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# Structure from motion

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### Structure from motion

Digital Visual Effects Yung-Yu Chuang

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

(2)

### Outline

• Epipolar geometry and fundamental matrix

• Structure from motion

• Factorization method

• Applications

(3)

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### The epipolar geometry

C,C’,x,x’ and X are coplanar

epipolar geometry demo

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### The epipolar geometry

What if only C,C’,x are known?

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### The epipolar geometry

All points on  project on l and l’

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### The epipolar geometry

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

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### 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

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C C’

T=C’-C

p R p’

### p  

Two reference frames are related via the extrinsic parameters

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essential matrix

###  





0 0

0

x y

x z

y z

T T

T T

T T

T

Multiply both sides by

T

T

T

T

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### Epp

Let M and M’ be the intrinsic matrices, then

1

1

1

1

1

### x

fundamental matrix

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### 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

T

T

### l0 

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

1. Transpose: if F is fundamental matrix for (P,P’), then FT is fundamental matrix for (P’,P)

2. Epipolar lines: l=Fx’ & l’=FTx

3. Epipoles: on all epipolar lines, thus eTFx’=0, x’

eTF=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)

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### The fundamental matrix F

• It can be used for

– Simplifies matching

– Allows to detect wrong matches

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### Estimation of F — 8-point algorithm

• The fundamental matrix F is defined by

### 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

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### 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.

f

1

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### 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.

3 2

1

0 0

0 0

0 0

Σ

0 0

0

0 0

0 0

Σ' 2

1

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### 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';

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### 8-point algorithm

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

• Cons: susceptible to noise

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### 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

(21)

### Normalized 8-point algorithm

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

i

i

'i

'i

' i i

Τ

1

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### 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]

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### 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);

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### 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;

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### 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

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### 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

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### 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

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### Matchmove

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

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### Structure from motion

2D feature

tracking 3D estimation optimization (bundle adjust)

geometry fitting

SFM pipeline

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### 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

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### KLT tracking

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

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### 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]

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### Structure from Motion

• Step 3: Refine estimates

– “Bundle adjustment” in photogrammetry – Other iterative methods

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### Structure from Motion

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

Good mesh

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### 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

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

 ij projective depth of qij

j i

ij

### q    

(x, y, z) (x / z, y / z)

ijz

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### 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

j

### p

i

• Start from a simpler projection model

(43)

### 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

(44)

### SFM under orthographic projection

2D image point

Orthographic projection

incorporating 3D rotation 3D scene point

image offset

### q  

1

2 23 31 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:

(45)

n 3 3

n 2

2

n

### 

1 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

measurement

motion

### S

shape

Key Observation: rank(W) <= 3

(46)

n 3 3 m 2 n

2m

### 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

1

n 3 3 m 2 n

2m

known solve for

(47)

### Metric constraints

• Orthographic Camera

– Rows of  are orthonormal:

• Enforcing “Metric” Constraints

– Compute A such that rows of M have these properties

### M ' 

 T 01 10

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

• Constraints are linear in AAT :

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

• Then G = AAT by SVD (since U = V)

T T T

T A A G where G AA

' ' ' '

1 0

0 1

(48)

n m 2 n

3 3 m 2 n

2m

### Factorization with noisy data

• SVD gives this solution

– Provides optimal rank 3 approximation W’ of W

n m n 2

2m n

2m

### 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

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(50)

### Extensions to factorization methods

• Projective projection

• With missing data

• Projective projection with missing data

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### 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.

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T

(54)

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### Levenberg-Marquardt method

• μ=0 → Newton’s method

• μ ∞ → → steepest descent method

• Strategy for choosing 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

(56)

• 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.

(57)

• n 3D points are seen in m views

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

• aj is the parameters for the j-th camera

• bi is the parameters for the i-th point

• BA attempts to minimize the projection error

Euclidean distance

predicted projection

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3 views and 4 points

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Multiplied by

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### Issues in SFM

• Nonlinear lens distortion

• Degeneracy and critical surfaces

• Prior knowledge and scene constraints

• Multiple motions

(65)

every 50th frame of a 800-frame sequence

(66)

lifetime of 3192 tracks from the previous sequence

(67)

track length histogram

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### Nonlinear lens distortion

effect of lens distortion

(70)

### Prior knowledge and scene constraints

add a constraint that several lines are parallel

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### Prior knowledge and scene constraints

add a constraint that it is a turntable sequence

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### 2d3 boujou

Enemy at the Gate, Double Negative

(75)

### 2d3 boujou

Enemy at the Gate, Double Negative

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### VideoTrace

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

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### 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

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### 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.

• Michael Grossberg, Shree Nayar, Determining the Camera Response from Images: What Is Knowable, PAMI 2003. • Michael Grossberg, Shree Nayar, Modeling the Space of Camera

• Michael Grossberg, Shree Nayar, Determining the Camera Response from Images: What Is Knowable, PAMI 2003. • Michael Grossberg, Shree Nayar, Modeling the Space of Camera

• The scene with depth variations and the camera has movement... Planar scene (or a

• The scene with depth variations and the camera has movement... Planar scene (or a

Not a simple transitory or rotary motion but several motions of different types, speeds, and amplitudes composing to make a resultant whole, just as one can compose colors, or

Spacelike distributions assumed identical in Euclidean and Minkowski space First calculation to work strictly in Euclidean space found no IR divergence.

Define instead the imaginary.. potential, magnetic field, lattice…) Dirac-BdG Hamiltonian:. with small, and matrix

A spiral curriculum whose structure allows for topics or skills to be revisited and repeated, each time in more detail or depth as the learner gains in knowledge, skills,