We present evidence that several higher-order statistical properties of natural images and signals can be explained by a stochastic model which simply varies scale of an otherwise stationary Gaussian process. We discuss two interesting consequences. The first is that a variety of natural signals can be related through a common 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 cases 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. 1 Introduction Recently, considerable attention has been paid to understanding and modeling the non-Gaussian or "higher-order" properties of natural signals, particularly images. Several non-Gaussian properties have been identified and studied.

We present evidence that several higher-order statistical properties of natural images and signals can be explained by a stochastic model which simply varies scale of an otherwise stationary Gaussian process. We discuss two interesting consequences. The first is that a variety of natural signals can be related through a common 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 cases 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. 1 Introduction Recently, considerable attention has been paid to understanding and modeling the non-Gaussian or "higher-order" properties of natural signals, particularly images. Several non-Gaussian properties have been identified and studied.

The first is that a variety of natural signals can be related through a common modelof 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 cases thenon-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. 1 Introduction Recently, considerable attention has been paid to understanding and modeling the non-Gaussian or "higher-order" properties of natural signals, particularly images. Several non-Gaussian properties have been identified and studied. For example, marginal densities of features have been shown to have high kurtosis or "heavy tails", indicating a non-Gaussian, sparse representation. Another example is the "bowtie" shape of conditional distributions of neighboring features, indicating dependence ofvariances [11]. These non-Gaussian properties have motivated a number of image and signal processing algorithms that attempt to exploit higher-order s tatistics of the signals, e.g., for blind source separation. In this paper we show that these previously observed higher-order phenomena are ubiquitous and can be accounted for by a model which simply varies the scale of an otherwise stationary Gaussianprocess. This enables us to relate a variety of natural signals to one another and to spherically invariant random processes, which are well-known in the signal processing literature [6, 3].

We formulate a model for probability distributions on image spaces. We show that any distribution of images can be factored exactly into conditional distributionsof 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 fix this, we introduce hiddenclass 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 likelihoodestimation using the EM algorithm. We have obtained encouraging preliminary results on the problems of detecting various objects inSAR images and target recognition in optical aerial images. 1 Introduction

In hyperspectral imagery one pixel typically consists of a mixture of the reflectance spectra of several materials, where the mixture coefficients correspond to the abundances of the constituting materials. Weassume linear combinations of reflectance spectra with some additive normal sensor noise and derive a probabilistic MAP framework for analyzing hyperspectral data. As the material reflectance characteristicsare not know a priori, we face the problem of unsupervised linear unmixing.

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 fix this, 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. 1 Introduction

In hyperspectral imagery one pixel typically consists of a mixture of the reflectance spectra of several materials, where the mixture coefficients correspond to the abundances of the constituting materials. We assume linear combinations of reflectance spectra with some additive normal sensor noise and derive a probabilistic MAP framework for analyzing hyperspectral data. As the material reflectance characteristics are not know a priori, we face the problem of unsupervised linear unmixing.

We have previously presented a coarse-to-fine hierarchical pyramid/neural network(HPNN) architecture which combines multiscale image processing techniques with neural networks.

The efficiency of image search can be greatly improved by using a coarse-to-fine search strategy with a multi-resolution image representation. However,if the resolution is so low that the objects have few distinguishing 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 represent high-frequencyinformation at low resolutions. The use of multiresolution searchtechniques 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 illustrate theseideas by training hierarchical systems of neural networks to find clusters of buildings in aerial photographs of farmland.

The efficiency of image search can be greatly improved by using a coarse-to-fine search strategy with a multi-resolution image representation. However, if the resolution is so low that the objects have few distinguishing 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 represent high-frequency information at low resolutions. The use of multiresolution 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 illustrate these ideas by training hierarchical systems of neural networks to find clusters of buildings in aerial photographs of farmland.