We present a new iterative approach to computing a median of a set of vectors. Like the standard approach, the proposed method minimizes the sum of the distances from the data points to the median. However, unlike the standard approach, the resulting median vector is not restricted to be a member of the data set. The proposed approach has several advantages over the standard definition, such as fast convergence, computational complexity that scales linearly with the number of exemplars, and a result that is closer to the center in an intuitive sense. As an example, we apply the method to color image filtering.

Probabilistic models of image statistics underlie many approaches in image analysis and processing. An important class of such models have variables whose dependency graph is a tree. If the hidden variables take values on a finite set, most computations with the model can be performed exactly, including the likelihood calculation, training with the EM algorithm, etc. Crouse et al. developed one such model, the hidden Markov tree (HMT). They took particular care to limit the complexity of their model. We argue that it is beneficial to allow more complex tree-structured models, describe the use of information theoretic penalties to choose the model complexity, and present experimental results to support these proposals. For these experiments, we use what we call the hierarchical image probability (HIP) model. The differences between the HIP and the HMT models include the use of multivariate Gaussians to model the distributions of local vectors of wavelet coefficients and the use of different numbers of hidden states at each resolution. We demonstrate the broad utility of image distributions by applying the HIP model to classification, synthesis, and compression, across a variety of image types, namely, electrooptical, synthetic aperture radar, and mammograms (digitized X-rays). In all cases, we compare with the HMT.

This paper describes a pattern recognition architecture, which we term hierarchical pyramid/neural network (HPNN), that learns to exploit image structure at multiple resolutions for detecting clinically significant features in digital/digitized mammograms. The HPNN architecture consists of a hierarchy of neural networks, each network receiving feature inputs at a given scale as well as features constructed by networks lower in the hierarchy. Networks are trained using a novel error function for the supervised learning of image search/detection tasks when the position of the objects to be found is uncertain or ill defined. We have evaluated the HPNN's ability to eliminate false positive (FP) regions of interest generated by the University of Chicago's (UofC) Computer-aided diagnosis (CAD) systems for microcalcification and mass detection. Results show that the HPNN architecture, trained using the uncertain object position (UOP) error function, reduces the FP rate of a mammographic CAD system by approximately 50% without significant loss in sensitivity. Investigation into the types of FPs that the HPNN eliminates suggests that the pattern recognizer is automatically learning and exploiting contextual information. Clinical utility is demonstrated through the evaluation of an integrated system in a clinical reader study. We conclude that the HPNN architecture learns contextual relationships between features at multiple scales and integrates these features for detecting microcalcifications and breast masses.

We formulate a model for probability distributions on image spaces. We show that any distribution of images can be factored exactly into conditional distributions of feature vectors at one resolution (pyramid level) conditioned on the image information at lower resolutions. We would like to factor this over positions in the pyramid levels to make it tractable, but such factoring may miss long-range dependencies. To capture long-range dependencies, we introduce hidden class labels at each pixel in the pyramid. The result is a hierarchical mixture of conditional probabilities, similar to a hidden Markov model on a tree. The model parameters can be found with maximum likelihood estimation using the EM algorithm. We have obtained encouraging preliminary results on the problems of detecting various objects in SAR images and target recognition in optical aerial images.

We present evidence that several higher-order statistical proper(cid:173) ties of natural images and signals can be explained by a stochastic model which simply varies scale of an otherwise stationary Gaus(cid:173) sian process. We discuss two interesting consequences. The first is that a variety of natural signals can be related through a com(cid:173) mon model of spherically invariant random processes, which have the attractive property that the joint densities can be constructed from the one dimensional marginal. The second is that in some cas(cid:173) es the non-stationarity assumption and only second order methods can be explicitly exploited to find a linear basis that is equivalent to independent components obtained with higher-order methods. This is demonstrated on spectro-temporal components of speech.

Acoustic signals recorded simultaneously in a reverberant environment can be described as sums of differently convolved sources. The task of source separation is to identify the multiple channels and possibly to invert those in order to obtain estimates of the underlying sources. We tackle the problem by explicitly exploiting the nonstationarity of the acoustic sources. Changing cross correlations at multiple times give a sufficient set of constraints for the unknown channels. A least squares optimization allows us to estimate a forward model, identifying thus the multipath channel. In the same manner we can find an FIR backward model, which generates well separated model sources. Furthermore, for more than three channels we have sufficient conditions to estimate underlying additive sensor noise powers. We show the good performance in a real room environments and demonstrate the algorithm's utility for automatic speech recognition.

An important problem in image analysis is finding small objects in large images. The problem is challenging because: 1) searching a large image is computationally expensive; and 2) small targets (on the order of a few pixels in size) have relatively few distinctive features which enable them to be distinguished from non-targets. To overcome these challenges the authors have developed a hierarchical neural network architecture which combines multiresolution pyramid processing with neural networks. Here the authors discuss the application of their hierarchical neural network architecture to the problem of detecting microcalcifications in digital mammograms. Microcalcifications are cues for breast tumors. 30% to 50% of breast carcinomas have microcalcifications visible in mammograms while 60% to 80% of all breast tumors eventually show microcalcifications via histology. Similar to the building/ATR problem, microcalcifications are generally very small point-like objects (<10 pixels in mammograms) which are hard to detect. Radiologists must often exploit other information in the imagery (e.g. location of blood vessels, ducts, etc.) in order to detect these microcalcifications. Here the authors examine how well their hierarchical neural network architecture learns and exploits contextual information in mammograms.

The efficiency of image search can be greatly improved by using a coarse-to-fine search strategy with a multi-resolution image representa(cid:173) tion. However, if the resolution is so low that the objects have few dis(cid:173) tinguishing features, search becomes difficult. We show that the performance of search at such low resolutions can be improved by using context information, i.e., objects visible at low-resolution which are not the objects of interest but are associated with them. The networks can be given explicit context information as inputs, or they can learn to detect the context objects, in which case the user does not have to be aware of their existence. We also use Integrated Feature Pyramids, which repre(cid:173) sent high-frequency information at low resolutions. The use of multi(cid:173) resolution search techniques allows us to combine information about the appearance of the objects on many scales in an efficient way. A natural fOlm of exemplar selection also arises from these techniques. We illus(cid:173) trate these ideas by training hierarchical systems of neural networks to find clusters of buildings in aerial photographs of farmland.