We introduce a novel parser based on a probabilistic version of a left-corner parser. The left-corner strategy is attractive because rule probabilities can be conditioned on both top-down goals and bottom-up derivations. We develop the underlying theory and explain how a grammar can be induced from analyzed data. We show that the left-corner approach provides an advantage over simple top-down probabilistic context-free grammars in parsing the Wall Street Journal using a grammar induced from the Penn Treebank. We also conclude that the Penn Treebank provides a fairly weak testbed due to the flatness of its bracketings and to the obvious overgeneration and undergeneration of its induced grammar.

We present a novel probabilistic model for the hierarchical structure of an image and its regions. We call this model spatial random tree grammars (SRTGs). We develop algorithms for the exact computation of likelihood and maximum a posteriori (MAP) estimates and the exact expectation-maximization (EM) updates for model-parameter estimation. We collectively call these algorithms the center-surround algorithm. We use the center-surround algorithm to automatically estimate the maximum likelihood (ML) parameters of SRTGs and classify images based on their likelihood and based on the MAP estimate of the associated hierarchical structure. We apply our method to the task of classifying natural images and demonstrate that the addition of hierarchical structure significantly improves upon the performance of a baseline model that lacks such structure.

A new class of multiscale multidimensional stochastic processes called spatial random trees was recently introduced in [9]. The model is based on multiscale stochastic trees with stochastic structure as well as stochastic states. In this work, we describe a method for estimating the parameters of the process.

Abstract—We develop a new class of hierarchical stochastic image models called spatial random trees (SRTs) which admit polynomial-complexity exact inference algorithms. Our framework of multitree dictionaries is the starting point for this construction. SRTs are stochastic hidden tree models whose leaves are associated with image data. The states at the tree nodes are random variables, and, in addition, the structure of the tree is random and is generated by a probabilistic grammar. We describe an efficient recursive algorithm for obtaining the maximum a posteriori estimate of both the tree structure and the tree states given an image. We also develop an efficient procedure for performing one iteration of the expectation-maximization algorithm and use it to estimate the model parameters from a set of training images. We address other inference problems arising in applications such as maximization of posterior marginals and hypothesis testing. Our models and algorithms are illustrated through several image classification and segmentation experiments, ranging from the segmentation of synthetic images to the classification of natural photographs and the segmentation of scanned documents. In each case, we show that our method substantially improves accuracy over a variety of existing methods. Index Terms—Dictionary, estimation, grammar, hierarchical model, image classification, probabilistic context-free grammar, segmentation, statistical image model, stochastic context-free grammar, tree model. I.

A new class of multiscale multidimensional stochastic processes called spatial random trees is introduced. The model is based on multiscale stochastic trees with stochastic structure as well as stochastic states. Procedures are developed for exact likelihood calculation, MAP estimation of the process, and estimation of the parameters of the process. The new framework is illustrated through a simple binary image classification problem.

A new class of multiscale stochastic processes called spatial random trees (SRTs) is introduced and studied. As with previous multiscale stochastic processes, SRTs model multidimensional signals using random processes on trees. Our key innovation, however, is that the tree structure itself is random and is generated by a probabilistic context-free grammar (PCFG) [26]. While PCFGs have been used to model 1-D signals, the generalization to multiple dimensions is not direct because the leaves of a tree generated by a PCFG cannot be naturally mapped to a multidimensional lattice. We solve this problem by defining a new class of PCFGs which can produce trees whose leaves are naturally arranged in a multidimensional lattice. We call such trees admissible and show that each of them generates a unique multidimensional signal. Based on this framework, procedures are developed for likelihood calculation, MAP estimation of the processes, and parameter estimation. The new framework is illustrated through simple detection problems.

We investigate the behaviour of a class of multitype Galton-Watson chains modelling the development of a genetic population. An algorithm for the computer generation of their trajectories is described and details concerning its implementation are given. We analyze through a simulation study the dependence of the extinction time of several populations on the initial gene frequencies and on the individual relative fertilities.

We present a method for imposing hierarchal structure on segmented images using probabilistic context-free grammars (PCFGs). The notion of PCFG, which has been used in the past to characterize 1D word strings, is extended to characterize 2D images as well. The inside-outside algorithm is then extended to support training, classification, and parsing on images with these extended PCFGs. The soundness and efficiency of these extensions rely on a novel notion of constituency that constrains the allowable ways to partition a parent segment into child subsegments during parsing. We successfully apply our method to the task of classifying natural images and also show that our method can learn the common hierarchal structure in such images, in an unsupervised fashion, from unlabeled training data.

Abstract—We present a novel probabilistic model for the hierarchical structure of an image and its regions. We call this model spatial random tree grammars (SRTGs). We develop algorithms for the exact computation of likelihood and maximum a posteriori (MAP) estimates and the exact expectation-maximization (EM) updates for model-parameter estimation. We collectively call these algorithms the center-surround algorithm. We use the center-surround algorithm to automatically estimate the maximum likelihood (ML) parameters of SRTGs and classify images based on their likelihood and based on the MAP estimate of the associated hierarchical structure. We apply our method to the task of classifying natural images and demonstrate that the addition of hierarchical structure significantly improves upon the performance of a baseline model that lacks such structure. Index Terms—Bayesian methods for image understanding, multiscale analysis. Ç