We give a lower bound for the error of any unitarily invariant algorithm learning half-spaces against the uniform or related distributions on the unit sphere. The bound is uniform in the choice of the target half-space and has an exponentially decaying deviation probability in the sample. The technique of proof is related to a proof of the Johnson Lindenstrauss Lemma. We argue that, unlike prev...

In this work, we present a segmentation algorithm for color images that uses the watershed algorithm to segment either the two-dimensional (2-D) or the three-dimensional (3-D) color histogram of an image. For compliance with the way humans perceive color, this segmentation has to take place in a perceptually uniform color space like the Luv space. To avoid oversegmentation, the watershed algori...

We present methods to estimate perceptual uniformity of color spaces and to derive a perceptually uniform RGB space using geometrical criteria defined in a logarithmic opponent color representation.

It is shown that associated with each metric space (X, d) there is a compactification udX of X that can be characterized as the smallest compactification of X to which each bounded uniformly continuous real-valued continuous function with domain X can be extended. Other characterizations of udX are presented, and a detailed study of the structure of udX is undertaken. This culminates in a topol...

Let (S,G, P ) be a probability space and let F be a set of measurable functions on S with an envelope function F finite everywhere. Let X1, X2, ... be a strictly stationary sequence of random variables with distribution P , and define the empirical measures Pn, based on {Xi}, as Pn = n−1 ∑n i=1 δXi . We say the uniform CLT holds over F , if n 1 2 (Pn − P ) converges in law, in the space l∞(F ) ...

The algorithmic theory of randomness is well developed when the underlying space is the set of finite or infinite sequences and the underlying probability distribution is the uniform distribution or a computable distribution. These restrictions seem artificial. Some progress has been made to extend the theory to arbitrary Bernoulli distributions (by Martin-Löf), and to arbitrary distributions (...

We study the properties of points in [0, 1]d generated by applying Hilbert’s space-filling curve to uniformly distributed points in [0, 1]. For deterministic sampling we obtain a discrepancy of O(n−1/d) for d ≥ 2. For random stratified sampling, and scrambled van der Corput points, we get a mean squared error of O(n−1−2/d) for integration of Lipshitz continuous integrands, when d ≥ 3. These rat...

We introduce and study the notions of a strongly completable and of a strongly complete quasi-uniform space. A quasi-uniform space (X,U) is said to be strongly complete if every Cauchy filter (in the sense of Sieber and Pervin) clusters in the uniform space (X,U ∨ U−1). An interesting motivation for the study of this notion of completeness is the fact, proved here, that the quasi-uniformity ind...