Advanced Computer Graphics
|
|
- Byron Lang
- 6 years ago
- Views:
Transcription
1 G , Fall 2010 Advanced Computer Graphics Project details and tools 1
2 Projects Details of each project are on the website under Projects Please review all the projects and come see me if you would like to discuss details. 2
3 Project Topics Computer Animation Interactive Shape Modeling Computational Photography Image Processing 3
4 For projects involving 2D only: Code Tips See links on the Resources webpage! You can use MATLAB and/or OpenCV Not much UI involved For projects in 3D I recommend OpenGL rendering with GLUT For GUI you can use AntTweakBar or GLUI If familiar: QT or other GUI APIs (more heavy weight) Mesh libraries: see course website. Again, preference for light weight (trimesh). 4
5 Optimization Almost all projects have a global optimization component: Define some energy functional E(x) x is the shape, or the image Compute the optimal x such that E(x) is minimized There are some constraints to the minimization 5
6 Computer Animation Character animation Create a system for rigging and posing of digital characters (shapes) Step 1: compute a skeleton 6
7 Computer Animation Character animation Create a system for rigging and posing of digital characters (shapes) Step 1: compute a skeleton Suggestion: implement the paper Skeleton Extraction by Mesh Contraction, SIGGRAPH 2008 also implement GUI to manually create a skeleton 7
8 Computer Animation Character animation Function E(x) measures how smooth the shape x is E( x) = n x i w ij i= 1 j N( i) Constraints: extremities (finger tips, ears, etc.) x j 2 8
9 Geometric Modeling Interactive shape editing system Implement one of the recent papers: PriMo, SGP 2006 The objective functional E(x) says the details of the shape should be preserved as much as possible The constraints for the minimization are the userdefined fixed positions 9
10 Interactive Shape Modeling Sketch based modeling Inflation of 2D sketches into a 3D shape User draws the silhouette System meshes the interior System inflates the shape Objective E(x) measures how smooth the surface x is The constraint is the silhouette curve 10
11 Interactive Shape Modeling Sketch based painting: diffusion curves on surfaces Implement a system for 3D painting on surfaces Color from a curve is diffused by means of the Poisson equation E(x) says the color field x should be smooth Constraints: colors of the curves 11
12 Computational Photography Shaky cam stabilization Implement Content Preserving Warps for 3D Video Stabilization, SIGGRAPH
13 Computational Photography Shaky cam stabilization Implement Content Preserving Warps for 3D Video Stabilization, SIGGRAPH 2009 E(x) measures: How well the quads are preserved (want all angles to be 90) Constraints: point to point (match new projected position to the old position) 13
14 Image Processing Image retargeting (resizing) The problem: resize an image to fit a display device with a different aspect ratio 14
15 Image Processing Image retargeting (resizing) The problem: resize an image to fit a display device with a different aspect ratio 15
16 Approach: Image Processing Image retargeting (resizing) Compute an importance map of the image Warp the image such that regions with high importance are preserved at the expense of unimportant regions importance map 16
17 Image Processing Image retargeting (resizing) Grid mesh, preserve the shape of the important quads quads with high importance: uniform scaling quads with low importance: allowed non uniform scaling Optimize the location of mesh vertices, interpolate image 17
18 Image Processing Image retargeting (resizing) Grid mesh, preserve the shape of the important quads Optimize the location of mesh vertices, interpolate image 18
19 Image Processing Image retargeting (resizing) Grid mesh, preserve the shape of the important quads Optimize the location of mesh vertices, interpolate image 19
20 E(x) measures: How well the quad shape is preserved How smooth the grid lines are Constraints: The boundary of the image (has to fit new dimensions) Cropping constraints: all important content should remain in the resulting image Image Processing Image retargeting (resizing) 20
21 Optimization In the projects: mostly linear optimization, i.e. linear least squares problems 21
22 Solving linear systems in LS sense Wish list of equations: i=1,, n: f (x i ) = f i Example: measuring surface smoothness E( x) = n x i w ij i= 1 j N( i) x j 2 22
23 Solving linear systems in LS sense Wish list of equations: i=1,, n: f (x i ) = f i Example: measuring surface smoothness E( x) = n x i w ij i= 1 j N( i) x j 2 i = 1, K, n: x i w ij j N( i) x j = 0 Wish equation 23
24 Solving linear systems in LS sense We have an over determined linear system k n: a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2 a k1 x 1 + a k2 x a kn x n = b k 24
25 Solving linear systems in LS sense In matrix form: = k kn k k n n a a a a a a a a a b b b M K M M M K K K x n x x 2 1 M 9/20/ Olga Sorkine, Courant Institute
26 Solving linear systems in LS sense In matrix form: Ax = b where A = (a ij ) is a rectangular k n matrix, k > n x = (x 1, x 2,, x n ) T b = (b 1, b 2,, b k ) T 26
27 Solving linear systems in LS sense More constrains than variables no exact solutions generally exist We want to find something that is an approximate solution : x opt = argmin Ax b 2 x min E(x) x 27
28 Finding the LS solution x R n Ax R k As we vary x, Ax varies over the linear subspace of R k spanned by the columns of A: Ax = A 1 A 2 A n x 1 x x x n x n 28
29 Finding the LS solution We want to find the closest Ax to b: min Ax b 2 x Ax closest to b b Subspace spanned by columns of A R k 29
30 Finding the LS solution The vector Ax closest to b satisfies: (Ax b) {subspace of A s columns} column A i : A i, Ax b = 0 These are called the normal equations i, A i T (Ax b) = 0 A T (Ax b) = 0 (A T A)x = A T b 30
31 Finding the LS solution We got a square symmetric system (A T A)x = A T b (n n) If A has full rank (the columns of A are linearly independent) then (A T A) is invertible. min x x = Ax b 2 ( T ) 1 T A A A b 31
32 Weighted least squares If each constraint has a weight in the energy: The weights w i > 0 and don t depend on x Then: min x n i= 1 wi (f i ( x i ) b min (Ax b) T W(Ax b) where W = (w i ) ii (A T W A) x = A T W b i 2 ) 32
33 Linear Systems Matrix is often fixed, rhs changes M x = r A T A A T b 33
34 Matrix Factorization LU decomposition M = L U Mx = r LUx= r 34
35 Matrix Factorization LU decomposition M = L U Mx = r L(Ux) = r 35
36 Matrix Factorization LU decomposition M = L U Mx = r L(Ux) = r Ly = r Ux= y This is backsubstitution. If L, U are sparse it is very fast. The hard work is computing L and U 36
37 Matrix Factorization LU decomposition M = L U Mx = r L(Ux) = r y = L 1 r x = U 1 y This is backsubstitution. If L, U are sparse it is very fast. The hard work is computing L and U 37
38 Matrix Factorization Cholesky decomposition M = L L T Cholesky factor exists if M is positive definite. It is even better than LU because we save memory. 38
39 Cholesky Decomposition M = LL T M is symmetric positive definite (SPD): x 0, Mx, x > 0 all M s eigenvalues > 0 M = L L T 39
40 Dense Cholesky Factorization M = LL T matrix 3500 nonzeros L 36k nonzeros Cholesky Factorization 40
41 Sparse Cholesky Factorization M = LL T matrix 3500 nonzeros L 36k nonzeros Reordering PMP T reverse Cuthill McKee algorithm Cholesky Factorization 41
42 Sparse Cholesky Factorization M = LL T matrix 3500 nonzeros L 36k nonzeros Reordering PMP T reverse Cuthill McKee algorithm Cholesky Factorization L 14k nonzeros 42
43 Sparse Cholesky Factorization M = LL T matrix 3500 nonzeros L 36k nonzeros Reordering PMP T nested dissection (parallelizable) Cholesky Factorization 43
44 Sparse Cholesky Factorization M = LL T matrix 3500 nonzeros L 36k nonzeros Reordering PMP T nested dissection (parallelizable) Cholesky Factorization L 7k nonzeros 44
45 Highly accurate Manipulate matrix structure No iterations, everything is closed form Easy to use Off the shelf library, no parameters Direct Solvers Discussion If M stays fixed, changing rhs (r) is cheap Just need to back substitute (factor precomputed) 45
46 High memory cost Direct Solvers Discussion Need to store the factor, which is typically denser than the matrix M If the matrix M changes, need to re compute the factor (expensive) 46
47 TAUCS tutorial TAUCS: a library of sparse linear solvers Has both iterative and direct solvers Direct (Cholesky and LU) use reordering and are very fast I provide a wrapper for TAUCS on the course homepage 47
48 TAUCS tutorial Basic operations: Define a sparse matrix structure Fill the matrix with its nonzero values (i, j, v) Factor A T A Provide an rhs and solve 48
49 TAUCS tutorial Basic operations: Define a sparse matrix structure InitTaucsInterface(); int ida; ida = CreateMatrix(4, 3); #rows #cols 49
50 TAUCS tutorial Basic operations: Fill the matrix A with its nonzero values (i, j, v) SetMatrixEntry(idA, i, j, v); 50
51 TAUCS tutorial Basic operations: Fill the matrix A with its nonzero values (i, j, v) SetMatrixEntry(idA, i, j, v); matrix ID, obtained in CreateMatrix 51
52 TAUCS tutorial Basic operations: Fill the matrix A with its nonzero values (i, j, v) SetMatrixEntry(idA, i, j, v); row index i, column index j, zero based 52
53 TAUCS tutorial Basic operations: Fill the matrix A with its nonzero values (i, j, v) SetMatrixEntry(idA, i, j, v); value of matrix entry ij for instance, w ij 53
54 TAUCS tutorial Basic operations: Factor the matrix A T A FactorATA(idA); 54
55 TAUCS tutorial Basic operations: Provide an rhs and solve taucstype b[4] = {3, 4, 5, 6}; taucstype x[3]; SolveATA(idA, b, x, 1); 55
56 TAUCS tutorial Basic operations: Provide an rhs and solve taucstype b[4] = {3, 4, 5, 6}; taucstype x[3]; SolveATA(idA, b, x, 1); typedef for double 56
57 TAUCS tutorial Basic operations: Provide an rhs and solve taucstype b[4] = {3, 4, 5, 6}; taucstype x[3]; SolveATA(idA, b, x, 1); ID of the A matrix 57
58 TAUCS tutorial Basic operations: Provide an rhs and solve taucstype b[4] = {3, 4, 5, 6}; taucstype x[3]; SolveATA(idA, b, x, 1); rhs for the LS system Ax = b 58
59 TAUCS tutorial Basic operations: Provide an rhs and solve taucstype b[4] = {3, 4, 5, 6}; taucstype x[3]; SolveATA(idA, b, x, 1); array for the solution 59
60 TAUCS tutorial Basic operations: Provide an rhs and solve A is 4x3 taucstype b[4] = {3, 4, 5, 6}; taucstype x[3]; SolveATA(idA, b, x, 1); number of rhs s 60
61 TAUCS tutorial Basic operations: Provide an rhs and solve A is 4x3 taucstype b2[8] = {3, 4, 5, 6, 7, 8, 9, 10}; taucstype xy[6]; SolveATA(idA, b2, xy, 2); number of rhs s 61
62 TAUCS tutorial If the matrix A is square a priori, no need to solve the LS system Then just use FactorA()and SolveA() 62
63 Further Reading Efficient Linear System Solvers for Mesh Processing Mario Botsch, David Bommes, Leif Kobbelt Invited paper at IMA Mathematics of Surfaces XI, Lecture Notes in Computer Science, Vol 3604, 2005, pp
64 Next week By September 27 you must: Read up on all the projects and decide on your priorities Discuss with me if you d like to do a customized project (your own ideas) E mail me your project preference (ranked list) No class meeting (do homework! ) 64
65 Next week and after On September 28 I will assign a project to everyone Next class: October 4; everyone presents their initial project plan Up to 20 minutes presentation (can be shorter) Explain the project in your words, with technical details Outline your work plan Bring up things that are unclear/challenges you foresee 65
Advanced Computer Graphics
G22.2274 001, Fall 2009 Advanced Computer Graphics Project details and tools 1 Project Topics Computer Animation Geometric Modeling Computational Photography Image processing 2 Optimization All projects
More information(Sparse) Linear Solvers
(Sparse) Linear Solvers Ax = B Why? Many geometry processing applications boil down to: solve one or more linear systems Parameterization Editing Reconstruction Fairing Morphing 2 Don t you just invert
More information(Sparse) Linear Solvers
(Sparse) Linear Solvers Ax = B Why? Many geometry processing applications boil down to: solve one or more linear systems Parameterization Editing Reconstruction Fairing Morphing 1 Don t you just invert
More informationSolving Sparse Linear Systems. Forward and backward substitution for solving lower or upper triangular systems
AMSC 6 /CMSC 76 Advanced Linear Numerical Analysis Fall 7 Direct Solution of Sparse Linear Systems and Eigenproblems Dianne P. O Leary c 7 Solving Sparse Linear Systems Assumed background: Gauss elimination
More informationAMS526: Numerical Analysis I (Numerical Linear Algebra)
AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 20: Sparse Linear Systems; Direct Methods vs. Iterative Methods Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Numerical Analysis I 1 / 26
More informationAMS526: Numerical Analysis I (Numerical Linear Algebra)
AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 5: Sparse Linear Systems and Factorization Methods Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical Analysis I 1 / 18 Sparse
More informationDef De orma f tion orma Disney/Pixar
Deformation Disney/Pixar Deformation 2 Motivation Easy modeling generate new shapes by deforming existing ones 3 Motivation Easy modeling generate new shapes by deforming existing ones 4 Motivation Character
More informationReal-Time Shape Editing using Radial Basis Functions
Real-Time Shape Editing using Radial Basis Functions, Leif Kobbelt RWTH Aachen Boundary Constraint Modeling Prescribe irregular constraints Vertex positions Constrained energy minimization Optimal fairness
More informationComputational Design. Stelian Coros
Computational Design Stelian Coros Schedule for presentations February 3 5 10 12 17 19 24 26 March 3 5 10 12 17 19 24 26 30 April 2 7 9 14 16 21 23 28 30 Send me: ASAP: 3 choices for dates + approximate
More informationEasy modeling generate new shapes by deforming existing ones
Deformation I Deformation Motivation Easy modeling generate new shapes by deforming existing ones Motivation Easy modeling generate new shapes by deforming existing ones Motivation Character posing for
More information10 - ARAP and Linear Blend Skinning
10 - ARAP and Linear Blend Skinning Acknowledgements: Olga Sorkine-Hornung As Rigid As Possible Demo Libigl demo 405 As-Rigid-As-Possible Deformation Preserve shape of cells covering the surface Ask each
More informationLarge Mesh Deformation Using the Volumetric Graph Laplacian
Large Mesh Deformation Using the Volumetric Graph Laplacian Kun Zhou1 Jin Huang2 John Snyder3 Xinguo Liu1 Hujun Bao2 Baining Guo1 Heung-Yeung Shum1 1 Microsoft Research Asia 2 Zhejiang University 3 Microsoft
More informationGeometric Modeling and Processing
Geometric Modeling and Processing Tutorial of 3DIM&PVT 2011 (Hangzhou, China) May 16, 2011 6. Mesh Simplification Problems High resolution meshes becoming increasingly available 3D active scanners Computer
More informationIN OUR LAST HOMEWORK, WE SOLVED LARGE
Y OUR HOMEWORK H A SSIGNMENT Editor: Dianne P. O Leary, oleary@cs.umd.edu FAST SOLVERS AND SYLVESTER EQUATIONS: BOTH SIDES NOW By Dianne P. O Leary IN OUR LAST HOMEWORK, WE SOLVED LARGE SPARSE SYSTEMS
More informationGraph Partitioning for High-Performance Scientific Simulations. Advanced Topics Spring 2008 Prof. Robert van Engelen
Graph Partitioning for High-Performance Scientific Simulations Advanced Topics Spring 2008 Prof. Robert van Engelen Overview Challenges for irregular meshes Modeling mesh-based computations as graphs Static
More informationEfficient Linear System Solvers for Mesh Processing
Efficient Linear System Solvers for Mesh Processing Mario Botsch, David Bommes, and Leif Kobbelt Computer Graphics Group, RWTH Aachen Technical University Abstract. The use of polygonal mesh representations
More informationLecture 7: Arnold s Cat Map
Lecture 7: Arnold s Cat Map Reference http://www.chaos.umd.edu/misc/catmap.html https://www.jasondavies.com/catmap/ Chaotic Maps: An Introduction A popular question in mathematics is how to analyze and
More informationLeast-Squares Fitting of Data with B-Spline Curves
Least-Squares Fitting of Data with B-Spline Curves David Eberly, Geometric Tools, Redmond WA 98052 https://www.geometrictools.com/ This work is licensed under the Creative Commons Attribution 4.0 International
More informationSubdivision. Outline. Key Questions. Subdivision Surfaces. Advanced Computer Graphics (Spring 2013) Video: Geri s Game (outside link)
Advanced Computer Graphics (Spring 03) CS 83, Lecture 7: Subdivision Ravi Ramamoorthi http://inst.eecs.berkeley.edu/~cs83/sp3 Slides courtesy of Szymon Rusinkiewicz, James O Brien with material from Denis
More informationAim. Structure and matrix sparsity: Part 1 The simplex method: Exploiting sparsity. Structure and matrix sparsity: Overview
Aim Structure and matrix sparsity: Part 1 The simplex method: Exploiting sparsity Julian Hall School of Mathematics University of Edinburgh jajhall@ed.ac.uk What should a 2-hour PhD lecture on structure
More informationLecture 11: Randomized Least-squares Approximation in Practice. 11 Randomized Least-squares Approximation in Practice
Stat60/CS94: Randomized Algorithms for Matrices and Data Lecture 11-10/09/013 Lecture 11: Randomized Least-squares Approximation in Practice Lecturer: Michael Mahoney Scribe: Michael Mahoney Warning: these
More informationParallel Computation of Spherical Parameterizations for Mesh Analysis. Th. Athanasiadis and I. Fudos University of Ioannina, Greece
Parallel Computation of Spherical Parameterizations for Mesh Analysis Th. Athanasiadis and I. Fudos, Greece Introduction Mesh parameterization is a powerful geometry processing tool Applications Remeshing
More informationIntroduction and Overview
CS 523: Computer Graphics, Spring 2009 Shape Modeling Introduction and Overview 1/28/2009 1 Geometric Modeling To describe any reallife object on the computer must start with shape (2D/3D) Geometry processing
More informationWhat is Multigrid? They have been extended to solve a wide variety of other problems, linear and nonlinear.
AMSC 600/CMSC 760 Fall 2007 Solution of Sparse Linear Systems Multigrid, Part 1 Dianne P. O Leary c 2006, 2007 What is Multigrid? Originally, multigrid algorithms were proposed as an iterative method to
More informationHomework # 2 Due: October 6. Programming Multiprocessors: Parallelism, Communication, and Synchronization
ECE669: Parallel Computer Architecture Fall 2 Handout #2 Homework # 2 Due: October 6 Programming Multiprocessors: Parallelism, Communication, and Synchronization 1 Introduction When developing multiprocessor
More informationLecture 15: More Iterative Ideas
Lecture 15: More Iterative Ideas David Bindel 15 Mar 2010 Logistics HW 2 due! Some notes on HW 2. Where we are / where we re going More iterative ideas. Intro to HW 3. More HW 2 notes See solution code!
More informationLecture 2 Unstructured Mesh Generation
Lecture 2 Unstructured Mesh Generation MIT 16.930 Advanced Topics in Numerical Methods for Partial Differential Equations Per-Olof Persson (persson@mit.edu) February 13, 2006 1 Mesh Generation Given a
More information03 - Reconstruction. Acknowledgements: Olga Sorkine-Hornung. CSCI-GA Geometric Modeling - Spring 17 - Daniele Panozzo
3 - Reconstruction Acknowledgements: Olga Sorkine-Hornung Geometry Acquisition Pipeline Scanning: results in range images Registration: bring all range images to one coordinate system Stitching/ reconstruction:
More information3D Regularization of Animated Surfaces
D Regularization of Animated Surfaces Simon Courtemanche Ensimag, Grenoble-INP Introduction Understanding the motion of living beings is a major research subject in computer animation. This kind of study
More informationSpace Filling Curves and Hierarchical Basis. Klaus Speer
Space Filling Curves and Hierarchical Basis Klaus Speer Abstract Real world phenomena can be best described using differential equations. After linearisation we have to deal with huge linear systems of
More informationIntroduction to Computer Graphics. Image Processing (1) June 8, 2017 Kenshi Takayama
Introduction to Computer Graphics Image Processing (1) June 8, 2017 Kenshi Takayama Today s topics Edge-aware image processing Gradient-domain image processing 2 Image smoothing using Gaussian Filter Smoothness
More informationCS 523: Computer Graphics, Spring Shape Modeling. Skeletal deformation. Andrew Nealen, Rutgers, /12/2011 1
CS 523: Computer Graphics, Spring 2011 Shape Modeling Skeletal deformation 4/12/2011 1 Believable character animation Computers games and movies Skeleton: intuitive, low-dimensional subspace Clip courtesy
More informationTopics in Computer Animation
Topics in Computer Animation Animation Techniques Artist Driven animation The artist draws some frames (keyframing) Usually in 2D The computer generates intermediate frames using interpolation The old
More informationEF432. Introduction to spagetti and meatballs
EF432 Introduction to spagetti and meatballs CSC 418/2504: Computer Graphics Course web site (includes course information sheet): http://www.dgp.toronto.edu/~karan/courses/418/ Instructors: L2501, T 6-8pm
More informationAdvanced Operations Research Techniques IE316. Quiz 1 Review. Dr. Ted Ralphs
Advanced Operations Research Techniques IE316 Quiz 1 Review Dr. Ted Ralphs IE316 Quiz 1 Review 1 Reading for The Quiz Material covered in detail in lecture. 1.1, 1.4, 2.1-2.6, 3.1-3.3, 3.5 Background material
More informationcalibrated coordinates Linear transformation pixel coordinates
1 calibrated coordinates Linear transformation pixel coordinates 2 Calibration with a rig Uncalibrated epipolar geometry Ambiguities in image formation Stratified reconstruction Autocalibration with partial
More informationSurface Reconstruction. Gianpaolo Palma
Surface Reconstruction Gianpaolo Palma Surface reconstruction Input Point cloud With or without normals Examples: multi-view stereo, union of range scan vertices Range scans Each scan is a triangular mesh
More informationNumerical Linear Algebra
Numerical Linear Algebra Probably the simplest kind of problem. Occurs in many contexts, often as part of larger problem. Symbolic manipulation packages can do linear algebra "analytically" (e.g. Mathematica,
More informationLecture 9. Introduction to Numerical Techniques
Lecture 9. Introduction to Numerical Techniques Ivan Papusha CDS270 2: Mathematical Methods in Control and System Engineering May 27, 2015 1 / 25 Logistics hw8 (last one) due today. do an easy problem
More informationThis work is about a new method for generating diffusion curve style images. Although this topic is dealing with non-photorealistic rendering, as you
This work is about a new method for generating diffusion curve style images. Although this topic is dealing with non-photorealistic rendering, as you will see our underlying solution is based on two-dimensional
More informationSpecial Topics in Visualization
Special Topics in Visualization Final Project Report Dual contouring of Hermite Data Submitted By S M Shahed Nejhum 8589-1199 May 19, 2008 Introduction Iso-surface extraction from 3D volumetric data is
More informationTopic 0. Introduction: What Is Computer Graphics? CSC 418/2504: Computer Graphics EF432. Today s Topics. What is Computer Graphics?
EF432 Introduction to spagetti and meatballs CSC 418/2504: Computer Graphics Course web site (includes course information sheet): http://www.dgp.toronto.edu/~karan/courses/418/ Instructors: L0101, W 12-2pm
More informationGeometry Processing & Geometric Queries. Computer Graphics CMU /15-662
Geometry Processing & Geometric Queries Computer Graphics CMU 15-462/15-662 Last time: Meshes & Manifolds Mathematical description of geometry - simplifying assumption: manifold - for polygon meshes: fans,
More informationGeostatistics 2D GMS 7.0 TUTORIALS. 1 Introduction. 1.1 Contents
GMS 7.0 TUTORIALS 1 Introduction Two-dimensional geostatistics (interpolation) can be performed in GMS using the 2D Scatter Point module. The module is used to interpolate from sets of 2D scatter points
More informationinterpolation, color, & light Outline HW I Announcements HW II--due today, 5PM HW III on the web later today
interpolation, color, & light Outline Announcements HW II--due today, 5PM HW III on the web later today HW I: Issues Structured vs. Unstructured Meshes Working with unstructured meshes Interpolation colormaps
More informationLecture VIII. Global Approximation Methods: I
Lecture VIII Global Approximation Methods: I Gianluca Violante New York University Quantitative Macroeconomics G. Violante, Global Methods p. 1 /29 Global function approximation Global methods: function
More informationAnimation. CS 4620 Lecture 32. Cornell CS4620 Fall Kavita Bala
Animation CS 4620 Lecture 32 Cornell CS4620 Fall 2015 1 What is animation? Modeling = specifying shape using all the tools we ve seen: hierarchies, meshes, curved surfaces Animation = specifying shape
More informationL15. POSE-GRAPH SLAM. NA568 Mobile Robotics: Methods & Algorithms
L15. POSE-GRAPH SLAM NA568 Mobile Robotics: Methods & Algorithms Today s Topic Nonlinear Least Squares Pose-Graph SLAM Incremental Smoothing and Mapping Feature-Based SLAM Filtering Problem: Motion Prediction
More informationStructure from Motion
11/18/11 Structure from Motion Computer Vision CS 143, Brown James Hays Many slides adapted from Derek Hoiem, Lana Lazebnik, Silvio Saverese, Steve Seitz, and Martial Hebert This class: structure from
More informationGeometric Modeling Assignment 3: Discrete Differential Quantities
Geometric Modeling Assignment : Discrete Differential Quantities Acknowledgements: Julian Panetta, Olga Diamanti Assignment (Optional) Topic: Discrete Differential Quantities with libigl Vertex Normals,
More informationEF432. Introduction to spagetti and meatballs
EF432 Introduction to spagetti and meatballs CSC 418/2504: Computer Graphics Course web site (includes course information sheet): http://www.dgp.toronto.edu/~karan/courses/418/fall2015 Instructor: Karan
More informationAA220/CS238 Parallel Methods in Numerical Analysis. Introduction to Sparse Direct Solver (Symmetric Positive Definite Systems)
AA0/CS8 Parallel ethods in Numerical Analysis Introduction to Sparse Direct Solver (Symmetric Positive Definite Systems) Kincho H. Law Professor of Civil and Environmental Engineering Stanford University
More informationComputer Vision Projective Geometry and Calibration. Pinhole cameras
Computer Vision Projective Geometry and Calibration Professor Hager http://www.cs.jhu.edu/~hager Jason Corso http://www.cs.jhu.edu/~jcorso. Pinhole cameras Abstract camera model - box with a small hole
More informationHomework #2 and #3 Due Friday, October 12 th and Friday, October 19 th
Homework #2 and #3 Due Friday, October 12 th and Friday, October 19 th 1. a. Show that the following sequences commute: i. A rotation and a uniform scaling ii. Two rotations about the same axis iii. Two
More informationGraph drawing in spectral layout
Graph drawing in spectral layout Maureen Gallagher Colleen Tygh John Urschel Ludmil Zikatanov Beginning: July 8, 203; Today is: October 2, 203 Introduction Our research focuses on the use of spectral graph
More information0. Introduction: What is Computer Graphics? 1. Basics of scan conversion (line drawing) 2. Representing 2D curves
CSC 418/2504: Computer Graphics Course web site (includes course information sheet): http://www.dgp.toronto.edu/~elf Instructor: Eugene Fiume Office: BA 5266 Phone: 416 978 5472 (not a reliable way) Email:
More informationSkeletal deformation
CS 523: Computer Graphics, Spring 2009 Shape Modeling Skeletal deformation 4/22/2009 1 Believable character animation Computers games and movies Skeleton: intuitive, low dimensional subspace Clip courtesy
More informationImage Manipulation in MATLAB Due Monday, July 17 at 5:00 PM
Image Manipulation in MATLAB Due Monday, July 17 at 5:00 PM 1 Instructions Labs may be done in groups of 2 or 3 (i.e., not alone). You may use any programming language you wish but MATLAB is highly suggested.
More information2D rendering takes a photo of the 2D scene with a virtual camera that selects an axis aligned rectangle from the scene. The photograph is placed into
2D rendering takes a photo of the 2D scene with a virtual camera that selects an axis aligned rectangle from the scene. The photograph is placed into the viewport of the current application window. A pixel
More informationGeometric Features for Non-photorealistiic Rendering
CS348a: Computer Graphics Handout # 6 Geometric Modeling and Processing Stanford University Monday, 27 February 2017 Homework #4: Due Date: Mesh simplification and expressive rendering [95 points] Wednesday,
More informationGRAPHING WORKSHOP. A graph of an equation is an illustration of a set of points whose coordinates satisfy the equation.
GRAPHING WORKSHOP A graph of an equation is an illustration of a set of points whose coordinates satisfy the equation. The figure below shows a straight line drawn through the three points (2, 3), (-3,-2),
More informationName: SID: LAB Section: Lab 6 - Part 1: Pixels and Triangle Rasterization
Name: SID: LAB Section: Lab 6 - Part 1: Pixels and Triangle Rasterization 1. Coordinate space conversions. In OpenGL, the coordinates of a vertex in model-view space can be converted to NPC coordinates,
More informationFiberMesh. Due: 4/12/10, 11:59 PM
CS224: Interactive Computer Graphics FiberMesh Due: 4/12/10, 11:59 PM Least Squares Solver 5 Mesh triangulation 10 Optimization Problem 1: Curve Dragging 25 Optimization Problem 2: Surface Optimization
More informationReport of Linear Solver Implementation on GPU
Report of Linear Solver Implementation on GPU XIANG LI Abstract As the development of technology and the linear equation solver is used in many aspects such as smart grid, aviation and chemical engineering,
More informationGraded Assignment 2 Maple plots
Graded Assignment 2 Maple plots The Maple part of the assignment is to plot the graphs corresponding to the following problems. I ll note some syntax here to get you started see tutorials for more. Problem
More informationSHORTEST PATHS ON SURFACES GEODESICS IN HEAT
SHORTEST PATHS ON SURFACES GEODESICS IN HEAT INF555 Digital Representation and Analysis of Shapes 28 novembre 2015 Ruoqi He & Chia-Man Hung 1 INTRODUCTION In this project we present the algorithm of a
More informationFree-Form Deformation and Other Deformation Techniques
Free-Form Deformation and Other Deformation Techniques Deformation Deformation Basic Definition Deformation: A transformation/mapping of the positions of every particle in the original object to those
More informationCIS 580, Machine Perception, Spring 2014: Assignment 4 Due: Wednesday, April 10th, 10:30am (use turnin)
CIS 580, Machine Perception, Spring 2014: Assignment 4 Due: Wednesday, April 10th, 10:30am (use turnin) Solutions (hand calculations, plots) have to be submitted electronically as a single pdf file using
More informationA Random Variable Shape Parameter Strategy for Radial Basis Function Approximation Methods
A Random Variable Shape Parameter Strategy for Radial Basis Function Approximation Methods Scott A. Sarra, Derek Sturgill Marshall University, Department of Mathematics, One John Marshall Drive, Huntington
More informationPROGRAMMING OF MULTIGRID METHODS
PROGRAMMING OF MULTIGRID METHODS LONG CHEN In this note, we explain the implementation detail of multigrid methods. We will use the approach by space decomposition and subspace correction method; see Chapter:
More information2D & 3D Finite Element Method Packages of CEMTool for Engineering PDE Problems
2D & 3D Finite Element Method Packages of CEMTool for Engineering PDE Problems Choon Ki Ahn, Jung Hun Park, and Wook Hyun Kwon 1 Abstract CEMTool is a command style design and analyzing package for scientific
More informationCSE 554 Lecture 7: Deformation II
CSE 554 Lecture 7: Deformation II Fall 2011 CSE554 Deformation II Slide 1 Review Rigid-body alignment Non-rigid deformation Intrinsic methods: deforming the boundary points An optimization problem Minimize
More informationCVPR 2014 Visual SLAM Tutorial Efficient Inference
CVPR 2014 Visual SLAM Tutorial Efficient Inference kaess@cmu.edu The Robotics Institute Carnegie Mellon University The Mapping Problem (t=0) Robot Landmark Measurement Onboard sensors: Wheel odometry Inertial
More informationLecture 7: Image Morphing. Idea #2: Align, then cross-disolve. Dog Averaging. Averaging vectors. Idea #1: Cross-Dissolving / Cross-fading
Lecture 7: Image Morphing Averaging vectors v = p + α (q p) = (1 - α) p + α q where α = q - v p α v (1-α) q p and q can be anything: points on a plane (2D) or in space (3D) Colors in RGB or HSV (3D) Whole
More informationNonphotorealism. Christian Miller CS Fall 2011
Nonphotorealism Christian Miller CS 354 - Fall 2011 Different goals Everything we ve done so far has been working (more or less) towards photorealism But, you might not want realism as a stylistic choice
More informationA Global Laplacian Smoothing Approach with Feature Preservation
A Global Laplacian Smoothing Approach with Feature Preservation hongping Ji Ligang Liu Guojin Wang Department of Mathematics State Key Lab of CAD&CG hejiang University Hangzhou, 310027 P.R. China jzpboy@yahoo.com.cn,
More information1.2 Numerical Solutions of Flow Problems
1.2 Numerical Solutions of Flow Problems DIFFERENTIAL EQUATIONS OF MOTION FOR A SIMPLIFIED FLOW PROBLEM Continuity equation for incompressible flow: 0 Momentum (Navier-Stokes) equations for a Newtonian
More informationIterative Algorithms I: Elementary Iterative Methods and the Conjugate Gradient Algorithms
Iterative Algorithms I: Elementary Iterative Methods and the Conjugate Gradient Algorithms By:- Nitin Kamra Indian Institute of Technology, Delhi Advisor:- Prof. Ulrich Reude 1. Introduction to Linear
More informationAs a consequence of the operation, there are new incidences between edges and triangles that did not exist in K; see Figure II.9.
II.4 Surface Simplification 37 II.4 Surface Simplification In applications it is often necessary to simplify the data or its representation. One reason is measurement noise, which we would like to eliminate,
More information1 2 (3 + x 3) x 2 = 1 3 (3 + x 1 2x 3 ) 1. 3 ( 1 x 2) (3 + x(0) 3 ) = 1 2 (3 + 0) = 3. 2 (3 + x(0) 1 2x (0) ( ) = 1 ( 1 x(0) 2 ) = 1 3 ) = 1 3
6 Iterative Solvers Lab Objective: Many real-world problems of the form Ax = b have tens of thousands of parameters Solving such systems with Gaussian elimination or matrix factorizations could require
More informationRobot Mapping. Least Squares Approach to SLAM. Cyrill Stachniss
Robot Mapping Least Squares Approach to SLAM Cyrill Stachniss 1 Three Main SLAM Paradigms Kalman filter Particle filter Graphbased least squares approach to SLAM 2 Least Squares in General Approach for
More informationGraphbased. Kalman filter. Particle filter. Three Main SLAM Paradigms. Robot Mapping. Least Squares Approach to SLAM. Least Squares in General
Robot Mapping Three Main SLAM Paradigms Least Squares Approach to SLAM Kalman filter Particle filter Graphbased Cyrill Stachniss least squares approach to SLAM 1 2 Least Squares in General! Approach for
More informationOn the regularizing power of multigrid-type algorithms
On the regularizing power of multigrid-type algorithms Marco Donatelli Dipartimento di Fisica e Matematica Università dell Insubria www.mat.unimi.it/user/donatel Outline 1 Restoration of blurred and noisy
More informationContents. Implementing the QR factorization The algebraic eigenvalue problem. Applied Linear Algebra in Geoscience Using MATLAB
Applied Linear Algebra in Geoscience Using MATLAB Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data Two-Dimensional Plots Programming in
More informationCS 248 Assignment 2 Polygon Scan Conversion. CS248 Presented by Michael Green Stanford University October 20, 2004
CS 248 Assignment 2 Polygon Scan Conversion CS248 Presented by Michael Green Stanford University October 20, 2004 Announcements First thing: read README.animgui.. It should tell you everything you need
More informationSparse matrices, graphs, and tree elimination
Logistics Week 6: Friday, Oct 2 1. I will be out of town next Tuesday, October 6, and so will not have office hours on that day. I will be around on Monday, except during the SCAN seminar (1:25-2:15);
More information12 - Spatial And Skeletal Deformations. CSCI-GA Computer Graphics - Fall 16 - Daniele Panozzo
12 - Spatial And Skeletal Deformations Space Deformations Space Deformation Displacement function defined on the ambient space Evaluate the function on the points of the shape embedded in the space Twist
More informationAnnouncements. Image Matching! Source & Destination Images. Image Transformation 2/ 3/ 16. Compare a big image to a small image
2/3/ Announcements PA is due in week Image atching! Leave time to learn OpenCV Think of & implement something creative CS 50 Lecture #5 February 3 rd, 20 2/ 3/ 2 Compare a big image to a small image So
More informationInterpolation - 2D mapping Tutorial 1: triangulation
Tutorial 1: triangulation Measurements (Zk) at irregular points (xk, yk) Ex: CTD stations, mooring, etc... The known Data How to compute some values on the regular spaced grid points (+)? The unknown data
More informationSparse matrices: Basics
Outline : Basics Bora Uçar RO:MA, LIP, ENS Lyon, France CR-08: Combinatorial scientific computing, September 201 http://perso.ens-lyon.fr/bora.ucar/cr08/ 1/28 CR09 Outline Outline 1 Course presentation
More informationScalar Field Visualization I
Scalar Field Visualization I What is a Scalar Field? The approximation of certain scalar function in space f(x,y,z). Image source: blimpyb.com f What is a Scalar Field? The approximation of certain scalar
More informationCSE452 Computer Graphics
CSE452 Computer Graphics Lecture 19: From Morphing To Animation Capturing and Animating Skin Deformation in Human Motion, Park and Hodgins, SIGGRAPH 2006 CSE452 Lecture 19: From Morphing to Animation 1
More informationSketch Based Image Deformation
Sketch Based Image Deformation Mathias Eitz Olga Sorkine Marc Alexa TU Berlin Email: {eitz,sorkine,marc}@cs.tu-berlin.de Abstract We present an image editing tool that allows to deform and composite image
More informationNumerical Methods in Physics Lecture 2 Interpolation
Numerical Methods in Physics Pat Scott Department of Physics, Imperial College November 8, 2016 Slides available from http://astro.ic.ac.uk/pscott/ course-webpage-numerical-methods-201617 Outline The problem
More informationImage Blending and Compositing NASA
Image Blending and Compositing NASA CS194: Image Manipulation & Computational Photography Alexei Efros, UC Berkeley, Fall 2016 Image Compositing Compositing Procedure 1. Extract Sprites (e.g using Intelligent
More informationTechnical Report. Removing polar rendering artifacts in subdivision surfaces. Ursula H. Augsdörfer, Neil A. Dodgson, Malcolm A. Sabin.
Technical Report UCAM-CL-TR-689 ISSN 1476-2986 Number 689 Computer Laboratory Removing polar rendering artifacts in subdivision surfaces Ursula H. Augsdörfer, Neil A. Dodgson, Malcolm A. Sabin June 2007
More information7.3 3-D Notes Honors Precalculus Date: Adapted from 11.1 & 11.4
73 3-D Notes Honors Precalculus Date: Adapted from 111 & 114 The Three-Variable Coordinate System I Cartesian Plane The familiar xy-coordinate system is used to represent pairs of numbers (ordered pairs
More information2D Shape Deformation Using Nonlinear Least Squares Optimization
2D Shape Deformation Using Nonlinear Least Squares Optimization Paper ID: 20 Abstract This paper presents a novel 2D shape deformation algorithm based on nonlinear least squares optimization. The algorithm
More informationGeometric Modeling in Graphics
Geometric Modeling in Graphics Part 10: Surface reconstruction Martin Samuelčík www.sccg.sk/~samuelcik samuelcik@sccg.sk Curve, surface reconstruction Finding compact connected orientable 2-manifold surface
More informationEfficient Minimization of New Quadric Metric for Simplifying Meshes with Appearance Attributes
Efficient Minimization of New Quadric Metric for Simplifying Meshes with Appearance Attributes (Addendum to IEEE Visualization 1999 paper) Hugues Hoppe Steve Marschner June 2000 Technical Report MSR-TR-2000-64
More information