The document discusses geometry processing and sparse matrices. It provides an overview of geometry processing, which uses applied math and computer science concepts to design algorithms for acquiring, reconstructing, analyzing, manipulating, simulating and transmitting complex 3D models. It then discusses representing shapes using triangle meshes and differential coordinates. It introduces discrete and continuous Laplace-Beltrami operators for geometry processing. Finally, it discusses representing the Laplacian matrix of a mesh as a sparse matrix using scipy and different sparse matrix storage formats like CSR and CSC for efficient computations.

1. GEOMETRY PROCESSINGで学ぶ
SPARSE MATRIX
2012/3/18 Tokyo.SciPy #3
齊藤 淳 Jun Saito @dukecyto

2. 本日の概要
Laplacian Mesh Processingを通じて
sparse matrix一般、scipy.sparseの理解を深める
!! = ! − !L !

3. Computer Graphics Animation
(c) copyright 2008, Blender Foundation / www.bigbuckbunny.org

4. Computer Graphics Animation
Modeling Animation Rendering
• 作る
• 動かす
• 絵にする

5. SIGGRAPH 2011 Papers Categories
Modeling Animation Rendering Images
• Understanding • Capturing & • Stochastic • Drawing, Painting
Shapes Modeling Humans Rendering & and Stylization
• Mapping & • Facial Animation Visibility • Tone Editing
Warping Shapes • Call Animal Control! • Volumes & Photons • By-Example Image
• Capturing • Contact and • Real-Time Synthesis
Geometry and Constraints Rendering • Image Processing
Appearance • Example-Based Hardware • Video Resizing &
• Geometry Simulation • Sampling & Noise Stabilization
Processing • Fluid Simulation • Stereo & Disparity
• Discrete Differential • Fast Simulation • Interactive Image
Geometry Editing
• Geometry • Colorful
Acquisition
• Surfaces
• Procedural &
Interactive Modeling
• Fun With Shapes

6. SIGGRAPH 2011 Papers Categories
Modeling Animation Rendering Images
• Understanding • Capturing & • Stochastic • Drawing, Painting
Shapes Modeling Humans Rendering & and Stylization
• Mapping & • Facial Animation Visibility • Tone Editing
Warping Shapes • Call Animal Control! • Volumes & Photons • By-Example Image
• Capturing • Contact and • Real-Time Synthesis
Geometry and Constraints Rendering • Image Processing
Appearance • Example-Based Hardware • Video Resizing &
• Geometry Simulation • Sampling & Noise Stabilization
Processing • Fluid Simulation • Stereo & Disparity
• Discrete Differential • Fast Simulation • Interactive Image
Geometry Editing
• Geometry • Colorful
Acquisition
• Surfaces
• Procedural &
Interactive Modeling
• Fun With Shapes

7. Geometry Processingとは
Geometry processing, or mesh processing, is a fast-growing
area of research that uses concepts from applied mathematics,
computer science and engineering to design efficient
algorithms for the
¨ acquisition
¨ reconstruction
¨ analysis
¨ manipulation
¨ simulation
¨ transmission
of complex 3D models.
http://en.wikipedia.org/wiki/Geometry_processing

8. Triangle Meshによる形状の表現

9. Differential Coordinates
¨ 形状の局所的な特徴
¤ 方向は法線、大きさは平均曲率の近似
γ
1 1
δi = ∑ ( vi − v ) ∫γ ( vi − v ) ds
di v∈N ( i ) len(γ ) v∈
Discrete Laplace-Beltrami operator
Continuous Laplace-Beltrami operator
[Sorkine 2005]

10. Discrete Laplace Operator
“Majority of
Applications contemporary geometry
processing tools rely on
Mesh Parameterization discrete Laplace
Fairing / Smoothing operators”
Denoising [Alexa 2011]
Manipulation / Editing
Compression
Shape Analysis
Physical Simulation

11. Laplacian Mesh Processing

12. Graph Laplacian
(a.k.a. Uniform Laplacian, Topological Laplacian)
http://en.wikipedia.org/wiki/Laplacian_matrix

13. Cotan-Weighted Laplacian
緑: Graph Laplacian
赤: Cotan-Weighted Laplacian
1 1
!!! = cot !!" + cot !!" (!! − !! )
4!(!! ) 2
!∈! !
[Sorkine 2005]

14. Stanford 3D Scan Repository
¨ Stanford Bunny
¤ 頂点数: 35,947
¤ 35,947 x 35,947 =
1.2 G

15. Stanford 3D Scan Repository
¨ Dragon
¤ 頂点数: 566,098
¤ 566,098 x 566,098 =
298G

16. 疎行列 Sparse Matrix
成分のほとんどがゼロである
ことを活用した形式で行列を
記憶・演算
“Sparse matrices are widely
used in scientific computation,
especially in large-scale
optimization, structural and circuit
analysis, computational fluid
dynamics, and, generally, the
numerical solution of partial
differential equations.”
Sparse Matrices in MATLAB: Design and
Implementation

17. 疎行列 Sparse Matrix
成分のほとんどがゼロである
ことを活用した形式で行列を
記憶・演算
Laplacian Matrixは
¨ sparse
¨ symmetric
¨ positive semi-definite

18. Python Sparse Matrix Packages
SciPy Sparse
PySparse
CVXOPT

19. Python Sparse Matrix Packages
SciPy Sparse
PySparse
CVXOPT

20. From OpenOpt doc...
“Unfortunately, sparse matrices still
remains one of most weak features in
Python usage for scientific purposes”

21. From OpenOpt doc...
“Unlike MATLAB, Octave, and a number of other
software, there is not standard Python library for
sparse matrices: someone uses scipy.sparse, someone
PySparse, someone (as CVXOPT) uses its own library
and/or BLAS, someone just uses 3 columns (for the
number indexes and value). SciPy developers refused
to author of scipy.sparse to include it into NumPy, I
think it's a big mistake, probably now it would been a
unified standard. Still I hope in future numpy versions
difference in API for handling sparse and dense
arrays will be removed. “

22. 基本的な流れ
構築に適した形 演算に適した形式
式で行列を作る
に変換し演算を行う
A = lil_matrix((N,N))! A = A.tocsr()!
A[i,j] = a! A = A.dot(D)!
! f = factorized(A)!
x = f.solve(b)!

23. Graph Laplacian
(a.k.a. Uniform Laplacian, Topological Laplacian)
http://en.wikipedia.org/wiki/Laplacian_matrix

24. Graph Laplacian in scipy.sparse
from scipy import sparse!
!
N = len(points)!
L = sparse.lil_matrix((N, N))!
for vids in faces:!
L[vids, vids] = 1!
D = sparse.spdiags(L.sum(axis=0), 0, N, N)!
L = L.tocsc()-D!

25. Graph Laplacian in scipy.sparse
from scipy import sparse!
!
N = len(points)!
L = sparse.lil_matrix((N, N))!
for vids in faces:!
L[vids, vids] = 1!
D = sparse.spdiags(L.sum(axis=0), 0, N, N)!
L = L.tocsc()-D!

26. points and faces from Maya
!
import numpy as np!
import pymel.core as pm!
!
mesh = pm.PyNode(‘bunny’)!
points = np.array(mesh.getPoints())!
faces = np.array(mesh.getVertices()!
! ! [1]).reshape(mesh.numFaces(),3)!

27. Graph Laplacian in scipy.sparse
from scipy import sparse!
!
N = len(points)!
L = sparse.lil_matrix((N, N))!
for vids in faces:!
L[vids, vids] = 1!
D = sparse.spdiags(L.sum(axis=0), 0, N, N)!
L = L.tocsc()-D!

28. Sparse Matrix Storage Formats
構築に適した形式 演算に適した形式
¨ LIst of Lists ¨ Compressed Sparse
(LIL) Row (CSR)
¨ Compressed Sparse
¨ COOrdinate lists
(COO) Column (CSC)
¨ Block Sparse Row
¨ Dictionary Of Keys
(BSR)
(DOK)
¨ DIAgonal (DIA)

29. sparse.lil_matrix
LIst of Lists (LIL) 形式
¨ データ構造 ¨ 利点
¤ 行毎に非ゼロ要素の ¤ 柔軟なslice
ソートされた列インデッ ¤ 他のmatrix形式への変
クスを保持するarray 換が速い
¤ 対応する非ゼロ要素の
¨ 欠点
値も同様に保持
¤ LIL+ LIL が遅い (CSR/
¨ 用途 CSC推奨)
¤ 行列の構築用 ¤ column slicingが遅い
¤ 巨大な行列を構築する (CSC推奨)
際にはCOOも検討する ¤ 行列ベクトル積が遅い
と良い (CSR/CSC推奨)

30. sparse.csr_matrix
Compressed Sparse Rows (CSR)形式
¨ CSC: 行と列が逆 ¨ 利点
¨ データ構造 ¤ 高速な演算 CSR + CSR,
CSR * CSR,等.
¤ Row slicingが速い
¤ 行列ベクトル積が速い
¨ 欠点
¤ Column
slicingが遅い
A = [1 2 3 1 2 2 1] 非ゼロ要素リスト!
(CSC推奨)
(IA = [1 1 1 2 3 3 4] i行)!
IA' = [1 4 5 7] ! ¤ 他のmatrix形式への変
JA = [1 2 3 4 1 4 4] j列! 換が遅い
(そうでもない？後述ベンチマーク参照)
A, IA’, JAを保持
http://ja.wikipedia.org/wiki/%E7%96%8E%E8%A1%8C%E5%88%97

31. sparse.dia_matrix
DIAgonal (DIA)形式
¨ 帯行列に適した形式
¨ オフセットを指定可能
>>> data = array([[1,2,3,4]]).repeat(3,axis=0)!
>>> offsets = array([0,-1,2])!
>>> dia_matrix( (data,offsets),
shape=(4,4)).todense()!
matrix([[1, 0, 3, 0],!
[1, 2, 0, 4],!
[0, 2, 3, 0],!
[0, 0, 3, 4]])!

32. Sparse Matrix Format Benchmark:
Construction
Benchmark by modified bench_sparse.py in SciPy distribution

33. Sparse Matrix Format Benchmark:
Conversion
Benchmark by modified bench_sparse.py in SciPy distribution

34. Sparse Matrix Format Benchmark:
Matrix Vector Product
Benchmark by modified bench_sparse.py in SciPy distribution

35. Graph Laplacian in scipy.sparse
from scipy import sparse!
!
N = len(points)!
L = sparse.lil_matrix((N, N))!
for vids in faces:!
L[vids, vids] = 1!
D = sparse.spdiags(L.sum(axis=0), 0, N, N)!
L = L.tocsc()-D!

36. Fancy indexingの仕様の違い
sparse.lil_matrix
numpy.array
>>> L = sparse.lil_matrix((N,N))! >>> Ldense = np.zeros((N,N))!
>>> ix = [1,3,4]! >>> ix = [1,3,4]!
>>> L[ix,ix] = 1! >>> Ldense[ix,ix] = 1!
>>> L.todense()! >>> Ldense!
matrix([[ 0., 0., 0., 0., 0.],! array([[ 0., 0., 0., 0., 0.],!
[ 0., 1., 0., 1., 1.],! [ 0., 1., 0., 0., 0.],!
[ 0., 0., 0., 0., 0.],! [ 0., 0., 0., 0., 0.],!
[ 0., 1., 0., 1., 1.],! [ 0., 0., 0., 1., 0.],!
[ 0., 1., 0., 1., 1.]])! [ 0., 0., 0., 0., 1.]])!

37. 補足: numpy.ix_()
>>> Ldense = np.zeros((N,N))!
>>> ix = [1,3,4]!
>>> Ldense[np.ix_(ix,ix)] = 1!
>>> Ldense!
array([[ 0., 0., 0., 0., 0.],!
[ 0., 1., 0., 1., 1.],!
[ 0., 0., 0., 0., 0.],!
[ 0., 1., 0., 1., 1.],!
[ 0., 1., 0., 1., 1.]])!

38. Graph Laplacian in scipy.sparse
from scipy import sparse!
!
N = len(points)!
L = sparse.lil_matrix((N, N))!
for vids in faces:!
L[vids, vids] = 1!
D = sparse.spdiags(L.sum(axis=0), 0, N, N)!
L = L.tocsc()-D!

39. 細かい演算の仕様
!
# LがCSC,CSRだとNotImplementedError!
L -= D!
!
# OK!
L = L-D!

40. Laplacian Mesh Fairing
!
! = ! − !L !
A Signal Processing Approach To Fair Surface Design
[Taubin 95]

41. Mesh Fairing in scipy.sparse
from scipy.sparse import linalg as spla!
!
A = sparse.identity(L.shape[0]) - _lambda * L!
solve = spla.factorized(A.tocsc())!
x = solve(points[:,0])!
y = solve(points[:,1])!
z = solve(points[:,2])!
new_points = np.array([x, y, z]).T!
!
# Set result back to Maya mesh!
mesh.setPoints(new_points)!

42. Mesh Fairing in scipy.sparse
from scipy.sparse import linalg as spla!
!
A = sparse.identity(L.shape[0]) - _lambda * L!
solve = spla.factorized(A.tocsc())!
x = solve(points[:,0])!
y = solve(points[:,1])!
z = solve(points[:,2])!
new_points = np.array([x, y, z]).T!
!
# Set result back to Maya mesh!
mesh.setPoints(new_points)!

43. Solving sparse linear system
¨ sparse.linalg.factorized(A)
¤ AのLU分解結果を保持したlinear system solver関数を
返す
¤ AはCSC形式で渡す必要がある
SparseEfficiencyWarning: splu requires CSC matrix
format

44. Mesh Fairing in scipy.sparse
from scipy.sparse import linalg as spla!
!
A = sparse.identity(L.shape[0]) - _lambda * L!
solve = spla.factorized(A.tocsc())!
x = solve(points[:,0])!
y = solve(points[:,1])!
z = solve(points[:,2])!
new_points = np.array([x, y, z]).T!
!
# Set result back to Maya mesh!
mesh.setPoints(new_points)!

45. Solving sparse linear system
# Error! 右辺はベクトルのみ!
new_points = solve(points)!
!
# 仕方がないので一列ずつ!
x = solve(points[:,0])!
y = solve(points[:,1])!
z = solve(points[:,2])!
new_points = np.array([x, y, z]).T!

46. Cotan-Weighted Laplacian
緑: Graph Laplacian
赤: Cotan-Weighted Laplacian
1 1
!!! = cot !!" + cot !!" (!! − !! )
4!(!! ) 2
!∈! !
[Sorkine 2005]

47. Cotan Laplacian in scipy.sparse (1/2)
def laplacian_cotan(points, faces):!
EDGE_LOOP = [(0,1,2),(0,2,1),(1,2,0)]!
N = len(points)!
point_area = np.zeros(N)!
L = sparse.lil_matrix((N, N))!
for vids in faces:!
point_area[vids] += area_triangle(points[vids]) /
3.0!
for i in EDGE_LOOP:!
v0 = vids[i[0]]!
v1 = vids[i[1]]!
v2 = vids[i[2]]!
e1 = points[v1] - points[v2]!
e2 = points[v0] - points[v2]!

48. Cotan Laplacian in scipy.sparse (2/2)
cosa = np.dot(e1, e2) / math.sqrt(np.dot(e1,
e1) * np.dot(e2, e2))!
sina = math.sqrt(1 - cosa * cosa)!
cota = cosa / sina!
w = 0.5 * cota!
L[v0, v0] -= w!
L[v0, v1] += w!
L[v1, v1] -= w!
L[v1, v0] += w!
D = sparse.spdiags(0.25/point_area, 0, N, N)!
return D.dot(L.tocsc())!

49. Cotan Laplacian in scipy.sparse (2/2)
cosa = np.dot(e1, e2) / math.sqrt(np.dot(e1,
e1) * np.dot(e2, e2))!
sina = math.sqrt(1 - cosa * cosa)!
cota = cosa / sina!
w = 0.5 * cota!
L[v0, v0] -= w!
L[v0, v1] += w!
L[v1, v1] -= w!
L[v1, v0] += w!
D = sparse.spdiags(0.25/point_area, 0, N, N)!
return D.dot(L.tocsc())!

50. 行列積について
# Denseに変換されてしまう!
np.dot(D, L)!
!
# Sparseのまま行列積!
D.dot(L)!

51. Laplacian Matrix構築
Cythonによる高速化
¨ Python: 2.6秒
¨ Cython: 1.8秒
¨ Cython + Dense: 0.8秒！
¤ あまり大きくないMeshならばDenseで
Laplacianを作った方が速い

52. ファイル入出力
¨ numpy.savetxt() >>> io.savemat('sparsetest.mat',
{'lil': L.tolil(),
¤ Sparse Matrixでは使えない 'csr': L.tocsr()},
oned_as='row')
¨ scipy.io.savemat() / >>> M = io.loadmat(‘sparsetest.mat')
loadmat() >>> M['lil']
¤ MATLAB .mat形式で保存 <5x5 sparse matrix of type '<type
'numpy.float64'>'
¤ Sparse Matrixは強制的に
CSCで保存される with 9 stored elements in Compressed
Sparse Column format>
>>> M['csr']
<5x5 sparse matrix of type '<type
'numpy.float64'>'
with 9 stored elements in Compressed
Sparse Column format>

53. MATLAB vs. SciPy – Sparse Matrices
MATLAB
SciPy Sparse
¨ sparse()で行列を作る ¨ 内部形式(denseも含
だけ！ め)を意識して使う
¨ 後の演算はdenseと全 ¨ 形式によって同じ演
く同じ 算、関数が使えない
¨ 内部形式はCSC
ケースがある

54. まとめ
¨ Sparse matrix一般
¤ よく使われるデータ構造を理解した
¨ scipy.sparse
¤ 使い方の基本がわかった
¤ いいところ
n やりたいことはできる。実用可能。
n Mesh
Processingが省メモリで計算可能。〜1000倍のオーダーで
速くなる。
¤ 悪いところ
n ドキュメントされていない仕様が多過ぎ。
n ndarrayと透過的に使えるようにしてください L
¨ Laplacian Mesh Processingはおもしろいですよ J

55. Future Work
¨ scipy.sparse.linalg の調査
¤ Iterative sparse linear system solver系
¨ Mesh Processingのもうちょっと深いネタを紹介

56. Further Readings
[Sorkine 2005] Sorkine, Olga.
“Laplacian Mesh Processing” Eurographics 2005
http://igl.ethz.ch/projects/Laplacian-mesh-processing/STAR/STAR-Laplacian-mesh-processing.pdf
¤ Mesh Laplacianの基礎と応用に関するまとめ

57. Further Readings
Levy, Bruno and Zhang, Hao.
“Spectral Mesh Processing” ACM SIGGRAPH 2010
http://alice.loria.fr/WIKI/index.php/Graphite/SpectralMeshProcessing
¤ Laplacianの特異値分解を用いてMeshの周波数解析

58. Further Readings
[Alexa 2011] Alexa, Marc and Wardetzky, Max.
“Discrete Laplacians on General Polygonal Meshes”
ACM SIGGRAPH 2011
http://cybertron.cg.tu-berlin.de/polymesh/ (ソースコード有)
¤ Discrete Laplacianを一般的なポリゴンメッシュに対し定義