The estimation of velocity vector fields from time-varying image sequences

Changes in successive images from a time-varying image sequence of a scene can be characterized by velocity vector fields. The estimate of the velocity vector field is determined as a compromise between optical flow and directional smoothness constraints. The optical flow constraints relate the values of the time-varying image function at the corresponding points of the successive images of the sequence. The directional smoothness constraints relate the values of neighboring velocity vectors. To achieve the compromise, we introduce a system of nonlinear equations of the unknown estimate of the velocity vector field using a novel variational principle applied to the weighted average of the optical flow and the directional smoothness constraints. A stable iterative method for solving this system is developed. The optical flow and the directional smoothness constraints are selectively suppressed in the neighborhoods of the occluding boundaries by implicitly adjusting their weights. These adjustments are based on the spatial variations of the estimates of the velocity vectors and the spatial variations of the time-varying image function. The system of nonlinear equations is defined in terms of the time-varying image function and its derivatives. The initial image functions are in general discontinuous and cannot be directly differentiated. These difficulties are overcome by treating the initial image functions as generalized functions and their derivatives as generalized derivatives. These generalized functions are evaluated (observed) on the parametric family of testing (smoothing) functions to obtain parametric families of secondary images, which are used in the system of nonlinear equations. The parameter specifies the degree of smoothness of each secondary image. The secondary images with progressively higher degrees of smoothness are sampled with progressively lower resolutions. Then coarse-to-fine control strategies are used to obtain the estimate.     

Complex random fields

The fundamental properties of complex random fields are derived directly in an n-dimensional setting and are not inferred as generalizations of the one-dimensional case. In particular, fields with orthogonal increments and stochastic integrals with respect to such fields are defined and their elementary properties analyzed. The spectral representation theorem for homogeneous fields is proved, and various second order properties resulting from the application of linear difference and differential operators to such fields are deduced. The specialization to isotropic fields is considered. Finally, white fields are defined and their characteristic property exploited.     

Accelerated isosurface extraction in time-varying fields

For large time-varying data sets, memory and disk limitations can lower the performance of visualization applications. Algorithms and data structures must be explicitly designed to handle these data sets in order to achieve more interactive rates. The Temporal Branch-on-Need Octree (T-BON) extends the three-dimensional branch-on-need octree for time-varying isosurface extraction. This data structure minimizes the impact of the I/O bottleneck by reading from disk only those portions of the search structure and data necessary to construct the current isosurface. By performing a minimum of I/O and exploiting the hierarchical memory found in modern CPUs, the T-BON algorithm achieves high performance isosurface extraction in time-varying fields. The paper extends earlier work on the T-BON data structure by including techniques for better memory utilization, out-of-core isosurface extraction, and support for nonrectilinear grids. Results from testing the T-BON algorithm on large data sets show that its performance is similar to that of the three-dimensional branch-on-need octree for static data sets while providing substantial advantages for time varying fields     

Hidden conditional random fields

We present a discriminative latent variable model for classification problems in structured domains where inputs can be represented by a graph of local observations. A hidden-state Conditional Random Field framework learns a set of latent variables conditioned on local features. Observations need not be independent and may overlap in space and time.     

Inducing features of random fields

We present a technique for constructing random fields from a set of training samples. The learning paradigm builds increasingly complex fields by allowing potential functions, or features, that are supported by increasingly large subgraphs. Each feature has a weight that is trained by minimizing the Kullback-Leibler divergence between the model and the empirical distribution of the training data. A greedy algorithm determines how features are incrementally added to the field and an iterative scaling algorithm is used to estimate the optimal values of the weights. The random field models and techniques introduced in this paper differ from those common to much of the computer vision literature in that the underlying random fields are non-Markovian and have a large number of parameters that must be estimated. Relations to other learning approaches, including decision trees, are given. As a demonstration of the method, we describe its application to the problem of automatic word classification in natural language processing     

The stochastic analysis of dynamic systems moving through random fields

In this paper we consider dynamic systems that move along specified trajectories across random fields, where the field acts as a driving force to the dynamic system. For a specific class of random fields we develop equations for the evolution of the covariance of the state of the dynamic system, and in the special case in which the trajectory is a straight line path followed by a 180° turn (i.e., an "over-and-back" trajectory) we develop a Markovian model that involves a change in the dimension of the state after the turn. For this case we also briefly discuss the estimation problem using recently developed results on "real-time smoothing."     

Finite-time stability of neutral-type neural networks with random time-varying delays

This paper is devoted to the finite-time stability analysis of neutral-type neural networks with random time-varying delays. The randomly time-varying delays are characterised by Bernoulli stochastic variable. This result can be extended to analysis and design for neutral-type neural networks with random time-varying delays. On the basis of this paper, we constructed suitable Lyapunov–Krasovskii functional together and established a set of sufficient linear matrix inequalities approach to guarantee the finite-time stability of the system concerned. By employing the Jensen's inequality, free-weighting matrix method and Wirtinger's double integral inequality, the proposed conditions are derived and two numerical examples are addressed for the effectiveness of the developed techniques.     

Generalized isotonic conditional random fields

Conditional random fields are one of the most popular structured prediction models. Nevertheless, the problem of incorporating domain knowledge into the model is poorly understood and remains an open issue. We explore a new approach for incorporating a particular form of domain knowledge through generalized isotonic constraints on the model parameters. The resulting approach has a clear probabilistic interpretation and efficient training procedures. We demonstrate the applicability of our framework with an experimental study on sentiment prediction and information extraction tasks.     

Tree approximations to Markov random fields

Methods for approximately computing the marginal probability mass functions and means of a Markov random field (MRF) by approximating the lattice by a tree are described. Applied to the a posteriori MRF these methods solve Bayesian spatial pattern classification and image restoration problems. The methods are described, several theoretical results concerning fixed-point problems are proven, and four numerical examples are presented, including comparison with optimal estimators and the iterated conditional mode estimator and including two agricultural optical remote sensing problems     

Mean fields and two-dimensional Markov random fields in image analysis

In this paper we compare two iterative approaches to the problem of pixel-level image restoration when the model contains unknown parameters. Pairwise interaction models are assumed to represent the local associations in the true scene. The first approach is a variation on the EM algorithm in which Mean-field approximations are used in the E-step and a variational approximation is used in the M-step. In the second approach, each iteration involves first restoring the image using the Iterated Conditional Modes (ICM) algorithm and then updating the parameter estimates by maximising the so-called pseudolikelihood. In addition, refinemenrs are made to the Mean-field approximation, and these are also used for restoration. The methods are compared empirically using both artificial and real noise-corrupted binary scenes. Within the comparisons the effects of using different convergence criteria for deciding when to stop the algorithms are also investigated.     

Control problem for diffusion-type random fields

Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 62–76, January–February, 1995.     

Semi-supervised clustering with discriminative random fields

Semi-supervised clustering exploits a small quantity of supervised information to improve the accuracy of data clustering. In this paper, a framework for semi-supervised clustering is proposed. This framework is capable of integrating with a traditional clustering algorithm seamlessly, and particularly useful for the application where a traditional clustering is designated to use.In the proposed framework, discriminative random fields (DRFs) are employed to model the consistency between the result of a traditional clustering algorithm and the supervised information with the assumption of semi-supervised learning. The semi-supervised clustering problem is thus formulated as finding the label configuration with the maximum a posteriori (MAP) probability of the DRF. A procedure based on the iterated conditional modes algorithm and a metric-learning algorithm is developed to find a suboptimal MAP solution of the DRF. The proposed approach has been tested against various data sets. Experimental results demonstrate that our approach can enhance the clustering accuracy, and thus prove the feasibility of the proposed approach.     

Conditional-mean least-squares fitting of Gaussian Markov random fields to Gaussian fields

This article discusses the following problem, often encountered when analyzing spatial lattice data. How can one construct a Gaussian Markov random field (GMRF), on a lattice, that reflects well the spatial-covariance properties present either in data or in prior knowledge? The Markov property on a spatial lattice implies spatial dependence expressed conditionally, which allows intuitively appealing site-by-site model building. There are also cases, such as in biological network analysis, where the Markov property has a deep scientific significance. Moreover, the model is often important for computational efficiency of Markov chain Monte Carlo algorithms. In this article, we introduce a new criterion to fit a GMRF to a given Gaussian field, where the Gaussian field is characterized by its spatial covariances. We establish that this criterion is computationally appealing, it can be used on both regular and irregular lattices, and both stationary and nonstationary fields can be fitted.     

High-quality and interactive animations of 3D time-varying vector fields

In this paper, we present an interactive texture-based method for visualizing three-dimensional unsteady vector fields. The visualization method uses a sparse and global representation of the flow, such that it does not suffer from the same perceptual issues as is the case for visualizing dense representations. The animation is made by injecting a collection of particles evenly distributed throughout the physical domain. These particles are then tracked along their path lines. At each time step, these particles are used as seed points to generate field lines using any vector field such as the velocity field or vorticity field. In this way, the animation shows the advection of particles while each frame in the animation shows the instantaneous vector field. In order to maintain a coherent particle density and to avoid clustering as time passes, we have developed a novel particle advection strategy which produces approximately evenly-spaced field lines at each time step. To improve rendering performance, we decouple the rendering stage from the preceding stages of the visualization method. This allows interactive exploration of multiple fields simultaneously, which sets the stage for a more complete analysis of the flow field. The final display is rendered using texture-based direct volume rendering     

Orientation of anisotropic random fields and images

Random field models are often used for characterizing two-dimensional data, i.e. images. Commonly, they exhibit some orientational variability, but, unlike one-dimensional random processes, are lacking in any obvious causality direction. This paper describes how their orientation may be recognized.     

Linear sequential coding of random space-time fields

Optimal (least-mean-square-error) linear preprocessing of observations of a random space-time signal preparatory to transmission over a sampled-data channel may be partitioned into a conventional “state-variable” filter operating on all current and past observations available to the primary transducer, and a linear coder which selects the combination of state estimates to be transmitted at each sampling instant. The post-sampling reconstruction filter is of conventional design, as if its objective were to reproduce not the original field but its prefiltered estimate. Both prefilter and postfilter incorporate dynamical models of the underlying physical process, with appropriate feedback in the two situations.Design of the coder requires further elaboration of the criterion of optimality: in general, a weighted space-time averaged mean-square error of the reconstructed signal. For the special case of pseudo-observations on a periodic spatial lattice, in which the error structure of the prefiltered field is homogeneous, the optimal coder is ideally wave-number-limited to a single spectral cell. The location and shape of this cell, and the relative weightings applied to the components of the prefiltered field over the cell, may vary from one sampling instant to the next.Examples are presented for (1) a one-dimensional “Rossby wave” field; (2) a two-dimensional diffusion process, sampled on a 60° rhombic lattice; and (3) a second-order wave field in one spatial dimension.     

Moment inequalities for random variables in computational geometry

LetX ₁,...,X _n be independent identically distributedR ^d -valued random vectors, and letA _n =A(X ₁,...,X _n ) be a subset of {X ₁,...,X _n }, invariant under permutations of the data, and possessing the inclusion property (X ₁ ∈A _n impliesX ₁ ∈A _i for alli≤n). For example, the convex hull, the collection of all maximal vectors, the set of isolated points and other structures satisfy these conditions. LetN _n be the cardinality ofA _n . We show that for allp≥1, there exists a universal constantC _p >0 such thatE(N _n ^p )≤C _p max (1,E ^p ) where . This complements Jensen's lower bound for thep-th moment:E(N _n ^p )≥E ^p (N _n ). The inequality is applied to the expected time analysis of algorithms in computational geometry. We also give necessary and sufficient conditions onE(N _n ) for linear expected time behaviour of divide-and-conquer methods for findingA _n .