Unnormalized Haar spectra and Ordered Binary Decision Diagrams (OBDDs) are two standard representations of Boolean functions used in logic design. In this article, mutual relationships between those two representations have been derived. The method of calculating the Haar spectrum from OBDD has been presented. The decomposition of the Haar spectrum, in terms of the cofactors of Boolean function...

Face detection systems based on boosted Haar features in a cascade architecture [3] perform very well both in terms of accuracy and speed. However, they are vulnerable to variations in illumination conditions. On the other hand, face detection systems based on the Local Binary Pattern (LBP) [2] or its variants [1] show robustness to illumination conditions but do not take advantage of the richn...

We prove that in Polish, abelian, non-locally-compact groups the family of Haar null sets of Christensen does not fulfil the countable chain condition, that is, there exists an uncountable family of pairwise disjoint universally measurable sets which are not Haar null. (Dougherty, answering an old question of Christensen, showed earlier that this was the case for some Polish, abelian, non-local...

An orthogonal Haar scattering transform is a deep network computed with a hierarchy of additions, subtractions and absolute values over pairs of coefficients. Unsupervised learning optimizes Haar pairs to obtain sparse representations of training data with an algorithm of polynomial complexity. For signals defined on a graph, a Haar scattering is computed by cascading orthogonal Haar wavelet tr...

Abstract The concept of Rademacher type p (1 ≤ p ≤ 2) plays an important role in the local theory of Banach spaces. In [3] Mascioni considers a weakening of this concept and shows that for a Banach space X weak Rademacher type p implies Rademacher type r for all r < p. As with Rademacher type p and weak Rademacher type p, we introduce the concept of Haar type p and weak Haar type p by replacing...

We analyze entropic uncertainty relations for two orthogonal measurements on a N-dimensional Hilbert space, performed in two generic bases. It is assumed that the unitary matrix U relating both bases is distributed according to the Haar measure on the unitary group. We provide lower bounds on the average Shannon entropy of probability distributions related to both measurements. The bounds are s...

Nadir Matringe has indicated to me that the paper Conducteur des Représentations du groupe linéaire ([JPSS81a], citeError2) contains an error. I correct the error in this note. The correct proof is actually simpler than the erroneous proof. Separately, Matringe has given a different, interesting proof of the result in question ([Mat11]). We recall the result in question. Let F be a local field....

The efficient and discriminating feature extraction is a significant problem in pattern recognition and computer vision. This paper presents a novel Discriminating Haar (D-Haar) features for eye detection. The DHaar feature extraction starts with a Principal Component Analysis (PCA) followed by a whitening transformation on the Haar feature space. A discriminant analysis is then performed on th...

The classical Haar wavelet system of L2(R) is commonly considered to be very local in space. We introduce and study in this paper piecewise-constant framelets (PCF) that include the Haar system as a special case. We show that any bi-framelet pair consisting of PCFs provides the same Besov space characterizations as the Haar system. In particular, it has Jackson-type performance sJ = 1 and Berns...

This chapter shows that combining Haar-Hilbert and Log-Gabor improves iris recognition performance leading to a less ambiguous biometric decision landscape in which the overlap between the experimental intraand interclass score distributions diminishes or even vanishes. Haar-Hilbert, Log-Gabor and combined Haar-Hilbert and Log-Gabor encoders are tested here both for single and dual iris approac...