Structure from Motion Introduction to Computer Vision CS223B, Winter 2005 Richard Szeliski.
Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion
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Transcript of Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion
![Page 1: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/1.jpg)
Stanford CS223B Computer Vision, Winter 2007
Lecture 8 Structure From Motion
Professors Sebastian Thrun and Jana Košecká
CAs: Vaibhav Vaish and David Stavens
Slide credit: Gary Bradski, Stanford SAIL
![Page 2: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/2.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Summary SFM
Problem– Determine feature locations (=structure)– Determine camera extrinsic (=motion)
Two Principal Solutions– Bundle adjustment (nonlinear least squares, local
minima)– SVD (through orthographic approximation, affine
geometry) Correspondence
– (RANSAC)– Expectation Maximization
![Page 3: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/3.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Structure From Motion
camera
features
Recover: structure (feature locations), motion (camera extrinsics)
![Page 4: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/4.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
SFM = Holy Grail of 3D Reconstruction
Take movie of object Reconstruct 3D model
Would be
commercially
highly viable
live.com
![Page 5: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/5.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Structure From Motion (1)
[Tomasi & Kanade 92]
![Page 6: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/6.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Structure From Motion (2)
[Tomasi & Kanade 92]
![Page 7: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/7.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Structure From Motion (3)
[Tomasi & Kanade 92]
![Page 8: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/8.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Structure From Motion (4a): Images
Marc Pollefeys
![Page 9: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/9.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Structure From Motion (4b)
Marc Pollefeys
![Page 10: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/10.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Structure From Motion (5)
http://www.cs.unc.edu/Research/urbanscape
![Page 11: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/11.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Structure From Motion
Problem 1:– Given n points pij =(xij, yij) in m images
– Reconstruct structure: 3-D locations Pj =(xj, yj, zj)
– Reconstruct camera positions (extrinsics) Mi=(Aj, bj)
Problem 2:– Establish correspondence: c(pij)
![Page 12: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/12.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Structure From Motion
camera
features
Recover: structure (feature locations), motion (camera extrinsics)
![Page 13: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/13.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Recovery Problems
1 image 2+ images
Location known calibration stereo
Location unknown
SFM, stitching
![Page 14: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/14.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
SFM: General Formulation
iz
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![Page 15: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/15.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
SFM: Bundle Adjustment
min
cossin0
sincos0
001
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010
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![Page 16: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/16.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Bundle Adjustment
SFM = Nonlinear Least Squares problem Minimize through
– Gradient Descent– Conjugate Gradient– Gauss-Newton– Levenberg Marquardt common method
Prone to local minima
![Page 17: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/17.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Count # Constraints vs #Unknowns
m camera poses n points 2mn point constraints 6m+3n unknowns
Suggests: need 2mn 6m + 3n But: Can we really recover all parameters???
![Page 18: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/18.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
How Many Parameters Can’t We Recover?
0 3 6 7 8 10 12 n m nm
Place Your Bet!
We can recover all but…
m = #camera posesn = # feature points
![Page 19: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/19.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Count # Constraints vs #Unknowns
m camera poses n points 2mn point constraints 6m+3n unknowns
Suggests: need 2mn 6m + 3n But: Can we really recover all parameters???
– Can’t recover origin, orientation (6 params)– Can’t recover scale (1 param)
Thus, we need 2mn 6m + 3n - 7
![Page 20: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/20.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Are we done?
No, bundle adjustment has many local minima.
![Page 21: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/21.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
The “Trick Of The Day”
Replace Perspective by Orthographic Geometry
Replace Euclidean Geometry by Affine Geometry
Solve SFM linearly via PCA (“closed” form, globally optimal)
Post-Process to make solution Euclidean
Post-Process to make solution perspective
By Tomasi and Kanade, 1992
![Page 22: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/22.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Orthographic Camera Model
Orthographic = Limit of Pinhole Model:
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x
z
y
x
b
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b
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333231
232221
131211
Extrinsic Parameters
Rotation
Orthographic Projection bAPb
b
P
P
P
a
a
a
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a
p
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y
x
Z
Y
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23
13
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![Page 23: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/23.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Orthographic Projection
Limit of Pinhole Model:
Orthographic Projection
1||
1||
0
22
21
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a
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rotation is
333231
232221
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ijiij bPAp
featurejcamerai
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![Page 24: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/24.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
The Orthographic SFM Problem
}{ and },{recover jPii bA
ijiij bPAp featurejcamerai 1||
1||
0
22
21
21
a
a
aa
subject to
![Page 25: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/25.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
The Affine SFM Problem
}{ and },{recover jPii bA
ijiij bPAp featurejcamerai 1||
1||
0
22
21
21
a
a
aa
subject todrop theconstraints
![Page 26: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/26.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Count # Constraints vs #Unknowns
m camera poses n points 2mn point constraints 8m+3n unknowns
Suggests: need 2mn 8m + 3n But: Can we really recover all parameters???
ijiij bPAp featurejcamerai
![Page 27: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/27.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
How Many Parameters Can’t We Recover?
0 3 6 7 8 10 12 n m nm
Place Your Bet!
We can recover all but…
![Page 28: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/28.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
The Answer is (at least): 12
ijiij bPAp
iijiij bdAdCPCCAp ))(( :Proof 11
iji bPA
iiiji bdAdAPA
''' ijiij bPAp
dCPCP jj11'
iii bdAb 'singular-non , Cd CAA ii '
![Page 29: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/29.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Points for Solving Affine SFM Problem
m camera poses n points
Need to have: 2mn 8m + 3n-12
![Page 30: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/30.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Affine SFM
jiij PAp
Fix coordinate systemby making pi0=P0=origin
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p
p
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A
A 1
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ADQn :points
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p
Q
1
1
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ijiij bPAp
Proof:
3m2 size has A
Rank Theorem: Q has rank 3
nD 3 size has
![Page 31: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/31.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
The Rank Theorem
3rank has
1
1
1
1
11
11
Nyy
Nxx
Nyy
Nxx
MM
MM
pp
pp
pp
pp
n elements
2m
ele
me
nts
![Page 32: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/32.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Singular Value Decomposition
T
Nyy
Nxx
Nyy
Nxx
VWU
pp
pp
pp
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MM
MM
1
1
1
1
11
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n332 m 33
![Page 33: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/33.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Affine Solution to Orthographic SFM
structure affine TWV
positions camera affine U
Gives also the optimal affine reconstruction under noise
![Page 34: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/34.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Back To Orthographic Projection
1||
1||
0
sConstraint
22
21
21
a
a
aa
matrix singular -non , vector Cd
with
Find C for which constraints are metSearch in 9-dim space (instead of 8m + 3n-12)
''' ijiij bPAp
dCPCP jj11'
ii CAA '
iii bdAb '
![Page 35: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/35.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Back To Projective Geometry
Orthographic (in the limit)
Projective
![Page 36: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/36.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Back To Projective Geometry
min
cossin0
sincos0
001
cos0sin
010
sin0cos
100
cossin0
sincos0
001
cos0sin
010
sin0cos
0cossin
0sincos
2
,
,
,
,
,
,
,
,
,
,
,
,
ji
iz
jz
jy
jx
ii
ii
ii
ii
iy
ix
jz
jy
jx
ii
ii
ii
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jy
jx
b
P
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b
b
P
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fp
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fZ Z
fXx
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Optimize
Using orthographic solution as starting point
![Page 37: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/37.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
The “Trick Of The Day”
Replace Perspective by Orthographic Geometry
Replace Euclidean Geometry by Affine Geometry
Solve SFM linearly via PCA (“closed” form, globally optimal)
Post-Process to make solution Euclidean
Post-Process to make solution perspective
By Tomasi and Kanade, 1992
![Page 38: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/38.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Structure From Motion
Problem 1:– Given n points pij =(xij, yij) in m images
– Reconstruct structure: 3-D locations Pj =(xj, yj, zj)
– Reconstruct camera positions (extrinsics) Mi=(Aj, bj)
Problem 2:– Establish correspondence: c(pij)
![Page 39: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/39.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
The Correspondence Problem
View 1 View 3View 2
![Page 40: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/40.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Correspondence: Solution 1
Track features (e.g., optical flow)
…but fails when images taken from widely different poses
![Page 41: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/41.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Correspondence: Solution 2
Start with random solution A, b, P Compute soft correspondence: p(c|A,b,P) Plug soft correspondence into SFM Reiterate
See Dellaert/Seitz/Thorpe/Thrun, Machine Learning Journal, 2003
![Page 42: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/42.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Example
![Page 43: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/43.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Results: Cube
![Page 44: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/44.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Animation
![Page 45: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/45.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Tomasi’s Benchmark Problem
![Page 46: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/46.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Reconstruction with EM
![Page 47: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/47.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
3-D Structure
![Page 48: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/48.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Correspondence: Alternative Approach
Ransac [Fisher/Bolles]
= Random sampling and consensus
Will be discussed Wednesday
![Page 49: Stanford CS223B Computer Vision, Winter 2007 Lecture 8 Structure From Motion](https://reader035.fdocuments.us/reader035/viewer/2022062422/56812ed3550346895d94735e/html5/thumbnails/49.jpg)
Sebastian Thrun and Jana Košecká CS223B Computer Vision, Winter 2007
Summary SFM
Problem– Determine feature locations (=structure)– Determine camera extrinsic (=motion)
Two Principal Solutions– Bundle adjustment (nonlinear least squares, local
minima)– SVD (through orthographic approximation, affine
geometry) Correspondence
– (RANSAC)– Expectation Maximization