Computational Optimal Transport
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Classical and Modern Diffraction Theory
Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3701993/frontmatter.pdf by guest on 29 September 2021 Classical and Modern Diffraction Theory Edited by Kamill Klem-Musatov Henning C. Hoeber Tijmen Jan Moser Michael A. Pelissier SEG Geophysics Reprint Series No. 29 Sergey Fomel, managing editor Evgeny Landa, volume editor Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3701993/frontmatter.pdf by guest on 29 September 2021 Society of Exploration Geophysicists 8801 S. Yale, Ste. 500 Tulsa, OK 74137-3575 U.S.A. # 2016 by Society of Exploration Geophysicists All rights reserved. This book or parts hereof may not be reproduced in any form without permission in writing from the publisher. Published 2016 Printed in the United States of America ISBN 978-1-931830-00-6 (Series) ISBN 978-1-56080-322-5 (Volume) Library of Congress Control Number: 2015951229 Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3701993/frontmatter.pdf by guest on 29 September 2021 Dedication We dedicate this volume to the memory Dr. Kamill Klem-Musatov. In reading this volume, you will find that the history of diffraction We worked with Kamill over a period of several years to compile theory was filled with many controversies and feuds as new theories this volume. This volume was virtually ready for publication when came to displace or revise previous ones. Kamill Klem-Musatov’s Kamill passed away. He is greatly missed. new theory also met opposition; he paid a great personal price in Kamill’s role in Classical and Modern Diffraction Theory goes putting forth his theory for the seismic diffraction forward problem. -
Wasserstein Distance W2, Or to the Kantorovich-Wasserstein Distance W1, with Explicit Constants of Equivalence
THE EQUIVALENCE OF FOURIER-BASED AND WASSERSTEIN METRICS ON IMAGING PROBLEMS G. AURICCHIO, A. CODEGONI, S. GUALANDI, G. TOSCANI, M. VENERONI Abstract. We investigate properties of some extensions of a class of Fourier-based proba- bility metrics, originally introduced to study convergence to equilibrium for the solution to the spatially homogeneous Boltzmann equation. At difference with the original one, the new Fourier-based metrics are well-defined also for probability distributions with different centers of mass, and for discrete probability measures supported over a regular grid. Among other properties, it is shown that, in the discrete setting, these new Fourier-based metrics are equiv- alent either to the Euclidean-Wasserstein distance W2, or to the Kantorovich-Wasserstein distance W1, with explicit constants of equivalence. Numerical results then show that in benchmark problems of image processing, Fourier metrics provide a better runtime with respect to Wasserstein ones. Fourier-based Metrics; Wasserstein Distance; Fourier Transform; Image analysis. AMS Subject Classification: 60A10, 60E15, 42A38, 68U10. 1. Introduction In computational applied mathematics, numerical methods based on Wasserstein distances achieved a leading role over the last years. Examples include the comparison of histograms in higher dimensions [6,9,22], image retrieval [21], image registration [4,11], or, more recently, the computations of barycenters among images [7, 15]. Surprisingly, the possibility to identify the cost function in a Wasserstein distance, together with the possibility of representing images as histograms, led to the definition of classifiers able to mimic the human eye [16, 21, 24]. More recently, metrics which are able to compare at best probability distributions were introduced and studied in connection with machine learning, where testing the efficiency of new classes of loss functions for neural networks training has become increasingly important. -
Image-Space Caustics and Curvatures
Image-space Caustics and Curvatures Xuan Yu Feng Li Jingyi Yu Department of Computer and Information Sciences University of Delaware Newark, DE 19716, USA fxuan,feng,[email protected] Abstract Caustics are important visual phenomena, as well as challenging global illumination effects in computer graph- ics. Physically caustics can be interpreted from one of two perspectives: in terms of photons gathered on scene geom- etry, or in terms of a pair of caustic surfaces. These caustic surfaces are swept by the foci of light rays. In this paper, we develop a novel algorithm to approximate caustic sur- faces of sampled rays. Our approach locally parameterizes rays by their intersections with a pair of parallel planes. We show neighboring ray triplets are constrained to pass simul- taneously through two slits, which rule the caustic surfaces. We derive a ray characteristic equation to compute the two slits, and hence, the caustic surfaces. Using the characteris- tic equation, we develop a GPU-based algorithm to render the caustics. Our approach produces sharp and clear caus- tics using much fewer ray samples than the photon mapping method and it also maintains high spatial and temporal co- herency. Finally, we present a normal-ray surface repre- Figure 1. We use our caustic-surface-based sentation that locally parameterizes the normals about a algorithm to render the refraction caustics surface point as rays. Computing the normal ray caustic cast by a crystal bunny of 69473 triangles. On surfaces leads to a novel real-time discrete shape operator. an NVidia GeForce7800, our method renders at 115 fps at an image resolution of 512x512. -
Caustic Architecture Article
Tricks'of'the'light' An#extended#version#of#an#article#in#New$Scientist,#30#January#2013# # Two#men#enter#the#darkened#stage,#apparently#carrying#a#thick#slab#of#badly# made#glass,#like#the#stuff#in#the#windows#of#old#houses#that#turns#the#world# outside#wobbly.#They#hold#it#up,#and#computer#scientist#Mark#Pauly#shines#a# torch#at#it.#The#crowd#in#this#Parisian#auditorium#gasps#and#then#breaks#into# spontaneous#applause.# For#there#on#the#screen#behind,#conjured#out#of#this#piece#of#nearFfeatureless# material#–#not#glass,#in#fact,#but#transparent#acrylic#plastic#(Perspex)#–#is#a# projected#image#of#Alan#Turing,#the#computer#pioneer#whose#centenary#is# celebrated#this#year.#Every#thread#of#his#thick#tweed#jacket#is#picked#out#in#light# and#shadow.#But#where#is#the#image#coming#from?#It#can#only#be#the#transparent# slab,#but#there#seems#to#be#nothing#there#to#produce#it,#nothing#but#a#slightly# uneven#surface.# # An#image#of#Alan#Turing#conjured#from#light#passing#through#a#slab#of#Perspex#at#the# Advances#in#Architectural#Geometry#conference#in#Paris,#September#2012.# This#image#is#made#from#rays#refracted,#folded#and#focused#by#the#slightly# uneven#surface#of#the#acrylic#block.#It’s#similar#to#the#filigree#of#bright#bands#seen# on#the#bottom#of#a#swimming#pool#in#the#sunlight,#called#a#caustic#and#caused#by# the#way#the#wavy#surface#refracts#and#focuses#light.#Caustics#are#familiar#enough,# but#they#never#looked#like#this#before.#Those#made#by#sunlight#shining#through# an#empty#glass#are#a#random#mass#of#cusps#and#squiggles.#Pauly,#a#specialist#in# -
Real-Time Caustics
EUROGRAPHICS 2003 / P. Brunet and D. Fellner Volume 22 (2003), Number 3 (Guest Editors) Real-Time Caustics M. Wand and W. Straßer WSI/GRIS, University of Tübingen Abstract We present a new algorithm to render caustics. The algorithm discretizes the specular surfaces into sample points. Each of the sample points is treated as a pinhole camera that projects an image of the incoming light onto the diffuse receiver surfaces. Anti-aliasing is performed by considering the local surface curvature at the sample points to filter the projected images. The algorithm can be implemented using programmable texture mapping hardware. It allows to render caustics in fully dynamic scenes in real-time on current PC hardware. Categories and Subject Descriptors: I.3.3 [Computer Graphics]: Picture / Image Generation – Display Algo- rithms; I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism 1. Introduction tion step has to be performed that is often even more ex- pensive. Thus, the technique is usually not very efficient. A real-time simulation of the interaction of light with Although the implementation techniques for raytracing complex, dynamically changing scenery is still one of the queries have made impressive advances in the last few major challenges in computer graphics. In this paper, we years29, raytracing based algorithms still need a consider- look at a special global illumination problem, rendering of able amount of computational power (such as a cluster of caustics. Caustics occur if light is reflected (or refracted) at several high end CPUs) to calculate global illumination one or more specular surfaces, focused into ray bundles of solutions in real time30. -
Adaptive Spectral Mapping for Real-Time Dispersive Refraction By
Adaptive Spectral Mapping for Real-Time Dispersive Refraction by Damon Blanchette A Thesis Submitted to the Faculty of the WORCESTER POLYTECHNIC INSTITUTE In partial fulfillment of the requirements for the Degree of Master of Science in Computer Science by ___________________________________ January 2012 APPROVED: ___________________________________ Professor Emmanuel Agu, Thesis Adviser ___________________________________ Professor Matthew Ward, Thesis Reader ___________________________________ Professor Craig Wills, Head of Department Abstract Spectral rendering, or the synthesis of images by taking into account the wavelengths of light, allows effects otherwise impossible with other methods. One of these effects is dispersion, the phenomenon that creates a rainbow when white light shines through a prism. Spectral rendering has previously remained in the realm of off-line rendering (with a few exceptions) due to the extensive computation required to keep track of individual light wavelengths. Caustics, the focusing and de-focusing of light through a refractive medium, can be interpreted as a special case of dispersion where all the wavelengths travel together. This thesis extends Adaptive Caustic Mapping, a previously proposed caustics mapping algorithm, to handle spectral dispersion. Because ACM can display caustics in real-time, it is quite amenable to be extended to handle the more general case of dispersion. A method is presented that runs in screen-space and is fast enough to display plausible dispersion phenomena in real-time at interactive frame rates. i Acknowledgments I would like to thank my adviser, Professor Emmanuel Agu, for his guidance, laughs, and answering my hundreds of questions over the year it took to complete this thesis. The beautiful video card helped, too. -
Free Complete Wasserstein Algebras
FREE COMPLETE WASSERSTEIN ALGEBRAS RADU MARDARE, PRAKASH PANANGADEN, AND GORDON D. PLOTKIN Abstract. We present an algebraic account of the Wasserstein distances Wp on complete metric spaces, for p ≥ 1. This is part of a program of a quantitative algebraic theory of effects in programming languages. In particular, we give axioms, parametric in p, for algebras over metric spaces equipped with probabilistic choice operations. The axioms say that the operations form a barycentric algebra and that the metric satisfies a property typical of the Wasserstein distance Wp. We show that the free complete such algebra over a complete metric space is that of the Radon probability measures with finite moments of order p, equipped with the Wasserstein distance as metric and with the usual binary convex sums as operations. 1. Introduction The denotational semantics of probabilistic programs generally makes use of one or another monad of probability measures. In [18, 10], Lawvere and Giry provided the first example of such a monad, over the category of measurable spaces; Giry also gave a second example, over Polish spaces. Later, in [15, 14], Jones and Plotkin provided a monad of evaluations over directed complete posets. More recently, in [12], Heunen, Kammar, et al provided such a monad over quasi-Borel spaces. Metric spaces (and nonexpansive maps) form another natural category to consider. The question then arises as to what metric to take over probability measures, and the Kantorovich-Wasserstein, or, more generally, the Wasserstein distances of order p > 1 (generalising from p = 1), provide natural choices. A first result of this kind was given in [2], where it was shown that the Kantorovich distance provides a monad over the sub- category of 1-bounded complete separable metric spaces. -
Statistical Aspects of Wasserstein Distances
Statistical Aspects of Wasserstein Distances Victor M. Panaretos∗and Yoav Zemely June 8, 2018 Abstract Wasserstein distances are metrics on probability distributions inspired by the problem of optimal mass transportation. Roughly speaking, they measure the minimal effort required to reconfigure the probability mass of one distribution in order to recover the other distribution. They are ubiquitous in mathematics, with a long history that has seen them catal- yse core developments in analysis, optimization, and probability. Beyond their intrinsic mathematical richness, they possess attractive features that make them a versatile tool for the statistician: they can be used to de- rive weak convergence and convergence of moments, and can be easily bounded; they are well-adapted to quantify a natural notion of pertur- bation of a probability distribution; and they seamlessly incorporate the geometry of the domain of the distributions in question, thus being useful for contrasting complex objects. Consequently, they frequently appear in the development of statistical theory and inferential methodology, and have recently become an object of inference in themselves. In this review, we provide a snapshot of the main concepts involved in Wasserstein dis- tances and optimal transportation, and a succinct overview of some of their many statistical aspects. Keywords: deformation map, empirical optimal transport, Fr´echet mean, goodness-of-fit, inference, Monge{Kantorovich problem, optimal coupling, prob- ability metric, transportation of measure, warping and registration, Wasserstein space AMS subject classification: 62-00 (primary); 62G99, 62M99 (secondary) arXiv:1806.05500v3 [stat.ME] 9 Apr 2019 1 Introduction Wasserstein distances are metrics between probability distributions that are inspired by the problem of optimal transportation. -
Advanced Computer Graphics CS 563: Adaptive Caustic Maps Using Deferred Shading
Advanced Computer Graphics CS 563: Adaptive Caustic Maps Using Deferred Shading Frederik Clinckemaillie Computer Science Dept. Worcester Polytechnic Institute (WPI) ItIntrod ucti on: CtiCaustics Reflective Caustics Refractive Caustics IdiIntroduction: CiCaustic MiMapping Much faster than path tracing algorithms Two‐pass process Similar to photon mapping Creates a caustic intensity map CtiCaustic MiMapping Three Step Process 1) Photon Emission 2) Rearrangement into Caustic Map 3) Caustic Map Projection CtiCaustic MiMapping Phot on EiiEmission Rasterizes from light view to generate grid of photons Creates photon buffer 2D image storing final photon hit points In conjunction with shadow maps Allows quick lookups to determine Indirect lighting from caustics CtiCaustic Map CtiCreation Controls lighting quality and cost Splatting photons into caustic map becomes bo ttlenec k Crisp noise‐free images require millions of phthotons Not feasible in interactive time Hierarchical caustic maps: discard unimportant parts of photon buffer to improve speed Uses multi‐resolution caustic map to reduce splatting costs Problems Poor photon sampling due to rasterization leads to under and over sampling Proper sampling resolution cannot be determined Millions of photons are required for high‐quality caustics. Processing each is too expensive Photon sampling location can change between frames, leading to coherency problems Photon sampling is not dynamic DfDeferre d Shadi ng Good sampling rates cannot be computed Ideally, number of photons is determined adtildaptively Hierarchical Caustic Map(HCM) Has a maximum number of photons, not all are processed Pho ton EiiEmission and CtiCaustic Map GtiGeneration should be coupled DfDeferre d Shadi ng Postpones final illumination computations until visible fragments are identified HCMs generates grid of phthotons and only processes relevant ones DfDeferre d Sha ding never generates ilirrelevan t photons. -
Controlling Caustics
Controlling Caustics Mark Pauly1, Michael Eigensatz, Philippe Bompas, Florian Rist2, Raimund Krenmuller2 1 EPFL 2 TU Wien Keywords 1 = Caustic 2 = Refraction 3 = Refl ection 4 = Design 5 = Milling 6 = Slumping Abstract Caustics are captivating light patterns created by materials focusing or diverting Figure 1 Left: A typical caustic created by a curved glass object appears chaotic and random. Right: Our light by refraction or refl ection. We know method creates glass objects that cast controlled caustics by optimization of the glass surface geometry alone. caustics as random side effects, appearing, The portrait of Alan Turing emerges from a beam of uniform light that is refracted by the circular glass piece. for example, at the bottom of a swimming pool, or generated by many glass objects, like drinking glasses or bottles. In this paper we show that it is possible to control caustic patterns to form almost any desired shape by optimizing the geometry of the refl ective or refractive surface generating the caustic. A seemingly fl at glass window, for example, can produce the image of a person as a caustic pattern on the fl oor, generated solely by the sunlight entering through that window. We demonstrate how this surprising result offers a new perspective on light control and the use of caustics as an inspiring design element in architecture, product design and beyond. Several produced samples illustrate that physical realizations of such optimized geometry are feasible. 1 Introduction The interaction of light with glass plays an important role for the perception and functionality of many glass products. One of the most fascinating light phenomena with glass objects are caustics: light gets focused and diverted when passing through the glass, creating an intriguing pattern of varying light intensity. -
Optimal Transportation: Duality Theory
Optimal Transportation: Duality Theory David Gu Yau Mathematics Science Center Tsinghua University Computer Science Department Stony Brook University [email protected] August 15, 2020 David Gu (Stony Brook University) Computational Conformal Geometry August 15, 2020 1 / 56 Motivation David Gu (Stony Brook University) Computational Conformal Geometry August 15, 2020 2 / 56 Why dose DL work? Problem 1 What does a DL system really learn ? 2 How does a DL system learn ? Does it really learn or just memorize ? 3 How well does a DL system learn ? Does it really learn everything or have to forget something ? Till today, the understanding of deep learning remains primitive. David Gu (Stony Brook University) Computational Conformal Geometry August 15, 2020 3 / 56 Why does DL work? 1. What does a DL system really learn? Probability distributions on manifolds. 2. How does a DL system learn ? Does it really learn or just memorize ? Optimization in the space of all probability distributions on a manifold. A DL system both learns and memorizes. 3. How well does a DL system learn ? Does it really learn everything or have to forget something ? Current DL systems have fundamental flaws, mode collapsing. David Gu (Stony Brook University) Computational Conformal Geometry August 15, 2020 4 / 56 Manifold Distribution Principle We believe the great success of deep learning can be partially explained by the well accepted manifold distribution and the clustering distribution principles: Manifold Distribution A natural data class can be treated as a probability distribution defined on a low dimensional manifold embedded in a high dimensional ambient space. Clustering Distribution The distances among the probability distributions of subclasses on the manifold are far enough to discriminate them. -
Data Assimilation in Optimal Transport Framework
Data Assimilation in Optimal Transport Framework (DRAFT) Raktim Bhattacharya Aerospace Engineering, Texas A&M University. May 19, 2020 1 Mathematical Preliminaries n Definition 1.1. (Wasserstein distance) Consider the vectors x1, x2 ∈ R . Let P2(p1,p2) denote the collection of all probability measures p supported on the product space R2n, having finite second moment, with first marginal p1 and second marginal p2. The Wasserstein distance of order 2, denoted as W2, between two probability measures p1,p2, is defined as 1 2 2 , n W2(p1,p2) inf k x1 − x2 kℓ2(R ) dp(x1, x2) . (1) p∈P2(p1,p2) ZR2n Remark 1. Intuitively, Wasserstein distance equals the least amount of work needed to morph one distribu- tional shape to the other, and can be interpreted as the cost for Monge-Kantorovich optimal transportation plan [1]. The particular choice of ℓ2 norm with order 2 is motivated by [2]. Further, one can prove (p. 208, [1]) that W2 defines a metric on the manifold of PDFs. Proposition 1. The Wasserstein distance W2 between two multivariate Gaussians N (µ1, Σ1) and N (µ2, Σ2) in Rn is given by 1 2 2 2 W2 (N (µ1, Σ1), N (µ2, Σ2)) = kµ1 − µ2k + tr Σ1 +Σ2 − 2 Σ1Σ2 Σ1 . (2) p p Corollary 1. The Wasserstein distance between Gaussian distribution N (µ, Σ) and the Dirac delta function δ(x − µc) is given by, 2 2 W2 (N (µ, Σ),δ(x − µc)) = kµ − µck + tr (Σ) . (3) arXiv:2005.08670v1 [eess.SY] 15 May 2020 Proof. Defining the Dirac delta function as (see e.g., p.