Rectangular Layouts and Contact Graphs

Contact graphs of isothetic rectangles unify many concepts from applications including VLSI and architectural design, computational geometry, and GIS. Minimizing the area of their corresponding {\em rectangular layouts} is a key problem. We study the area-optimization problem and show that it is NP-hard to find a minimum-area rectangular layout of a given contact graph. We present O(n)-time algorithms that construct $O(n^2)$-area rectangular layouts for general contact graphs and $O(n\log n)$-area rectangular layouts for trees. (For trees, this is an $O(\log n)$-approximation algorithm.) We also present an infinite family of graphs (rsp., trees) that require $\Omega(n^2)$ (rsp., $\Omega(n\log n)$) area. We derive these results by presenting a new characterization of graphs that admit rectangular layouts using the related concept of {\em rectangular duals}. A corollary to our results relates the class of graphs that admit rectangular layouts to {\em rectangle of influence drawings}.Comment: 28 pages, 13 figures, 55 references, 1 appendi

On the maximum order of graphs embedded in surfaces

The maximum number of vertices in a graph of maximum degree $\Delta\ge 3$ and fixed diameter $k\ge 2$ is upper bounded by $(1+o(1))(\Delta-1)^{k}$. If we restrict our graphs to certain classes, better upper bounds are known. For instance, for the class of trees there is an upper bound of $(2+o(1))(\Delta-1)^{\lfloor k/2\rfloor}$ for a fixed $k$. The main result of this paper is that graphs embedded in surfaces of bounded Euler genus $g$ behave like trees, in the sense that, for large $\Delta$, such graphs have orders bounded from above by begin{cases} c(g+1)(\Delta-1)^{\lfloor k/2\rfloor} & \text{if $k$ is even} c(g^{3/2}+1)(\Delta-1)^{\lfloor k/2\rfloor} & \text{if $k$ is odd}, \{cases} where $c$ is an absolute constant. This result represents a qualitative improvement over all previous results, even for planar graphs of odd diameter $k$. With respect to lower bounds, we construct graphs of Euler genus $g$, odd diameter $k$, and order $c(\sqrt{g}+1)(\Delta-1)^{\lfloor k/2\rfloor}$ for some absolute constant $c>0$. Our results answer in the negative a question of Miller and \v{S}ir\'a\v{n} (2005).Comment: 13 pages, 3 figure

FEAFA: A Well-Annotated Dataset for Facial Expression Analysis and 3D Facial Animation

Facial expression analysis based on machine learning requires large number of well-annotated data to reflect different changes in facial motion. Publicly available datasets truly help to accelerate research in this area by providing a benchmark resource, but all of these datasets, to the best of our knowledge, are limited to rough annotations for action units, including only their absence, presence, or a five-level intensity according to the Facial Action Coding System. To meet the need for videos labeled in great detail, we present a well-annotated dataset named FEAFA for Facial Expression Analysis and 3D Facial Animation. One hundred and twenty-two participants, including children, young adults and elderly people, were recorded in real-world conditions. In addition, 99,356 frames were manually labeled using Expression Quantitative Tool developed by us to quantify 9 symmetrical FACS action units, 10 asymmetrical (unilateral) FACS action units, 2 symmetrical FACS action descriptors and 2 asymmetrical FACS action descriptors, and each action unit or action descriptor is well-annotated with a floating point number between 0 and 1. To provide a baseline for use in future research, a benchmark for the regression of action unit values based on Convolutional Neural Networks are presented. We also demonstrate the potential of our FEAFA dataset for 3D facial animation. Almost all state-of-the-art algorithms for facial animation are achieved based on 3D face reconstruction. We hence propose a novel method that drives virtual characters only based on action unit value regression of the 2D video frames of source actors.Comment: 9 pages, 7 figure

DeepCoder: Semi-parametric Variational Autoencoders for Automatic Facial Action Coding

Human face exhibits an inherent hierarchy in its representations (i.e., holistic facial expressions can be encoded via a set of facial action units (AUs) and their intensity). Variational (deep) auto-encoders (VAE) have shown great results in unsupervised extraction of hierarchical latent representations from large amounts of image data, while being robust to noise and other undesired artifacts. Potentially, this makes VAEs a suitable approach for learning facial features for AU intensity estimation. Yet, most existing VAE-based methods apply classifiers learned separately from the encoded features. By contrast, the non-parametric (probabilistic) approaches, such as Gaussian Processes (GPs), typically outperform their parametric counterparts, but cannot deal easily with large amounts of data. To this end, we propose a novel VAE semi-parametric modeling framework, named DeepCoder, which combines the modeling power of parametric (convolutional) and nonparametric (ordinal GPs) VAEs, for joint learning of (1) latent representations at multiple levels in a task hierarchy1, and (2) classification of multiple ordinal outputs. We show on benchmark datasets for AU intensity estimation that the proposed DeepCoder outperforms the state-of-the-art approaches, and related VAEs and deep learning models.Comment: ICCV 2017 - accepte

Revision of Madagascar's Dwarf Lemurs (Cheirogaleidae:Cheirogaleus): Designation of Species, Candidate Species Status and Geographic Boundaries Based on Molecular and Morphological Data

The genus Cheirogaleus, the dwarf lemurs, is a radiation of strepsirrhine primates endemic to the island of Madagascar. The dwarf lemurs are taxonomically grouped in the family Cheirogaleidae (Infraorder: Lemuriformes) along with the genera Microcebus, Mirza, Allocebus, and Phaner. The taxonomic history of the genus Cheirogaleus has been controversial since its inception due to a paucity of evidence in support of some proposed species. In this study, we addressed this issue by expanding the geographic breadth of samples by 91 individuals and built upon existing mitochondrial (cytb and COII) and nuclear (FIBA and vWF) DNA datasets to better resolve the phylogeny of Cheirogaleus. The mitochondrial gene fragments D-loop and PAST as well as the CFTR-PAIRB nuclear loci were also sequenced. In agreement with previous genetic studies, numerous deep divergences were resolved in the C. major, C. minor and C. medius lineages. Four of these lineages were segregated as new species, seven were identified as confirmed candidate species, and four were designated as unconfirmed candidate species based on comparative mitochondrial DNA sequence data gleaned from the literature or this study. Additionally, C. thomasi was resurrected. Given the widespread distribution of the genus Cheirogaleus throughout Madagascar, the methodology employed in this study combined all available lines of evidence to standardize investigative procedures in a genus with limited access to type material and a lack of comprehensive sampling across its total distribution. Our results highlighted lineages that likely represent new species and identified localities that may harbor an as-yet undescribed cryptic species diversity pending further field and laboratory work.We are most grateful to the Ahmanson Foundation, the Theodore F. and Claire M. Hubbard Family Foundation, the Primate Action Fund / Conservation International, the Margot Marsh Biodiversity Foundation, and the National Geographic Society, for financial assistance