Structure from motion

Structure from motion

Digital Visual Effects Yung-Yu Chuang

with slides by Richard Szeliski, Steve Seitz, Zhengyou Zhang and Marc Pollefyes

Outline

• Epipolar geometry and fundamental matrix

• Structure from motion

• Factorization method

• Bundle adjustment

• Applications

Epipolar geometry &

fundamental matrix

The epipolar geometry

C,C’,x,x’ and X are coplanar

epipolar geometry demo

The epipolar geometry

What if only C,C’,x are known?

The epipolar geometry

All points on  project on l and l’

The epipolar geometry

Family of planes  and lines l and l’ intersect at e and e’

The epipolar geometry

epipolar plane = plane containing baseline

epipolar line = intersection of epipolar plane with image epipolar pole

= intersection of baseline with image plane

= projection of projection center in other image

epipolar geometry demo

The fundamental matrix F

C C’

T=C’-C

p R p’

T Rp'

p  

Two reference frames are related via the extrinsic parameters

The fundamental matrix F

0 ' 



Ep

p

T Rp'

p  

 























 

0 0

0

x y

x z

y z

T T

T T

T T

T

Multiply both sides by

^p

  ^T

  ^{T p p}   ^T  ^{Rp' T} 

p





  ^{T Rp'}

p



0

The fundamental matrix F

0 ' 



Ep p

Let M and M’ be the intrinsic matrices, then

x M

p 

p '  M '

x ' 0

) ' '

( )

( M

x

E M

x  0 '

'









M EM x

x

0 ' 



Fx

x

The fundamental matrix F

• The fundamental matrix is the algebraic representation of epipolar geometry

• The fundamental matrix satisfies the condition that for any pair of corresponding points x x’ ↔ in the two images

0 Fx'

x

  ^x

^{l  0} 

F is the unique 3x3 rank 2 matrix that satisfies x^TFx’=0 for all x x’↔

1. Transpose: if F is fundamental matrix for (P,P’), then F^T is fundamental matrix for (P’,P)

2. Epipolar lines: l=Fx’ & l’=F^Tx

3. Epipoles: on all epipolar lines, thus e^TFx’=0, x’

e^TF=0, similarly Fe’=0

4. F has 7 d.o.f. , i.e. 3x3-1(homogeneous)-1(rank2)

5. F is a correlation, projective mapping from a point x to a line l=Fx’ (not a proper correlation, i.e. not

invertible)

The fundamental matrix F

The fundamental matrix F

• It can be used for

– Simplifies matching

– Allows to detect wrong matches

Estimation of F — 8-point algorithm

• The fundamental matrix F is defined by

x

Fx '  0

for any pair of matches x and x’ in two images.

• Let x=(u,v,1)^T and x’=(u’,v’,1)^T,



















33 32

31

23 22

21

13 12

11

f f

f

f f

f

f f

f F

each match gives a linear equation

0 '

' '

' '

' f₁₁  uv f₁₂  uf₁₃  vu f₂₁  vv f₂₂  vf₂₃  u f₃₁  v f₃₂  f₃₃  uu

8-point algorithm

0 1

´

´

´

´

´

´

1

´

´

´

´

´

´

1

´

´

´

´

´

´

33 32 31 23 22 21 13 12 11

2 2

2 2

2 2

2 2

2 2 2

2

1 1

1 1

1 1

1 1

1 1 1

1



















































f f f f f f f f f

v u

v v

v u

v u

v u u

u

v u

v v

v u

v u

v u u

u

v u

v v

v u

v u

v u u

u

n n

n n

n n

n n

n n n

n



















• In reality, instead of solving , we seek f to minimize subj. . Find the vector corresponding to the least singular value.

 0

Af

Af

8-point algorithm

• To enforce that F is of rank 2, F is replaced by F’

that minimizes subject to .

F  F ' det F '  0

• It is achieved by SVD. Let , where , let

then is the solution.

 U V

F Σ



















3 2

1

0 0

0 0

0 0

Σ

























0 0

0

0 0

0 0

Σ' ₂

1





 U V

F ' Σ'

8-point algorithm

% Build the constraint matrix

A = [x2(1,:)‘.*x1(1,:)' x2(1,:)'.*x1(2,:)' x2(1,:)' ...

x2(2,:)'.*x1(1,:)' x2(2,:)'.*x1(2,:)' x2(2,:)' ...

x1(1,:)' x1(2,:)' ones(npts,1) ];

[U,D,V] = svd(A);

% Extract fundamental matrix from the column of V

% corresponding to the smallest singular value.

F = reshape(V(:,9),3,3)';

% Enforce rank2 constraint [U,D,V] = svd(F);

F = U*diag([D(1,1) D(2,2) 0])*V';

8-point algorithm

• Pros: it is linear, easy to implement and fast

• Cons: susceptible to noise

Problem with 8-point algorithm

~10000 ~100 00

~10000 ~10000

~100 ~1

00 1

~100~100

!

Orders of magnitude difference between column of data matrix

 least-squares yields poor results

0 1

´

´

´

´

´

´

1

´

´

´

´

´

´

1

´

´

´

´

´

´

33 32 31 23 22 21 13 12 11

2 2

2 2

2 2

2 2

2 2 2

2

1 1

1 1

1 1

1 1

1 1 1

1



















































f f f f f f f f f

v u

v v

v u

v u

v u u

u

v u

v v

v u

v u

v u u

u

v u

v v

v u

v u

v u u

u

n n

n n

n n

n n

n n n

n



















Normalized 8-point algorithm

1. Transform input by , 2. Call 8-point on to obtain 3.

i

i

Tx

x ˆ  x ˆ

 Tx

' i i

x x ˆ ˆ , T

F T

F  '

ˆ

Fˆ

 0



Fx x'

ˆ 0 ˆ

T '

FT

x  x'

Fˆ

Normalized 8-point algorithm

(0,0)

(700,500)

(700,0) (0,500)

(1,-1) (0,0)

(1,1) (-1,1)

(-1,-1)

























1 500 1

2

1 700 0

2

normalized least squares yields good results Transform image to ~[-1,1]x[-1,1]

Normalized 8-point algorithm

A = [x2(1,:)‘.*x1(1,:)' x2(1,:)'.*x1(2,:)' x2(1,:)' ...

x2(2,:)'.*x1(1,:)' x2(2,:)'.*x1(2,:)' x2(2,:)' ...

x1(1,:)' x1(2,:)' ones(npts,1) ];

[U,D,V] = svd(A);

F = reshape(V(:,9),3,3)';

[U,D,V] = svd(F);

F = U*diag([D(1,1) D(2,2) 0])*V';

% Denormalise F = T2'*F*T1;

[x1, T1] = normalise2dpts(x1);

[x2, T2] = normalise2dpts(x2);

Normalization

function [newpts, T] = normalise2dpts(pts) c = mean(pts(1:2,:)')'; % Centroid

newp(1,:) = pts(1,:)-c(1); % Shift origin to centroid.

newp(2,:) = pts(2,:)-c(2);

meandist = mean(sqrt(newp(1,:).^2 + newp(2,:).^2));

scale = sqrt(2)/meandist;

T = [scale 0 -scale*c(1) 0 scale -scale*c(2) 0 0 1 ];

newpts = T*pts;

RANSAC

repeat

select minimal sample (8 matches) compute solution(s) for F

determine inliers

until (#inliers,#samples)>95% or too many times compute F based on all inliers

Results (ground truth)

Results (8-point algorithm)

Results (normalized 8-point algorithm)

Structure from motion

Structure from motion

structure for motion: automatic recovery of camera motion and scene structure from two or more images. It is a self calibration technique and called automatic camera tracking or matchmoving.

Unknown Unknown

camera camera viewpoints viewpoints

Applications

• For computer vision, multiple-view shape reconstruction, novel view synthesis and autonomous vehicle navigation.

• For film production, seamless insertion of CGI into live-action backgrounds

Matchmove

example #1 example #2 example #3 example #4

Structure from motion

2D feature

tracking 3D estimation optimization (bundle adjust)

geometry fitting

SFM pipeline

Structure from motion

• Step 1: Track Features

– Detect good features, Shi & Tomasi, SIFT – Find correspondences between frames

• Lucas & Kanade-style motion estimation

• window-based correlation

• SIFT matching

KLT tracking

http://www.ces.clemson.edu/~stb/klt/

Structure from Motion

• Step 2: Estimate Motion and Structure

– Simplified projection model, e.g., [Tomasi 92]

– 2 or 3 views at a time [Hartley 00]

Structure from Motion

• Step 3: Refine estimates

– “Bundle adjustment” in photogrammetry – Other iterative methods

Structure from Motion

• Step 4: Recover surfaces (image-based triangulation, silhouettes, stereo…)

Good mesh

Factorization methods

Problem statement

Notations

• n 3D points are seen in m views

• q=(u,v,1): 2D image point

• p=(x,y,z,1): 3D scene point

 : projection matrix

 : projection function

• q_ij is the projection of the i-th point on image j

 _ij projective depth of q_ij

) (

ij

p

q    ^

ij  z



Structure from motion

• Estimate and to minimize

) );

( ( log )

, ,

, ,

, (

1 1 1

1 j i ij

m j

n i

ij n

m p p w P Π p q

Π

Π 





 

 



otherwise

j in view visible

is if

0

1 _i

ij

w p



 

• Assume isotropic Gaussian noise, it is reduced to

2 1 1

1

1, , , , , ) ( )

( _{j i ij}

m j

n i

ij n

m p p w Π p q

Π

Π 



 



 _{ }



p

• Start from a simpler projection model

Orthographic projection

• Special case of perspective projection

– Distance from the COP to the PP is infinite

– Also called “parallel projection”: (x, y, z) (x, y)→

Image World

SFM under orthographic projection

2D image point

Orthographic projection

incorporating 3D rotation 3D scene point

image offset

t Πp

q  

1

2 23 31 21

• Trick

– Choose scene origin to be centroid of 3D points – Choose image origins to be centroid of 2D points – Allows us to drop the camera translation:

Πp

q 

factorization (Tomasi & Kanade)

  ^{ }

n 3 3

n 2

2  



 ⁿ



2

1 q q p p p

q  

projection of n features in one image:

 

n 3 3

n 2m 2m

2 1

2 1

2 1

2 22

21

1 12

11













































n

mn m m

m

n n

p p

p Π

Π Π

q q

q

q q

q

q q

q

 















projection of n features in m images

W

M

S

Key Observation: rank(W) <= 3

n 3 3 m 2 n

2m

' '

 



 M S

W

• Factorization Technique

– W is at most rank 3 (assuming no noise)

– We can use singular value decomposition to factor W:

Factorization

– S’ differs from S by a linear transformation A:

– Solve for A by enforcing metric constraints on M

) )(

( '

' S MA AS

M

W  

n 3 3 m 2 n

2m

W

 M

S

known solve for

Metric constraints

• Orthographic Camera

– Rows of  are orthonormal:

• Enforcing “Metric” Constraints

– Compute A such that rows of M have these properties

M A

M ' 



 



 

 ^T ₀¹ ₁⁰

Trick (not in original Tomasi/Kanade paper, but in followup work)

• Constraints are linear in AA^T :

• Solve for G first by writing equations for every _i in M

• Then G = AA^T by SVD (since U = V)

 ^{T T T}

T   A A   G where G  AA





 



 



 ' ' ' '

1 0

0 1

n m 2 n

3 3 m 2 n

2m

W

 M

S

 E

Factorization with noisy data

• SVD gives this solution

– Provides optimal rank 3 approximation W’ of W

n m n 2

2m n

2m

'

 



 W  E

W

• Approach

– Estimate W’, then use noise-free factorization of W’

as before

– Result minimizes the SSD between positions of image features and projection of the reconstruction

Results

Extensions to factorization methods

• Projective projection

• With missing data

• Projective projection with missing data

Bundle adjustment

Levenberg-Marquardt method

• LM can be thought of as a combination of steepest descent and the Newton method.

When the current solution is far from the correct one, the algorithm behaves like a

steepest descent method: slow, but guaranteed to converge. When the current solution is close to the correct solution, it becomes a Newton’s method.

Nonlinear least square

).

ˆ ( with ˆ ,

Here, minimal.

is distance

squared

that the so

vector parameter

best the

find try to

, ts measuremen of

set a

Given

p x

x x

p

x

f

T





 





Levenberg-Marquardt method

Levenberg-Marquardt method

• μ=0 → Newton’s method

• μ ∞ → → steepest descent method

• Strategy for choosing μ

– Start with some small μ

– If error is not reduced, keep trying larger μ until it does

– If error is reduced, accept it and reduce μ for the next iteration

Bundle adjustment

• Bundle adjustment (BA) is a technique for simultaneously refining the 3D structure and camera parameters

• It is capable of obtaining an optimal

reconstruction under certain assumptions on image error models. For zero-mean Gaussian image errors, BA is the maximum likelihood estimator.

Bundle adjustment

• n 3D points are seen in m views

• x_ij is the projection of the i-th point on image j

• a_j is the parameters for the j-th camera

• b_i is the parameters for the i-th point

• BA attempts to minimize the projection error

Euclidean distance

predicted projection

Bundle adjustment

Bundle adjustment

3 views and 4 points

Typical Jacobian

Block structure of normal equation

Bundle adjustment

Bundle adjustment

Multiplied by

Issues in SFM

• Track lifetime

• Nonlinear lens distortion

• Degeneracy and critical surfaces

• Prior knowledge and scene constraints

• Multiple motions

Track lifetime

every 50th frame of a 800-frame sequence

Track lifetime

lifetime of 3192 tracks from the previous sequence

Track lifetime

track length histogram

Nonlinear lens distortion

Nonlinear lens distortion

effect of lens distortion

Prior knowledge and scene constraints

add a constraint that several lines are parallel

Prior knowledge and scene constraints

add a constraint that it is a turntable sequence

Applications of

matchmove

Jurassic park

2d3 boujou

Enemy at the Gate, Double Negative

2d3 boujou

Enemy at the Gate, Double Negative

Photo Tourism

VideoTrace

http://www.acvt.com.au/research/videotrace/

Video stabilization

Project #3 MatchMove

• It is more about using tools in this project

• You can choose either calibration or structure from motion to achieve the goal

• Calibration

• Voodoo/Icarus

• Examples from previous classes, #1, #2

References

• Richard Hartley, In Defense of the 8-point Algorithm, ICCV, 1995.

• Carlo Tomasi and Takeo Kanade, Shape and Motion from Image Streams: A Factorization Method, Proceedings of Natl. Acad. Sci., 1993.

• Manolis Lourakis and Antonis Argyros, The Design and

Implementation of a Generic Sparse Bundle Adjustment Software Package Based on the Levenberg-Marquardt Algorithm, FORTH- ICS/TR-320 2004.

• N. Snavely, S. Seitz, R. Szeliski, Photo Tourism: Exploring Photo Collections in 3D, SIGGRAPH 2006.

• A. Hengel et. al., VideoTrace: Rapid Interactive Scene Modelling from Video, SIGGRAPH 2007.