Computer and Robot Vision II
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Transcript of Computer and Robot Vision II
Computer and Robot Vision II
Chapter 15Motion and Surface Structure from
Time Varying Image Sequences
Presented by: 傅楸善 & 王林農0917 533843
[email protected]指導教授 : 傅楸善 博士
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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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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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15.1 Introduction (cont’)
object motion and surface structure recovery from: observed optic flow point correspondences
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15.2 The Fundamental Optic Flow Equation
(x, y, z): 3D point on moving rigid body (u, v): perspective projection on the image pla
ne f: camera constant (u, v): velocity of the point (u, v)
. .
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15.2 The Fundamental Optic Flow Equation (cont’)
take time derivatives of both sides
yields the fundamental optic flow equation:
y
x
z
f
v
u
yzzy
xzzx
z
f
v
u2
z
y
x
v
u
f
f
zv
u 0
0
1
.
.. .. .
.
.
.
.
.
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15.2 The Fundamental Optic Flow Equation (cont’)
general solution: (λ is a free variable)
f
v
u
v
u
f
z
z
y
x
0
.
.
.
.
.
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15.2.1 Translational Motion
Known: 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
Nnnnnn vuvu 1)},,,{(
. .
Nnnnn zyx 1)},,{(
. . .
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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
0vu. .
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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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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:
),,(),,,( 222111 zyxzyx
. . .
)()( 22221111 v,u,,vu,v,u,,vu
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15.2.3 Moving Line Segment (cont’)
Unknown: : two unknown 3D points
common velocity: (x, y, z)
),,(),,,( 222111 zyxzyx
. . .
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15.2.3 Moving Line Segment (cont’)
From the perspective projection equations:
From the optic flow equation:
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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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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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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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15.2.4 Optic Flow Acceleration Invariant
Since differentiating general solution in Sec 15.2 and s
olve for (x, y, z)
zf .
.. .. ..
f
v
u
f
zv
u
f
zv
u
f
z
z
y
x
0
2
0
..
..
..
..
..
.
.. .
.
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joke
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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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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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15.3 Rigid-Body Motion (cont’)
Rigid-body motion in displacement vectors:
velocity vector: time derivative of its position:
)()0()()( tTptRtP
)()0()()( tTptRtP .. .
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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
)()()()()()()()(
)]()()[()0(11
1
tTtTtRtRtptRtRtp
tTtptRp
. . .
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15.3 Rigid-Body Motion (cont’)
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15.3 Rigid-Body Motion (cont’)
we can describe rigid-body motion in instantaneous velocity by
)()()()( tktpttp .
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15.3 Rigid-Body Motion (cont’)
: angular velocities in three axes : translational velocities in three axes from rigid-body-motion equation
zyx ,,
zyx kkk ,,
zyx
yxz
xzy
z
y
x
z
y
x
kxy
kzx
kyz
k
k
k
z
y
x
k
z
y
x
z
y
x
.
.
.
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15.3 Rigid-Body Motion (cont’)
and perspective projection equation
we can determine an expression for z:
f
v
u
f
z
z
y
x
.
zyx kuvf
zz )(
.
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15.3 Rigid-Body Motion (cont’)
after simplification
z
y
x
z
y
x
k
k
k
u
v
f
uv
f
uf
f
vf
f
uv
z
uz
u
z
fz
f
v
u
22
22
0
0
..
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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
),,( zyx ),,( zyx ttt
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15.3 Rigid-Body Motion (cont’)
Rigid-body motion in displacement vectors:
z
y
x
t
t
t
z
y
x
RRR
z
y
x
yxyxy
zxzyxzxzyxzy
zxzyxzxzyxzy
z
y
x
xyz
coscoscossinsin
sinsincossincoscoscossinsinsinsincos
sinsincossincossincoscossinsincoscos
)()()(
'
'
'
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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,
0sinsin,1cos,sin
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15.3 Rigid-Body Motion (cont’)
first order approximation: when time=1 thus x=(x’ - x)/1 first order approximation:
.
zzyyxxzzyyxx tktktk ,,,,,
zxy
yzx
xyz
z
y
x
xy
xz
yz
tyxz
txzy
tzyx
t
t
t
z
y
x
z
y
x
1
1
1
'
'
'
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joke
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15.4 Linear Algorithms for Motion and Surface Structure from Optic Flow
15.4.1 The Planar Patch Case : arbitrary object point on planar
patch at time t : central projective coordinates of p(t) o
nto image plane z= f
)]'(),(),([)( tztytxtp
)](),([ tvtu
]'),(),([)(
)(
)(
)()(
)(
)()(
ftvtuf
tztp
tz
tyftv
tz
txftu
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15.4.1 The Planar Patch Case
: instantaneous velocity of moving image point
: optic flow image point : instantaneous rotational angular
velocity : instantaneous translational velocity
)](),([ tvtu)](),([ tvtu
. .
)](),(),(),([ tvtutvtu . .
)(t
)(tk
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15.4.1 The Planar Patch Case (cont’)
unit vector n(t): orthogonal to moving planar patch rigid planar patch motion represented by rigid-moti
on constraint:
1)()'(
)()()()(
tptn
tktptwtp.
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15.4.1 The Planar Patch Case (cont’)
from above two equations:
Let
Rigid-motion constraint could be written as
pknpknpp
pknpp
)'('
0
0
0
' ,,
'
12
13
23
321
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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
'kn',',' 321
''' nkknWW
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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
vzzv
uzzu
fy
x
v
u
f
z
y
x
z
y
x
W
z
y
x
1..
..
.
.
.
.
.
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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
v
u
f
v
u
v
u
f
f
v
u
f
zz
f
v
u
z
fz
vzzv
uzzu
32
1
3
3
2
1
'1
'
'
'
'
'
'
. .
. ..
.
.
.
.
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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
n
nn
n
n
n
v
u
f
v
u
v
u
f 32
1 '1
'
'
.
.
nk,,
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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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15.4.3 A Linear Algorithm for Solving Optic FlowMotion Equations
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15.5.4 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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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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joke
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15.5 The Two View-Linear Motion Algorithm
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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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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
),(),,(
)',,(
)',,(
2211
2222
1111
vuvu
zyxp
zyxp
21, pp
2
22
2
22
1
11
1
11
,
,
zyfvz
xfu
zyfvz
xfu
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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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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
10002
10
0102
)'(
1'
pntRp
pn
tpRp
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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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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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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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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 000 and,, rtR
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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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15.5.3 Determining Translational Orientation
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15.5.4 Determining Mode of Motion and Relative Depths
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15.5.5 A Simplified Two View-Motion Linear Algorithm
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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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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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Shimon Ullman, The Interpretation of Visual Motion
The MIT Press, Cambridge MA. 1979
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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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15.6.1 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
321 ,, ttt
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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
1313
3333
)(,)(
)(,)(
sisrir
ijij
sTtT
sSrR
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15.6.1 Problem Formulation (cont’)
Known: four image-point correspondences
Unkown:
4,3,2,1),,,(
),(),,(
4,3,2,1),","()','(),(
izyx
TSTR
iuuuuuu
iii
sr
iiiiii
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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:
)","()","(
)','()','(
),(),(
yxvu
yxvu
yxvu
4,3,2,,2,1,,,, 1 izzittSR isiri
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15.6.2 Determining ),,,(),,,,(,, 32312313323123133333 ssssrrrrsr
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15.6.3 Solving a Unique Orthonormal Matrix R
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15.6.4 Linear Algorithm to Uniquely Solve R, s, a3
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15.6.5 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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Joke
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15.7 Developing a Highly Robust Estimator for General Regression
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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
relatively small performance degradation with larger deviations from assumed distribution
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15.7.2 Partially Modeling Log Likelihood Function by Using Heuristics
MF-estimator: robust regression more appropriate model-fitting
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15.7.3 Discussion
M-, L-, R and MF-estimator: all residual based
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15.7.4 MF-Estimator
MF-estimator: combine Bayes statistical decision rule with heuristics
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15.8 Optic Flow-Instantaneous Rigid-Motion Segmentation and Estimation
formulate optic flow-single rigid-motion estimation into general regression
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15.8.1 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)
)]'(),(),([)( tztytxtp
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15.8.1 Single Rigid Motion (cont’)
]'),(),([)(
)(
)(
)()(
)(
)()(
ftYtXf
tztp
tz
tyftY
tz
txftX
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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:
)]'(),(),([)(
)]'(),(),([)(
)()()()(
)]}(),([)],(),({[
321
321
tktktktk
tttt
tktpttp
tvtutYtX
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15.8.1 Single Rigid Motion (cont’)
Differentiating above equation
where for simplicity the time variable t has been omitted
)'0,,()',,( vuf
zfYX
f
zp
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15.8.1 Single Rigid Motion (cont’)
Combine above two equations:
)'0,,()',,()',,( vuf
zfYX
f
zkfYX
f
z
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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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joke
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15.9 Experimental Protocol
Simulate simplest location estimation Simulate Optic flow-rigid-motion
segmentation and estimation
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15.9 Experimental Protocol (cont’)
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15.9 Experimental Protocol (cont’)
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15.9 Experimental Protocol (cont’)
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15.10 Motion and Surface Structure from Line Correspondences
Discussion concerns only the general rigid motion of straight-line structure
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15.10.1 Problem Formulation
Cartesian reference system-central projection
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15.10.1 Problem Formulation (cont’)
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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’ : lines moved by rigid motion (R” , T”)” at time t”
},...,{
},...,{
},...,{
""1
''1
1
k
k
k
ll
ll
ll
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15.10.1 Problem Formulation (cont’)
: projections of lines
; respective projective planes
}"{}{}{
}{}{}{
k1k1k1
k1k1k1
",...,,',...,',,...,
L",...,L",L',...,L',L,...,L
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15.10.1 Problem Formulation (cont’)
Known: K triples of line correspondences in three views
Unkown: rotations and translations: 3D lines
kiLLL iii ,...,1,"'
",',",' TTRR
Kili ,...,1,
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15.10.2 Solving Rotation Matrices R’, R” and Translations T’,R”
: the normals of the ith projective planes
Then
",',
",',
iii
iii nnn
0)""(
0)''(
0
"
'
TpRpn
TpRpn
pn
ti
ti
ti
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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
2''
Rn
nRank
ti
ti
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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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15.11 Multiple Rigid Motions from Two Perspective Views
15.11.1 Problem Statement imaging geometry for two-view-motion
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15.11.1 Problem Statement
How many good point correspondences are needed in order to apply the nonlinear least-squares estimator?
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15.11.2 Simulated Experiments
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15.11.2 Simulated Experiments (cont’)
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15.11.2 Simulated Experiments (cont’)
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15.12 Rigid Motion from Three Orthographic Views
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15.12.1 Problem Formulation and Algorithm
same as Sec. 15.6, instead of linear algorithms, formulate model-fitting problem
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15.12.2 Simulated Experiments
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15.12.2 Simulated Experiments (cont’)
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15.12.2 Simulated Experiments (cont’)
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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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difficulty of motion and shape recovery: ambiguity of displacement field
Fuh. Ph.D. Thesis, Fig 4.1
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15.13 Literature Review
15.13.1 Inferring Motion and Surface Structure
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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 p
oint use of the entire optic flow field
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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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Joke
DC & CV Lab.DC & CV Lab.CSIE NTU
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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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
DC & CV Lab.DC & CV Lab.CSIE NTU
DC & CV Lab.DC & CV Lab.CSIE NTU
15.13.2 Computing Optic Flow or Image-Point Correspondences (cont’)
impart time dimension to image data spatiotemporal image data block
DC & CV Lab.DC & CV Lab.CSIE NTU
DC & CV Lab.DC & CV Lab.CSIE NTU
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
DC & CV Lab.DC & CV Lab.CSIE NTU
DC & CV Lab.DC & CV Lab.CSIE NTU
DC & CV Lab.DC & CV Lab.CSIE NTU
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)
DC & CV Lab.DC & CV Lab.CSIE NTU
DC & CV Lab.DC & CV Lab.CSIE NTU
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 righ
t of the figure
DC & CV Lab.DC & CV Lab.CSIE NTU
DC & CV Lab.DC & CV Lab.CSIE NTU
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
DC & CV Lab.DC & CV Lab.CSIE NTU
DC & CV Lab.DC & CV Lab.CSIE NTU
joke
DC & CV Lab.DC & CV Lab.CSIE NTU
B.K.P. Horn, Robot Vision, The MIT Press, Cambridge, MA, 1986
Chapter 12 Motion Field & Optical Flow optic flow: apparent motion of brightness patt
erns during relative motion
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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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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 correspondi
ng image point displacement
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12.1 Motion Field (cont’)
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12.1 Motion Field (cont’)
Velocities:
where ro and ri are related by
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12.1 Motion Field (cont’)
differentiation of this perspective projection equation yields
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joke
DC & CV Lab.DC & CV Lab.CSIE NTU
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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DC & CV Lab.DC & CV Lab.CSIE NTU
12.2 Optical Flow (cont’)
not easy to decide which P’ on contour C’ corresponds to P on C
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DC & CV Lab.DC & CV Lab.CSIE NTU
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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12.2 Optical Flow (cont’)
assumption: irradiance the same at time
fact: motion field continuous almost everywhere
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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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12.2 Optical Flow (cont’)
which is actually just the expansion of the equation
abbreviations:
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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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DC & CV Lab.DC & CV Lab.CSIE NTU
12.2 Optical Flow (cont’)
rewrite constraint equation:
aperture problem: cannot determine optical flow along isobrightness contour
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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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12.3 Smoothness of the Optical Flow (cont’)
error in optical flow constraint equation should be small
overall, to minimize
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12.3 Smoothness of the Optical Flow (cont’)
large if brightness measurements are accurate
small if brightness measurements are noisy
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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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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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joke
DC & CV Lab.DC & CV Lab.CSIE NTU
12.6 The Discrete Case
first partial derivatives of u, v: can be estimated using difference
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DC & CV Lab.DC & CV Lab.CSIE NTU
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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12.6 The Discrete Case (cont’)
dieffrentiating e with respect to
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12.6 The Discrete Case (cont’)
where are local average of u, v (9 neighbors? )
extremum occurs where the above derivatives of e are zero:
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12.6 The Discrete Case (cont’)
determinant of 2x2 coefficient matrix:
so that
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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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DC & CV Lab.DC & CV Lab.CSIE NTU
12.6 The Discrete Case (cont’)
first derivatives estimated using first differences in 2x2x2 cube
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DC & CV Lab.DC & CV Lab.CSIE NTU
12.6 The Discrete Case (cont’) consistent estimates of three first partial deriv
atives:
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12.6 The Discrete Case (cont’)
four successive synthetic images of rotating sphere
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DC & CV Lab.DC & CV Lab.CSIE NTU
12.6 The Discrete Case (cont’)
estimated optical flow after 1, 4, 16, and 64 iterations
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DC & CV Lab.DC & CV Lab.CSIE NTU
12.6 The Discrete Case (cont’)
(a) estimated optical flow after several more iterations
(b) computed motion field
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DC & CV Lab.DC & CV Lab.CSIE NTU
12.7 Discontinuities in Optical Flow
discontinuities in optical flow: on silhouettes where occlusion occurs
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Joke
DC & CV Lab.DC & CV Lab.CSIE NTU
Project due May 2
implementing Horn & Schunck optical flow estimation as above
synthetically translate lena.im one pixel to the right and downward
Try 10 1, 0.1, of