Advanced Computer Graphics

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