Image quantization and dithering are fundamental image processing problems in computer vision and graphics. Both steps are generally performed sequentially and, in most cases, independent of each other. Color quantization with a pixel-wise defined distortion measure and the dithering process with its local neighborhood typically optimize different quality criteria or, frequently, follow a heuristic approach without reference to any quality measure. In this paper we propose a new model to simultaneously quantize and dither color images. The method is based on a rigorous cost-function approach which optimizes a quality criterion derived from a simplified model of human perception. Optimizations are performed by an efficient multiscale procedure which substantially alleviates the computational burden. The quality criterion and the optimization algorithms are evaluated on a representative set of artificial and real-world images thereby showing a significant image quality improvement over standard color reduction approaches.

Clusters in data clustering should be robust to sample fluctuation, i.e., the estimate of cluster parameters on a second sample set should yield qualitatively similar results. This robustness requirement can be quantified by large deviation arguments from statistical learning theory. We use the principle of Empirical Risk Approximation to determine an optimal number of clusters for the case of histogram clustering. The analysis validates stochastic approximation algorithms like Markov Chain Monte Carlo which maximize the entropy for fixed optimization costs.

Unsupervised learning algorithms are designed to extract structure from data samples on the basis of a cost function for structures. For a reliable and robust inference process, the unsupervised learning algorithm has to guarantee that the extracted structures are typical for the data source. In particular, it has to reject all structures where the inference is dominated by the arbitrariness of the sample noise and which, consequently, can be characterized as overfitting in unsupervised learning. This paper summarizes an inference principle called Empirical Risk Approximation which allows us to quantitatively measure the overfitting effect and to derive a criterion as a saveguard against it. The crucial condition for learning is met if (i) the empirical risk of learning uniformly converges towards the expected risk and if (ii) the hypothesis class retains a minimal variety for consistent inference. Parameter selection of learnable data structures is demonstrated for the case of k-means clustering and Monte Carlo simulations are presented to support the selection principle.

Cluster analysis is a fundamental principle in exploratory data analysis, providing the user with a description of the group structure of given data, A key problem in this context is the interpretation and visualization of clustering solutions in high-dimensional or abstract data spaces. In particular, probabilistic descriptions of the group structure, essential to capture inter-cluster relationships, are hardly assessable by simple inspection of the probabilistic assignment variables. We present a novel approach to the visualization of group structure. It is based on a statistical model of the object assignments which have been observed or estimated by a probabilistic clustering procedure. The objects or data points are embedded in a low dimensional Euclidean space by approximating the observed data statistics with a Gaussian mixture model. The algorithm provides a new approach to the visualization of the inherent structure for a broad variety of data types, e.g. histogram data, proximity data and co-occurrence data. To demonstrate the power of the approach, histograms of textured images are visualized as an example of a large-scale data mining application.

Image quantization and digital halftoning are fundamental problems in computer graphics, which arise when displaying high‐color images on non‐truecolor devices. Both steps are generally performed sequentially and, in most cases, independent of each other. Color quantization with a pixel‐wise defined distortion measure and the dithering process with its local neighborhood optimize different quality criteria or, frequently, follow a heuristic without reference to any quality measure.In this paper we propose a new method to simultaneously quantize and dither color images. The method is based on a rigorous cost‐function approach which optimizes a quality criterion derived from a generic model of human perception. A highly efficient algorithm for optimization based on a multiscale method is developed for the dithered color quantization cost function. The quality criterion and the optimization algorithms are evaluated on a representative set of artificial and real‐world images as well as on a collection of icons. A significant image quality improvement is observed compared to standard color reduction approaches.

An adaptive on-line algorithm is proposed to estimate hierarchical data structures for non-stationary data sources. The approach is based on the principle of minimum cross entropy to derive a decision tree for data clustering and it employs a metalearning idea (learning to learn) to adapt to changes in data characteristics. Its efficiency is demonstrated by grouping non-stationary artifical data and by hierarchical segmentation of LANDSAT images.