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As it is known, the convergence rate of the multidimensional Whittaker-Kotelnikov-Shannon (WKS) sampling series is slow due to the slow decay of the sinc function. In this paper, we incorporate a convergence factor from the Bernstein space into the multidimensional WKS sampling series to establish regularized sampling and a corresponding improved convergence rate. The convergence rate of this regularized series depends on the decay of the convergence factor. Various bounds for the truncation of the regularized sampling series are investigated depending on the convergence factor. Furthermore, we estimate two types of perturbation errors associated with this series. Some numerical experiments are presented.

This paper is devoted to the establishment of two-dimensional sampling theorems for discrete transforms, whose kernels arise from second order partial difference equations. We define a discrete type partial difference operator and investigate its spectral properties. Green’s function is constructed and kernels that generate orthonormal basis of eigenvectors are defined. A discrete Kramer-type lemma is introduced and two sampling theorems of Lagrange interpolation type are proved. Several illustrative examples are depicted. The theory is extendible to higher order settings.

In the present paper we study the pointwise and uniform convergence properties of a family of multidimensional sampling Kantorovich type operators. Moreover, besides convergence, quantitative estimates and a Voronovskaja type theorem have been established.

International audience; Error Whitenoise DartThrowing Sobol Ours White noise Dart throwing Sobol Ours Fig. 1. Sliced Optimal Transport Sampling. Global illumination of a scene (top left, San Miguel) requires integrating radiance over a high-dimensional space of light paths. The projective variant of our sliced optimal transport (SOT) sampling technique, leveraging the particular nature of integral evaluation in rendering and further combined with a micro-Cranley-Patterson rotation per pixel, outperforms standard Monte Carlo and Quasi-Monte Carlo techniques, exhibiting less noise and no structured artifact (top right, 32spp) while offering a better spatial distribution of error (bottom right, errors from blue (small) to red (large)). Moreover, our projective SOT sampling produces better convergence of the mean absolute error for the central 7×7 zone of the highlighted reference window as a function of the number of samples per pixel (from 4spp to 4096spp, bottom-left graph) in the case of indirect lighting with one bounce. In this paper, we introduce a numerical technique to generate sample distributions in arbitrary dimension for improved accuracy of Monte Carlo integration. We point out that optimal transport offers theoretical bounds on Monte Carlo integration error, and that the recently-introduced numerical framework of sliced optimal transport (SOT) allows us to formulate a novel and efficient approach to generating well-distributed high-dimensional pointsets. The resulting sliced optimal transport sampling, solely involving repeated 1D solves, is particularly simple and efficient for the common case of a uniform density over a d-dimensional ball. We also construct a volume-preserving map from a d-ball to a d-cube (generalizing the Shirley-Chiu mapping to arbitrary dimensions) to offer fast SOT sampling over d-cubes. We provide ample numerical evidence of the improvement in Monte Carlo integration accuracy that SOT sampling brings compared to existing QMC techniques, and derive a projective variant for rendering which rivals, and at times outperforms, current sampling strategies using low-discrepancy sequences or optimized samples.

Generalized Kaiser–Bessel radial basis functions are frequently used in tomographic reconstructions, and the parameter selection can significantly impact both the qualitative and quantitative image characteristics. Currently, the blob parameters are simply selected based on the first-order approximation of a uniform object. To obtain optimal images, a detailed investigation and optimization of blob parameters is needed. In this paper, we aim at optimizing the parameters of the basis functions for optimal image representation and image reconstruction. We first represent the unknown radiotracer activity distribution as the coefficients of blob basis functions on a Bravais lattice from crystallography. We point out that the optimal sampling lattice is just the reciprocal of the lattice that achieves highest packing of equally sized spheres. Based on multidimensional sampling theorem and the Kepler conjecture (proven by Thomas C. Hales), we obtain that the body-centered cubic (BCC) and hexagonal close-packing lattices are optimal in terms of either reducing computational cost at similar image quality or improving image quality at the same computational cost. To optimize the blob parameters, we formulate two new image quality metrics—total harmonic distortion (THD) and minimum approximation error (MAE) based on the theory of approximation—to analytically characterize the blob representation errors of a nonzero uniform object and an arbitrary object, respectively. The MAE is formulated in terms of the power spectral density of an imaging object, and we use it for object-dependent blob parameter optimization. We validate the THD and MAE through numerical examples using homogeneous and heterogeneous objects, respectively. We also present 3-D blob-based image reconstructions using BCC lattice with different shape parameters to show the usefulness of the proposed optimization.

A one-dimensional projection (ODP) technique is introduced to overcome the challenges in multidimensional sampling. With the help of a projected "advancement of reaction (ξ)" control parameter, it was demonstrated that multidimensional samplings could be performed with a single parameter, thus dramatically reducing the computational overhead. In our test studies, the ODP technique successfully yielded the free energy surface of the double proton transfer reaction of the acetic acid dimer and the hydrolysis of the methyl diazonium ion. In short, the ODP can be a promising protocol for the samplings of complex chemical reaction dynamics.

In high-dimensional magnetic resonance imaging applications, time-consuming, sequential acquisition of data samples in the spatial frequency domain (k-space) can often be accelerated by accounting for dependencies along imaging dimensions other than space in linear reconstruction, at the cost of noise amplification that depends on the sampling pattern. Examples are support-constrained, parallel, and dynamic MRI, and k-space sampling strategies are primarily driven by image-domain metrics that are expensive to compute for arbitrary sampling patterns. It remains challenging to provide systematic and computationally efficient automatic designs of arbitrary multidimensional Cartesian sampling patterns that mitigate noise amplification, given the subspace to which the object is confined. To address this problem, this work introduces a theoretical framework that describes local geometric properties of the sampling pattern and relates these properties to a measure of the spread in the eigenvalues of the information matrix described by its first two spectral moments. This new criterion is then used for very efficient optimization of complex multidimensional sampling patterns that does not require reconstructing images or explicitly mapping noise amplification. Experiments with in vivo data show strong agreement between this criterion and traditional, comprehensive image-domain- and k-space-based metrics, indicating the potential of the approach for computationally efficient (on-thefly), automatic, and adaptive design of sampling patterns.

This chapter discusses the topic of emerging light field displays from a signal processing perspective. Light field displays are defined as devices which deliver continuous parallax along with the focus and binocular visual cues acting together in rivalry-free manner. In order to ensure such functionality, one has to deal with the light field, conceptualized by the plenoptic function and its adequate parametrization, sampling and reconstruction. The light field basics and the corresponding display technologies are overviewed in order to address the fundamental problems of analyzing light field displays as signal processing channels, and of capturing and representing light field visual content for driving such displays. Spectral analysis of multidimensional sampling operators is utilized to profile the displays in question, and modern sparsification approaches are employed to develop methods for high-quality light field reconstruction and rendering. acceptedVersion

A new lower bound on the average reconstruction error variance of multidimensional sampling and reconstruction is presented. It applies to sampling on arbitrary lattices in arbitrary dimensions, assuming a stochastic process with constant, isotropically bandlimited spectrum and reconstruction by the best linear interpolator. The lower bound is exact for any lattice at sufficiently high and low sampling rates. The two threshold rates where the error variance deviates from the lower bound gives two optimality criteria for sampling lattices. It is proved that at low rates, near the first threshold, the optimal lattice is the dual of the best sphere-covering lattice, which for the first time establishes a rigorous relation between optimal sampling and optimal sphere covering. A previously known result is confirmed at high rates, near the second threshold, namely, that the optimal lattice is the dual of the best sphere-packing lattice. Numerical results quantify the performance of various lattices for sampling and support the theoretical optimality criteria.

SummaryObjectives: A major problem associated with the irradiation of thoracic and abdominal tumors is respiratory motion. In clinical practice, motion compensation approaches are frequently steered by low-dimensional breathing signals (e.g., spirometry) and patient-specific correspondence models, which are used to estimate the sought internal motion given a signal measurement. Recently, the use of multidimensional signals derived from range images of the moving skin surface has been proposed to better account for complex motion patterns. In this work, a simulation study is carried out to investigate the motion estimation accuracy of such multidimensional signals and the influence of noise, the signal dimensionality, and different sampling patterns (points, lines, regions).Methods: A diffeomorphic correspondence modeling framework is employed to relate multidimensional breathing signals derived from simulated range images to internal motion patterns represented by diffeomorphic non-linear transformations. Furthermore, an automatic approach for the selection of optimal signal combinations/patterns within this framework is presented.Results: This simulation study focuses on lung motion estimation and is based on 28 4D CT data sets. The results show that the use of multidimensional signals instead of one-dimensional signals significantly improves the motion estimation accuracy, which is, however, highly affected by noise. Only small differences exist between different multidimensional sampling patterns (lines and regions). Automatically determined optimal combinations of points and lines do not lead to accuracy improvements compared to results obtained by using all points or lines.Conclusions: Our results show the potential of multidimensional breathing signals derived from range images for the model-based estimation of respiratory motion in radiation therapy.

This paper addresses the problem of multidimensional signal reconstruction from generalized samples in fractional Fourier domain including the deterministic case and the stochastic case. The generalized sampling expansion is investigated for the case where the fractional bandlimited input depends on N real variable, i.e., f ( t ) = f ( t 1 , ⋯ , t N ) and is used as a common input to a parallel bank of m independent N dimensional linear fractional Fourier filters Hα,k(u), k = 1 , ⋯ , m . For the deterministic input, the input is assumed to have its N dimensional fractional Fourier transform bandlimited to the frequency rang |ui|≤Ωi, for i = 1 , ⋯ , N . If m, the number of fractional Fourier filters, is written as a product of positive integers in the form m = m 1 m 2 ⋯ m N , and if the fractional bandlimited input f(t) is processed by fractional Fourier filter Hα,k(u)resulting m outputs gk(t), then f(t) can be reconstructed in terms of the samples gk(nT), each output being sampled at the identical rates of Ω 1 csc α / m 1 π , Ω 2 csc α / m 2 π , ⋯ , Ω N csc α / m N π samples/second in t 1 , ⋯ , t N respectively. This contrasts with the rates of Ω 1 csc α / π , Ω 2 csc α / π , ⋯ , Ω N csc α / π in t 1 , ⋯ , t N needed for reconstruction of the unfiltered input f(t). Input sampling expansions in terms of samples of the output filters are given for both deterministic and stochastic inputs, the generalized sampling expansion for random input having the same form as for the deterministic case but interpreted in the mean-square sense. Our formulation and results are general and include derivative sampling and periodic nonuniform sampling in the fractional Fourier domain for multidimensional signals as special case. Finally, the potensional application of the multidimensional generalized sampling is presented to show the advantage of the theory. Especially, the application of multidimensional generalized sampling in the context of the image scaling about image super-resolution is also discussed. The simulation results of image scaling are also presented.

As a key sampling scheme in Markov chain Monte Carlo (MCMC) methods, Gibbs sampling is widely used in various research fields due to its elegant univariate conditional sampling, especially in tacking with multidimensional sampling systems. In this paper, a Gibbs-based sampler named as symmet- ric Metropolis-within-Gibbs (SMWG) algorithm is proposed for lattice Gaussian sampling. By adopting a symmetric Metropolis- Hastings (MH) step into the Gibbs update, we show the Markov chain arising from it is geometrically ergodic, which converges exponentially fast to the stationary distribution. Moreover, by optimizing its symmetric proposal distribution, the convergence efficiency can be further enhanced.

In this paper, we study the problem of generating uniform random point samples on a domain of d dimensional space based on a minimum distance criterion between point samples (Poisson-disk sampling or PDS). First, we formally define PDS via the pair correlation function (PCF) to quantitatively evaluate properties of the sampling process. Surprisingly, none of the existing PDS techniques satisfy both uniformity and minimum distance criterion, simultaneously. These approaches typically create an approximate PDS with high regularity, and inherently present high risk for sample aliasing. Our new formulation based on PCF introduces a new approach to evaluate PDS properties which leads to theoretical bounds on the size of a PDS in arbitrary dimensions as well as a faster algorithm to create better quality samplings than the current PDS approaches.

The optimal description of the reaction coordinate in chemical systems is of great importance in simulating condensed phase reactions. In the current work, we present a collective reaction coordinate which is composed of several geometric coordinates which represent structural progress during the course of a hydride transfer reaction: the antisymmetric reactive stretch coordinate, the donor-acceptor distance (DAD) coordinate, and an orbital rehybridization coordinate. In this approach, the former coordinate serves as a distinguished reaction coordinate, while the latter two serve as environmental, Marcus-type inner-sphere reorganization coordinates. The classical free energy surface is obtained from multidimensional quantum mechanics-molecular mechanics (QM/MM) potential of mean force (PMF) simulations in conjunction with a general and efficient multidimensional weighted histogram method implementation. The minimum free energy path, or the collective reaction coordinate, connecting the dividing hypersurface to reactants and products, is obtained using an iterative scheme. In this approach, the string method is used to find the minimum free energy path. This path guides the multidimensional sampling, while the path is adaptively refined until convergence is achieved. As a model system, we choose the hydride transfer reaction in Escherichia coli dihydrofolate reductase (ecDHFR) using a recently developed accurate semiempirical potential energy surface. To estimate the advantages of the collective reaction coordinate, we perform activated dynamics simulations to obtain the reaction transmission coefficient. The results show that the combination of a distinguished reaction coordinate and an inner-sphere reorganization coordinate considerably reduces the dividing surface recrossing.

The issue of how to efficiently represent the data collected by a network of microphones recording spatio-temporal acoustic wave fields is addressed. Each sensor node in the network samples the sound field, quantizes the samples and transmits the encoded samples to some central unit, which computes an estimate of the original sound field based on the information received from all the microphones. Our analysis is based on the spectral properties of the sound field, which are induced by the physics of wave propagation and have a significant impact on the efficiency of the chosen sampling lattice and coding scheme. As field acquisition by a sensor network typically implies spatio-temporal sampling of the field, a multidimensional sampling theorem for homogeneous random fields with compactly supported spectral measures is proved. To assess the loss of information implied by source coding, rate distortion functions for various coding schemes and sampling lattices are determined. In particular, centralized coding, independent coding and some multiterminal schemes are compared. Under the assumption of spectral whiteness of the sound field, it is shown that sampling with a quincunx lattice followed by independent coding is optimal as it achieves the lower bound given by centralized coding.

Time-varying illumination on the focal plane is a three-dimensional signal. Multidimensional sampling theory proves that the temporal resolution can be optimally improved by a factor of \(\sqrt 2\) while the spatial resolution is reserved by changing the sampling scheme. Based on the theory, a prototype multi-field CMOS image sensor (CIS) is designed for a 0.35-μm 2P4M CMOS process. Corresponding pixels in 4×4-pixel clusters are assembled into 16 fields over the whole array. Control pins (resets and shutters) of pixels are separated which provides the ability of sampling the illumination with the optimal sampling scheme.

We address the optimal sensor arrangement problem, which is the determination of a geometric configuration of sensors such that the mean-squared error (MSE) in the estimation of an unknown trigonometric polynomial is minimum. Unsurprisingly, an arrangement in which sensors are spaced uniformly in each dimension is optimal. However, for multidimensional problems the minimum MSE is achieved with a much larger class of configurations that we call generalized regular arrangements. These arrangements are not necessarily generated by lattices and may exhibit great nonuniformity locally.

We present an accurate and efficient method to generate samples based on a Poisson-disk distribution. This type of distribution, because of its blue noise spectral properties, is useful for image sampling. It is also useful for multidimensional Monte Carlo integration and as part of a procedural object placement function. Our method extends trivially from 2D to 3D or to any higher dimensional space. We demonstrate results for up to four dimensions, which are likely to be the most useful for computer graphics applications. The method is accurate because it generates distributions with the same statistical properties of those generated with the brute-force dart-throwing algorithm, the archetype against which all other Poisson-disk sampling methods are compared. The method is efficient because it employs a spatial subdivision data structure that signals the regions of space where the insertion of new samples is allowed. The method has O ( N log N ) time and space complexity relative to the total number of samples. The method generates maximal distributions in which no further samples can be inserted at the completion of the algorithm. The method is only limited in the number of samples it can generate and the number of dimensions over which it can work by the available physical memory.

Sharp upper bounds for interpolation remainders of the multidimensional Paley-Wiener class functions by finite regular Whittaker-Kotel’nikov-Shannon sampling sum are obtained. The extremal functions are given for which the derived bounds are attained. Truncation error analysis and convergence rate is provided in weak Cramer class random fields. The historical background, the development, and extensive reference list are given concerning truncation error upper bounds for deterministic and random signal functions. Finally, new research directions are posed and discussed.

In this paper, we consider the problem of sampling signals that are nonband-limited but have finite number of degrees of freedom per unit of time and call this number the rate of innovation. Streams of Diracs and piecewise polynomials are the examples of such signals, and thus are known as signals with finite rate of innovation (FRI). We know that the classical ("band-limited sine") sampling theory does not enable perfect reconstruction of such signals from their samples since they are not band-limited. However, the recent results on FRI sampling suggest that it is possible to sample and perfectly reconstruct such nonband-limited signals using a rich class of kernels. In this paper, we extend those results in higher dimensions using compactly supported kernels that reproduce polynomials (satisfy Strang-Fix conditions). In fact, the polynomial reproduction property of the kernel makes it possible to obtain the continuous moments of the signal from its samples. Using these moments and the annihilating filter method (Prony's method), the innovative part of the signal, and therefore, the signal itself is perfectly reconstructed. In particular, we present local (directional-derivatives-based) and global (complex-moments-based, Radon-transform-based) sampling schemes for classes of FRI signals such as sets of Diracs, bilevel, and planar polygons, quadrature domains (e.g., circles, ellipses, and cardioids), 2D polynomials with polygonal boundaries, and n-dimensional Diracs and convex polytopes. This work has been explored in a promising way in super-resolution algorithms and distributed compression, and might find its applications in photogrammetry, computer graphics, and machine vision.

Visual tomography is a relatively new method for 3D scene reconstruction. It is adopted from medical tomography and based on multiple images from different viewpoints of a scene. In this context, multidimensional spectra and filtering techniques are the key technology for the reconstruction process. Visual tomography differs from classical tomography in several aspects which leads to new challenges with respect to mathematical description. The present paper examines the influence of image background on reconstruction quality. This background problem does not appear in classical medical tomography applications. In particular, the influence of multidimensional sampling and restrictions with respect to the number of view angles can be analyzed by using multidimensional signal theoretical concepts. The differences between ideal (no background) and real acquisition conditions are examined. Visual tomography has the potential for innovative new fields of applications, where Enabling Technologies for Societal Challenges are the focus of our considerations. Demographic change leads to a high interest for enabling mobility for elderly people with physical disabilities. Walking frames equipped with such technologies will be able to assist such people in real day environments.

Time shifted aliasing error upper bound extremals for the sampling reconstruction procedure are fully characterized. Sharp upper bounds are found on the aliasing error of truncated cardinal series and the corresponding extremals are described for entire functions from certain specific L p , p > 1 , classes. Analogous results are obtained in multidimensional regular sampling. Truncation error analysis is provided in all cases considered. Moreover, sharpness of bounding inequalities, convergence rates and various sufficient conditions are discussed.

This paper addresses the problem of multidimensional signal reconstruction from nonuniform or generalized samples. Typical solutions in the literature for this problem utilize continuous filtering. The key result of the current paper is a multidimensional "interpolation" identity, which establishes the equivalence of two multidimensional processing operations. One of these uses continuous domain filters, whereas the other uses discrete processing. This result has obvious benefits in the context of the afore mentioned problem. The results here expand and generalize earlier work by other authors on the one-dimensional (1-D) case. Potential applications include two-dimensional (2-D) images and video signals.

The paper establishes the commutativity conditions of upsampling and downsampling for multidimensional signals defined on discrete Abelian groups (lattices). The general condition includes results available in the literature as particular cases. Complete generality is achieved by giving an abstract definition of up/downsampling and by working in the signal domain instead of the traditional approach based on z and Fourier transforms. Examples of applications are outlined 1) for the one-dimensional (1-D) case verifying existing results, 2) for television scanning, 3) for degenerate (reduced-dimensionality) lattices, and 4) for multiplicative groups.

The truncation error associated with a given sampling representation is defined as the difference between the signal and an approximating sumutilizing a finite number of terms. In this paper we give uniform bound for truncation error of bandlimited functions in the n dimensional Lebesgue spaceL p (R n ) associated with multidimensional Shannon sampling representation.

We present sampling results for certain classes of two-dimensional (2-D) signals that are not bandlimited but have a parametric representation with a finite number of degrees of freedom. While there are many such parametric signals, it is often difficult to propose practical sampling schemes; therefore, we will concentrate on those classes for which we are able to give exact sampling algorithms and reconstruction formulas. We analyze in detail a set of 2-D Diracs and extend the results to more complex objects such as lines and polygons. Unlike most multidimensional sampling schemes, the methods we propose perfectly reconstruct such signals from a finite number of samples in the noiseless case. Some of the techniques we use are already encountered in the context of harmonic retrieval and error correction coding. In particular, singular value decomposition (SVD)-based methods and the annihilating filter approach are both explored as inherent parts of the developed algorithms. Potentials and limitations of the algorithms in the noisy case are also pointed out. Applications of our results can be found in astronomical signal processing, image processing, and in some classes of identification problems.

Let Bυ(ℝn), % % MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyXduNaey % ypa0Jaaiikaiabew8a1naaBaaaleaacaaIXaaabeaakiaacYcacaGG % UaGaaiOlaiaac6cacaGGSaGaeqyXdu3aaSbaaSqaaiaad6gaaeqaaO % GaaiykaiabgIGiolabl2riHoaaDaaaleaacqGHRaWkaeaacaWGUbaa % aaaa!4830! $$\upsilon = ({\upsilon _1},...,{\upsilon _n}) \in {\cal R}_ + ^n$$ , be the set of entire functions of exponential type υ bounded on ℝn and % % MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqamaaDa % aaleaacaWGWbWaaWbaaWqabeaacqaH4oqCaaaaleaacaWGYbaaaOWa % aeWaaeaacqWIDesOdaahaaWcbeqaaiaad6gaaaaakiaawIcacaGLPa % aacaGGSaGaaGPaVlaaigdacqGHKjYOcaWGWbGaeyizImQaeyOhIuQa % aiilaiaaigdacqGHKjYOcqaH4oqCcqGHKjYOcqGHEisPaaa!4FB4! $$B_{{p^\theta }}^r\left( {{{\cal R}^n}} \right),\,1 \leqslant p \leqslant \infty, 1 \leqslant \theta \leqslant \infty $$ , be the anisotropic Besov classes of functions. The uniform bounds of truncation error is estimated for f ∈ Bυ(ℝn) and satisfying the following decay condition: % % MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabeGaaa % qaamaaemaabaGaamOzaiaacIcacaWG4bGaaiykaaGaay5bSlaawIa7 % aiabgsMiJoaalaaabaGaamyqaaqaaiaacIcacaaIXaGaey4kaSYaaq % WaaeaacaWG4baacaGLhWUaayjcSdWaaWbaaSqabeaacqaH0oazaaGc % caGGPaaaaiaacYcaaeaacaWG4bGaeyypa0JaaiikaiaadIhadaWgaa % WcbaGaaGymaaqabaGccaGGSaGaaiOlaiaac6cacaGGUaGaaiilaiaa % dIhadaWgaaWcbaGaamOBaaqabaGccaGGPaGaeyicI4SaeSyhHe6aaW % baaSqabeaacaWGUbaaaOGaaiilaaaaaaa!5846! $${\left| {f(x)} \right| \leqslant \frac{A}{{(1 + {{\left| x \right|}^\delta })}},} \;\; {x = ({x_1},...,{x_n}) \in {{\cal R}^n},}$$ associated with the Shannon multidimensional sampling representation. The bounds of aliasing error for % % MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaayk % W7cqGHiiIZcaaMc8UaamOqamaaDaaaleaacqGHEisPdaahaaadbeqa % aiabeI7aXbaaaSqaaiaadkhaaaGcdaqadaqaaiabl2riHoaaCaaale % qabaGaamOBaaaaaOGaayjkaiaawMcaaaaa!44F3! $$f\, \in \,B_{{\infty ^\theta }}^r\left( {{{\cal R}^n}} \right)$$ and satisfying the same decay condition as above is also evaluated.

Image synthesis often requires the Monte Carlo estimation of integrals. Based on a generalized concept of stratification we present an efficient sampling scheme that consistently outperforms previous techniques. This is achieved by assembling sampling patterns that are stratified in the sense of jittered sampling and N-rooks sampling at the same time. The faster convergence and improved anti-aliasing are demonstrated by numerical experiments. Categories and Subject Descriptors (according to ACM CCS): G.3 [Probability and Statistics]: Probabilistic Algorithms (including Monte Carlo); I.3.2 [Computer Graphics]: Picture/Image Generation; I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism.

Missing data cases are a problem in all types of statistical analyses and arise in almost all application domains. Several schemes have been studied in this paper to overcome the drawbacks produced by missing values in data mining tasks, one of the most well known is based on pre processing, formerly known as imputation. In this work, we propose a new multiple imputation approach based on sampling techniques to handle missing values problems, in order to improving the quality and efficiency of data mining process. The proposed method is favourably compared with some imputation techniques and outperforms the existing approaches using an experimental benchmark on a large scale, waveform dataset taken from machine learning repository and different rate of missing values (till 95%).

International audience; Efficient sampling from high-dimensional distribu- tions is a challenging issue that is encountered in many large data recovery problems. In this context, sampling using Hamil- tonian dynamics is one of the recent techniques that have been proposed to exploit the target distribution geometry. Such schemes have clearly been shown to be efficient for multidimensional sam- pling but, rather, are adapted to distributions from the exponential family with smooth energy functions. In this paper, we address the problem of using Hamiltonian dynamics to sample from probabil- ity distributions having non-differentiable energy functions such as those based on the l1 norm. Such distributions are being used intensively in sparse signal and image recovery applications. The technique studied in this paper uses a modified leapfrog transform involving a proximal step. The resulting nonsmooth Hamiltonian Monte Carlo method is tested and validated on a number of exper- iments. Results show its ability to accurately sample according to various multivariate target distributions. The proposed technique is illustrated on synthetic examples and is applied to an image denoising problem.