CVPR2016 Fitting Surface Models to Data 抜粋
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This document provides a summary of a CVPR 2016 tutorial on fitting surface models to data. The tutorial covered several applications of surface fitting including curve and surface fitting, parameter estimation, bundle adjustment, and more. It discussed fitting subdivision surfaces and polygon meshes to 2D and video data. Specific examples of fitting hand models to data for applications like hand tracking were presented. The tutorial aimed to teach attendees how to solve hard vision problems using tools that may seem inelegant but are smarter than they appear for fitting models to data.
![17
[3D Scanning Deformable Objects with a Single RGBD Sensor, Dou et al, CVPR15]
Input Kinect Stream KinectFusion Deformable Fusion](https://image.slidesharecdn.com/cvpr2016fittingsurfacemodelstodataaccuracyspeedrobustnessv1-160926042728/85/CVPR2016-Fitting-Surface-Models-to-Data-17-320.jpg)
![IMAGEDENOISING 19[STRANDMARK & AGARWAL, 2014, arXiv:1403.5590]](https://image.slidesharecdn.com/cvpr2016fittingsurfacemodelstodataaccuracyspeedrobustnessv1-160926042728/85/CVPR2016-Fitting-Surface-Models-to-Data-19-320.jpg)
![MATRIXFACTORIZATION[HONG&F.,ICCV15]](https://image.slidesharecdn.com/cvpr2016fittingsurfacemodelstodataaccuracyspeedrobustnessv1-160926042728/85/CVPR2016-Fitting-Surface-Models-to-Data-20-320.jpg)
![MATRIXFACTORIZATION[HONG&F.,ICCV15]](https://image.slidesharecdn.com/cvpr2016fittingsurfacemodelstodataaccuracyspeedrobustnessv1-160926042728/85/CVPR2016-Fitting-Surface-Models-to-Data-21-320.jpg)
![GOAL
Given function
𝑓 𝑥 : ℝ 𝑑
↦ ℝ,
Devise strategies for finding 𝑥 which minimizes 𝑓
• Gradient descent++: Stochastic, Block, Minibatch
• Coordinate descent++: Block
• Newton++:Gauss, Quasi, Damped, Levenberg Marquardt, dogleg,Trust
region, Doublestep LM, [L-]BFGS, NonlinCG
• Not covered
• Proximal methods: Nesterov, ADMM…](https://image.slidesharecdn.com/cvpr2016fittingsurfacemodelstodataaccuracyspeedrobustnessv1-160926042728/85/CVPR2016-Fitting-Surface-Models-to-Data-26-320.jpg)
![USESECONDDERIVATIVES…
Choose 𝛿 so that 𝑓 𝑥 + 𝛿 is better than 𝑓(𝑥)?
Compute
min
𝜹
𝑓 + 𝜹⊤ 𝑔 + 1
2 𝜹⊤ 𝐻 𝜹
[derive]](https://image.slidesharecdn.com/cvpr2016fittingsurfacemodelstodataaccuracyspeedrobustnessv1-160926042728/85/CVPR2016-Fitting-Surface-Models-to-Data-44-320.jpg)
![ Surface: mapping 𝑆 𝒖 from ℝ2
↦ ℝ3
E.g. cylinder 𝑆 𝑢, 𝑣 = cos 𝑢 , sin 𝑢 , 𝑣
Probably not all of ℝ2
, but a subset Ω
E.g. square Ω = 0,2𝜋 × [0, 𝐻]
But also any union of patch domains Ω = 𝑝
Ω 𝑝
SURFACE 62
*the surface is actually the set {𝑀 𝑢; Θ |𝑢 ∈ Ω}
𝑢
𝑣](https://image.slidesharecdn.com/cvpr2016fittingsurfacemodelstodataaccuracyspeedrobustnessv1-160926042728/85/CVPR2016-Fitting-Surface-Models-to-Data-62-320.jpg)
![ Surface: mapping 𝑆 𝒖 from ℝ2
↦ ℝ3
E.g. cylinder 𝑆 𝑢, 𝑣 = cos 𝑢 , sin 𝑢 , 𝑣
Probably not all of ℝ2
, but a subset Ω
E.g. square Ω = 0,2𝜋 × [0, 𝐻]
But also any union of patch domains Ω = 𝑝
Ω 𝑝
And we’ll look at parameterised surfaces 𝑆 𝒖; Θ
E.g. Cylinder 𝑆 𝑢, 𝑣; 𝑅, 𝐻 = 𝑅 cos 𝑢 , 𝑅 sin 𝑢 , 𝐻𝑣
with Ω = 0,2𝜋 × 0,1
E.g. subdivision surface 𝑆 𝒖; 𝑋
where Θ = 𝑋 ∈ ℝ3×𝑛
is matrix of control vertices
SURFACE 63
*the surface is actually the set {𝑀 𝑢; Θ |𝑢 ∈ Ω}
𝑢
𝑣](https://image.slidesharecdn.com/cvpr2016fittingsurfacemodelstodataaccuracyspeedrobustnessv1-160926042728/85/CVPR2016-Fitting-Surface-Models-to-Data-63-320.jpg)
CVPR2016 Fitting Surface Models to Data 抜粋
- 1. CVPR 2016 TUTORIAL FITTING SURFACE MODELS TO DATA のご紹介 2016/10/02 @sumisumith
- 3. Fitting Surface Models to Data 3 CVPR 2016Tutorial Andrew Fitzgibbon, Microsoft JonathanTaylor, PerceptiveIO
- 4. PEOPLE Finding Nemo: Deformable Object Class Modelling using Curve Matching CVPR ’10 Mukta Prasad,Andrew Fitzgibbon,AndrewZisserman, LucVan Gool KinÊtre: Animating theWorld with the Human Body UIST ’12 Jiawen (Kevin) Chen, Shahram Izadi, Fitzgibbon TheVitruvian Manifold: Inferring dense correspondences for one-shot human pose estimation CVPR ’12 JonathanTaylor, Jamie Shotton,TobySharp, Fitzgibbon What shape are dolphins? Building 3D morphable models from 2D images PAMI ’13 Tom Cashman, Fitzgibbon User-Specific Hand Modeling from Monocular Depth Sequences CVPR ’14 Taylor, Richard Stebbing,Varun Ramakrishna, Cem Keskin, Shotton, Izadi, Fitzgibbon,Aaron Hertzmann Real-Time Non-Rigid Reconstruction Using an RGB-D Camera SIGGRAPH ’14 Michael Zollhöfer, Matthias Nießner, Izadi, Christoph Rhemann, Christopher Zach, Matthew Fisher, ChengleiWu, Fitzgibbon,Charles Loop, ChristianTheobalt, Marc Stamminger Learning an Efficient Model of Hand ShapeVariation from Depth Images CVPR ’15 Sameh Khamis,Taylor,Shotton, Keskin, Izadi, Fitzgibbon Efficient and Precise Interactive Hand Tracking through Joint, Continuous Optimization of Pose SIGGRAPH ‘16 and Correspondences Taylor, Lucas Bordeaux,Cashman, Bob Corish, Keskin, Sharp, Eduardo Soto, David Sweeney, JulienValentin, Ben Luff, ArranTopalian, ErrollWood, Khamis, Kohli, Izadi, Richard Banks, Fitzgibbon, Shotton. Fits Like a Glove: Rapid and Reliable Hand Shape Personalization. CVPR ’16 David JosephTan,Cashman,Taylor, Fitzgibbon, DanielTarlow, Khamis, Izadi, Shotton.
- 5. LEARN HOWTO SOLVE HARDVISION PROBLEMS, USINGTOOLSTHAT MAY APPEAR INELEGANT, BUT ARE MUCH SMARTERTHANTHEY LOOK. Goal 5
- 6. APPLICATIONS Curve/surface fitting Parameter estimation “Bundle adjustment” (Video from our friends at Google)
- 7. KINÊTRE 7
- 8. KINÊTRE 8
- 9. KINÊTRE 9
- 15. 15
- 17. 17 [3D Scanning Deformable Objects with a Single RGBD Sensor, Dou et al, CVPR15] Input Kinect Stream KinectFusion Deformable Fusion
- 18. HANDTRACKING • Hand Shape Personalization: • CVPR 2014,CVPR 2015,CVPR 2016 • Discriminative Hand Pose Reinitialization • ICCV 2015,CHI 2015 • Hand Pose Estimation via Model Fitting (read “HandTracking”) • CHI 2015, SIGGRAPH 2016
- 19. IMAGEDENOISING 19[STRANDMARK & AGARWAL, 2014, arXiv:1403.5590]
- 23. Write energy describing the image collection 𝑓=1 𝐹 𝐸data 𝐼𝑓, 𝜽 𝑓 + 𝐸reg 𝜽 𝑓, 𝜽core Where: 𝜽 𝑓 are (unknown) parameters of surface model in frame 𝑓 𝜽core are (unknown) parameters of some shape model (e.g. linear combination) and 𝐸reg measures distance, e.g. ARAP And optimize it using Levenberg-Marquardt (i.e. any Newton-like algorithm, making maximum use of problem structure) FOREACHTASK,THEMETHODISTHESAME 23
- 24. So, you can do lots of things by “fitting models to data”. How do you do it right? Let’s look at some examples. 24
- 26. GOAL Given function 𝑓 𝑥 : ℝ 𝑑 ↦ ℝ, Devise strategies for finding 𝑥 which minimizes 𝑓 • Gradient descent++: Stochastic, Block, Minibatch • Coordinate descent++: Block • Newton++:Gauss, Quasi, Damped, Levenberg Marquardt, dogleg,Trust region, Doublestep LM, [L-]BFGS, NonlinCG • Not covered • Proximal methods: Nesterov, ADMM…
- 27. CLASSESOFFUNCTIONS quadratic convex quasiconvex multi- extremum noisy horrible Given function 𝑓 𝑥 : ℝ 𝑑 ↦ ℝ Devise strategies for finding 𝑥 which minimizes 𝑓
- 28. CLASSESOFFUNCTIONS quadratic convex quasiconvex multi- extremum noisy horrible Given function 𝑓 𝑥 : ℝ 𝑑 ↦ ℝ Devise strategies for finding 𝑥 which minimizes 𝑓
- 31. DERIVATIVES Fast minimization depends on derivatives Gradient 𝑓: ℝ 𝑛 ↦ ℝ When 𝑓 𝒙 = 𝑭 𝒙 2 𝑭: ℝ 𝑛 ↦ ℝ 𝑚 use Jacobian 𝜕𝑭 𝜕𝒙 31
- 34. GRADIENTDESCENT Alternation is slow because valleys may not be axis aligned So try gradient descent? -5 0 5 10 15 -5 0 5 10 15
- 35. GRADIENTDESCENT Alternation is slow because valleys may not be axis aligned So try gradient descent?
- 36. GRADIENTDESCENT Alternation is slow because valleys may not be axis aligned So try gradient descent? Note that convergence proofs are available for both of the above But so what?
- 38. USEABETTERALGORITHM (Nonlinear) conjugate gradients Uses 1st derivatives only Avoids “undoing” previous work
- 39. USEABETTERALGORITHM (Nonlinear) conjugate gradients Uses 1st derivatives only And avoids “undoing” previous work 101 iterations on this problem
- 40. BUTWE CAN DO BETTER…
- 41. USESECONDDERIVATIVES… Starting with 𝒙 how can I choose 𝜹 so that 𝑓 𝒙 + 𝜹 is better than 𝑓(𝒙)? So compute min 𝜹∈ℝ 𝑑 𝑓 𝒙 + 𝜹 But hang on, that’s the same problem we were trying to solve?
- 42. USESECONDDERIVATIVES… Starting with 𝑥 how can I choose 𝛿 so that 𝑓 𝑥 + 𝛿 is better than 𝑓(𝑥)? So compute min 𝛿 𝑓 𝑥 + 𝛿 ≈ min 𝛿 𝑓 𝑥 + 𝛿⊤ 𝑔(𝑥) + 1 2 𝛿⊤ 𝐻 𝑥 𝛿 𝑔 𝑥 = 𝛻𝑓 𝑥 𝐻 𝑥 = 𝛻𝛻⊤ 𝑓(𝑥)
- 43. USESECONDDERIVATIVES… How does it look? 𝑓 𝑥 + 𝛿⊤ 𝑔(𝑥) + 1 2 𝛿⊤ 𝐻 𝑥 𝛿 𝑔 𝑥 = 𝛻𝑓 𝑥 𝐻 𝑥 = 𝛻𝛻⊤ 𝑓(𝑥)
- 44. USESECONDDERIVATIVES… Choose 𝛿 so that 𝑓 𝑥 + 𝛿 is better than 𝑓(𝑥)? Compute min 𝜹 𝑓 + 𝜹⊤ 𝑔 + 1 2 𝜹⊤ 𝐻 𝜹 [derive]
- 45. USESECONDDERIVATIVES… Choose 𝛿 so that 𝑓 𝑥 + 𝛿 is better than 𝑓(𝑥)? Compute min 𝜹 𝑓 + 𝜹⊤ 𝑔 + 1 2 𝜹⊤ 𝐻 𝜹 𝜹 = −𝐻−1 𝑔
- 46. USESECONDDERIVATIVES… Choose 𝛿 so that 𝑓 𝑥 + 𝛿 is better than 𝑓(𝑥)? Updates: 𝜹Newton = −𝐻−1 𝑔 𝜹GradientDescent = −𝜆𝑔
- 47. USESECONDDERIVATIVES… Updates: 𝜹Newton = −𝐻−1 𝑔 𝜹GradientDescent = −𝜆𝑔 So combine them: 𝜹DampedNewton = − 𝐻 + 𝜆−1 𝐼 𝑑 −1 𝑔 = −𝜆 𝜆𝐻 + 𝐼 𝑑 −1 𝑔 𝜆 small ⇒conservative gradient step 𝜆 large ⇒Newton step
- 48. UPDATING 𝜆 𝜆 = 10−3; 𝜆′ = 3; while 𝜆 < 109 𝑓, 𝒈, 𝑯 = error_function(𝒙 𝑘) % Perhaps Gauss-Newton for H 𝜹 = − 𝑯 + 𝜆𝑰 𝒈 % Many ways to do this efficiently 𝒙 𝑛𝑒𝑤 = 𝒙 𝑘 + 𝜹 if error_function(𝒙 𝑛𝑒𝑤) < 𝑓: 𝒙 𝑘 = 𝒙 𝑛𝑒𝑤 % Decreased error, accept the new 𝑥 𝜆 = 𝜆/𝜆′; 𝜆′ = 3 % Doing well—decrease 𝜆 else 𝜆 = 𝜆𝜆′; 𝜆′ = 3𝜆′ % Doing badly—increase 𝜆 quick
- 49. 1ST DERIVATIVESAGAIN Levenberg-Marquardt Just damped Newton with approximate 𝐻 For a special form of 𝑓 𝑓 𝑥 = 𝑖 𝑓𝑖 𝑥 2 where 𝑓𝑖(𝑥) are zero-mean small at the optimum
- 50. BACKTOFIRSTDERIVATIVES Levenberg Marquardt Just damped Newton with approximate 𝐻 For a special form of 𝑓 𝑓 𝑥 = 𝑖 𝑓𝑖 𝑥 2 𝛻𝑓 𝑥 = 𝛻𝛻⊤ 𝑓 𝑥 =
- 51. BACKTOFIRSTDERIVATIVES Levenberg Marquardt Just damped Newton with approximate 𝐻 For a special form of 𝑓 𝑓 𝑥 = 𝑖 𝑓𝑖 𝑥 2 𝛻𝑓 𝑥 = 𝑖 2𝑓𝑖 𝑥 𝛻𝑓𝑖(𝑥) 𝛻𝛻⊤ 𝑓 𝑥 = 2 𝑖 𝑓𝑖 𝑥 𝛻𝛻⊤ 𝑓𝑖 𝑥 + 𝛻𝑓𝑖(𝑥)𝛻⊤ 𝑓𝑖 𝑥
- 52. ORDERNCUBED? Not 𝑂 𝑛3 if you exploit sparsity of Hessian or Jacobian J = 𝛻𝑓1(𝑥) ⋮ 𝛻𝑓𝑛(𝑥)
- 55. CONCLUSION:YMMV
- 56. CONCLUSION:YMMV GIRAFFE 500 runs for k=1:500 𝑥0 = 𝑟𝑎𝑛𝑑𝑛 𝑛, 1 ; 𝑥∗ = 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑓, 𝑥0 ; 𝐸 𝑘 = 𝑓 𝑥∗ end plot(sort(E));
- 58. On many problems, alternation is just fine Indeed always start with a couple of alternation steps Computing 2nd derivatives is a pain But you don’t need to for LM But just alternation is not Unless you’re willing to problem-select Convergence guarantees are fine, but practice is what matters Inverting the Hessian is rarely 𝑂(𝑛3 ) There is no universal optimizer
- 59. WHAT IS A SURFACE? 59
- 60. 0 0.2 0.4 0.6 0.8 1 1.9 2 2.1 2.2 2.3 2.4 CURVES function y(x::Interval)::Real = .3*x + 2 function C(t::Interval)::Point2D = Point2D(t^2 + 2, t^2 – t + 1) function S(u::Interval, v::Real)::Point3D = Point3D(cos(u), sin(u), v) 60 2 2.2 2.4 2.6 2.8 30.75 0.8 0.85 0.9 0.95 1
- 61. Surface: mapping 𝑆 𝒖 from ℝ2 ↦ ℝ3 E.g. cylinder 𝑆 𝑢, 𝑣 = cos 𝑢 , sin 𝑢 , 𝑣 SURFACE 61 *the surface is actually the set {𝑀 𝑢; Θ |𝑢 ∈ Ω} 𝑢 𝑣
- 62. Surface: mapping 𝑆 𝒖 from ℝ2 ↦ ℝ3 E.g. cylinder 𝑆 𝑢, 𝑣 = cos 𝑢 , sin 𝑢 , 𝑣 Probably not all of ℝ2 , but a subset Ω E.g. square Ω = 0,2𝜋 × [0, 𝐻] But also any union of patch domains Ω = 𝑝 Ω 𝑝 SURFACE 62 *the surface is actually the set {𝑀 𝑢; Θ |𝑢 ∈ Ω} 𝑢 𝑣
- 63. Surface: mapping 𝑆 𝒖 from ℝ2 ↦ ℝ3 E.g. cylinder 𝑆 𝑢, 𝑣 = cos 𝑢 , sin 𝑢 , 𝑣 Probably not all of ℝ2 , but a subset Ω E.g. square Ω = 0,2𝜋 × [0, 𝐻] But also any union of patch domains Ω = 𝑝 Ω 𝑝 And we’ll look at parameterised surfaces 𝑆 𝒖; Θ E.g. Cylinder 𝑆 𝑢, 𝑣; 𝑅, 𝐻 = 𝑅 cos 𝑢 , 𝑅 sin 𝑢 , 𝐻𝑣 with Ω = 0,2𝜋 × 0,1 E.g. subdivision surface 𝑆 𝒖; 𝑋 where Θ = 𝑋 ∈ ℝ3×𝑛 is matrix of control vertices SURFACE 63 *the surface is actually the set {𝑀 𝑢; Θ |𝑢 ∈ Ω} 𝑢 𝑣
- 65. CONTROLMESH 65 Control mesh vertices 𝑋 ∈ ℝ3×𝑚 Here 𝑚 = 16
- 66. LIMITSURFACE 66 Control mesh vertices 𝑋 ∈ ℝ3×𝑚 Here 𝑚 = 16
- 71. LIMITSURFACE 71 Control mesh vertices 𝑉 ∈ ℝ3×𝑚 Here 𝑚 = 16 Blue surface is 𝑀 𝒖; 𝑉 | 𝒖 ∈ Ω Ω is the grey surface
- 72. CONTROLVERTICESDEFINETHESHAPE 72 Control mesh vertices 𝑉 ∈ ℝ3×𝑛 Here 𝑛 = 16 Blue surface is 𝑀 𝒖; 𝑉 | 𝒖 ∈ Ω Ω is the grey surface
- 73. Mostly, 𝑀 is quite simple: 𝑀 𝒖; 𝑋 = 𝑀 𝑡, 𝑢, 𝑣; 𝒙1, … , 𝒙 𝑛 = 𝑖+𝑗≤4 𝑘=1..𝑛 𝐴𝑖𝑗𝑘 𝑡 𝑢 𝑖 𝑣 𝑗 𝒙 𝑘 Integer triangle id 𝑡 Quartic in 𝑢, 𝑣 Linear in 𝑋 Easy derivatives But… 2nd Derivatives unbounded although normals well defined Piecewise parameter domain SUBDIVISIONSURFACE:PARAMETRICFORM 73
- 74. EXAMPLES 74
- 76. WHAT SHAPE ARE DOLPHINS? BUILDING 3D MORPHABLE MODELS FROM 2D IMAGES 勝手な通称：イルカの論文 76
- 77. 77 • 概要 • Loop Subdivision Surface で表現された、基本3Dモデル：Bを用意 • シルエット制約などの各種制約を設計し、バランスを取れた モデルができるように、エネルギー関数を設計 • Bを変形させるルール／パラメタに従い、変形させ、 エネルギーが最小／最適な3Dモデルを推定 • 入力： • ヒューリスティックに与えた、キーポイント数個 • 一般 Non-rigid 物体の基本3Dモデル B • シルエット既知の画像 • 各種変形に用いる、パラメタ値 • 出力： • 入力されたシルエット画像に合う、変形された3Dモデル イルカ ( 一般 Non-rigid 物体 ) の姿勢推定
- 78. MODELREPRESENTATION 𝑋 𝑛 = 𝑘=0 𝐾 𝛼𝑖𝑘ℬ 𝑘 𝛼𝑖1 ℬ1 𝛼𝑖2 ℬ2+ +𝑋𝑖 = Linear blend shapes: Image 𝑖 represented by coefficient vector 𝜶𝑖 = 𝛼𝑖1, … , 𝛼𝑖𝐾 ℬ0 78
- 79. 79
- 80. 80
- 81. 81
- 82. 𝒔𝑖𝑗 2D point 𝒏𝑖𝑗 2D normal DATATERMS Image 𝑖 𝒖𝑖𝑗 Contour generator preimage in 𝛀 (unknown) c.g. point in 3D is 𝑀 𝒖𝑖𝑗; 𝑿𝑖 82
- 83. DATATERMS Image 𝑖 𝒔𝑖𝑗, 𝒏𝑖𝑗 83 Camera position Silhouette: 𝐸𝑖 𝑠𝑖𝑙 = 𝑗=1 𝑆𝑖 𝒔𝑖𝑗 − 𝜋 𝜃𝑖, 𝑀 𝑢𝑖𝑗, 𝑿𝑖 2 Normal: 𝐸𝑖 𝑠𝑖𝑙 = 𝑗=1 𝑆𝑖 𝒏𝑖𝑗 0 − 𝑅 𝜃𝑖 𝑁 𝑢𝑖𝑗, 𝑿𝑖 2 Projection e.g. Perspective
- 84. DATATERMS Image 𝑖 𝒔𝑖𝑗, 𝒏𝑖𝑗 Linear Blend Shapes (PCA) Model: 𝑿𝑖 = 𝑘 𝛼𝑖𝑘 𝑩 𝑘 84 Silhouette: 𝐸𝑖 𝑠𝑖𝑙 = 𝑗=1 𝑆𝑖 𝒔𝑖𝑗 − 𝜋 𝜃𝑖, 𝑀 𝑢𝑖𝑗, 𝑿𝑖 2 Normal: 𝐸𝑖 𝑠𝑖𝑙 = 𝑗=1 𝑆𝑖 𝒏𝑖𝑗 0 − 𝑅 𝜃𝑖 𝑁 𝑢𝑖𝑗, 𝑿𝑖 2
- 85. Data fidelity terms 𝑝 𝐼 𝑋𝑖; 𝑈 Gaussian shape weights Smooth contour Smooth Basis 𝑝 𝚯 85
- 86. 86
- 87. CONTINOUSOPTIMIZATION Can focus on this term to understand entire optimization. Total number of residuals 𝑛 = number of silhouette points. Say 300𝑁 (𝑁 = number of images) ≈ 10,000 Total number of unknowns 2𝑛 + 𝐾𝑁 + 𝑚 where 𝑚 ≈ 3𝐾 × number of vertices ≈ 3,000 87
- 88. INITIALESTIMATEFORMEANSHAPE This is true, but misleading 88
- 89. INITIALESTIMATEFORMEANSHAPE True initial estimate: only the topology is really important. But the easiest way to get the topology is to build a rough template. 89
- 90. INITIALESTIMATEFORMEANSHAPE True initial estimate: only the topology is really important. But the easiest way to get the topology is to build a rough template. 90
- 91. 91
- 94. OPTIMIZATION 94
- 95. NUMBEROFIMAGES 95 8 16 32
- 96. PARAMETERSENSITIVITY “Pixel” terms: noise level params “Dimensionless” terms “Smoothness” terms 𝐸 = 𝑖=1 𝑛 𝐸𝑖 sil + 𝐸𝑖 norm + 𝐸𝑖 con + 𝑖=1 𝑛 𝐸𝑖 cg + 𝐸𝑖 reg + 𝝃 𝟎 𝟐 𝐸0 tp + 𝝃 𝐝𝐞𝐟 𝟐 𝑖=1 𝑛 𝐸 𝑚 tp 96
- 99. 99
- 100. 100
- 101. 101
- 102. 102 Fitting Surface Models to Data について、抜粋して、ご紹介 最適化には多種の手法があるが、研究段階や対象のモデルに 合わせて、適宜、利用するものを変えることが望ましい 対象課題についての、なるべく単純なモデルを作れると、 最適化の戦略も練りやすい 実行速度をリアルタイムにするため、モデル制約や行列の スパース性を利用してやり、実現することも可能 まとめ