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In biomedical research with longitudinal designs, missing values due to intermittent nonresponse or premature withdrawal are usually ’nonignorable’ in the sense that un- observed values are related to the patterns of missingness. When missing values are simply ignored, analyses based on observed-data likelihood may yield biased estimates or invalid inferences. By drawing the framework of a shared-parameter mechanism, the process yielding the repeated count measures and the process yielding missing val- ues can be modelled separately, conditionally on a group of shared parameters. For chronic diseases, Markov transition models can be used to study the transitional fea- tures of the pathologic processes. In this paper, Markov chain Monte Carlo (MCMC) algorithms are developed to fit a random-effects Markov transition model (REMTM) for incomplete count repeated measures, within which random effects are shared by the counting process and the missing-data mechanism. Assuming a Poisson distribu- tion for the count measures, the transition probabilities are estimated using a Poisson linear regression model. The missingness mechanism is modeled with a multinomial- logit regression to calculate the transition probabilities of the missingness indicators. The method is demonstrated using both simulated data sets and a practical data set from a smoking cessation clinical trial

ChIP-chip (or ChIP-on-chip) is a technology for isolation and identification of genomic sites occupied by specific DNA binding proteins in living cells. The ChIP-chip data can be obtained over the whole genome by tiling arrays, where a peak in the signal is generally observed at a protein binding site. In this paper, we describe and present a probability theory for modelling ChIP-chip data. We then propose a model-based computa- tional method for locating and testing peaks for the purpose of identifying potential protein binding sites.

Following Marr’s insight, we propose a generative image representation called primal sketch, which integrates two modeling components. The first component explains the structural part of an image, such as ob ject boundaries, by a hidden layer of image primitives. The second component models the remaining textural part without distinguishable elements by Markov random fields that interpolate the structural part of the image. We adopt an artist’s notion by calling the two components “sketchable” and “non-sketchable” parts respectively. A dictionary of image primitives are used for modeling structures in natural images, and each primitive is specified by variables for its photometric, geometric, and topological attributes. The primitives in the image representation are not independent but organized in an sketch graph. This sketch graph is modeled by a spatial Markov model that enforces Gestalt organizations. The inference of the sketch graph consists of two phases. Phase I sequentially adds the most prominent image primitives in a procedure similar to matching pursuit. Phase II edits the sketch graph by a number of graph operators to achieve good Gestalt organizations. Experiments show that the primal sketch model produces satisfactory results for a large number of generic images. The primal sketch model is not only a parsimonious image representation for lossy image coding, but also provides a meaningful mid-level generic representation for other vision tasks.

Vision can be posed as a statistical learning and inference problem. As an over-simplified account, let W be a description of the outside scene in terms of ìwhat is where,î let I be the retina image, and let p(W, I) be the joint distribution of W and I. 1 Then visual learning is to learn p(W, I) from training data, and visual perception is to infer W from I based on p(W|I).There are two major schools on visual learning and perception. One school is operation-oriented and learns the inferential process defined by p(W|I) directly, often in the form of an explicit transformation W ? F(I). This scheme is mostly used in supervised learning, where W is object category, and is given in training data. The other school is representation-oriented and learns the generative process p(W) and p(I|W) explicitly, then perception is to invert the generative process by maximizing or sampling p(W|I) ? p(W)p(I|W). In this scheme, p(W) may also be accounted for by a regularization term such as smoothness or sparsity. This scheme is often used in unsupervised learning where W is not available in training data.

Computer vision can be considered a highly specialized data collection and data analysis problem. We need to understand the special properties of image data in order to construct statistical models for representing the wide variety of image patterns. One special property of vision that distinguishes itself from other sensory data such as speech data is that distance or scale plays a profound role in image data. More specifically, visual objects and patterns can appear at a wide range of distances or scales, and the same visual pattern appearing at different distances or scales produces different image data with different statistical properties, thus entails different regimes of statistical models. In particular, we show that the entropy rate of the image data changes over the viewing distance (as well as the camera resolution). Moreover, the inferential uncertainty changes with viewing distance too. We call these changes information scaling. From this perspective, we examine both empirically and theoretically two prominent and yet largely isolated research themes in image modeling literature, namely, wavelet sparse coding and Markov random fields. Our results indicate that the two models are appropriate on two different entropy regimes: sparse coding targets the low entropy regime, whereas the random fields are suitable for the high entropy regime. Because of information scaling, both models are necessary for representing and interpreting image intensity patterns in the whole entropy range, and information scaling triggers transitions between these two regimes of models. This motivates us to propose a full-zoom primal sketch model that integrates both sparse coding and Markov random fields. In this model, local image intensity patterns are classified into “sketchable regime” and “non-sketchable regime” by a sketchability criterion. In the sketchable regime, the image data are represented deterministically by highly parametrized sketch primitives. In the non-sketchable regime, the image data are characterized by Markov random fields whose sufficient statistics summarize computational results from failed attempts of sparse coding. The contribution of our work is two folded. First, information scaling provides a dimension to chart the space of natural images. Second, the full-zoom modeling scheme provides a natural integration of sparse coding and Markov random fields, thus enables us to develop a new and richer class of statistical models.

Textons refer to fundamental micro-structures in generic natural images and thus constitute the basic elements in early (pre-attentive) visual perception. However, the word "textong" remains a vague concept in the literature of computer vision and visual perception, and a precist mathematical definition has yet to be found. In this article, we argue that the definition of texton should be governed by a sound mathematical model of images, and the set of textons must be learned from, or best tuned to, an image ensemble. We adopt a generative image model that an image is a superposition of bases from an over-complete dictionary, then a texton is defined as a mini-template that consists of a varying number of image bases with some geometric and photometric configurations. By analogy to physics, if image bases are like protons, neutrons and electrons, then textons are like atoms. Then a small number of textons can be learned from training images as repeating micro-structures. We report four experiments for comparison. The first experiment computes clusters in feature space of filter responses. The second transformed component analysis in both feature space and image patches. The third adopts a two-layer generative model where an image is generated by image bases and image bases are generated by textons. The fourth experiment shows textons from motion image sequences, which we call movetons.

Recent results on sparse coding and independent component analysis suggest that human vision first represents a visual image by a linear superposition of a relatively small number of localized elongate, oriented image bases. With this representation, the sketch of an image consists of the locations, orientations, and elongations of the images bases, and the sketch can be visually illustrated by depicting each image base by a linelet of the same length and orientation. Built on the insight of sparse and independent component analysis, we propose a two-level generative model for textures. At the bottom-level, the texture image is represented by a linear superposition of image bases. At the top-level, a Markov model is assumed for the placement of the image bases or the sketch, and the model is characterized by a set of simple geometrical feature statistics.

This paper presents a class of statistical models that integrate two statistical modeling paradigms in the literature: I) Descriptive methods, such as Markov random fields and minimax entropy learning [41] and II) Generative methods, such as principal component analysis, independent component analysis [2] transformed component analysis [11], wavelet coding [27, 5], and sparse coding [30, 24]. In this paper, we demonstrate the integrated framwork by constructing a class of hierarchical models for texton patterns ) the term "texton" was coined by psychologist Julez in the early 80's). At the bottom level of the model, we assume that an observed texture image is generated by multiple hidden "texton maps", and textons on each map are translated, scaled, stretched, and oriented versions of a window function, like mini-templates or wavelet bases. The texton maps generate the observed image by occlusion or linear superposition. this bottom level of the model is generative in nature. At the top level of the model, the spatial arrangements of the textons in the texton maps are characterized by minimax entropy principle, which leads to embellished versions of Gibbs point rocess [34]. The top level of the model is descriptive in nature. We demonstrate the integrated model by a set of experiments.

Dynamic textures are sequences of images of moving scenes that exhibit certain stationarity properties in time; these include sea-waves, smoke, foliage, whirlwind etc. We present a novel characterization of dynamic textures that poses the problems of modeling, learning, recognizing and synthesizing dynamic textures on a firm analytical footing. We borrow tools from system identification to capture the “essence” of dynamic textures; we do so by learning (i.e. identifying) models that are optimal in the sense of maximum likelihood or minimum prediction error variance. For the special case of second-order stationary processes, we identify the model sub-optimally in closed-form. Once learned, a model has predictive power and can be used for extrapolating synthetic sequences to infinite length with negligible computa- tional cost. We present experimental evidence that, within our framework, even low-dimensional models can capture very complex visual phenomena.

Independent component analysis (ICA) is an important method for blind source separation and unsupervised learning. Recently, the method has been extended to the overcomplete situation where the number of sources is greater than the number of receivers. Comparing complete ICA and overcomplete ICA in existing literature, one can notice that complete ICA does not assume noise in observations, and the sources can be ex- plicitly solved from the receivers, whereas the overcomplete ICA in gen- eral assumes noise in observations and the sources are implicitly solved by gradient type algorithms. In this paper, we present an explicit null- space representation for overcomplete ICA in the noiseless situation based on singular value decomposition (SVD), and develop an algorithm for estimating mixing matrix and recovering the sources. The null-space representation makes the connection between complete ICA and over- complete ICA more apparent, and leads to a mathematical explanation of lateral inhibition in the context of overcomplete linear model. It also appears to work well on the experimental examples. Moreover, it can be extended to the situation where there is noise in observations, and may lead to more efficient algorithms in this situation.

We have learned a lot from studying the sequence of artful works of the two authors on EM/data augmentation. In this note, we will discuss some of our thoughts (or rather speculations) on the problem of vision from the perspective of missing data modeling and data augmentation.

This paper presents a mathematical framework for visual learning that integrates two popular statistical learning paradigms in the literature: I) Descriptive learning, such as Markov random fields and minimax entropy learning, and II) Generative learning, such as PCA, ICA, TCA, and HMM. We apply the integrated learning framework to texture modeling, and we assume that an observed texture image is generated by multiple layers of hidden stochastic processes with various texture elements called "textons". Each texton is expressed as a window function like a mini-template or a wavelet, and each hidden stochastic process is a spatial pattern with a number of textons subject to affine transfomrations. The hidden layers are characterized by minimax entropy models, and they generate images by occlusion or linear addition. Thus given a raw input image, the learning framework achieves four goals: i) Computing the appearance of textons. ii) Inferring the hidden stochastic processes iii) Learning Gibbs models for each hidden stochastic process and iv) Verifying the learnt textons and models through random sampling. The integrated framework subsumes the minimax entropy learning paradigm and creates a richer class of probability models for visual patterns. Furthermore we show that the integration of descriptive and generative methods is a natural path of visual learning. We demonstrate the proposed framework and algorithms on many real images.