DOCSLIB.ORG

Geometry-Aware Face Completion and Editing

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Geometry-Aware Face Completion and Editing The Thirty-Third AAAI Conference on Artificial Intelligence (AAAI-19) Geometry-Aware Face Completion and Editing Linsen Song, Jie Cao, Lingxiao Song, Yibo Hu, Ran He∗ National Laboratory of Pattern Recognition, CASIA Center for Research on Intelligent Perception and Computing, CASIA Center for Excellence in Brain Science and Intelligence Technology, CAS University of Chinese Academy of Sciences, Beijing 100190, China [email protected], fjie.cao,[email protected], flingxiao.song,[email protected] Abstract Face completion is a challenging generation task because it requires generating visually pleasing new pixels that are se- mantically consistent with the unmasked face region. This paper proposes a geometry-aware Face Completion and Edit- ing NETwork (FCENet) by systematically studying facial ge- ometry from the unmasked region. Firstly, a facial geome- try estimator is learned to estimate facial landmark heatmaps and parsing maps from the unmasked face image. Then, an encoder-decoder structure generator serves to complete a face image and disentangle its mask areas conditioned on both the masked face image and the estimated facial geometry im- ages. Besides, since low-rank property exists in manually la- beled masks, a low-rank regularization term is imposed on the disentangled masks, enforcing our completion network to manage occlusion area with various shape and size. Further- more, our network can generate diverse results from the same masked input by modifying estimated facial geometry, which provides a flexible mean to edit the completed face appear- Figure 1: Face completion and attributes editing results. The ance. Extensive experimental results qualitatively and quanti- first and third row show the visual quality results of face tatively demonstrate that our network is able to generate visu- completion under the guidance of inferred facial geome- ally pleasing face completion results and edit face attributes try. In the second row, the girl’s eyes become smaller af- as well. ter changing its facial geometry. In the fourth row, the boy’s mouth becomes much bigger by replacing the mouth patch Introduction in the facial geometry images with a bigger mouth patch. Face completion, also known as face inpainting, aims to complete a face image with a masked region or missing con- Yeh et al. 2017; Li et al. 2017). These methods set a masked tent. As a common face image editing technique, it can also image as the network input to learn context features, then be used to edit face attributes. The generated face image can restore the missing content from the learned features. Some either be as accurate as the original face image, or content challenges still exist in these methods when they are applied coherent to the context so that the completed image looks to real-world face completion problems. First of all, differ- visually realistic. Most traditional methods (Barnes et al. ent from general objects, human faces have distinct geome- 2009; Huang et al. 2014; Darabi et al. 2012) rely on low- try distribution, and hence face geometry prior is more likely level cues to search for patches to synthesize missing con- to facilitate face completion (Yeh et al. 2017). Few existing tent. These image completion algorithms are skilled in fill- methods utilize the facial prior knowledge and well incorpo- ing backgrounds which contain similar textures but not excel rate it into an end-to-end network. Second, these algorithms at specific image object like face since prior domain knowl- are incapable of modifying the face attributes of the filled edge isn’t well incorporated. region (Li et al. 2017), e.g., editing the eye shape or mouth Recently, CNN based image completion methods with ad- size in figure 1. versarial training strategy have already significantly boosted To address these two problems, this paper studies the image completion performance (Pathak et al. 2016; a geometry-aware face completion and editing network ∗Ran He is the corresponding author. (FCENet) by exploring facial geometry priors including Copyright c 2019, Association for the Advancement of Artificial facial landmark heatmaps and parsing maps. Landmark Intelligence (www.aaai.org). All rights reserved. heatmap is an image composed of 68 points labeled by dif- 2506 ferent values to distinguish different face components. Sim- Radford et al. propose a deep convolutional GAN (DCGAN) ilarly, facial parsing map is an image with different val- (Radford, Metz, and Chintala 2016) to generate high-quality ues representing different face components (e.g., eyes, nose, images on multiple datasets. It is the first time to integrate mouth, hair, cheek, and background). They can not only pro- the deep models into GAN. To address training instability vide a hint of target face’s geometry information for face of GAN, Arjovsky et al. (Arjovsky, Chintala, and Bottou completion but also point out a novel way to modify face 2017) analyze the causes of training instability theoretically attributes of masked regions. The FCENet consists of three and propose Wasserstein GAN. Mao et al. (Mao et al. 2017) stages. In the first stage, facial parsing maps and landmark put forward LSGAN to avoid vanishing gradients problem heatmaps are inferred from masked face images. In the sec- during the learning process. Recently, improved GAN archi- ond stage, we concatenate the masked image, inferred land- tectures have achieved a great success on super-resolution mark heatmaps and parsing maps together as the input of (Ledig et al. 2017), style transfer (Li and Wand 2016), image a face completion generator to restore the uncorrupted face inpainting (Pathak et al. 2016), face attribute manipulation images and mask. In the third stage, two discriminators dis- (Shen and Liu 2017), and face rotation (Huang et al. 2017; tinguish generated face images and real ones globally and Hu et al. 2018). Motivated by these successful solutions, we locally to force the generated face images as realistic as develop the FCENet based on GAN. possible. Furthermore, the low-rank regularization boosts Image Completion. The image completion techniques can the completion network to disentangle more complex mask be broadly divided into three categories, The first cate- from the corrupted face image. Our FCENet is efficient to gory exploits the diffusion equation to propagate the low- generate face images with a variety of attributes depending level feature from the context region to the missing area on the facial geometry images concatenated with masked along the boundaries iteratively (Bertalmio et al. 2000; face image. A few face completion and editing examples of Elad et al. 2005). These methods excel at inpainting small FCENet are shown in figure 1. holes and superimposed text or lines but having limitations The main contributions of this paper are summarized as to the reproduction of large textured regions. The second follows: category is patch-based methods, which observe the con- 1. We design a novel network called facial geometry estima- text of the missing content and search similar patch from the tor to estimate reasonable facial geometry from masked same image or external image databases (Darabi et al. 2012; face images. Such facial geometry is leveraged to guide Bertalmio et al. 2003; Criminisi, Perez,´ and Toyama 2004; the face completion task. Several experiments systemat- Hays and Efros 2007). These methods can achieve ideal ically demonstrate the performance improvements from completion results on backgrounds like sky and grass, but different facial geometry, including facial landmarks, fa- they fail to generate semantic new pixels if these patches cial parsing maps and both together. don’t exist in the databases. The third category is CNN- based, which train an encoder-decoder architecture network 2. The FCENet allows interactive facial attributes editing to extract image features from the context and generate miss- of the generated face image by simply modifying its in- ing content according to the image features (Pathak et al. ferred facial geometry images. Therefore, face comple- 2016; Iizuka, Simo-Serra, and Ishikawa 2017). Deepak et tion and face attributes editing are integrated into one uni- al. (Pathak et al. 2016) present an unsupervised visual fea- fied framework. ture learning algorithm called context encoder to produce a 3. Our face completion generator simultaneously accom- plausible hypothesis for the missing part conditioned on its plishes face completion and mask disentanglement from surroundings. A pixel-wise reconstruction loss and an adver- the masked face image. A novel low-rank loss regular- sarial loss regularize the filling content to bear some resem- izes the disentangled mask to further enhance similarity blance to its surrounding area both on appearance and se- between the disentangled mask and the original mask, mantic segmentation. Satoshi et al. (Iizuka, Simo-Serra, and which enables our method to handle various masks with Ishikawa 2017) put forward an approach which can gener- different shapes and sizes. ate inpainting images that are both locally and globally con- 4. Experiments on the CelebA (Liu et al. 2015b) and Multi- sistent by training a global and local context discriminator. PIE (Gross et al. 2010) dataset demonstrate the superior- Their method is able to complete images of arbitrary res- ity of our approach over the existing approaches. olutions and various obstructed shapes by training a fully- convolutional neural network. Face Completion. Human face completion is much more Related Work challenging than general image completion tasks because fa- Generative Adversarial Network (GAN). As one of the cial components (e.g., eyes, nose, and mouth) are of highly most significant image generation techniques, GAN (Good- structurization and contain large appearance variations. Be- fellow et al. 2014) utilizes the mini-max adversarial game to sides, the symmetrical structure is reflected on human faces train a generator and a discriminator alternatively. The ad- like many natural objects. Second, compared to general ob- versarial training method encourages the generator to gen- ject image completion, face completion need to pay more erate realistic outputs to fool the discriminator.
Load more

Recommended publications
  • 1 Lifts of Polytopes
    Lecture 5: Lifts of polytopes and non-negative rank CSE 599S: Entropy optimality, Winter 2016 Instructor: James R. Lee Last updated: January 24, 2016 1 Lifts of polytopes 1.1 Polytopes and inequalities Recall that the convex hull of a subset X n is defined by ⊆ conv X λx + 1 λ x0 : x; x0 X; λ 0; 1 : ( ) f ( − ) 2 2 [ ]g A d-dimensional convex polytope P d is the convex hull of a finite set of points in d: ⊆ P conv x1;:::; xk (f g) d for some x1;:::; xk . 2 Every polytope has a dual representation: It is a closed and bounded set defined by a family of linear inequalities P x d : Ax 6 b f 2 g for some matrix A m d. 2 × Let us define a measure of complexity for P: Define γ P to be the smallest number m such that for some C s d ; y s ; A m d ; b m, we have ( ) 2 × 2 2 × 2 P x d : Cx y and Ax 6 b : f 2 g In other words, this is the minimum number of inequalities needed to describe P. If P is full- dimensional, then this is precisely the number of facets of P (a facet is a maximal proper face of P). Thinking of γ P as a measure of complexity makes sense from the point of view of optimization: Interior point( methods) can efficiently optimize linear functions over P (to arbitrary accuracy) in time that is polynomial in γ P . ( ) 1.2 Lifts of polytopes Many simple polytopes require a large number of inequalities to describe.
    [Show full text]
  • Can Every Face of a Polyhedron Have Many Sides ?
    Can Every Face of a Polyhedron Have Many Sides ? Branko Grünbaum Dedicated to Joe Malkevitch, an old friend and colleague, who was always partial to polyhedra Abstract. The simple question of the title has many different answers, depending on the kinds of faces we are willing to consider, on the types of polyhedra we admit, and on the symmetries we require. Known results and open problems about this topic are presented. The main classes of objects considered here are the following, listed in increasing generality: Faces: convex n-gons, starshaped n-gons, simple n-gons –– for n ≥ 3. Polyhedra (in Euclidean 3-dimensional space): convex polyhedra, starshaped polyhedra, acoptic polyhedra, polyhedra with selfintersections. Symmetry properties of polyhedra P: Isohedron –– all faces of P in one orbit under the group of symmetries of P; monohedron –– all faces of P are mutually congru- ent; ekahedron –– all faces have of P the same number of sides (eka –– Sanskrit for "one"). If the number of sides is k, we shall use (k)-isohedron, (k)-monohedron, and (k)- ekahedron, as appropriate. We shall first describe the results that either can be found in the literature, or ob- tained by slight modifications of these. Then we shall show how two systematic ap- proaches can be used to obtain results that are better –– although in some cases less visu- ally attractive than the old ones. There are many possible combinations of these classes of faces, polyhedra and symmetries, but considerable reductions in their number are possible; we start with one of these, which is well known even if it is hard to give specific references for precisely the assertion of Theorem 1.
    [Show full text]
  • 15 BASIC PROPERTIES of CONVEX POLYTOPES Martin Henk, J¨Urgenrichter-Gebert, and G¨Unterm
    15 BASIC PROPERTIES OF CONVEX POLYTOPES Martin Henk, J¨urgenRichter-Gebert, and G¨unterM. Ziegler INTRODUCTION Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their im- portance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry to linear and combinatorial optimiza- tion. In this chapter we try to give a short introduction, provide a sketch of \what polytopes look like" and \how they behave," with many explicit examples, and briefly state some main results (where further details are given in subsequent chap- ters of this Handbook). We concentrate on two main topics: • Combinatorial properties: faces (vertices, edges, . , facets) of polytopes and their relations, with special treatments of the classes of low-dimensional poly- topes and of polytopes \with few vertices;" • Geometric properties: volume and surface area, mixed volumes, and quer- massintegrals, including explicit formulas for the cases of the regular simplices, cubes, and cross-polytopes. We refer to Gr¨unbaum [Gr¨u67]for a comprehensive view of polytope theory, and to Ziegler [Zie95] respectively to Gruber [Gru07] and Schneider [Sch14] for detailed treatments of the combinatorial and of the convex geometric aspects of polytope theory. 15.1 COMBINATORIAL STRUCTURE GLOSSARY d V-polytope: The convex hull of a finite set X = fx1; : : : ; xng of points in R , n n X i X P = conv(X) := λix λ1; : : : ; λn ≥ 0; λi = 1 : i=1 i=1 H-polytope: The solution set of a finite system of linear inequalities, d T P = P (A; b) := x 2 R j ai x ≤ bi for 1 ≤ i ≤ m ; with the extra condition that the set of solutions is bounded, that is, such that m×d there is a constant N such that jjxjj ≤ N holds for all x 2 P .
    [Show full text]
  • Reconstructing Detailed Dynamic Face Geometry from Monocular Video
    Reconstructing Detailed Dynamic Face Geometry from Monocular Video Pablo Garrido1 ∗ Levi Valgaerts1 † Chenglei Wu1,2 ‡ Christian Theobalt1 § 1Max Planck Institute for Informatics 2Intel Visual Computing Institute Figure 1: Two results obtained with our method. Left: The input video. Middle: The tracked mesh shown as an overlay. Right: Applying texture to the mesh and overlaying it with the input video using the estimated lighting to give the impression of virtual face make-up. Abstract Links: DL PDF WEB VIDEO 1 Introduction Detailed facial performance geometry can be reconstructed using dense camera and light setups in controlled studios. However, Optical performance capture methods can reconstruct faces of vir- a wide range of important applications cannot employ these ap- tual actors in videos to deliver detailed dynamic face geometry. proaches, including all movie productions shot from a single prin- However, existing approaches are expensive and cumbersome as cipal camera. For post-production, these require dynamic monoc- they can require dense multi-view camera systems, controlled light ular face capture for appearance modification. We present a new setups, active markers in the scene, and recording in a controlled method for capturing face geometry from monocular video. Our ap- studio (Sec. 2.2). At the other end of the spectrum are computer proach captures detailed, dynamic, spatio-temporally coherent 3D vision methods that capture face models from monocular video face geometry without the need for markers. It works under un- (Sec. 2.1). These captured models are extremely coarse, and usually controlled lighting, and it successfully reconstructs expressive mo- only contain sparse collections of 2D or 3D facial landmarks rather tion including high-frequency face detail such as folds and laugh than a detailed 3D shape.
    [Show full text]
  • 1 More Background on Polyhedra
    CS 598CSC: Combinatorial Optimization Lecture date: 26 January, 2010 Instructor: Chandra Chekuri Scribe: Ben Moseley 1 More Background on Polyhedra This material is mostly from [3]. 1.1 Implicit Equalities and Redundant Constraints Throughout this lecture we will use affhull to denote the affine hull, linspace to be the linear space, charcone to denote the characteristic cone and convexhull to be the convex hull. Recall n that P = x Ax b is a polyhedron in R where A is a m n matrix and b is a m 1 matrix. f j ≤ g × × An inequality aix bi in Ax b is an implicit equality if aix = bi x P . Let I 1; 2; : : : ; m be the index set of≤ all implicit≤ equalities in Ax b. Then we can partition8 2 A into⊆A= fx b= andg A+x b+. Here A= consists of the rows of A with≤ indices in I and A+ are the remaining≤ rows of A. Therefore,≤ P = x A=x = b=;A+x b+ . In other words, P lies in an affine subspace defined by A=x = b=. f j ≤ g Exercise 1 Prove that there is a point x0 P such that A=x0 = b= and A+x0 < b+. 2 Definition 1 The dimension, dim(P ), of a polyhedron P is the maximum number of affinely independent points in P minus 1. n Notice that by definition of dimension, if P R then dim(P ) n, if P = then dim(P ) = 1, and dim(P ) = 0 if and only if P consists of a⊆ single point.
    [Show full text]
  • Designing Modular Sculpture Systems
    Bridges 2017 Conference Proceedings Designing Modular Sculpture Systems Christopher Carlson Wolfram Research, Inc 100 Trade Centre Drive Champaign, IL, 61820, USA E-mail: [email protected] Abstract We are interested in the sculptural possibilities of closed chains of modular units. One such system is embodied in MathMaker, a set of wooden pieces that can be connected end-to-end to create a fascinating variety of forms. MathMaker is based on a cubic honeycomb. We explore the possibilities of similar systems based on octahedral- tetrahedral, rhombic dodecahedral, and truncated octahedral honeycombs. Introduction The MathMaker construction kit consists of wooden parallelepipeds that can be connected end-to-end to make a great variety of sculptural forms [1]. Figure 1 shows on the left an untitled sculpture by Koos Verhoeff based on that system of modular units [2]. MathMaker derives from a cubic honeycomb. Each unit connects the center of one cubic face to the center of an adjacent face (Figure 1, center). Units connect via square planar faces which echo the square faces of cubes in the honeycomb. The square connecting faces can be matched in 4 ways via 90° rotation, so from any position each adjacent cubic face is accessible. A chain of MathMaker units jumps from cube to adjacent cube, adjacent units never traversing the same cube. A variant of MathMaker called “turned cross mitre” has units whose square cross-sections are rotated 45° about their central axes relative to the standard MathMaker units (Figure 1, right). Although the underlying cubic honeycomb structure is the same, the change in unit yields structures with strikingly different visual impressions.
    [Show full text]
  • Arxiv:1501.06198V1 [Math.MG]
    EMBEDDED FLEXIBLE SPHERICAL CROSS-POLYTOPES WITH NON-CONSTANT VOLUMES ALEXANDER A. GAIFULLIN Abstract. We construct examples of embedded flexible cross-polytopes in the spheres of all dimensions. These examples are interesting from two points of view. First, in dimensions 4 and higher, they are the first examples of embedded flexible polyhedra. Notice that, unlike in the spheres, in the Euclidean spaces and the Lobachevsky spaces of dimensions 4 and higher, still no example of an embedded flexible polyhedron is known. Second, we show that the volumes of the constructed flexible cross-polytopes are non- constant during the flexion. Hence these cross-polytopes give counterexamples to the Bellows Conjecture for spherical polyhedra. Earlier a counterexample to this conjecture was built only in dimension 3 (Alexandrov, 1997), and was not embedded. For flexible polyhedra in spheres we suggest a weakening of the Bellows Conjecture, which we call the Modified Bellows Conjecture. We show that this conjecture holds for all flexible cross-polytopes of the simplest type among which there are our counterexamples to the usual Bellows Conjecture. By the way, we obtain several geometric results on flexible cross-polytopes of the simplest type. In particular, we write relations on the volumes of their faces of codimensions 1 and 2. To Nicolai Petrovich Dolbilin on the occasion of his seventieth birthday 1. Introduction Let Xn be one of the three n-dimensional spaces of constant curvature, that is, the Euclidean space En or the sphere Sn or the Lobachevsky space Λn. For convenience, we shall always normalize metrics on the sphere Sn and on the Lobachevsky space Λn so that their curvatures are equal to 1 and 1 respectively.
    [Show full text]
  • A Recursive Algorithm for the K-Face Numbers of Wythoffian-N-Polytopes Constructed from Regular Polytopes
    Journal of Information Processing Vol.25 528–536 (Aug. 2017) [DOI: 10.2197/ipsjjip.25.528] Invited Paper A recursive Algorithm for the k-face Numbers of Wythoffian-n-polytopes Constructed from Regular Polytopes Jin Akiyama1,a) Sin Hitotumatu2 Motonaga Ishii3,b) Akihiro Matsuura4,c) Ikuro Sato5,d) Shun Toyoshima6,e) Received: September 6, 2016, Accepted: May 25, 2017 Abstract: In this paper, an n-dimensional polytope is called Wythoffian if it is derived by the Wythoff construction from an n-dimensional regular polytope whose finite reflection group belongs to An,Bn,Cn,F4,G2,H3,H4 or I2(p). Based on combinatorial and topological arguments, we give a matrix-form recursive algorithm that calculates the num- ber of k-faces (k = 0, 1,...,n) of all the Wythoffian-n-polytopes using Schlafli-Wytho¨ ff symbols. The correctness of the algorithm is reconfirmed by the method of exhaustion using a computer. Keywords: Wythoff construction, Wythoffian polytopes, k-face numbers, recursive algorithm, Coxeter group ( c ) volumes and surface areas. 1. Introduction Our purpose has been completed and already implemented High-dimensional polytopes have been studied mainly by the as computer programs. We will introduce a new methodology following methodologies: to study volumes, surface areas, number of k-faces sharing a (1) Combinatorics and Topology, (2) Graph Theory, (3) Group vertex, etc. of Wythoffian-n-polytopes constructed from regular Theory and (4) Metric Geometry. See [1], [2], [8], [9], [10], [11], polytopes in the forthcoming papers. In this paper, as a first [14], [15], [18], [19] for some background researches on tilings, step, we develop a recursive algorithm that computes the f-vector lattices and polytopes.
    [Show full text]
  • A Statistical Model for Synthesis of Detailed Facial Geometry
    A Statistical Model for Synthesis of Detailed Facial Geometry Aleksey Golovinskiy Wojciech Matusik Hanspeter Pfister Szymon Rusinkiewicz Thomas Funkhouser Princeton University MERL MERL Princeton University Princeton University (a) Original low-resolution face (b) Synthesized detail (c) Aged Figure 1: Synthesis of facial detail. (a) We begin with a low-resolution mesh obtained from a commercial scanner. (b) Then, we synthesize detail on it using statistics extracted from high-resolution meshes in our database. (c) Finally, we age the face by adjusting the statistics to match those of an elderly man. Note that all figures in this paper use Exaggerated Shading [Rusinkiewicz et al. 2006] to emphasize small geometric details. Abstract 1 Introduction Detailed surface geometry contributes greatly to the visual realism Creating realistic models of human faces is an important problem of 3D face models. However, acquiring high-resolution face geom- in computer graphics. Face models are widely used in computer etry is often tedious and expensive. Consequently, most face mod- games, commercials, movies, and for avatars in virtual reality ap- els used in games, virtual reality, or computer vision look unrealisti- plications. The goal is to capture all aspects of a person’s face in cally smooth. In this paper, we introduce a new statistical technique a digital model – i.e., “Digital Face Cloning” [Pighin and Lewis for the analysis and synthesis of small three-dimensional facial fea- 2005]. tures, such as wrinkles and pores. We acquire high-resolution face Ideally, a face model should be indistinguishable from a real geometry for people across a wide range of ages, genders, and face, easy to acquire, and intuitive to edit.
    [Show full text]
  • Lecture 2 - Introduction to Polytopes
    Lecture 2 - Introduction to Polytopes Optimization and Approximation - ENS M1 Nicolas Bousquet 1 Reminder of Linear Algebra definitions n Pm Let x1; : : : ; xm be points in R and λ1; : : : ; λm be real numbers. Then x = i=1 λixi is said to be a: • Linear combination (of x1; : : : ; xm) if the λi are arbitrary scalars. • Conic combination if λi ≥ 0 for every i. Pm • Convex combination if i=1 λi = 1 and λi ≥ 0 for every i. In the following, λ will still denote a scalar (since we consider in real spaces, λ is a real number). The linear space spanned by X = fx1; : : : ; xmg (also called the span of X), denoted by Span(X), is the set of n n points x of R which can be expressed as linear combinations of x1; : : : ; xm. Given a set X of R , the span of X is the smallest vectorial space containing the set X. In the following we will consider a little bit further the other types of combinations. Pm A set x1; : : : ; xm of vectors are linearly independent if i=1 λixi = 0 implies that for every i ≤ m, λi = 0. The dimension of the space spanned by x1; : : : ; xm is the cardinality of a maximum subfamily of x1; : : : ; xm which is linearly independent. The points x0; : : : ; x` of an affine space are said to be affinely independent if the vectors x1−x0; : : : ; x`− x0 are linearly independent. In other words, if we consider the space to be “centered” on x0 then the vectors corresponding to the other points in the vectorial space are independent.
    [Show full text]
  • Nets and Tiling Michael O'keeffe
    Nets and Tiling Michael O'Keeffe Introduction to tiling theory and its application to crystal nets Start with tiling in two dimensions. Surface of sphere and plane Sphere is two-dimensional. We require only two coordinates to specify position on the surface of a sphere: The coordinates of Berkeley 37.9 N, 122.3 W Tempe 33.4 N, 121.9 W As far as we are concerned Tilings in two dimensions are edge-to-edge (each edge is common to just two tiles) In three dimensions face-to-face (each face common to just two tiles) Three different embeddings of the same abstract tiling brick wall honeycomb herringbone c2mm p6mm p2gg Again, two embeddings of the same abstract tiling double brick Cairo tiling Both have same symmetry, p4gm. The Cairo conformation is the minimum density for equal edges. Recall Steinitz theorem Planar 3-connected graph is graph of a convex polyhedron 33 34 43 35 53 Tilings of the sphere (polyhedra) - regular polyhedra. one kind of vertex, one kind of edge, one kind of face 3.4.3.4 3.5.3.5 Quasiregular polyhedra: one kind of vertex, one kind of edge Tiling of the plane - regular tilings one kind of vertex, one kind of edge, one kind of face 36 44 63 hexagonal lattice square lattice honeycomb net quasiregular one kind of vertex, one kind of edge 3.6.3.6 kagome net honey comb net is not a lattice A lattice is a set of points related by translations honeycombnet is actually a lattice complex - a set of symmetry-related points related by translations cubic Archimedean polyhedra - one kind of vertex icosahedral Archimedean poyhedra
    [Show full text]
  • Vertex-, Face-, Point-, Schläfli-, and Delaney-Symbols in Nets, Polyhedra
    HIGHLIGHT www.rsc.org/crystengcomm | CrystEngComm Vertex-, face-, point-, Schlafli-,€ and Delaney-symbols in nets, polyhedra and tilings: recommended terminology V. A. Blatov,*a M. O’Keeffe*b and D. M. Proserpio*c DOI: 10.1039/b910671e We review the various kinds of symbols used to characterize the topology of vertices in 3-periodic nets, tiles and polyhedra, and symbols for tilings, making a recommendation for uniform nomenclature where there is some confusion and misapplication of terminology. The recent explosion of interest in with the inorganic anologues (if any)2 and edges, and no vertex other than the periodic nets and tilings as applied to the many cases of wrong assignment of beginning and ending one occurs more description of the topology of materials topology appeared in the literature (for than once in the sequence. such as coordination polymers/networks example the confusion between NbO and The sum of two cycles is the set of edges 1 3 and metal–organic frameworks (MOFs) CdSO4 related topologies). For recent (and their vertices) contained in just one points to the need for clarification applications derived from the geometrical or the other, but not both, cycles. A and agreement on the definition and study of periodic nets,4 see the reticular straightforward generalization is to the terminology for certain commonly-used chemistry approach5 and many references sum of a number of cycles as the set of all topological indices. Since the early work reported in ref. 1. edges that occur only an odd number of on coordination networks it has been In this highlight we review the various times in the set of cycles.
    [Show full text]