Texture characterization is a central element in many image processing applications. Multifractal analysis is a useful signal and image processing tool, yet, the accurate estimation of multifractal parameters for image texture remains a challenge. This is due in the main to the fact that current estimation procedures consist of performing linear regressions across frequency scales of the 2D dyadic wavelet transform, for which only a few such scales are computable for images. The strongly non-Gaussian nature of multifractal processes, combined with their complicated dependence structure, makes it difficult to develop suitable models for parameter estimation. Here, we propose a Bayesian procedure that addresses the difficulties in the estimation of the multifractality parameter. The originality of the procedure is threefold. The construction of a generic semiparametric statistical model for the logarithm of wavelet leaders; the formulation of Bayesian estimators that are associated with this model and the set of parameter values admitted by multifractal theory; the exploitation of a suitable Whittle approximation within the Bayesian model which enables the otherwise infeasible evaluation of the posterior distribution associated with the model. Performance is assessed numerically for several 2D multifractal processes, for several image sizes and a large range of process parameters. The procedure yields significant benefits over current benchmark estimators in terms of estimation performance and ability to discriminate between the two most commonly used classes of multifractal process models. The gains in performance are particularly pronounced for small image sizes, notably enabling for the first time the analysis of image patches as small as @math pixels.
As far as MMC processes are concerned, @cite_8 proposes an ML approach in the time domain for one specific process. However, the method relies strongly on the particular construction of this process and cannot easily accommodate more general model classes. Moreover, the method is formulated for 1D signals only. Finally, a Bayesian estimation procedure for the parameter @math of multifractal time series has recently been proposed in @cite_3 . Unlike the methods mentioned above, it does not rely on specific assumptions but instead employs a heuristic semi-parametric model for the statistics of the logarithm of wavelet leaders associated with univariate MMC processes. Yet, it is designed for and can only be applied to univariate time series of small sample size.
@cite_3 @cite_8
2183406623 2151246884 2141789091 1989307896
We introduce the Gaussian Process Convolution Model (GPCM), a two-stage non-parametric generative procedure to model stationary signals as the convolution between a continuous-time white-noise process and a continuous-time linear filter drawn from Gaussian process. The GPCM is a continuous-time nonparametric-window moving average process and, conditionally, is itself a Gaussian process with a nonparametric kernel defined in a probabilistic fashion. The generative model can be equivalently considered in the frequency domain, where the power spectral density of the signal is specified using a Gaussian process. One of the main contributions of the paper is to develop a novel variational free-energy approach based on inter-domain inducing variables that efficiently learns the continuous-time linear filter and infers the driving white-noise process. In turn, this scheme provides closed-form probabilistic estimates of the covariance kernel and the noise-free signal both in denoising and prediction scenarios. Additionally, the variational inference procedure provides closed-form expressions for the approximate posterior of the spectral density given the observed data, leading to new Bayesian nonparametric approaches to spectrum estimation. The proposed GPCM is validated using synthetic and real-world signals. This paper addresses the problem of source localization and waveform estimation in array processing applications. We present two nonparametric user parameter free algorithms, namely the iterative adaptive approach (IAA) and the maximum likelihood based IAA (IAA-ML). Both IAA and IAA-ML can work with arbitrary array geometries and uncorrelated as well as coherent sources. We extend IAA and IAA-ML to yield sparse results by using the Bayesian information criterion (BIC). Further improvements in performance can be achieved by employing the last step of the parametric RELAX algorithm initialized by the estimates of IAA and IAA-ML with BIC. Our simulations demonstrate that IAA and IAA-ML show better performance than the most competitive sparse signal representation approaches in the literature: M-FOCUSS, M-SBL and lscr1-SVD. We consider the problem of autonomously estimating position and orientation of an object from tactile data. When initial uncertainty is high, estimation of all six parameters precisely is computationally expensive. We propose an efficient Bayesian approach that is able to estimate all six parameters in both unimodal and multimodal scenarios. The approach is termed scaling series sampling as it estimates the solution region by samples. It performs the search using a series of successive refinements, gradually scaling the precision from low to high. Our approach can be applied to a wide range of manipulation tasks. We demonstrate its portability on two applications: (1) manipulating a box and (2) grasping a door handle In this paper we propose a general framework for the uncertainty quantification of quantities of interest for high-contrast single-phase flow problems. It is based on the generalized multiscale finite element method (GMsFEM) and multilevel Monte Carlo (MLMC) methods. The former provides a hierarchy of approximations of different resolution, whereas the latter gives an efficient way to estimate quantities of interest using samples on different levels. The number of basis functions in the online GMsFEM stage can be varied to determine the solution resolution and the computational cost, and to efficiently generate samples at different levels. In particular, it is cheap to generate samples on coarse grids but with low resolution, and it is expensive to generate samples on fine grids with high accuracy. By suitably choosing the number of samples at different levels, one can leverage the expensive computation in larger fine-grid spaces toward smaller coarse-grid spaces, while retaining the accuracy of the final Monte Carlo estimate. Further, we describe a multilevel Markov chain Monte Carlo method, which sequentially screens the proposal with different levels of approximations and reduces the number of evaluations required on fine grids, while combining the samples at different levels to arrive at an accurate estimate. The framework seamlessly integrates the multiscale features of the GMsFEM with the multilevel feature of the MLMC methods following the work in [26], and our numerical experiments illustrate its efficiency and accuracy in comparison with standard Monte Carlo estimates.