is proposed. In the routine of the proposed method, the optimal DT-RADWT basis is constructed dynamically and adaptively based on the input signal. Additionally, the sensitive wavelet subband is chosen using kurtosis maximization principle to reveal the potential weak fault features. The proposed method is applied on engineering applications for defects detection of the rolling bearing and gearbox. The results show that the proposed method performs better in extracting the fault features than dyadic WT and empirical mode decomposition (EMD), especially when the incipient fault features are embedded in the frequency transition bands of the dyadic WT.

Global $L^p$ continuity of Fourier integral operators

In this paper we establish global Lp regularity properties of Fourier integral operators. The orders of decay of the amplitude are determined for operators to be bounded on $L^p(\Rn)$, $1<p<\infty$, as well as to be bounded from Hardy space $H^1(\Rn)$ to $L^1(\Rn)$. The obtained results extend local $L^p$ regularity properties of Fourier integral operators established by Seeger, Sogge and Stein (1991) as well as global $L^2(\Rn)$ results of Asada and Fujiwara (1978) and Ruzhansky and Sugimoto (2006), to the global setting of $L^p(\Rn)$. Global boundedness in weighted Sobolev spaces $W^{\sigma,p}_s(\Rn)$ is also established. The techniques used in the proofs are the space dependent dyadic decomposition and the global calculi developed by Ruzhansky and Sugimoto (2006) and Coriasco (1999).

Spectrum Decomposition for Image/Signal Coding

In conventional subband/wavelet image coding, the subband decomposition is performed on the spatial-domain image. Here, we introduce a novel decomposition where the subband decomposition is performed on the global DCT spectrum of the image. That is, the two-dimensional spectrum rather than the image is represented by a sum of basis functions, each weighted by the transform coefficients. The distinct features of this decomposition are analyzed from a transform perspective. This spectral subband decomposition is then used as the basis for a new image coder, building on the condensed wavelet packet (CWP) algorithm. Ironically, this new method is shown to have lower arithmetic complexity than conventional subband/wavelet coders that directly decompose a time or spatial domain signal. Comparisons of the new method against conventional subband/wavelet coders that use the popular 9/7 dyadic decomposition, condensed wavelet packets, and generalized lapped orthogonal transforms, show that the new method has lower complexity and higher compression performance.