Multiscale Likelihood Research Articles

Synthetic aperture radar image segmentation using non‐linear diffusion‐based hierarchical triplet Markov fields model

Triplet Markov fields (TMF) model is widely used to deal with non-stationary synthetic aperture radar (SAR) images. However, its ability to capture global information remains limited due to the non-causal property. A hierarchical TMF model is proposed in this study based on the non-linear diffusion (ND) strategy, which is denoted as ND-hierarchical TMF (HTMF). ND is adopted to generate multiscale decomposition according to local image content, and that is superior to traditional wavelet decomposition in reflecting hierarchical nature of image structure and detailed features. The auxiliary field in ND-HTMF is redefined and initialised on the finest scale to characterise edge information and that enhances the prior modelling ability for non-stationary local image features. The multiscale likelihood and multiscale causal prior energy functions are then defined respectively in bottom-up and top-down procedures to capture local and global information for performing segmentation. Segmentation experiments on simulated and real SAR images demonstrate the effectiveness of ND-HTMF in both edge characterisation accuracy and robustness against speckle noise.

Unsupervised SAR Image Segmentation Based on a Hierarchical TMF Model in the Discrete Wavelet Domain for Sea Area Detection

Unsupervised synthetic aperture radar (SAR) image segmentation is a fundamental preliminary processing step required for sea area detection in military applications. The purpose of this step is to classify large image areas into different segments to assist with identification of the sea area and the ship target within the image. The recently proposed triplet Markov field (TMF) model has been successfully used for segmentation of nonstationary SAR images. This letter presents a hierarchical TMF model in the discrete wavelet domain of unsupervised SAR image segmentation for sea area detection, which we have named the wavelet hierarchical TMF (WHTMF) model. The WHTMF model can precisely capture the global and local image characteristics in the two-pass computation of posterior distribution. The multiscale likelihood and the multiscale energy function are constructed to capture the intrascale and intrascale dependencies in a random field (X,U). To model the SAR data related to radar backscattering sources, the Gaussian distribution is utilized. The effectiveness of the proposed model for SAR image segmentation is evaluated using synthesized and real SAR data.

SAR Image Multiclass Segmentation Using a Multiscale TMF Model in Wavelet Domain

The triplet Markov field (TMF) model recently proposed is suitable for dealing with nonstationary synthetic aperture radar (SAR) image segmentation. In this letter, we propose a multiscale TMF model in wavelet domain, named as the wavelet-domain TMF (WTMF) model. In the WTMF model, a multiscale causal WTMF energy function is constructed to capture the intra- and interscale dependences in random fields <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$(X, U)$</tex> </formula> . Moreover, multiscale likelihoods of the WTMF model are derived based on a wavelet hidden Markov tree to capture the statistical properties of wavelet coefficients. The proposed model can integrate the global and local information in terms of spatial configuration and image features in a more complete manner. The coarser scale information is utilized to guide the finer scale segmentation, and the coarse-to-fine causal interactions are considered using a Markov chain. Experimental results prove that the proposed model can segment SAR images better than several models previously proposed.

Multiscale Poisson Data Smoothing

SummaryThe paper introduces a framework for non-linear multiscale decompositions of Poisson data that have piecewise smooth intensity curves. The key concept is conditioning on the sum of the observations that are involved in the computation of a given multiscale coefficient. Within this framework, most classical wavelet thresholding schemes for data with additive homoscedastic noise can be used. Any family of wavelet transforms (orthogonal, biorthogonal or second generation) can be incorporated in this framework. Our second contribution is to propose a Bayesian shrinkage approach with an original prior for coefficients of this decomposition. As such, the method combines the advantages of the Haar–Fisz transform with wavelet smoothing and (Bayesian) multiscale likelihood models, with additional benefits, such as extendability towards arbitrary wavelet families. Simulations show an important reduction in average squared error of the output, compared with the present techniques of Anscombe or Fisz variance stabilization or multiscale likelihood modelling.

Multiscale likelihood analysis and complexity penalized estimation

We describe here a framework for a certain class of multiscale likelihood factorizations wherein, in analogy to a wavelet decomposition of an L2 function, a given likelihood function has an alternative representation as a product of conditional densities reflecting information in both the data and the parameter vector localized in position and scale. The framework is developed as a set of sufficient conditions for the existence of such factorizations, formulated in analogy to those underlying a standard multiresolution analysis for wavelets, and hence can be viewed as a multiresolution analysis for likelihoods. We then consider the use of these factorizations in the task of nonparametric, complexity penalized likelihood estimation. We study the risk properties of certain thresholding and partitioning estimators, and demonstrate their adaptivity and near-optimality, in a minimax sense over a broad range of function spaces, based on squared Hellinger distance as a loss function. In particular, our results provide an illustration of how properties of classical wavelet-based estimators can be obtained in a single, unified framework that includes models for continuous, count and categorical data types.

Multiscale Statistical Models for Hierarchical Spatial Aggregation

Scale dependency is an inherent property of geographic phenomena since most geographic patterns under observation vary with scale. Across numerous disciplines, including geography, various types of so‐called “multiscale” models have been used for the task of modeling and understanding the effects of scale. However, most of these models are descriptive—as opposed to inferential—in nature, and few of them (particularly outside geography) are well adapted to the wide variety of data structures typically encountered in geography. In this paper, we introduce a new, general framework for multiscale statistical modeling and inference that is explicitly designed for a broad class of geographic data. The key structural assumption underlying these models is that of a set of hierarchically defined partitions, corresponding to successive aggregations of an initial data space. Within our framework the effects of scale associated with such aggregation are captured through a fundamental decomposition of the data likelihood, directly induced by the hierarchical nature of the partitions, into individual components of local information at all possible spatial resolutions. Upon combining these multiscale likelihoods with an appropriately defined Bayesian prior probability structure, a powerful inferential framework results. We describe in detail how this framework may be used for the tasks of statistical estimation and classification, and illustrate its usage with an analysis of data from census geography.