Heat Kernel Signature

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Heat Kernel Signature"

Transcription

1 INTODUCTION Heat Kernel Signature Thomas Hörmann Informatics - Technische Universität ünchen Abstract Due to the increasing computational power and new low cost depth cameras the analysis of 3D shapes has become more popular during the last view years. A special challenge is provided by the comparison of non-rigid shapes. The Heat Kernel Signature is an approach, which is well suited for this kind of problem. Intoduction 3D shape matching is a common field in modern geometry processing. It is based on computational understanding geometry. A lot of methods has been proposed by various scientist to solve this problem, but there is no default choice algorithm yet, like the SIFT algorithm[] in the 2D Image processing. The Heat Kernel Signature (HKS)[2] is one of the concepts that face this challenge. The HKS has received a lot of attention in the last years. In 200 the SHREC benchmark proposed the heat kernel based algorithms as the methods with the best performance[3].. Point Signatures A point signature is a data set that represents specific features of a given point. They can be used to define some feature points on a manifold, detect similarities of two different manifolds or to describe the geometry. In 996 one of the first point signatures was proposed by Chua and Jarvis[4] for effictient feature point mapping. The idea was to directly map two feature point of different 3D shape, instead of the two step process, first detecting all feature points and afterwards map them by a matching algorithm like the RANSAC[5], known from image matching. This point signature describes for each point x a one dimensional function f : [0, 2π] R using the intersection of the manifold and a sphere with constant radius. Unfortunately this representation has some disadvantages. It is not stable against perturbations and it does not describe the shape or the point of the shape in a unique manner. Another approach is the global point signature given by Rustamov[6]. By using the eigenvalues λ i and eigenfunctions φ i of the Laplace Beltrami operator Given the ordering of the eigenvalues the Global Point Signature GPS is defined as GP S(p) = ( φ (p), λ φ i = λ i φ i. () 0 = λ 0 < λ < λ 2 <... (2) λ2 φ 2 (p), λ3 φ 3 (p),...). (3) The GPS has some useful properties like invariance under isometric mappings and uniqueness of the signature for every point in the manifold. But it is not stable under perturbations. This can result in a change of the sign for some eigenfunctions as shown on Figure. The last point signature that is mentioned here is the Wave Kernel Signature[7] that has a lot of similarities the the HKS, but is based on Schrödingers equation ψ (x, t) = i ψ(x, t). (4) t The Wave Kernel Signature is defined as WKS(E, x) = lim T T T 0 ψ E (x, t) 2 (5)

2 2 APPROACH Figure : This figure shows the 9th eigenfunction of the shape Victoria in different poses. The coloring goes from blue negative values to red positive values. We can clearly see the change of the sign of the two models. 2 Approach The heat operator H t is a function that maps an initial heat distribution f : R to the heat distribution for any time t. lim H t(f) = f (6) t 0 The function k t that satisfies the following equation is called the heat kernel. H t f(x) = k t (x, y)f(y)dy (7) In order to find a solution for k t we have to solve the heat equation first. u(x, t) = u(x, t). (8) t First of all we define the scalar product of the function space on manifolds as follows f, g := f(x)g(x)dx. (9) We know that the eigenfunctions of the Laplace Beltrami Operator are orthogonal so we can decompose every function on the manifold as u(x, t) = u(t, ), φ i ( ) φ i (x) (0) Solving the heat equation by using the decomposition we get the following formulas t u(x, t) = t u(t, ), φ i( ) φ i (x) () u(x, t) = u(t, ), φ i ( ) φ i (x) = u(t, ), φ i ( ) λ i φ i (x) (2) Comparing the coefficients brings us to the first order ordinary differential equation t u(t, ), φ i( ) φ i (x) = u(t, ), φ i ( ) λ i φ i (x) (3) 2

3 2 APPROACH Figure 2: The figure shows the heat distribution on a cat for two different initial values f = δ x and f 2 = δ x2 where x x 2 on the left. The right side shows the remaining heat at point x i over time to the correspondent cat from the left. which results in the solution u(t, ), φ i ( ) = d i e λit (4) and a closed form for u u(x, t) = d i e λit φ i (x). (5) To define the unknown d i we use the initial value of f for t 0 = 0. So we get lim u(x, t) = d i φ i (x) = f (6) t 0 d i = f, φ i (7) ( ) φ i (y)f(y)dy e λit φ i (x) = e λit φ i (x)φ i (y)f(y)dy = By comparision we get a solution for the heat kernel k t (x, y) = k t (x, y)f(y)dy (8) e λit φ i (x)φ i (y) (9) The heat kernel itself can be thought of the heat distribution over the manifold with put a single heat source onto the point y at time t 0 = 0. Figure 2 shows two different heat distributions computed with the heat kernel. Also the heat kernel fully characterizes the shape up to isoemetry[8] but holds a lot of redundant information[2]. 3

4 2. Heat Kernel Signature 3 EXPERIENTS 2. Heat Kernel Signature The Heat Kernel Signature is defined as the remaining heat at the point x after time t and initial distribution f = δ x. HKS(x, t) = k t (x, x) = e λit φ 2 i (x) (20) By removing most of the information and only keeping the remaining heat at one point over the time the Heat Kernel Signature stays informative[2]. Because the heat kernel describes the heat diffusion the HKS is also stable against noise and change of topology to some extent. In comparison to the GPS the HKS has additional advantages. The HKS is a weighted sum over the squares of the eigenfunctions, so it is neither sensitive to the order of the eigenfunctions nor the sign. By using only predefined time intervals the HKS can also be used for multiscale matching. This can be easily explained by the fact that the heat only distribute to a finite surface at small time intervals. Figure 3: Color plot of the difference between the HKS using eulcedean distance. The difference increases as the color changes from blue to green to red. left: 0 sampling points right: 50 sampling points Figure 4: Color plot of the difference between the HKS using the scaled HKS. The difference increases as the color changes from blue to green to red. left: 5 sampling points right: 0 sampling points 3 Experiments The eigenfunctions of the Laplace beltrami Operator are invariant under isometric transformations despite their sign. Figure show the 9th eigenfunction of Victoria in two different poses. Since the heat kernel signature is a weighted sum of the square of the eigenfunctions it is not sensitive to the sign or the order of the eigenfunctions. The HKS also represents the curvature of the shape in some sense. The heat distributes slower on regions with positive curvature and faster on regions with negative curvature. So we can approximate for some t the curvature of the shape[2]. 3. Comparing two Signatures In computer science we can compare two signatures by sampling them into discrete function values and compare those vectors by some norm. The first approach is using for the HKS a vector with 4

5 3.2 ulti Scale Feature 4 SUARY equidistant sample values and compare them by a norm like the Euclidean distance. ( N ) d(x, x ) = k ti (x, x) k ti (x, x 2 ), tk+ t k = const (2) Figure 3 shows the comparison of one point on the surface to each other. Since the HKS is a monotonic function and the derivate is decreasing equidistant samplings may not be the best way to compare two signatures. There is less change of the HKS for bigger t and also the difference of two samples decreases, because the HKS converges to lim t k t (x, x) = area() for all x. ( N d(x, x k ti (x, x) k ti (x, x ) 2 ) ) =, k log(tk+ ) log(t k ) = const (22) t(y, y)dy where k t(x, x)dx is called the heat trace and can be computed as k t (x, x)dx = e λit. (23) The function SHKS based on the scaled distance function is called scaled Heat Kernel Signature. SHKS(x, t) = k t(x, x) e λit (24) One of the big advantages of the scaled Heat Kernel Signature is, that we need less sampling points. Figure 5: The figure shows two scaled signatures of of a cat. Figure 5 shows the distance using the scaled HKS from one point to each other. Compared to the Euclidean distance we achieve almost the same results with 5 sampling points of the SHKS against the Euclidean distance with 50 sampling points. 3.2 ulti Scale Feature Another property of the heat kernel signature is, that it can compare multi scale shapes. Since the HKS is based on heat distribution it is well suited for comparing local structure. Using for small t the HKS describes the local structure of the shape. For a partial shape the eigenfunctions can be completely different but the HKS stays the same for a small interval [t, t 2 ]. On Figure 6 we can see that the heat distribution for the full and the partial shape looks almost the same. Also the error plot indicates a rising error only for large t. 4 Summary The HKS can be used for multi scale shape matching among different shapes. It has been shown, that the HKS is powerful tool for multi scale shape matching. For many different types in shape geometry Heat Kernel Signature based methods can be used[3]. However there are still some disadvantages. The HKS is not invariant under different scales of the same shape. Also the eigenfunctions with low eigenvalues are taken more into account, but eigenfunctions with high eigenvalues still carry some informations about the shape. 5

6 REFERENCES REFERENCES Figure 6: The figure shows the heat distribution of the same point for time t = 400. We can see, that the error rises when the heat arrives the edge. left: full shape, middle: only left hand, right: error plot References [] Lowe, D.G.: Distinctive image features from scale-invariant keypoints. International journal of computer vision 60 (2004) 9 0 [2] Sun, J., Ovsjanikov,., Guibas, L.: A concise and provably informative multi-scale signature based on heat diffusion. In: Computer Graphics Forum. Volume 28., Wiley Online Library (2009) [3] Bronstein, A., Bronstein,., Bustos, B., Castellani, U., Crisani,., Falcidieno, B., Guibas, L., Kokkinos, I., urino, V., Ovsjanikov,., et al.: Shrec 200: robust feature detection and description benchmark. Proc. 3DOR 2 (200) 6 [4] Chua, C.S., Jarvis, R.: Point signatures: A new representation for 3d object recognition. International Journal of Computer Vision 25 (996) [5] Bolles, R.C., Fischler,.A.: A ransac-based approach to model fitting and its application to finding cylinders in range data. In: IJCAI. Volume 98. (98) [6] Rustamov, R..: Laplace-beltrami eigenfunctions for deformation invariant shape representation. In: Proceedings of the fifth Eurographics symposium on Geometry processing, Eurographics Association (2007) [7] Aubry,., Schlickewei, U., Cremers, D.: The wave kernel signature: A quantum mechanical approach to shape analysis. In: Computer Vision Workshops (ICCV Workshops), 20 IEEE International Conference on, IEEE (20) [8] Hsu, E.P.: Stochastic analysis on manifolds. Volume 38. American athematical Soc. (2002) 6

The Wave Kernel Signature: A Quantum Mechanical Approach to Shape Analysis

The Wave Kernel Signature: A Quantum Mechanical Approach to Shape Analysis The Wave Kernel Signature: A Quantum Mechanical Approach to Shape Analysis Mathieu Aubry, Ulrich Schlickewei and Daniel Cremers Department of Computer Science, TU München Abstract We introduce the Wave

More information

Shape Google: a computer vision approach to invariant shape retrieval

Shape Google: a computer vision approach to invariant shape retrieval Shape Google: a computer vision approach to invariant shape retrieval Maks Ovsjanikov Dept. of Computer Science Stanford University maks@stanford.edu Alexander M. Bronstein Dept. of Computer Science Technion

More information

3D Human Face Recognition Using Point Signature

3D Human Face Recognition Using Point Signature 3D Human Face Recognition Using Point Signature Chin-Seng Chua, Feng Han, Yeong-Khing Ho School of Electrical and Electronic Engineering Nanyang Technological University, Singapore 639798 ECSChua@ntu.edu.sg

More information

THE DIFFUSION EQUATION

THE DIFFUSION EQUATION THE DIFFUSION EQUATION R. E. SHOWALTER 1. Heat Conduction in an Interval We shall describe the diffusion of heat energy through a long thin rod G with uniform cross section S. As before, we identify G

More information

Centroid Distance Function and the Fourier Descriptor with Applications to Cancer Cell Clustering

Centroid Distance Function and the Fourier Descriptor with Applications to Cancer Cell Clustering Centroid Distance Function and the Fourier Descriptor with Applications to Cancer Cell Clustering By, Swati Bhonsle Alissa Klinzmann Mentors Fred Park Department of Mathematics Ernie Esser Department of

More information

Clustering and Data Mining in R

Clustering and Data Mining in R Clustering and Data Mining in R Workshop Supplement Thomas Girke December 10, 2011 Introduction Data Preprocessing Data Transformations Distance Methods Cluster Linkage Hierarchical Clustering Approaches

More information

Section 1.1. Introduction to R n

Section 1.1. Introduction to R n The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

More information

BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBERT SPACE REVIEW BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

More information

Efficient visual search of local features. Cordelia Schmid

Efficient visual search of local features. Cordelia Schmid Efficient visual search of local features Cordelia Schmid Visual search change in viewing angle Matches 22 correct matches Image search system for large datasets Large image dataset (one million images

More information

Fourier Series. A Fourier series is an infinite series of the form. a + b n cos(nωx) +

Fourier Series. A Fourier series is an infinite series of the form. a + b n cos(nωx) + Fourier Series A Fourier series is an infinite series of the form a b n cos(nωx) c n sin(nωx). Virtually any periodic function that arises in applications can be represented as the sum of a Fourier series.

More information

arxiv:math/0010057v1 [math.dg] 5 Oct 2000

arxiv:math/0010057v1 [math.dg] 5 Oct 2000 arxiv:math/0010057v1 [math.dg] 5 Oct 2000 HEAT KERNEL ASYMPTOTICS FOR LAPLACE TYPE OPERATORS AND MATRIX KDV HIERARCHY IOSIF POLTEROVICH Preliminary version Abstract. We study the heat kernel asymptotics

More information

Trace Formula Lite: All you need is the Pythagorean theorem and the volume of a cylinder

Trace Formula Lite: All you need is the Pythagorean theorem and the volume of a cylinder Trace Formula Lite: All you need is the Pythagorean theorem and the volume of a cylinder Peter G. Doyle DRAFT Version dated 23 July 2004 UNDER CONSTRUCTION GNU FDL The Laplace spectrum via elementary geometry

More information

1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is:

1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is: CONVERGENCE OF FOURIER SERIES 1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is: with coefficients given by: a n = 1 π f(x) a 0 2 + a n cos(nx) + b n sin(nx), n 1 f(x) cos(nx)dx

More information

Determine whether the following lines intersect, are parallel, or skew. L 1 : x = 6t y = 1 + 9t z = 3t. x = 1 + 2s y = 4 3s z = s

Determine whether the following lines intersect, are parallel, or skew. L 1 : x = 6t y = 1 + 9t z = 3t. x = 1 + 2s y = 4 3s z = s Homework Solutions 5/20 10.5.17 Determine whether the following lines intersect, are parallel, or skew. L 1 : L 2 : x = 6t y = 1 + 9t z = 3t x = 1 + 2s y = 4 3s z = s A vector parallel to L 1 is 6, 9,

More information

APPLICATIONS AND REVIEW OF FOURIER TRANSFORM/SERIES (David Sandwell, Copyright, 2004)

APPLICATIONS AND REVIEW OF FOURIER TRANSFORM/SERIES (David Sandwell, Copyright, 2004) 1 APPLICATIONS AND REVIEW OF FOURIER TRANSFORM/SERIES (David Sandwell, Copyright, 2004) (Reference The Fourier Transform and its Application, second edition, R.N. Bracewell, McGraw-Hill Book Co., New York,

More information

Numerical Methods for Differential Equations

Numerical Methods for Differential Equations Numerical Methods for Differential Equations Course objectives and preliminaries Gustaf Söderlind and Carmen Arévalo Numerical Analysis, Lund University Textbooks: A First Course in the Numerical Analysis

More information

Research Problems in Finite Element Theory: Analysis, Geometry, and Application

Research Problems in Finite Element Theory: Analysis, Geometry, and Application Research Problems in Finite Element Theory: Analysis, Geometry, and Application Andrew Gillette Department of Mathematics University of Arizona Research Tutorial Group Presentation Slides and more info

More information

α = u v. In other words, Orthogonal Projection

α = u v. In other words, Orthogonal Projection Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v

More information

ART Extension for Description, Indexing and Retrieval of 3D Objects

ART Extension for Description, Indexing and Retrieval of 3D Objects ART Extension for Description, Indexing and Retrieval of 3D Objects Julien Ricard, David Coeurjolly, Atilla Baskurt LIRIS, FRE 2672 CNRS, Bat. Nautibus, 43 bd du novembre 98, 69622 Villeurbanne cedex,

More information

From Fourier Series to Fourier Integral

From Fourier Series to Fourier Integral From Fourier Series to Fourier Integral Fourier series for periodic functions Consider the space of doubly differentiable functions of one variable x defined within the interval x [ L/2, L/2]. In this

More information

Fourier Series Chapter 3 of Coleman

Fourier Series Chapter 3 of Coleman Fourier Series Chapter 3 of Coleman Dr. Doreen De eon Math 18, Spring 14 1 Introduction Section 3.1 of Coleman The Fourier series takes its name from Joseph Fourier (1768-183), who made important contributions

More information

MATH 461: Fourier Series and Boundary Value Problems

MATH 461: Fourier Series and Boundary Value Problems MATH 461: Fourier Series and Boundary Value Problems Chapter III: Fourier Series Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2015 fasshauer@iit.edu MATH 461 Chapter

More information

Introduction to the Finite Element Method

Introduction to the Finite Element Method Introduction to the Finite Element Method 09.06.2009 Outline Motivation Partial Differential Equations (PDEs) Finite Difference Method (FDM) Finite Element Method (FEM) References Motivation Figure: cross

More information

Subspace Analysis and Optimization for AAM Based Face Alignment

Subspace Analysis and Optimization for AAM Based Face Alignment Subspace Analysis and Optimization for AAM Based Face Alignment Ming Zhao Chun Chen College of Computer Science Zhejiang University Hangzhou, 310027, P.R.China zhaoming1999@zju.edu.cn Stan Z. Li Microsoft

More information

AB2.14: Heat Equation: Solution by Fourier Series

AB2.14: Heat Equation: Solution by Fourier Series AB2.14: Heat Equation: Solution by Fourier Series Consider the boundary value problem for the one-dimensional heat equation describing the temperature variation in a bar with the zero-temperature ends:

More information

Automatic 3D Reconstruction via Object Detection and 3D Transformable Model Matching CS 269 Class Project Report

Automatic 3D Reconstruction via Object Detection and 3D Transformable Model Matching CS 269 Class Project Report Automatic 3D Reconstruction via Object Detection and 3D Transformable Model Matching CS 69 Class Project Report Junhua Mao and Lunbo Xu University of California, Los Angeles mjhustc@ucla.edu and lunbo

More information

Chapter 41. One Dimensional Quantum

Chapter 41. One Dimensional Quantum Chapter 41. One Dimensional Quantum Mechanics Quantum effects are important in nanostructures such as this tiny sign built by scientists at IBM s research laboratory by moving xenon atoms around on a metal

More information

Face Recognition using SIFT Features

Face Recognition using SIFT Features Face Recognition using SIFT Features Mohamed Aly CNS186 Term Project Winter 2006 Abstract Face recognition has many important practical applications, like surveillance and access control.

More information

Inner product. Definition of inner product

Inner product. Definition of inner product Math 20F Linear Algebra Lecture 25 1 Inner product Review: Definition of inner product. Slide 1 Norm and distance. Orthogonal vectors. Orthogonal complement. Orthogonal basis. Definition of inner product

More information

11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space

11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space 11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of

More information

Stable Topological Signatures for Points on 3D Shapes

Stable Topological Signatures for Points on 3D Shapes Eurographics Symposium on Geometry Processing 2015 Mirela Ben-Chen and Ligang Liu (Guest Editors) Volume 34 (2015), Number 5 Stable Topological Signatures for Points on 3D Shapes Mathieu Carrière 1 and

More information

Midterm Exam I, Calculus III, Sample A

Midterm Exam I, Calculus III, Sample A Midterm Exam I, Calculus III, Sample A 1. (1 points) Show that the 4 points P 1 = (,, ), P = (, 3, ), P 3 = (1, 1, 1), P 4 = (1, 4, 1) are coplanar (they lie on the same plane), and find the equation of

More information

2. Norm, distance, angle

2. Norm, distance, angle L. Vandenberghe EE133A (Spring 2016) 2. Norm, distance, angle norm distance angle hyperplanes complex vectors 2-1 Euclidean norm (Euclidean) norm of vector a R n : a = a 2 1 + a2 2 + + a2 n = a T a if

More information

Fourier Series and Fejér s Theorem

Fourier Series and Fejér s Theorem wwu@ocf.berkeley.edu June 4 Introduction : Background and Motivation A Fourier series can be understood as the decomposition of a periodic function into its projections onto an orthonormal basis. More

More information

POISSON AND LAPLACE EQUATIONS. Charles R. O Neill. School of Mechanical and Aerospace Engineering. Oklahoma State University. Stillwater, OK 74078

POISSON AND LAPLACE EQUATIONS. Charles R. O Neill. School of Mechanical and Aerospace Engineering. Oklahoma State University. Stillwater, OK 74078 21 ELLIPTICAL PARTIAL DIFFERENTIAL EQUATIONS: POISSON AND LAPLACE EQUATIONS Charles R. O Neill School of Mechanical and Aerospace Engineering Oklahoma State University Stillwater, OK 74078 2nd Computer

More information

A numerical solution of Nagumo telegraph equation by Adomian decomposition method

A numerical solution of Nagumo telegraph equation by Adomian decomposition method Mathematics Scientific Journal Vol. 6, No. 2, S. N. 13, (2011), 73-81 A numerical solution of Nagumo telegraph equation by Adomian decomposition method H. Rouhparvar a,1 a Department of Mathematics, Islamic

More information

Lorentzian Quantum Einstein Gravity

Lorentzian Quantum Einstein Gravity Lorentzian Quantum Einstein Gravity Stefan Rechenberger Uni Mainz 12.09.2011 Phys.Rev.Lett. 106 (2011) 251302 with Elisa Manrique and Frank Saueressig Stefan Rechenberger (Uni Mainz) Lorentzian Quantum

More information

Image Segmentation and Registration

Image Segmentation and Registration Image Segmentation and Registration Dr. Christine Tanner (tanner@vision.ee.ethz.ch) Computer Vision Laboratory, ETH Zürich Dr. Verena Kaynig, Machine Learning Laboratory, ETH Zürich Outline Segmentation

More information

ADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB. Sohail A. Dianat. Rochester Institute of Technology, New York, U.S.A. Eli S.

ADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB. Sohail A. Dianat. Rochester Institute of Technology, New York, U.S.A. Eli S. ADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB Sohail A. Dianat Rochester Institute of Technology, New York, U.S.A. Eli S. Saber Rochester Institute of Technology, New York, U.S.A. (g) CRC Press Taylor

More information

Inner Product Spaces

Inner Product Spaces Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

More information

Improved Billboard Clouds for Extreme Model Simplification

Improved Billboard Clouds for Extreme Model Simplification Improved Billboard Clouds for Extreme Model Simplification I.-T. Huang, K. L. Novins and B. C. Wünsche Graphics Group, Department of Computer Science, University of Auckland, Private Bag 92019, Auckland,

More information

Structural Matching of 2D Electrophoresis Gels using Graph Models

Structural Matching of 2D Electrophoresis Gels using Graph Models Structural Matching of 2D Electrophoresis Gels using Graph Models Alexandre Noma 1, Alvaro Pardo 2, Roberto M. Cesar-Jr 1 1 IME-USP, Department of Computer Science, University of São Paulo, Brazil 2 DIE,

More information

Quantum Fields in Curved Spacetime

Quantum Fields in Curved Spacetime Quantum Fields in Curved Spacetime Lecture 1: Introduction Finn Larsen Michigan Center for Theoretical Physics Yerevan, August 20, 2016. Intro: Fields Setting: many microscopic degrees of freedom interacting

More information

Math 241, Exam 1 Information.

Math 241, Exam 1 Information. Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

More information

A Simple Introduction to Finite Element Analysis

A Simple Introduction to Finite Element Analysis A Simple Introduction to Finite Element Analysis Allyson O Brien Abstract While the finite element method is extensively used in theoretical and applied mathematics and in many engineering disciplines,

More information

Geometric Algebra Computing Analysis of point clouds 27.11.2014 Dr. Dietmar Hildenbrand

Geometric Algebra Computing Analysis of point clouds 27.11.2014 Dr. Dietmar Hildenbrand Geometric Algebra Computing Analysis of point clouds 27.11.2014 Dr. Dietmar Hildenbrand Technische Universität Darmstadt Literature Book Foundations of Geometric Algebra Computing, Dietmar Hildenbrand

More information

Mean-Shift Tracking with Random Sampling

Mean-Shift Tracking with Random Sampling 1 Mean-Shift Tracking with Random Sampling Alex Po Leung, Shaogang Gong Department of Computer Science Queen Mary, University of London, London, E1 4NS Abstract In this work, boosting the efficiency of

More information

Differential Geometry

Differential Geometry )NPUT $ATA 2ANGE 3CAN #!$ 4OMOGRAPHY 2EMOVAL OF TOPOLOGICAL AND GEOMETRICAL ERRORS!NALYSIS OF SURFACE QUALITY 3URFACE SMOOTHING FOR NOISE REMOVAL Differential Geometry 0ARAMETERIZATION 3IMPLIFICATION FOR

More information

Multimedia Databases. Wolf-Tilo Balke Philipp Wille Institut für Informationssysteme Technische Universität Braunschweig http://www.ifis.cs.tu-bs.

Multimedia Databases. Wolf-Tilo Balke Philipp Wille Institut für Informationssysteme Technische Universität Braunschweig http://www.ifis.cs.tu-bs. Multimedia Databases Wolf-Tilo Balke Philipp Wille Institut für Informationssysteme Technische Universität Braunschweig http://www.ifis.cs.tu-bs.de 14 Previous Lecture 13 Indexes for Multimedia Data 13.1

More information

Metropolis Light Transport. Samuel Donow, Mike Flynn, David Yan CS371 Fall 2014, Morgan McGuire

Metropolis Light Transport. Samuel Donow, Mike Flynn, David Yan CS371 Fall 2014, Morgan McGuire Metropolis Light Transport Samuel Donow, Mike Flynn, David Yan CS371 Fall 2014, Morgan McGuire Overview of Presentation 1. Description of necessary tools (Path Space, Monte Carlo Integration, Rendering

More information

CHAPTER 5 THE HARMONIC OSCILLATOR

CHAPTER 5 THE HARMONIC OSCILLATOR CHAPTER 5 THE HARMONIC OSCILLATOR The harmonic oscillator is a model which has several important applications in both classical and quantum mechanics. It serves as a prototype in the mathematical treatment

More information

IEEE floating point numbers. Floating-Point Representation and Approximation. floating point numbers with base 10. Errors

IEEE floating point numbers. Floating-Point Representation and Approximation. floating point numbers with base 10. Errors EE103 (Shinnerl) Floating-Point Representation and Approximation Errors Cancellation Instability Simple one-variable examples Swamping IEEE floating point numbers floating point numbers with base 10 floating

More information

3. INNER PRODUCT SPACES

3. INNER PRODUCT SPACES . INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

More information

A Introduction to Matrix Algebra and Principal Components Analysis

A Introduction to Matrix Algebra and Principal Components Analysis A Introduction to Matrix Algebra and Principal Components Analysis Multivariate Methods in Education ERSH 8350 Lecture #2 August 24, 2011 ERSH 8350: Lecture 2 Today s Class An introduction to matrix algebra

More information

Face Recognition in Low-resolution Images by Using Local Zernike Moments

Face Recognition in Low-resolution Images by Using Local Zernike Moments Proceedings of the International Conference on Machine Vision and Machine Learning Prague, Czech Republic, August14-15, 014 Paper No. 15 Face Recognition in Low-resolution Images by Using Local Zernie

More information

We call this set an n-dimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.

We call this set an n-dimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P. Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to

More information

15.062 Data Mining: Algorithms and Applications Matrix Math Review

15.062 Data Mining: Algorithms and Applications Matrix Math Review .6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop

More information

APPLICATIONS AND REVIEW OF FOURIER TRANSFORM/SERIES (Copyright 2001, David T. Sandwell)

APPLICATIONS AND REVIEW OF FOURIER TRANSFORM/SERIES (Copyright 2001, David T. Sandwell) 1 APPLICATIONS AND REVIEW OF FOURIER TRANSFORM/SERIES (Copyright 2001, David T. Sandwell) (Reference The Fourier Transform and its Application, second edition, R.N. Bracewell, McGraw-Hill Book Co., New

More information

Sharp counterexamples in unique continuation for second order elliptic equations

Sharp counterexamples in unique continuation for second order elliptic equations Sharp counterexamples in unique continuation for second order elliptic equations Herbert Koch Institut für Angewandte Mathemati Universität Heidelberg Daniel Tataru Department of Mathematics Northwestern

More information

Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model

Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written

More information

Interactive Math Glossary Terms and Definitions

Interactive Math Glossary Terms and Definitions Terms and Definitions Absolute Value the magnitude of a number, or the distance from 0 on a real number line Additive Property of Area the process of finding an the area of a shape by totaling the areas

More information

EECS 556 Image Processing W 09. Interpolation. Interpolation techniques B splines

EECS 556 Image Processing W 09. Interpolation. Interpolation techniques B splines EECS 556 Image Processing W 09 Interpolation Interpolation techniques B splines What is image processing? Image processing is the application of 2D signal processing methods to images Image representation

More information

Sine and Cosine Series; Odd and Even Functions

Sine and Cosine Series; Odd and Even Functions Sine and Cosine Series; Odd and Even Functions A sine series on the interval [, ] is a trigonometric series of the form k = 1 b k sin πkx. All of the terms in a series of this type have values vanishing

More information

Mesh Smoothing. Mark Pauly )NPUT $ATA 2EMOVAL OF TOPOLOGICAL AND GEOMETRICAL ERRORS !NALYSIS OF SURFACE QUALITY 3URFACE SMOOTHING FOR NOISE REMOVAL

Mesh Smoothing. Mark Pauly )NPUT $ATA 2EMOVAL OF TOPOLOGICAL AND GEOMETRICAL ERRORS !NALYSIS OF SURFACE QUALITY 3URFACE SMOOTHING FOR NOISE REMOVAL )NPUT $ATA 2ANGE 3CAN #!$ 4OMOGRAPHY 2EMOVAL OF TOPOLOGICAL AND GEOMETRICAL ERRORS!NALYSIS OF SURFACE QUALITY 3URFACE SMOOTHING FOR NOISE REMOVAL Mesh Smoothing 0ARAMETERIZATION Mark Pauly 3IMPLIFICATION

More information

Mathematics (MAT) MAT 061 Basic Euclidean Geometry 3 Hours. MAT 051 Pre-Algebra 4 Hours

Mathematics (MAT) MAT 061 Basic Euclidean Geometry 3 Hours. MAT 051 Pre-Algebra 4 Hours MAT 051 Pre-Algebra Mathematics (MAT) MAT 051 is designed as a review of the basic operations of arithmetic and an introduction to algebra. The student must earn a grade of C or in order to enroll in MAT

More information

Expression Invariant 3D Face Recognition with a Morphable Model

Expression Invariant 3D Face Recognition with a Morphable Model Expression Invariant 3D Face Recognition with a Morphable Model Brian Amberg brian.amberg@unibas.ch Reinhard Knothe reinhard.knothe@unibas.ch Thomas Vetter thomas.vetter@unibas.ch Abstract We present an

More information

Vector Spaces; the Space R n

Vector Spaces; the Space R n Vector Spaces; the Space R n Vector Spaces A vector space (over the real numbers) is a set V of mathematical entities, called vectors, U, V, W, etc, in which an addition operation + is defined and in which

More information

Android Ros Application

Android Ros Application Android Ros Application Advanced Practical course : Sensor-enabled Intelligent Environments 2011/2012 Presentation by: Rim Zahir Supervisor: Dejan Pangercic SIFT Matching Objects Android Camera Topic :

More information

Some Comments on the Derivative of a Vector with applications to angular momentum and curvature. E. L. Lady (October 18, 2000)

Some Comments on the Derivative of a Vector with applications to angular momentum and curvature. E. L. Lady (October 18, 2000) Some Comments on the Derivative of a Vector with applications to angular momentum and curvature E. L. Lady (October 18, 2000) Finding the formula in polar coordinates for the angular momentum of a moving

More information

Chess Vision. Chua Huiyan Le Vinh Wong Lai Kuan

Chess Vision. Chua Huiyan Le Vinh Wong Lai Kuan Chess Vision Chua Huiyan Le Vinh Wong Lai Kuan Outline Introduction Background Studies 2D Chess Vision Real-time Board Detection Extraction and Undistortion of Board Board Configuration Recognition 3D

More information

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic

More information

Constructing Laplace Operator from Point Clouds in R d

Constructing Laplace Operator from Point Clouds in R d Constructing Laplace Operator from Point Clouds in R d Mikhail Belkin Jian Sun Yusu Wang Abstract We present an algorithm for approximating the Laplace-Beltrami operator from an arbitrary point cloud obtained

More information

AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS

AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS Revised Edition James Epperson Mathematical Reviews BICENTENNIAL 0, 1 8 0 7 z ewiley wu 2007 r71 BICENTENNIAL WILEY-INTERSCIENCE A John Wiley & Sons, Inc.,

More information

R U S S E L L L. H E R M A N

R U S S E L L L. H E R M A N R U S S E L L L. H E R M A N A N I N T R O D U C T I O N T O F O U R I E R A N D C O M P L E X A N A LY S I S W I T H A P P L I C AT I O N S T O T H E S P E C T R A L A N A LY S I S O F S I G N A L S R.

More information

Problems for Quiz 14

Problems for Quiz 14 Problems for Quiz 14 Math 3. Spring, 7. 1. Consider the initial value problem (IVP defined by the partial differential equation (PDE u t = u xx u x + u, < x < 1, t > (1 with boundary conditions and initial

More information

Lecture VI. Review of even and odd functions Definition 1 A function f(x) is called an even function if. f( x) = f(x)

Lecture VI. Review of even and odd functions Definition 1 A function f(x) is called an even function if. f( x) = f(x) ecture VI Abstract Before learning to solve partial differential equations, it is necessary to know how to approximate arbitrary functions by infinite series, using special families of functions This process

More information

JUST THE MATHS UNIT NUMBER 8.5. VECTORS 5 (Vector equations of straight lines) A.J.Hobson

JUST THE MATHS UNIT NUMBER 8.5. VECTORS 5 (Vector equations of straight lines) A.J.Hobson JUST THE MATHS UNIT NUMBER 8.5 VECTORS 5 (Vector equations of straight lines) by A.J.Hobson 8.5.1 Introduction 8.5. The straight line passing through a given point and parallel to a given vector 8.5.3

More information

Classifying Manipulation Primitives from Visual Data

Classifying Manipulation Primitives from Visual Data Classifying Manipulation Primitives from Visual Data Sandy Huang and Dylan Hadfield-Menell Abstract One approach to learning from demonstrations in robotics is to make use of a classifier to predict if

More information

AB2.2: Curves. Gradient of a Scalar Field

AB2.2: Curves. Gradient of a Scalar Field AB2.2: Curves. Gradient of a Scalar Field Parametric representation of a curve A a curve C in space can be represented by a vector function r(t) = [x(t), y(t), z(t)] = x(t)i + y(t)j + z(t)k This is called

More information

Recovering Primitives in 3D CAD meshes

Recovering Primitives in 3D CAD meshes Recovering Primitives in 3D CAD meshes Roseline Bénière a,c, Gérard Subsol a, Gilles Gesquière b, François Le Breton c and William Puech a a LIRMM, Univ. Montpellier 2, CNRS, 161 rue Ada, 34392, France;

More information

A Study on SURF Algorithm and Real-Time Tracking Objects Using Optical Flow

A Study on SURF Algorithm and Real-Time Tracking Objects Using Optical Flow , pp.233-237 http://dx.doi.org/10.14257/astl.2014.51.53 A Study on SURF Algorithm and Real-Time Tracking Objects Using Optical Flow Giwoo Kim 1, Hye-Youn Lim 1 and Dae-Seong Kang 1, 1 Department of electronices

More information

Epipolar Geometry and Stereo Vision

Epipolar Geometry and Stereo Vision 04/12/11 Epipolar Geometry and Stereo Vision Computer Vision CS 543 / ECE 549 University of Illinois Derek Hoiem Many slides adapted from Lana Lazebnik, Silvio Saverese, Steve Seitz, many figures from

More information

Computer Vision - part II

Computer Vision - part II Computer Vision - part II Review of main parts of Section B of the course School of Computer Science & Statistics Trinity College Dublin Dublin 2 Ireland www.scss.tcd.ie Lecture Name Course Name 1 1 2

More information

Liouville Quantum gravity and KPZ. Scott Sheffield

Liouville Quantum gravity and KPZ. Scott Sheffield Liouville Quantum gravity and KPZ Scott Sheffield Scaling limits of random planar maps Central mathematical puzzle: Show that the scaling limit of some kind of discrete quantum gravity (perhaps decorated

More information

Sequences and Convergence in Metric Spaces

Sequences and Convergence in Metric Spaces Sequences and Convergence in Metric Spaces Definition: A sequence in a set X (a sequence of elements of X) is a function s : N X. We usually denote s(n) by s n, called the n-th term of s, and write {s

More information

ACCURACY ASSESSMENT OF BUILDING POINT CLOUDS AUTOMATICALLY GENERATED FROM IPHONE IMAGES

ACCURACY ASSESSMENT OF BUILDING POINT CLOUDS AUTOMATICALLY GENERATED FROM IPHONE IMAGES ACCURACY ASSESSMENT OF BUILDING POINT CLOUDS AUTOMATICALLY GENERATED FROM IPHONE IMAGES B. Sirmacek, R. Lindenbergh Delft University of Technology, Department of Geoscience and Remote Sensing, Stevinweg

More information

3D Model based Object Class Detection in An Arbitrary View

3D Model based Object Class Detection in An Arbitrary View 3D Model based Object Class Detection in An Arbitrary View Pingkun Yan, Saad M. Khan, Mubarak Shah School of Electrical Engineering and Computer Science University of Central Florida http://www.eecs.ucf.edu/

More information

Computer Graphics. Geometric Modeling. Page 1. Copyright Gotsman, Elber, Barequet, Karni, Sheffer Computer Science - Technion. An Example.

Computer Graphics. Geometric Modeling. Page 1. Copyright Gotsman, Elber, Barequet, Karni, Sheffer Computer Science - Technion. An Example. An Example 2 3 4 Outline Objective: Develop methods and algorithms to mathematically model shape of real world objects Categories: Wire-Frame Representation Object is represented as as a set of points

More information

Hermitian Operators An important property of operators is suggested by considering the Hamiltonian for the particle in a box: d 2 dx 2 (1)

Hermitian Operators An important property of operators is suggested by considering the Hamiltonian for the particle in a box: d 2 dx 2 (1) CHAPTER 4 PRINCIPLES OF QUANTUM MECHANICS In this Chapter we will continue to develop the mathematical formalism of quantum mechanics, using heuristic arguments as necessary. This will lead to a system

More information

Practice Final Math 122 Spring 12 Instructor: Jeff Lang

Practice Final Math 122 Spring 12 Instructor: Jeff Lang Practice Final Math Spring Instructor: Jeff Lang. Find the limit of the sequence a n = ln (n 5) ln (3n + 8). A) ln ( ) 3 B) ln C) ln ( ) 3 D) does not exist. Find the limit of the sequence a n = (ln n)6

More information

Analecta Vol. 8, No. 2 ISSN 2064-7964

Analecta Vol. 8, No. 2 ISSN 2064-7964 EXPERIMENTAL APPLICATIONS OF ARTIFICIAL NEURAL NETWORKS IN ENGINEERING PROCESSING SYSTEM S. Dadvandipour Institute of Information Engineering, University of Miskolc, Egyetemváros, 3515, Miskolc, Hungary,

More information

Segmentation of building models from dense 3D point-clouds

Segmentation of building models from dense 3D point-clouds Segmentation of building models from dense 3D point-clouds Joachim Bauer, Konrad Karner, Konrad Schindler, Andreas Klaus, Christopher Zach VRVis Research Center for Virtual Reality and Visualization, Institute

More information

So which is the best?

So which is the best? Manifold Learning Techniques: So which is the best? Todd Wittman Math 8600: Geometric Data Analysis Instructor: Gilad Lerman Spring 2005 Note: This presentation does not contain information on LTSA, which

More information

SOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve

SOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve SOLUTIONS Problem. Find the critical points of the function f(x, y = 2x 3 3x 2 y 2x 2 3y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Partial derivatives

More information

Mean value theorem, Taylors Theorem, Maxima and Minima.

Mean value theorem, Taylors Theorem, Maxima and Minima. MA 001 Preparatory Mathematics I. Complex numbers as ordered pairs. Argand s diagram. Triangle inequality. De Moivre s Theorem. Algebra: Quadratic equations and express-ions. Permutations and Combinations.

More information

Persistent Heat Signature for Pose-oblivious Matching of Incomplete Models

Persistent Heat Signature for Pose-oblivious Matching of Incomplete Models Eurographics Symposium on Geometry Processing 2010 Olga Sorkine and Bruno Lévy (Guest Editors) Volume 29 (2010), Number 5 Persistent Heat Signature for Pose-oblivious Matching of Incomplete Models T. K.

More information

Learning Gaussian process models from big data. Alan Qi Purdue University Joint work with Z. Xu, F. Yan, B. Dai, and Y. Zhu

Learning Gaussian process models from big data. Alan Qi Purdue University Joint work with Z. Xu, F. Yan, B. Dai, and Y. Zhu Learning Gaussian process models from big data Alan Qi Purdue University Joint work with Z. Xu, F. Yan, B. Dai, and Y. Zhu Machine learning seminar at University of Cambridge, July 4 2012 Data A lot of

More information

An Introduction to Separation of Variables with Fourier Series Math 391w, Spring 2010 Tim McCrossen Professor Haessig

An Introduction to Separation of Variables with Fourier Series Math 391w, Spring 2010 Tim McCrossen Professor Haessig An Introduction to Separation of Variables with Fourier Series Math 391w, Spring 2010 Tim McCrossen Professor Haessig Abstract: This paper aims to give students who have not yet taken a course in partial

More information

Rate of convergence towards Hartree dynamics

Rate of convergence towards Hartree dynamics Rate of convergence towards Hartree dynamics Benjamin Schlein, LMU München and University of Cambridge Universitá di Milano Bicocca, October 22, 2007 Joint work with I. Rodnianski 1. Introduction boson

More information

ART Extension for Description, Indexing and Retrieval of a 3D Objects

ART Extension for Description, Indexing and Retrieval of a 3D Objects ART Extension for Description, Indexing and Retrieval of a 3D Objects Julien Ricard, David Coeurjolly, Atilla Baskurt To cite this version: Julien Ricard, David Coeurjolly, Atilla Baskurt. ART Extension

More information