Structure from Motion - CVG · 2013-01-15 · Calibrated structure from motion QEquations more complicated, but less degeneracies QFor calibrated cameras: – 5-point relative motion

Structure from Motion

Read Chapter 7 in Szeliski’s book

Video → 3D model

3D fro

2

Recorded at archaeological site of Sagalassos in Turkey

accuracy ~1/500 from DV video (i.e. 140kb jpegs 576x720

Schedule (tentative)

3

# date topic 1 Sep.19 Introduction and geometry 2 Sep.26 Camera models and calibration 3 Oct.3 Multiple-view geometry (Richard Hartley) 4 Oct.10 Invariant features 5 Oct.7 Model fitting (RANSAC, EM, …) 6 Oct.24 Segmentation (Jianbo Shi)

7 Oct.31 Stereo Matching 8 Nov.7 Shape from Silhouettes 9 Nov.14 Structure from motion 10 Nov.21 Optical Flow 11 Nov.28 Tracking (Kalman, particle filter) 12 Dec.5 Object recognition 13 Dec.12 Object category recognition 14 Dec.19 Applications and demos

Today’s class

  Structure from motion

–  factorization –  sequential

– bundle adjustment

Factorization

  Factorise observations in structure of the scene and motion/calibration of the camera

  Use all points in all images at the same time

ü Affine factorisation ü  Projective factorisation

Affine camera The affine projection equations are

1

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

j

j

j

yi

xi

ij

ij

ZYX

PP

yx

10001

1 ⎥

⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

j

j

j

yi

xi

ij

ij

ZYX

PP

yx

~~

4

4

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−

j

j

j

yi

xi

ij

ijyiij

xiij

ZYX

PP

yx

PyPx

how to find the origin? or for that matter a 3D reference point?

affine projection preserves center of gravity

∑−=i

ijijij xxx~ ∑−=i

ijijij yyy~

Orthographic factorization The orthographic projection equations are

where njmijiij ,...,1,,...,1,Mm === P

All equations can be collected for all i and j

where

[ ]n

mmnmm

n

n

M,...,M,M,,

mmm

mmmmmm

212

1

21

22221

11211

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

= M

P

PP

Pm

MPm =

M ~~

m⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=

j

j

j

jyi

xi

iij

ijij

ZYX

,PP,

yx

P

Note that P and M are resp. 2mx3 and 3xn matrices and therefore the rank of m is at most 3

(Tomasi Kanade’92)

Orthographic factorization Factorize m through singular value decomposition

An affine reconstruction is obtained as follows

TVUm Σ=

TVMUP Σ==~,~


[ ]n

mmnmm

n

n

M,...,M,M

mmm

mmmmmm

min 212

1

21

22221

11211

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

P

PP

Closest rank-3 approximation yields MLE!

A metric reconstruction is obtained as follows

Where A is computed from



TVUm Σ=

TVMUP Σ==~,~

MAMAPP ~,~ 1 == −

0

1

1

=

=

=

T

T

T

yi

xi

yi

yi

xi

xi

PP

PP

PP


0~~1~~1~~

1

1

1

=

=

=

−−

−−

−−

TT

TT

TT

yi

xi

yi

yi

xi

xi

PP

PP

PP

AA

AA

AA





TVUm Σ=

TVMUP Σ==~,~

MAMAPP ~,~ 1 == −


0~~1~~1~~

=

=

=

T

T

T

yi

xi

yi

yi

xi

xi

PP

PP

PP

C

C

C





TVUm Σ=

TVMUP Σ==~,~

MAMAPP ~,~ 1 == −

3 linear equations per view on symmetric matrix C (6DOF)

A can be obtained from C through Cholesky factorisation and inversion


Examples

Tomasi Kanade’92, Poelman & Kanade’94

Examples


Examples


Examples


Perspective factorization

The camera equations

for a fixed image i can be written in matrix form as

where

mjmijiijij ,...,1,,...,1,Mmλ === P

MPm iii =Λ

[ ] [ ]( )imiii

mimiii

λ,...,λ,λdiagM,...,M,M , m,...,m,m

21

2121

=Λ== Mm

Perspective factorization

All equations can be collected for all i as

where PMm =

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

Λ

Λ

Λ

=

mnn P

PP

P

m

mm

m...

,...

2

1

22

11

In these formulas m are known, but Λi,P and M are unknown

Observe that PM is a product of a 3mx4 matrix and a 4xn matrix, i.e. it is a rank-4 matrix

Perspective factorization algorithm

Assume that Λi are known, then PM is known. Use the singular value decomposition

PM=UΣ VT In the noise-free case

Σ=diag(σ1,σ2,σ3,σ4,0, … ,0) and a reconstruction can be obtained by setting:

P=the first four columns of UΣ.

M=the first four rows of V.

Iterative perspective factorization

When Λi are unknown the following algorithm can be used:

1. Set λij=1 (affine approximation).

2. Factorize PM and obtain an estimate of P and M. If σ5 is sufficiently small then STOP.

3. Use m, P and M to estimate Λi from the camera equations (linearly) mi Λi=PiM

4. Goto 2.

In general the algorithm minimizes the proximity measure P(Λ,P,M)=σ5

Note that structure and motion recovered up to an arbitrary projective transformation

Further Factorization work

Factorization with uncertainty Factorization for dynamic scenes

(Irani & Anandan, IJCV’’02)

(Costeira and Kanade ‘‘94) (Bregler et al. ‘‘00, Brand ‘‘01)

(Yan and Pollefeys, ‘‘05/’’06)

practical structure and motion recovery from images

  Obtain reliable matches using matching or tracking and 2/3-view relations   Compute initial structure and motion   Refine structure and motion   Auto-calibrate   Refine metric structure and motion

Initialize Motion (P1,P2 compatibel with F)

Sequential Structure and Motion Computation

Initialize Structure (minimize reprojection error)

Extend motion (compute pose through matches seen in 2 or more previous views)

Extend structure (Initialize new structure, refine existing structure)

Computation of initial structure and motion

according to Hartley and Zisserman “this area is still to some extend a

black-art”

All features not visible in all images ⇒  No direct method (factorization not applicable) ⇒  Build partial reconstructions and assemble (more views is more stable, but less corresp.)

1) Sequential structure and motion recovery

2) Hierarchical structure and motion recovery

Sequential structure and motion recovery

  Initialize structure and motion from two views   For each additional view

– Determine pose – Refine and extend structure

  Determine correspondences robustly by jointly estimating matches and epipolar geometry

Initial structure and motion

[ ][ ][ ]eeaFeP0IP

Tx +=

=

2

1

Epipolar geometry ↔ Projective calibration

012 =FmmT

compatible with F

Yields correct projective camera setup (Faugeras´92,Hartley´92)

Obtain structure through triangulation Use reprojection error for minimization Avoid measurements in projective space

Compute Pi+1 using robust approach (6-point RANSAC) Extend and refine reconstruction

)x,...,X(xPx 11 −= iii

2D-2D

2D-3D 2D-3D

mi mi+1

M

new view

Determine pose towards existing structure

Compute P with 6-point RANSAC

  Generate hypothesis using 6 points

  Count inliers   Projection error

( )( ) ?x,x...,,xXP 11 td iii <−

  Back-projection error ( ) ijtd jiij <∀< ?,x,xF  Re-projection error ( )( ) td iiii <− x,x,x...,,xXP 11

  3D error ( )( ) ?X,xP 3-1

Dii td <Λ

  Projection error with covariance ( )( ) td iii <−Λ x,x...,,xXP 11

  Expensive testing? Abort early if not promising   Verify at random, abort if e.g. P(wrong)>0.95

(Chum and Matas, BMVC’02)

Calibrated structure from motion

  Equations more complicated, but less degeneracies   For calibrated cameras:

–  5-point relative motion (5DOF) Nister CVPR03

–  3-point pose estimation (6DOF) Haralick et al. IJCV94

D. Nistér, An efficient solution to the five-point relative pose problem, In Proc. IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 2003), Volume 2, pages. 195-202, 2003.

R. Haralick, C. Lee, K. Ottenberg, M. Nolle. Review and Analysis of Solutions of the Three Point Perspective Pose Estimation Problem. Int’l Journal of Computer Vision, 13, 3, 331-356, 1994.

  Linear equations for 5 points

  Linear solution space

  Non-linear constraints

5-point relative motion

E = xX + yY + zZ + wW

detE = 0 10 cubic polynomials

w = 1scale does not matter, choose

(Nister, CVPR03)

!x1x1 !x1y1 !x11 !y1x1 !y1y1 !y1 x1 y1 1!x2x2 !x2y2 !x21 !y2x2 !y2y2 !y2 x2 y2 1!x3x3 !x3y3 !x31 !y3x3 !y3y3 !y3 x3 y3 1!x4x4 !x4y4 !x41 !y4x4 !y4y4 !y4 x4 y4 1!x5x5 !x5y5 !x51 !y5x5 !y5y5 !y5 x5 y5 1

"

#

$$$$$$$

%

&

'''''''

E11E12E13E21E22E23E31E32E33

"

#

$$$$$$$$$$$$$

%

&

'''''''''''''

= 0

5-point relative motion

  Perform Gauss-Jordan elimination on polynomials

-z

-z

-z

[n]represents polynomial of degree n in z

(Nister, CVPR03)

Three points perspective pose – p3p

(Haralick et al., IJCV94)

All techniques yield 4th order polynomial Haralick et al. recommends using Finsterwalder’s technique as it yields the best results numerically

1903 1841

Minimal solvers Lot’s of recent activity using Groebner bases:   Henrik Stewénius, David Nistér, Fredrik Kahl, Frederik

Schaffalitzky: A Minimal Solution for Relative Pose with Unknown Focal Length, CVPR 2005.

  H. Stewénius, D. Nistér, M. Oskarsson, and K. Åström. Solutions to minimal generalized relative pose problems. Omnivis 2005.

  D. Nistér, A Minimal solution to the generalised 3-point pose problem, CVPR 2004

  Martin Bujnak, Zuzana Kukelova, Tomás Pajdla: A general solution to the P4P problem for camera with unknown focal length. CVPR 2008.

  Brian Clipp, Christopher Zach, Jan-Michael Frahm and Marc Pollefeys, A New Minimal Solution to the Relative Pose of a Calibrated Stereo Camera with Small Field of View Overlap, ICCV 2009.

  Zuzana Kukelova, Martin Bujnak, Tomás Pajdla: Automatic Generator of Minimal Problem Solvers. ECCV 2008.

  F. Fraundorfer, P. Tanskanen and M. Pollefeys. A Minimal Case Solution to the Calibrated Relative Pose Problem for the Case of Two Known Orientation Angles. ECCV 2010.

  …

Minimal relative pose with know vertical

39

-g

Fraundorfer, Tanskanen and Pollefeys, ECCV2010

5 linear unknowns → linear 5 point algorithm 3 unknowns → quartic 3 point algorithm

Vertical direction can often be estimated   inertial sensor   vanishing point

Incremental Structure from Motion

Photo Tourism

Noah Snavely, Steve Seitz, Rick Szeliski University of Washington

Richard Szeliski

Incremental structure from motion

•  Automa�cally select an ini�al pair of images



Hierarchical structure and motion recovery

  Compute 2-view   Compute 3-view   Stitch 3-view reconstructions   Merge and refine reconstruction

F T

H

PM

Hierarchical structure from motion

  Farenzena, Fusiello, and Gherardi, Structure-and-motion Pipeline on a Hierarchical Cluster Tree, 3DIM 2009, October 2009

Richar

ICVSS 2010 45


Richar

ICVSS 2010 46


Richar

ICVSS 2010 47

Stitching 3-view reconstructions

Different possibilities 1. Align (P2,P3)with(P’1,P’2) ( ) ( )-123

-112

HHP',PHP',Pminarg AA dd +

2. Align X,X’ (and C’C’) ( )∑j

jjAd HX',XminargH

3. Minimize reproj. error ( )( )∑

∑+

jjj

jjj

d

d

x',HXP'

x,X'PHminarg 1-

H

4. MLE (merge) ( )∑j

jjd x,PXminargXP,

More issues

  Repetition ambiguities

  Large scale reconstructions

Marc Pollefeys ICVSS 2010 49

Disambiguating visual relations using loop constraints

50

(Zach et al CVPR‘10)

Exploiting symmetries in SfM

51

Detected symmetry matches

(Cohen et al 2012?)

57

Projective ambiguity

reconstruction from uncalibrated images only ⇒ projective ambiguity on reconstruction

´M´M))((Mm 1 PTPTP === − Similarity (7DOF)

Affine (12DOF)

Projective (15DOF)

58

Euclidean projection matrix

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

1yy

xx

cfcsf

K

Factorization of Euclidean projection matrix

Intrinsics:

Extrinsics: ( )t,RNote: every projection matrix can be factorized, but only meaningful for euclidean projection matrices

(camera geometry)

(camera motion)

[ ]tRRKP TT −=

59

The Absolute Quadric

Eliminate extrinsics from equation

TT KKPP =Ω

[ ]tRRK TT − TKR TRK TKK

)1110(diag* =Ω

Equivalent to projection of quadric

))(Ω)((Ω *1* TTTTT PTTTPTPPKK −−==

Absolute Quadric also exists in projective world

Absolute Quadric

T´Ω´´ *PP=Transforming world so that reduces ambiguity to metric

** ΩΩ´ →

2 2 2 0A B C+ + =

0AX BY CZ D+ + + =

60

ω*

Ω*

projection

constraints

Absolute Quadric = calibration object which is always present but can only be observed through constraints on the intrinsics

Tii

Tiii Ωω KKPP == ∗∗

Absolute quadric and self-calibration

Projection equation:

  Translate constraints on K   through projection equation   to constraints on Ω*

  Don’t treat all constraints equal

61

Practical linear self-calibration

( ) ( )( )( )( ) 0Ω

0Ω0Ω

0ΩΩ

23T13

T12

T22

T11

T

=

=

=

=−

∗

∗

∗

∗∗

PPPPPP

PPPP(Pollefeys et al., ECCV‘02)

2

* 2

ˆ 0 0ˆ0 0

0 0 1

f

f

⎡ ⎤⎢ ⎥

= Ω ≈ ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

P PT TKK

1ˆ ≈f( ) ( )( ) ( ) 0ΩΩ

0ΩΩ

33T

22T

33T

11T

=−

=−∗∗

∗∗

PPPPPPPP

(only rough aproximation, but still usefull to avoid degenerate configurations)

(relatively accurate for most cameras)

91

91

1.011.0101.01

2.01

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

1yy

xx

cfcsf

K

after normalization!

when fixating point at image-center not only absolute quadric diag(1,1,1,0) satisfies ICCV98 eqs., but also diag(1,1,1,a), i.e. real or imaginary spheres! Recent related work (Gurdjos et al. ICCV‘09)

62

Upgrade from projective to metric

  Locate ΩΩ* in projective reconstruction (self-calibration) Problem: not always unique (Critical Motion Sequences)

  Transform projective reconstruction   to bring ΩΩ* in canonical position

Refining structure and motion

  Minimize reprojection error

– Maximum Likelyhood Estimation (if error zero-mean Gaussian noise)

– Huge problem but can be solved efficiently (Bundle adjustment)

( )∑∑= =

m

k

n

iikD

ik 1 1

2

kiM̂,P̂

M̂P̂,mmin

Non-linear least-squares

 Newton iteration   Levenberg-Marquardt   Sparse Levenberg-Marquardt

(P)X f= (P)X argminP

f−

Newton iteration

Taylor approximation

f (P0 +Δ) ≈ f (P0 )+ JΔ J = ∂X∂P

Jacobian

X− f (P1) X− f (P1) ≈ X− f (P0 )− JΔ = e0 − JΔ

⇒ JTJΔ = JTe0 ⇒Δ = JTJ( )-1JTe0

Pi+1 = Pi +Δ Δ = JTJ( )-1JTe0

Δ = JTΣ-1J( )-1JTΣ-1e0

normal eq.

Levenberg-Marquardt

0TT JNJJ e=Δ=Δ

0TJN' e=Δ

Augmented normal equations

Normal equations

J)λdiag(JJJN' TT +=

30 10λ −=

10/λλ :success 1 ii =+

ii λ10λ :failure = solve again

accept

λ small ~ Newton (quadratic convergence)

λ large ~ descent (guaranteed decrease)

Levenberg-Marquardt

Requirements for minimization   Function to compute f   Start value P0   Optionally, function to compute J (but numerical ok, too)

Richard Szeliski

Bundle Adjustment

  What makes this non-linear minimization hard? – many more parameters: potentially slow – poorer conditioning (high correlation) – potentially lots of outliers – gauge (coordinate) freedom

Chained transformations

  Think of the optimization as a set of transformations along with their derivatives

Richard Szeliski

Richard Szeliski

Levenberg-Marquardt

  Iterative non-linear least squares [Press’92] – Linearize measurement equations

– Substitute into log-likelihood equation: quadratic cost function in ΔΔm

Richard Szeliski

Robust error models

  Outlier rejection – use robust penalty applied

to each set of joint measurements

–  for extremely bad data, use random sampling [RANSAC, Fischler & Bolles, CACM’81]

Richard Szeliski

  Iterative non-linear least squares [Press’92] – Solve for minimum

Hessian: error:

Levenberg-Marquardt

you may have to delete the image and then insert it again.

Chained transformations

  Only some derivatives are non-zero: j  th camera, i  th point

Richard Szeliski

Multi-camera rig

  Easy extension of generalized bundler:

Richard Szeliski

Richard Szeliski

Lots of parameters: sparsity

  Only a few entries in Jacobian are non-zero

Exploiting sparsity

  Schur complement to get reduced camera matrix

Richard Szeliski

Sparse Levenberg-Marquardt

  complexity for solving – prohibitive for large problems (100 views 10,000 points ~30,000 unknowns)

  Partition parameters – partition A – partition B (only dependent on A and itself)

0T-1 JN' e=Δ3N

Sparse bundle adjustment

residuals: normal equations: with note: tie points should be in partition A

The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been

then insert it again.

The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been

insert it again.


normal equations:

modified normal equations:

solve in two parts:


U1

U2

U3

WT

W

V

P1 P2 P3 M

Jacobian of has sparse block structure

=J == JJN T

12xm 3xn (in general

much larger)

im.pts. view 1

( )( )∑∑= =

m

k

n

iikD

1 1

2ki M̂P̂,m

Needed for non-linear minimization


  Eliminate dependence of camera/motion parameters on structure parameters Note in general 3n >> 11m

WT V

U-WV-1WT =×⎥⎦

⎤⎢⎣⎡ − −

NI0WVI 1

11xm 3xn

Allows much more efficient computations

e.g. 100 views,10000 points, solve ±1000x1000, not ±30000x30000

Often still band diagonal use sparse linear algebra algorithms


normal equations:

modified normal equations:

solve in two parts:


  Covariance estimation

-1WVY =( )

1a

Tb

1-a

VYYWWVU

−

+

+∑=∑

−=∑

Yaab ∑=∑ -

Direct vs. iterative solvers

  How do they work?   When should you use them?

Richard Szeliski

Large reduced camera matrices

  Use iterative preconditioned conjugate gradient –  [Jeong, Nister, Steedly, Szeliski, Kweon, CVPR 2009] –  [Agarwal, Snavely, Seitz, and Szeliski, ECCV 2009] –  Block diagonal works well –  Sometimes partial Cholesky does too –  Direct solvers for small problems

Richard Szeliski

Large-scale structure from motion

Richard Szeliski

Large-scale reconstruction

  Most of the models shown so far have had ~500 images

  How do we scale from 100s to 10,000s of images?

  Observation: Internet collections represent very non-uniform samplings of viewpoint

Richard Szeliski

Skeletal graphs for efficient structure from motion

Noah Snavely, Steve Seitz, Richard Szeliski

CVPR’2008

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appears, you may have to delete the image and then insert it again.

Skeletal set

  Goal: select a smaller set of images to reconstruct, without sacrificing quality of reconstruction

  Two problems: – How do we measure quality? – How do we find a subset of images with

bounded quality loss?

Skeletal set

Goal: given an image graph ,   select a small set S of important images to reconstruct, bounding the loss in quality of the reconstruction

  Reconstruct the skeletal set S   Estimate the remaining images with much faster pose estimation steps

Properties of the skeletal set

  Should touch all parts of Dominating set

  Should form a single reconstruction Connected dominating set

  Should result in an accurate reconstruction ?

Example

Full graph Skeletal graph

Representing information in a graph What kind of information?

– No absolute information about camera positions

– Each edge provides information about the relative positions of two images

– … but not all edges are equally informative We model information with the uncertainty (covariance) in pairwise camera positions

Estimating certainty

  Usually measured with covariance matrix

  SfM has thousands or millions of parameters   We only measure covariance in camera positions (3x3 matrix for every camera)

Pantheon

Full graph Skeletal graph (t=16)

Skeletal reconstruc�on 101 images

A�er adding leaves 579 images

A�er final op�miza�on 579 images

Pantheon

Trafalgar Square

2973 images registered (277 in skeletal set)

Trafalgar Square

Running time

0

20

40

60

80

100

120

140

160

180

Stonehenge St. Peters Pantheon Pisa Trafalgar

Full reconstruction

Skeletal reconstruction

Skeletal reconstruction + BA

(~10 days) (~30 days)

hours

Building Rome in a Day

Sameer Agarwal, Noah Snavely, Ian Simon, Steven M. Seitz,

Richard Szeliski ICCV’2009

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insert it again.

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then insert it again.

Distributed matching pipeline

Query expansion

Results: Dubrovnik

Results: Rome

Changchang Wu’’s SfM code for iconic graph   uses 5-point+RANSAC for 2-view initialization   uses 3-point+RANSAC for adding views   performs bundle adjustment For additional images   use 3-point+RANSAC pose estimation

Rome on a cloudless day

(Frahm et al. ECCV 2010)   GIST & clustering (1h35)

SIFT & Geometric verification (11h36)

SfM & Bundle (8h35)

Dense Reconstruction (1h58)

Some numbers   1PC   2.88M images   100k clusters   22k SfM with 307k images   63k 3D models   Largest model 5700 images   Total time 23h53

Structure from motion: limitations

  Very difficult to reliably estimate metric structure and motion unless: –  large (x or y) rotation or –  large field of view and depth variation

  Camera calibration important for Euclidean reconstructions   Need good feature tracker

Related problems

  On-line structure from motion and SLaM (Simultaneous Localization and Mapping) – Kalman filter (linear) – Particle filters (non-linear)

Visual SLAM

  Real-time Stereo Visual SLAM

113

(Clipp et al., IROS2010)

Real-Time Stereo Visual SLAM

Marc Pollefeys

(Lim et al., CVPR201

Collaboration with

Open challenges

  Large scale structure from motion – Complete building – Complete city

Open source vode available, e.g. Christopher Zach (SfM, Bundle adjustment) http://www.inf.ethz.ch/personal/chzach/opensource.html

Photo Tourism Bundler: http://phototour.cs.washington.edu/bundler/

117

Next week: Optical flow

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Structure from Motion - CVG · 2013-01-15 · Calibrated structure from motion QEquations more complicated, but less degeneracies QFor calibrated cameras: – 5-point relative motion