On several varieties of cacti and their relations

Motivated by string topology and the arc operad, we introduce the notion of quasi-operads and consider four (quasi)-operads which are different varieties of the operad of cacti. These are cacti without local zeros (or spines) and cacti proper as well as both varieties with fixed constant size one of the constituting loops. Using the recognition principle of Fiedorowicz, we prove that spineless cacti are equivalent as operads to the little discs operad. It turns out that in terms of spineless cacti Cohen's Gerstenhaber structure and Fiedorowicz' braided operad structure are given by the same explicit chains. We also prove that spineless cacti and cacti are homotopy equivalent to their normalized versions as quasi-operads by showing that both types of cacti are semi-direct products of the quasi-operad of their normalized versions with a re-scaling operad based on R>0. Furthermore, we introduce the notion of bi-crossed products of quasi-operads and show that the cacti proper are a bi-crossed product of the operad of cacti without spines and the operad based on the monoid given by the circle group S^1. We also prove that this particular bi-crossed operad product is homotopy equivalent to the semi-direct product of the spineless cacti with the group S^1. This implies that cacti are equivalent to the framed little discs operad. These results lead to new CW models for the little discs and the framed little discs operad.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-13.abs.htm

The State of the Art in Cartograms

Cartograms combine statistical and geographical information in thematic maps, where areas of geographical regions (e.g., countries, states) are scaled in proportion to some statistic (e.g., population, income). Cartograms make it possible to gain insight into patterns and trends in the world around us and have been very popular visualizations for geo-referenced data for over a century. This work surveys cartogram research in visualization, cartography and geometry, covering a broad spectrum of different cartogram types: from the traditional rectangular and table cartograms, to Dorling and diffusion cartograms. A particular focus is the study of the major cartogram dimensions: statistical accuracy, geographical accuracy, and topological accuracy. We review the history of cartograms, describe the algorithms for generating them, and consider task taxonomies. We also review quantitative and qualitative evaluations, and we use these to arrive at design guidelines and research challenges

2D-to-3D facial expression transfer

© 20xx IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.Automatically changing the expression and physical features of a face from an input image is a topic that has been traditionally tackled in a 2D domain. In this paper, we bring this problem to 3D and propose a framework that given an input RGB video of a human face under a neutral expression, initially computes his/her 3D shape and then performs a transfer to a new and potentially non-observed expression. For this purpose, we parameterize the rest shape --obtained from standard factorization approaches over the input video-- using a triangular mesh which is further clustered into larger macro-segments. The expression transfer problem is then posed as a direct mapping between this shape and a source shape, such as the blend shapes of an off-the-shelf 3D dataset of human facial expressions. The mapping is resolved to be geometrically consistent between 3D models by requiring points in specific regions to map on semantic equivalent regions. We validate the approach on several synthetic and real examples of input faces that largely differ from the source shapes, yielding very realistic expression transfers even in cases with topology changes, such as a synthetic video sequence of a single-eyed cyclops.Peer ReviewedPostprint (author's final draft