Models of discretized moduli spaces, cohomological field theories, and Gaussian means

We prove combinatorially the explicit relation between genus filtrated $s$-loop means of the Gaussian matrix model and terms of the genus expansion of the Kontsevich--Penner matrix model (KPMM). The latter is the generating function for volumes of discretized (open) moduli spaces $M_{g,s}^{\mathrm{disc}}$ given by $N_{g,s}(P_1,\dots,P_s)$ for $(P_1,\dots,P_s)\in{\mathbb Z}_+^s$. This generating function therefore enjoys the topological recursion, and we prove that it is simultaneously the generating function for ancestor invariants of a cohomological field theory thus enjoying the Givental decomposition. We use another Givental-type decomposition obtained for this model by the second authors in 1995 in terms of special times related to the discretisation of moduli spaces thus representing its asymptotic expansion terms (and therefore those of the Gaussian means) as finite sums over graphs weighted by lower-order monomials in times thus giving another proof of (quasi)polynomiality of the discrete volumes. As an application, we find the coefficients in the first subleading order for ${\mathcal M}_{g,1}$ in two ways: using the refined Harer--Zagier recursion and by exploiting the above Givental-type transformation. We put forward the conjecture that the above graph expansions can be used for probing the reduction structure of the Delgne--Mumford compactification $\overline{\mathcal M}_{g,s}$ of moduli spaces of punctured Riemann surfaces.Comment: 36 pages in LaTex, 6 LaTex figure

Stable pair compactification of moduli of K3 surfaces of degree 2

We prove that the universal family of polarized K3 surfaces of degree 2 can be extended to a flat family of stable slc pairs $(X,\epsilon R)$ over the toroidal compactification associated to the Coxeter fan. One-parameter degenerations of K3 surfaces in this family are described by integral-affine structures on a sphere with 24 singularities.Comment: v2: Updated reference

Deep Reflectance Maps

Undoing the image formation process and therefore decomposing appearance into its intrinsic properties is a challenging task due to the under-constraint nature of this inverse problem. While significant progress has been made on inferring shape, materials and illumination from images only, progress in an unconstrained setting is still limited. We propose a convolutional neural architecture to estimate reflectance maps of specular materials in natural lighting conditions. We achieve this in an end-to-end learning formulation that directly predicts a reflectance map from the image itself. We show how to improve estimates by facilitating additional supervision in an indirect scheme that first predicts surface orientation and afterwards predicts the reflectance map by a learning-based sparse data interpolation. In order to analyze performance on this difficult task, we propose a new challenge of Specular MAterials on SHapes with complex IllumiNation (SMASHINg) using both synthetic and real images. Furthermore, we show the application of our method to a range of image-based editing tasks on real images.Comment: project page: http://homes.esat.kuleuven.be/~krematas/DRM

Null twisted geometries

We define and investigate a quantisation of null hypersurfaces in the context of loop quantum gravity on a fixed graph. The main tool we use is the parametrisation of the theory in terms of twistors, which has already proved useful in discussing the interpretation of spin networks as the quantization of twisted geometries. The classical formalism can be extended in a natural way to null hypersurfaces, with the Euclidean polyhedra replaced by null polyhedra with space-like faces, and SU(2) by the little group ISO(2). The main difference is that the simplicity constraints present in the formalims are all first class, and the symplectic reduction selects only the helicity subgroup of the little group. As a consequence, information on the shapes of the polyhedra is lost, and the result is a much simpler, abelian geometric picture. It can be described by an Euclidean singular structure on the 2-dimensional space-like surface defined by a foliation of space-time by null hypersurfaces. This geometric structure is naturally decomposed into a conformal metric and scale factors, forming locally conjugate pairs. Proper action-angle variables on the gauge-invariant phase space are described by the eigenvectors of the Laplacian of the dual graph. We also identify the variables of the phase space amenable to characterize the extrinsic geometry of the foliation. Finally, we quantise the phase space and its algebra using Dirac's algorithm, obtaining a notion of spin networks for null hypersurfaces. Such spin networks are labelled by SO(2) quantum numbers, and are embedded non-trivially in the unitary, infinite-dimensional irreducible representations of the Lorentz group.Comment: 22 pages, 3 figures. v2: minor corrections, improved presentation in section 4, references update

Separation of line drawings based on split faces for 3D object reconstruction

© 2014 IEEE. Reconstructing 3D objects from single line drawings is often desirable in computer vision and graphics applications. If the line drawing of a complex 3D object is decomposed into primitives of simple shape, the object can be easily reconstructed. We propose an effective method to conduct the line drawing separation and turn a complex line drawing into parametric 3D models. This is achieved by recursively separating the line drawing using two types of split faces. Our experiments show that the proposed separation method can generate more basic and simple line drawings, and its combination with the example-based reconstruction can robustly recover wider range of complex parametric 3D objects than previous methods.This work was supported by grants from Science, Industry, Trade, and Information Technology Commission of Shenzhen Municipality (No. JC201005270378A), Guangdong Innovative Research Team Program (No. 201001D0104648280), Shenzhen Basic Research Program (JCYJ20120617114614438, JC201005270350A, JCYJ20120903092050890), Scientific Research Fund of Hunan Provincial Education Department (No. 13C073), Industrial Technology Research and Development Program of Hengyang Science and Technology Bureau (No.2013KG75), and the Construct Program of the Key Discipline in Hunan Provinc

Image collection pop-up: 3D reconstruction and clustering of rigid and non-rigid categories

© 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.This paper introduces an approach to simultaneously estimate 3D shape, camera pose, and object and type of deformation clustering, from partial 2D annotations in a multi-instance collection of images. Furthermore, we can indistinctly process rigid and non-rigid categories. This advances existing work, which only addresses the problem for one single object or, if multiple objects are considered, they are assumed to be clustered a priori. To handle this broader version of the problem, we model object deformation using a formulation based on multiple unions of subspaces, able to span from small rigid motion to complex deformations. The parameters of this model are learned via Augmented Lagrange Multipliers, in a completely unsupervised manner that does not require any training data at all. Extensive validation is provided in a wide variety of synthetic and real scenarios, including rigid and non-rigid categories with small and large deformations. In all cases our approach outperforms state-of-the-art in terms of 3D reconstruction accuracy, while also providing clustering results that allow segmenting the images into object instances and their associated type of deformation (or action the object is performing).Postprint (author's final draft