1. Computer and Robot Vision II
Chapter 15 Motion and Surface Structure from Time Varying Image Sequences
Presented by: 傅楸善 & 林衛成
指導教授: 傅楸善 博士

2. 15.1 Introduction
Motion analysis involves estimating the relative motion of objects with respect to each other and the camera given two or more perspective projection images in a time sequence.
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3. 15.1 Introduction (cont’) Real-world applications:
industrial automation and inspection, robot assembly, autonomous vehicle navigation, biomedical engineering, remote sensing, general 3D-scene understanding
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4. 15.1 Introduction (cont’)
object motion and surface structure recovery from:
observed optic flow
point correspondences
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5. 15.2 The Fundamental Optic Flow Equation
(x, y, z): 3D point on moving rigid body
(u, v): perspective projection on the image plane
f: camera constant
(u, v): velocity of the point (u, v) . .
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6. 15.2 The Fundamental Optic Flow Equation (cont’)
take time derivatives of both sides
yields the fundamental optic flow equation: . . . . . . . . . . .
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7. 15.2 The Fundamental Optic Flow Equation (cont’)
general solution: (λ is a free variable) . . . . .
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8. 15.2.1 Translational Motion . . . . . Known: Unknown:
N-point optic flow field:
Unknown:
corresponding unknown 3D points:
all points moving with same but unknown velocity
(x, y, z)
can be solved up to a multiplicative constant . . . . .
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9. 15.2.2 Focus of Expansion and Contraction
Known:
3D motion is translational
one 2D projected point (u, v) has no motion:
thus translational motion is in a direction along the ray of sight . .
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10. 15.2.2 Focus of Expansion and Contraction (cont’)
focus of expansion (FOE): if 3D point field moving toward camera
FOE: motion-field vectors radiate outward from that point
focus of contraction (FOC): if 3D point field moving away from camera
FOC: vectors radiate inward toward diametrically opposite point flow pattern of the motion field of a forward-moving observer
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12. 15.2.3 Moving Line Segment . . . Known:
fixed distance between two unknown 3D points
translational motion with common velocity
(x, y, z)
corresponding optic flow: . . .
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13. 15.2.3 Moving Line Segment (cont’)
Unknown:
: two unknown 3D points
common velocity: (x, y, z) . . .
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14. 15.2.3 Moving Line Segment (cont’)
From the perspective projection equations:
From the optic flow equation:
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15. 15.2.3 Moving Line Segment (cont’)
From the known length of the line segment:
The optic flow equation (15.9) permits us to obtain a least squares solution for z in terms of z1 and z2, from
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16. 15.2.3 Moving Line Segment (cont’)
We obtain
Substituting this back into the equation, we can solve z2 in terms of z1:z2=kz1
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17. 15.2.3 Moving Line Segment (cont’)
Substitute the relations for (x1, y1, z1) from equations into Eq.(15.10) to obtain
Hence:
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18. 15.2.4 Optic Flow Acceleration Invariant
Since
differentiating general solution in Sec 15.2 and solve for (x, y, z) .. .. .. . .. .. . .. . .. .. ..
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19. 15.3 Rigid-Body Motion
Rigid-body motion: no relative motion of points w.r.t. (with respect to) one another
Rigid-body motion: points maintain fixed position relative to one another
Rigid-body motion: all points move with the body as a whole
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20. 15.3 Rigid-Body Motion (cont’)
R(t): rotation matrix
T(t): translation vector
p(0): initial position of given point
R(0)=I, T(0)=0
p(t): position of given point at time t
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21. 15.3 Rigid-Body Motion (cont’)
Rigid-body motion in displacement vectors:
velocity vector: time derivative of its position: . . .
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22. 15.3 Rigid-Body Motion (cont’)
Since
(a) translational-motion field under projection onto hemispherical surface only translational-component motion useful in determining scene structure
(b) rotational-motion field under projection onto hemispherical surface rotational-motion field provides no information about scene structure . . .
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23. 15.3 Rigid-Body Motion (cont’)
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24. 15.3 Rigid-Body Motion (cont’)
we can describe rigid-body motion in instantaneous velocity by .
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25. 15.3 Rigid-Body Motion (cont’)
: angular velocities in three axes
: translational velocities in three axes
from rigid-body-motion equation . . .
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26. 15.3 Rigid-Body Motion (cont’)
and perspective projection equation
we can determine an expression for z: . .
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27. 15.3 Rigid-Body Motion (cont’)
after simplification . .
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28. 15.3 Rigid-Body Motion (cont’)
image velocity: expressed as sum of translational field and rotational field
(x, y, z): 3D coordinate before rigid-body motion in displacement vectors
(x’, y’, z’): 3D coordinate after rigid-body motion in displacement vectors
: rotation angles in three axes
: translation in three axes
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29. 15.3 Rigid-Body Motion (cont’)
Rigid-body motion in displacement vectors:
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30. 15.3 Rigid-Body Motion (cont’)
motion in displacement vector and instantaneous velocity is different:
e.g. moon encircling earth
instantaneous velocity: first order approximation of displacement vector
first order approximation: when small,
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31. 15.3 Rigid-Body Motion (cont’)
first order approximation: when time=1 thus
x=(x’ - x)/1
first order approximation: .
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32. 15.4 Linear Algorithms for Motion and Surface Structure from Optic Flow
The Planar Patch Case
: arbitrary object point on planar patch at time t
: central projective coordinates of p(t) onto image plane z= f
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33. The Planar Patch Case . .
: instantaneous velocity of moving image point
: optic flow image point
: instantaneous rotational angular velocity
: instantaneous translational velocity . .
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34. 15.4.1 The Planar Patch Case (cont’)
unit vector n(t): orthogonal to moving planar patch
rigid planar patch motion represented by rigid-motion constraint: .
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35. 15.4.1 The Planar Patch Case (cont’)
from above two equations:
Let
Rigid-motion constraint could be written as
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36. 15.4.1 The Planar Patch Case (cont’)
denote the 3 x 3 matrix by W and its three row vectors by
W: called planar motion parameter matrix
since skew symmetric
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37. 15.4.1 The Planar Patch Case (cont’)
above equation can be written as
from perspective projection equations:
taking time derivatives of these equations we have . . . . . . . . .
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38. 15.4.1 The Planar Patch Case (cont’)
substitute equations into above equations:
from third row
substitute z to obtain optical flow-planar motion equation . . . . . . . . .
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39. 15.4.1 The Planar Patch Case (cont’)
we have 2N linear equations: n=1,…,N:
optic flow-planar motion recovery: first solve W then find . .
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40. 15.4.2 General Case Optic Flow-Motion Equation
1. set up optic flow-motion equation not involving depth information
2. solve it by using linear least-squares technique
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41. 15.4.3 A Linear Algorithm for Solving Optic Flow-Motion Equations
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42. 15.4.4 Mode of Motion, Direction of Translation, and Surface Structure
mode of motion: whether translation k=0 or not
direction of translation: direction of k
surface structure: relative depth when k 0
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43. 15.4.5 Linear Optic Flow-Motion Algorithm and Simulation Results
motion and shape recovery algorithms should answer three questions:
minimum number of points to compute motion and shape
what set of optic flow points violate rank assumption e.g. collinearity…
What’s the accuracy of estimated motion from noisy optic flow?
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44. 15.5 The Two View-Linear Motion Algorithm
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45. 15.5.1 Planar Patch Motion Recovery from Two Perspective Views: A Brief Review
Two View-Planar Motion Equation
imaging geometry for two view-planar motion
rigid planar patch in motion in half-space z< 0
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46. DC & CV Lab.
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47. 15.5.1 Planar Patch Motion Recovery from Two Perspective Views: A Brief Review (cont’)
: arbitrary object point before motion
: same object point after motion
: central projective coordinates of
f : camera constant
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48. 15.5.1 Planar Patch Motion Recovery from Two Perspective Views: A Brief Review (cont’)
R0: 3 X 3 rotational matrix, R0’R0=I,|R0|=1
t0: 3 X 1 translational vector
n0: 3 X 1 normal vector
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49. 15.5.1 Planar Patch Motion Recovery from Two Perspective Views: A Brief Review (cont’)
Rigid-body-motion equation relates p1 to p2 as follows:
planarity constrains p1 by
combining two equations produces planar rigid-body-motion-equation
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50. 15.5.1 Planar Patch Motion Recovery from Two Perspective Views: A Brief Review (cont’)
Projecting the planar rigid-body motion onto the image plane z = f produces
Let the planar rigid –motion parameter matrix be defined by
Where bi, i=1,2,3, are three row vectors of B.
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51. 15.5.1 Planar Patch Motion Recovery from Two Perspective Views: A Brief Review (cont’)
Then above equation could be written as
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52. 15.5.1 Planar Patch Motion Recovery from Two Perspective Views: A Brief Review (cont’)
From above we derive the two view-planar motion equation
With the natural constraint
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53. 15.5.1 Planar Patch Motion Recovery from Two Perspective Views: A Brief Review (cont’)
Now the planar motion recovery problem involves first solving the planar rigid-motion parameter matrix B and then estimating
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54. 15.5.2 General Curved Patch Motion Recovery from Two Perspective Views A Simplified Linear Algorithm
discard planar patch assumption, consider general curved patch
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55. 15.5.3 Determining Translational Orientation
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56. 15.5.4 Determining Mode of Motion and Relative Depths
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57. 15.5.5 A Simplified Two View-Motion Linear Algorithm
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58. 15.5.6 Discussion and Summary
when no noise appears: algorithm extremely accurate
when small noise appears: it works well except mode of motion incorrect
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59. 15.6 Linear Algorithm for Motion and Structure from Three Orthographic Views
Ullman (1979) showed that for the orthographic case four-point correspondences over three views are sufficient to determine the motion and structure of the four-point rigid configuration
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60. Shimon Ullman, The Interpretation of Visual Motion
The MIT Press, Cambridge MA. 1979
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61. 15.6 Linear Algorithm for Motion and Structure from Three Orthographic Views
to infer depth information: translation needed in perspective projection
to infer depth information: rotation useless in perspective projection
to infer depth information: rotation needed in orthographic projection
to infer depth information translation useless in orthographic projection
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62. DC & CV Lab.
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63. Problem Formulation
image plane stationary three orthographic views at time
(x, y, z): object-space coordinates of point P at t1
(x’, y’, z’): object-space coordinates of point P at t2
(x”, y”, z”): object-space coordinates of point P at t3
(u, v): image-space coordinates of P at t1
(u’, v’): image-space coordinates of P at t2
(u”, v”): image-space coordinates of P at t3
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64. 15.6.1 Problem Formulation (cont’)
: rotation matrix
: translation vector
(x’, y’, z’)’ = R(x, y, z)’ + Tr
(x”, y”, z”)’ = S(x, y, z)’ + Ts
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65. 15.6.1 Problem Formulation (cont’)
Known:
four image-point correspondences
Unkown:
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66. 15.6.1 Problem Formulation (cont’)
note that with orthographic projections
therefore it is obvious that tr3, ts3 can never be determined
we are trying to determine:
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67. Determining
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68. 15.6.3 Solving a Unique Orthonormal Matrix R
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69. 15.6.4 Linear Algorithm to Uniquely Solve R, S, a3
a3 : (z2 – z1, z3 – z1, z4 – z1)
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70. Summary
Given two orthographic views, one cannot finitely determine the motion and structure of a rigid body, no matter how many point correspondences are used, as shown by Huang.
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71. 15.7 Developing a Highly Robust Estimator for General Regression
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72. 15.7.1 Inability of the Classical Robust M-Estimator to Render High Robustness
Classical robust estimator, such as M-, L-, or R-estimator:
1. optimal or nearly optimal at assumed noise distribution
2. relatively small performance degradation with small number of outliers
3. larger deviations from assumed distribution do not cause catastrophe
MF-estimator with new property much stronger than property 3
4. relatively small performance degradation with larger deviations from assumed distribution
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73. 15.7.2 Partially Modeling Log Likelihood Function by Using Heuristics
MF-estimator: robust regression more appropriate model-fitting
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74. 15.7.3 Discussion M-, L-, R and MF-estimator: all residual based
Residual: the difference between the sample and the observed sample mean or regressed (fitted) function
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75. MF-Estimator
MF-estimator: combine Bayes statistical decision rule with heuristics
Bayes statistical decision rule: maximizes the posterior expected value of a utility function or minimizes the posterior expected value of a loss function (also called posterior expected loss)
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76. 15.8 Optic Flow-Instantaneous Rigid-Motion Segmentation and Estimation
Formulate optic flow-single rigid-motion estimation into general regression
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77. Single Rigid Motion
P(t): position vector of an object point at the time t
[X(t), Y(t)]: central projective coordinate of P(t)
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78. 15.8.1 Single Rigid Motion (cont’)
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79. 15.8.1 Single Rigid Motion (cont’)
: noisy optic flow image point
Instantaneous representation of the rigid motion is described by
Instantaneous rotational angular velocity of rigid motion:
Instantaneous translational angular velocity of rigid motion:
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80. 15.8.1 Single Rigid Motion (cont’)
Differentiating above equation
where for simplicity the time variable t has been omitted
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81. 15.8.1 Single Rigid Motion (cont’)
Combine above two equations:
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82. 15.8.2 Multiple Rigid Motions
We turn the optic flow-multiple rigid –motion segmentation and estimation problem into a number of successive optic flow-single rigid-motion estimation problems.
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83. 15.9 Experimental Protocol
Simulate simplest location estimation
Simulate Optic flow-rigid-motion segmentation and estimation
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84. 15.9 Experimental Protocol (cont’)
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85. 15.9 Experimental Protocol (cont’)
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86. 15.9 Experimental Protocol (cont’)
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87. 15.10 Motion and Surface Structure from Line Correspondences
Discussion concerns only the general rigid motion of straight-line structure
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88. Problem Formulation
Cartesian reference system-central projection
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89. 15.10.1 Problem Formulation (cont’)
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90. 15.10.1 Problem Formulation (cont’)
l: line in 3D space
L: projection of the line on image plane z = f
z = f : image frame
: known plane line L is in; projective plane of l
: set of lines in 3D space
: lines moved by rigid motion
(R’ , T’)’ at time t’
(R” , T”)” at time t”
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91. 15.10.1 Problem Formulation (cont’)
: projections of lines
; respective projective planes
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92. 15.10.1 Problem Formulation (cont’)
Known:
K triples of line correspondences in three views
Unkown:
rotations and translations:
3D lines
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93. 15.10.2 Solving Rotation Matrices R’, R” and Translations T’,R”
: the normals of the ith projective planes
Then
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94. 15.10.2 Solving Rotation Matrices R’, R” and Translations T’,R”
Above two equations define a unique three-dimensional line solution if and only if
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95. 15.10.3 Solving Three-Dimensional Line Structure
Once rotation martrices R’, R” and translations T’, T” are solved, each three-dimensional line li can be determined
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96. 15.11 Multiple Rigid Motions from Two Perspective Views
Problem Statement
imaging geometry for two-view-motion
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97. DC & CV Lab.
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98. Problem Statement
How many good point correspondences are needed in order to apply the nonlinear least-squares estimator?
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99. 15.11.2 Simulated Experiments
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100. 15.11.2 Simulated Experiments (cont’)
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101. 15.11.2 Simulated Experiments (cont’)
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102. 15.12 Rigid Motion from Three Orthographic Views
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103. 15.12.1 Problem Formulation and Algorithm
same as Sec. 15.6, instead of linear algorithms, formulate model-fitting problem
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104. 15.12.2 Simulated Experiments
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105. 15.12.2 Simulated Experiments (cont’)
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106. 15.12.2 Simulated Experiments (cont’)
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107. 15.12.3 Further Research on the MF-Estimator
two problems to be solved for MF-estimator to be practically useful:
distance problem
requirement for a good initial approximation
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108. difficulty of motion and shape recovery: ambiguity of displacement field
Fuh. Ph.D. Thesis, Fig 4.1
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109. DC & CV Lab.
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110. 15.13 Literature Review 15.13.1 Inferring Motion and Surface Structure
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111. 15.13.1 Inferring Motion and Surface Structure
classifications for methods of inferring 3D motion and shape
use of individual sets of feature points
use of local optic flow information about a single point
use of the entire optic flow field
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112. 15.13.1 Inferring Motion and Surface Structure
Despite all the results obtained over the years, almost none of these inference techniques have been successfully applied to feature-point correspondences calculated from real imagery
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113. 15.13.2 Computing Optic Flow or Image-Point Correspondences
problem source contains abundant information
occlusion boundaries
specular points
near a focus of expansion
noise and digitization effects in image formation
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114. 15.13.2 Computing Optic Flow or Image-Point Correspondences (cont’)
motion parallax: apparent relative motion between objects and observer
points in observer’s direction of translation remain relatively unchanged information available to a moving observer
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115. DC & CV Lab.
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116. 15.13.2 Computing Optic Flow or Image-Point Correspondences (cont’)
impart time dimension to image data
spatiotemporal image data block
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117. DC & CV Lab.
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118. 15.13.2 Computing Optic Flow or Image-Point Correspondences (cont’)
motion field: assignment of vectors to image points representing motion
angular velocity of fixed scene: inversely proportional to distance pilot in straight-ahead level flight on an overcast day
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119. DC & CV Lab.
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120. DC & CV Lab.
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121. 15.13.2 Computing Optic Flow or Image-Point Correspondences (cont’)
motion field of pilot looking straight ahead in motion direction
zero image velocity: at approach point and at infinity (along horizon)
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122. DC & CV Lab.
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123. 15.13.2 Computing Optic Flow or Image-Point Correspondences (cont’)
motion field of pilot looking to the right in level flight
focus of expansion here: at infinity to the left
focus of contraction here: at infinity to the right of the figure
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124. DC & CV Lab.
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125. 15.13.2 Computing Optic Flow or Image-Point Correspondences (cont’)
spatiotemporal image data acquired by a camera,- caption -
straight streaks at block top due to translating parallel to image plane
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126. DC & CV Lab.
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127. B.K.P. Horn, Robot Vision, The MIT Press, Cambridge, MA, 1986
Chapter 12 Motion Field & Optical Flow
optic flow: apparent motion of brightness patterns during relative motion
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128. 12.1 Motion Field
motion field: assigns velocity vector to each point in the image
Po: some point on the surface of an object
Pi: corresponding point in the image
vo: object point velocity relative to camera
vi: motion in corresponding image point
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129. 12.1 Motion Field (cont’)
ri: distance between perspectivity center and image point
ro: distance between perspectivity center and object point
f’: camera constant
z: depth axis, optic axis
object point displacement causes corresponding image point displacement
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130. 12.1 Motion Field (cont’)
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131. 12.1 Motion Field (cont’) Velocities: where ro and ri are related by
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132. 12.1 Motion Field (cont’)
differentiation of this perspective projection equation yields
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133. 12.2 Optical Flow
optical flow need not always correspond to the motion field
(a) perfectly uniform sphere rotating under constant illumination:
no optical flow, yet nonzero motion field
(b) fixed sphere illuminated by moving light source:
nonzero optical flow, yet zero motion field
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134. DC & CV Lab.
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135. 12.2 Optical Flow (cont’)
not easy to decide which P’ on contour C’ corresponds to P on C
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136. DC & CV Lab.
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137. 12.2 Optical Flow (cont’)
optical flow: not uniquely determined by local information in changing
irradiance at time t at image point (x, y)
components of optical flow vector
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138. 12.2 Optical Flow (cont’) assumption: irradiance the same at time
fact: motion field continuous almost everywhere
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139. 12.2 Optical Flow (cont’) expand above equation in Taylor series
e: second- and higher-order terms in
cancelling E( x, y, t), dividing through by
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140. 12.2 Optical Flow (cont’)
which is actually just the expansion of the equation
abbreviations:
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141. 12.2 Optical Flow (cont’) we obtain optical flow constraint equation:
flow velocity (u, v): lies along straight line perpendicular to intensity gradient
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142. DC & CV Lab.
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143. 12.2 Optical Flow (cont’) rewrite constraint equation:
aperture problem: cannot determine optical flow along isobrightness contour
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144. 12.3 Smoothness of the Optical Flow
motion field: usually varies smoothly in most parts of image
try to minimize a measure of departure from smoothness
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145. 12.3 Smoothness of the Optical Flow (cont’)
error in optical flow constraint equation should be small
overall, to minimize
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146. 12.3 Smoothness of the Optical Flow (cont’)
large if brightness measurements are accurate
small if brightness measurements are noisy
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147. 12.4 Filling in Optical Flow Information
regions of uniform brightness: optical flow velocity cannot be found locally
brightness corners: reliable information is available
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148. 12.5 Boundary Conditions
Well-posed problem: solution exists and is unique
partial differential equation: infinite number of solution unless with boundary
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149. 12.6 The Discrete Case
first partial derivatives of u, v: can be estimated using difference
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150. DC & CV Lab.
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151. 12.6 The Discrete Case (cont’)
measure of departure from smoothness:
error in optical flow constraint equation:
to seek set of values that minimize
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152. 12.6 The Discrete Case (cont’)
dieffrentiating e with respect to
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153. 12.6 The Discrete Case (cont’)
where are local average of u, v (with 8 neighbors? )
extremum occurs where the above derivatives of e are zero:
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154. 12.6 The Discrete Case (cont’)
determinant of 2x2 coefficient matrix:
so that
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155. 12.6 The Discrete Case (cont’)
suggests iterative scheme such as
new value of (u, v): average of surrounding values minus adjustment
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156. DC & CV Lab.
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157. 12.6 The Discrete Case (cont’)
first derivatives estimated using first differences in 2x2x2 cube
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158. DC & CV Lab.
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159. 12.6 The Discrete Case (cont’)
consistent estimates of three first partial derivatives:
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160. 12.6 The Discrete Case (cont’)
four successive synthetic images of rotating sphere
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161. DC & CV Lab.
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162. 12.6 The Discrete Case (cont’)
estimated optical flow after 1, 4, 16, and 64 iterations
DC & CV Lab.
CSIE NTU

163. DC & CV Lab.
CSIE NTU

164. 12.6 The Discrete Case (cont’)
(a) estimated optical flow after several more iterations
(b) computed motion field
DC & CV Lab.
CSIE NTU

165. DC & CV Lab.
CSIE NTU

166. 12.7 Discontinuities in Optical Flow
discontinuities in optical flow: on silhouettes where occlusion occurs
DC & CV Lab.
CSIE NTU

167. Project due Apr. 28
implementing Horn & Schunck optical flow estimation as above
synthetically translate lena.im one pixel to the right and downward
Try
DC & CV Lab.
CSIE NTU