Self-Similarities and Invariant Densities for Model Sets

Model sets (also called cut and project sets) are generalizations of lattices. Here we show how the self-similarities of model sets are a natural replacement for the group of translations of a lattice. This leads us to the concept of averaging operators and invariant densities on model sets. We prove that invariant densities exist and that they produce absolutely continuous invariant measures in internal space. We study the invariant densities and their relationships to diffraction, continuous refinement operators, and Hutchinson measures.Comment: 15 pages, 2 figures, to appear in: Algebraic Methods and Theoretical Physics (ed. Y. St. Aubin

Invariant Submodules and Semigroups of Self-Similarities for Fibonacci Modules

The problem of invariance and self-similarity in Z-modules is investigated. For a selection of examples relevant to quasicrystals, especially Fibonacci modules, we determine the semigroup of self-similarities and encapsulate the number of similarity submodules in terms of Dirichlet series generating functions.Comment: 7 pages; to appear in: Aperiodic 97, eds. M. de Boissieu, J. L. Verger-Gaugry and R. Currat, World Scientific, Singapore (1998), in pres

Self-similarities in the frequency-amplitude space of a loss-modulated CO2_2 laser

We show the standard two-level continuous-time model of loss-modulated CO$_2$ lasers to display the same regular network of self-similar stability islands known so far to be typically present only in discrete-time models based on mappings. For class B laser models our results suggest that, more than just convenient surrogates, discrete mappings in fact could be isomorphic to continuous flows.Comment: (5 low-res color figs; for ALL figures high-res PDF: http://www.if.ufrgs.br/~jgallas/jg_papers.html

Cross-View Action Recognition from Temporal Self-Similarities

This paper concerns recognition of human actions under view changes. We explore self-similarities of action sequences over time and observe the striking stability of such measures across views. Building upon this key observation we develop an action descriptor that captures the structure of temporal similarities and dissimilarities within an action sequence. Despite this descriptor not being strictly view-invariant, we provide intuition and experimental validation demonstrating the high stability of self-similarities under view changes. Self-similarity descriptors are also shown stable under action variations within a class as well as discriminative for action recognition. Interestingly, self-similarities computed from different image features possess similar properties and can be used in a complementary fashion. Our method is simple and requires neither structure recovery nor multi-view correspondence estimation. Instead, it relies on weak geometric cues captured by self-similarities and combines them with machine learning for efficient cross-view action recognition. The method is validated on three public datasets, it has similar or superior performance compared to related methods and it performs well even in extreme conditions such as when recognizing actions from top views while using side views for training only