CSCE 643 Computer Vision: Structure from Motion Jinxiang Chai

CSCE 643 Computer Vision: Structure from Motion

Jinxiang Chai

Stereo reconstruction

Given two or more images of the same scene or object, compute a representation of its shape

knownknowncameracamera

viewpointsviewpoints

Stereo reconstruction

Given two or more images of the same scene or object, compute a representation of its shape

knownknowncameracamera

viewpointsviewpoints

How to estimate camera parameters?

- where is the camera?

- where is it pointing?

- what are internal parameters, e.g. focal length?

Calibration from 2D motion

Structure from motion (SFM) - track points over a sequence of images

- estimate for 3D positions and camera positions

- calibrate intrinsic camera parameters before hand

Self-calibration: - solve for both intrinsic and extrinsic camera parameters

SFM = Holy Grail of 3D Reconstruction

Take movie of object

Reconstruct 3D model

Would be

commercially

highly viable

How to Get Feature Correspondences

Feature-based approach

- good for images

- feature detection (corners or sift features)

- feature matching using RANSAC (epipolar line)

Pixel-based approach

- good for video sequences

- patch based registration with lucas-kanade algorithm

- register features across the entire sequence

A Brief Introduction on Feature-based Matching

Find a few important features (aka Interest Points)

Match them across two images

Compute image transformation function h

Feature Detection

-Two images taken at the same place with different angles

- Projective transformation H3X3

Feature Matching

?



Feature Matching

?



How do we match features across images? Any criterion?

Feature Matching

?



How do we match features across images? Any criterion?

Feature Matching

Intensity/Color similarity• The intensity of pixels around the corresponding features should

have similar intensity

Feature Matching

Feature similarity (Intensity or SIFT signature)• The intensity of pixels around the corresponding features should


• Cross-correlation, SSD

Feature Matching




Distance constraint• The displacement of features should be smaller than a given

threshold

Feature Matching





threshold

Epipolar line constraint• The corresponding pixels satisfy epipolar line constraints.

Feature Matching





threshold

Epipolar line constraint• The corresponding pixels satisfy epipolar line constraints.

Fundamental matrix H

Feature-space Outlier Rejection

bad

Good


Can we now compute H3X3 from the blue points?





• No! Still too many outliers…


Can we now compute H3X3 from the blue points?• No! Still too many outliers…

• What can we do?


Can we now compute H3X3 from the blue points?• No! Still too many outliers…

• What can we do?

Robust estimation!

Robust Estimation: A Toy Example

How to fit a line based on a set of 2D points?

RANSAC for Estimating Projective Transformation

RANSAC loop:Select four feature pairs (at random)

Compute the transformation matrix H (exact)

Compute inliers where SSD(pi’, H pi) < ε

Keep largest set of inliers

Re-compute least-squares H estimate on all of the inliers

For more detail, check

- http://research.microsoft.com/en-us/um/people/zhang/INRIA/software-FMatrix.html

- Philip H. S. Torr (1997). "The Development and Comparison of Robust Methods for Estimating

the Fundamental Matrix". International Journal of Computer Vision 24 (3): 271–300

Structure from Motion

Two Principal Solutions• Bundle adjustment (nonlinear optimization)

• Factorization (SVD, through orthographic approximation, affine geometry)

Projection Matrix

Perspective projection:

2D coordinates are just a nonlinear function of its 3D coordinates and camera parameters:

1100

0

1 3

2

1

3

2

1

0

0

i

i

i

T

T

T

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x

i

i

z

y

x

t

t

t

r

r

r

vf

uf

v

u

33

32302

33

30213021

)(

)(

tPr

ttfPrvrfv

tPr

tuttfPrurrfu

T

yTT

yi

Tx

TTTx

i

K

);,,( iPTRKf

);,,( iPTRKg

R T P

Nonlinear Approach for SFM

What’s the difference between camera calibration and SFM?


M

j

N

iijj

jiijj

ji

TRK

PTRKgvPTRKfujj

1 1

22

}{},{,

));,,(());,,((minarg


- camera calibration: known 3D and 2D


M

j

N

iijj

jiijj

ji

TRKP

PTRKgvPTRKfujji

1 1

22

}{},{,},{

)),,,(()),,,((minarg

M

j

N

iijj

jiijj

ji

TRK

PTRKgvPTRKfujj

1 1

22

}{},{,

));,,(());,,((minarg



- SFM: unknown 3D and known 2D


M

j

N

iijj

jiijj

ji

TRKP

PTRKgvPTRKfujji

1 1

22

}{},{,},{

)),,,(()),,,((minarg

M

j

N

iijj

jiijj

ji

TRK

PTRKgvPTRKfujj

1 1

22

}{},{,

));,,(());,,((minarg




- what’s 3D-to-2D registration problem?


M

j

N

iijj

jiijj

ji

TRKP

PTRKgvPTRKfujji

1 1

22

}{},{,},{

)),,,(()),,,((minarg

M

j

N

iijj

jiijj

ji

TRK

PTRKgvPTRKfujj

1 1

22

}{},{,

));,,(());,,((minarg




- what’s 3D-to-2D registration problem?

SFM: Bundle Adjustment

SFM = Nonlinear Least Squares problem

Minimize through• Gradient Descent

• Conjugate Gradient

• Gauss-Newton

• Levenberg Marquardt common method

Prone to local minima

M

j

N

iijj

jiijj

ji

TRKP

PTRKgvPTRKfujji

1 1

22

}{},{,},{

)),,,(()),,,((minarg

Count # Constraints vs #Unknowns

M camera poses

N points

2MN point constraints

6M+3N + 4 (unknowns)

Suggests: need 2mn 6m + 3n+4

But: Can we really recover all parameters???

M

j

N

iijj

jiijj

ji

TRKP

PTRKgvPTRKfujji

1 1

22

}{},{,},{

)),,,(()),,,((minarg

Count # Constraints vs #Unknowns

M camera poses

N points

2MN point constraints

6M+3N+4 unknowns (known intrinsic camera parameters)

Suggests: need 2mn 6m + 3n+4

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+4 - 7

M

j

N

iijj

jiijj

ji

TRKP

PTRKgvPTRKfujji

1 1

22

}{},{,},{

)),,,(()),,,((minarg

Are We Done?

No, bundle adjustment has many local minima.

SFM Using Factorization

12

1

2

1

i

i

i

T

T

i

i

z

y

x

t

t

r

r

v

u

Assume an orthographic camera

Image World


12

1

2

1

i

i

i

T

T

i

i

z

y

x

t

t

r

r

v

u

Assume orthographic camera

Image World

i

i

i

T

T

N

ii

i

N

ii

i

z

y

x

r

r

N

vv

N

uu

2

1

1

1

Subtract the mean


N

N

N

T

T

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N

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...

...

...

~

~

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2

2

2

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1

2

1

2

2

1

1

Stack all the features from the same frame:


N

N

N

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T

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2

2

2

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2

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2

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1

1

N

N

N

TF

TF

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T

NF

NF

F

F

F

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NF

NF

F

F

F

F

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2,

2,

1,

1,


Stack all the features from all the images:

W


N

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2,

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,

,

2,

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1,

,

,

2,

2,

1,

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NFW 2

~



W

32 FM NS 3


N

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TF

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NF

NF

F

F

F

F

NF

NF

F

F

F

F

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NFW 2

~32 FM


W

NS 3

Factorize the matrix into two matrix using SVD:

NFW 2

~

TNF

TNF VSUMVUW 2

1

32

1

322

~~~


N

N

N

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TF

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NF

NF

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NF

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NFW 2

~32 FM


NS 3


NFW 2

~

TNF

TNF VSUMVUW 2

1

32

1

322

~~~

NNFF SQSQMM

31

333333232

~~


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TF

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T

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NF

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NF

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F

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2,

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2,

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1,

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,

2,

2,

1,

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NFW 2

~32 FM


W

NS 3


NFW 2

~

TNF

TNF VSUMVUW 2

1

32

1

322

~~~

NNFF SQSQMM

31

333333232

~~

How to compute the matrix ? 33Q


2,2,2,11,1

2,

1,

2,1

1,1

3232 FF

TF

TF

T

T

TFF rrrr

r

r

r

r

MM

M is the stack of rotation matrix:

2,2,

2,1,

1,2,

1,1,

2,12,1

2,11,1

1,12,1

1,11,1

FTF

FTF

FTF

FTF

T

T

T

T

rr

rr

rr

rr

rr

rr

rr

rr


2,2,2,11,1

2,

1,

2,1

1,1

3232 FF

TF

TF

T

T

TFF rrrr

r

r

r

r

MM


2,2,

2,1,

1,2,

1,1,

2,12,1

2,11,1

1,12,1

1,11,1

FTF

FTF

FTF

FTF

T

T

T

T

rr

rr

rr

rr

rr

rr

rr

rr

1 010

1 010

Orthogonal constraints from rotation matrix


2,2,2,11,1

2,

1,

2,1

1,1

3232 FF

TF

TF

T

T

TFF rrrr

r

r

r

r

MM

2,2,

2,1,

1,2,

1,1,

2,12,1

2,11,1

1,12,1

1,11,1

FTF

FTF

FTF

FTF

T

T

T

T

rr

rr

rr

rr

rr

rr

rr

rr


1 010

1 010


TF

TF MQQM 32333332

~~


TF

TF MQQM 32333332

~~

2,2,

2,1,

1,2,

1,1,

2,12,1

2,11,1

1,12,1

1,11,1

FTF

FTF

FTF

FTF

T

T

T

T

rr

rr

rr

rr

rr

rr

rr

rr

1 010

1 010

Orthogonal constraints from rotation matrices:


TF

TF MQQM 32333332

~~

2,2,

2,1,

1,2,

1,1,

2,12,1

2,11,1

1,12,1

1,11,1

FTF

FTF

FTF

FTF

T

T

T

T

rr

rr

rr

rr

rr

rr

rr

rr

1 010

1 010


QQ: symmetric 3 by 3 matrix


TF

TF MQQM 32333332

~~

2,2,

2,1,

1,2,

1,1,

2,12,1

2,11,1

1,12,1

1,11,1

FTF

FTF

FTF

FTF

T

T

T

T

rr

rr

rr

rr

rr

rr

rr

rr

1 010

1 010


How to compute QQT?

least square solution

- 4F linear constraints, 9 unknowns (6 independent due to symmetric matrix)



TF

TF MQQM 32333332

~~

2,2,

2,1,

1,2,

1,1,

2,12,1

2,11,1

1,12,1

1,11,1

FTF

FTF

FTF

FTF

T

T

T

T

rr

rr

rr

rr

rr

rr

rr

rr

1 010

1 010


How to compute QQT?

least square solution

- 4F linear constraints, 9 unknowns (6 independent due to symmetric matrix) How to compute Q from QQT:

SVD again: 2

1

UQVUQQ T



2,2,2,11,1

2,

1,

2,1

1,1

3232 FF

TF

TF

T

T

TFF rrrr

r

r

r

r

MM

2,2,

2,1,

1,2,

1,1,

2,12,1

2,11,1

1,12,1

1,11,1

FTF

FTF

FTF

FTF

T

T

T

T

rr

rr

rr

rr

rr

rr

rr

rr


1 010

1 010


TF

TF MQQM 32333332

~~

QQT: symmetric 3 by 3 matrix

Computing QQT is easy:

- 3F linear equations

- 6 independent unknowns


1. Form the measurement matrix

2. Decompose the matrix into two matrices and using SVD

3. Compute the matrix Q with least square and SVD

4. Compute the rotation matrix and shape matrix:

and

NFW 2

~

NS 3

~ 32

~FM

QMM F 32

~ 32

1 ~

FSQS

Weak-perspective Projection

Factorization also works for weak-perspective projection (scaled orthographic projection):

d z0

12

1

2

1

i

i

i

T

T

i

i

z

y

x

t

t

r

r

v

u

Factorization for Full-perspective Cameras

[Han and Kanade]

SFM for Deformable Objects

For detail, click here

http://faculty.cs.tamu.edu/jchai/papers/IJCV_nsfm.pdf

SFM for Articulated Objects

For video, click here


Bundle adjustment (nonlinear optimization) - work with perspective camera model - work with incomplete data - prone to local minima

Factorization: - closed-form solution for weak perspective camera - simple and efficient - usually need complete data - becomes complicated for full-perspective camera model

Phil Torr’s structure from motion toolkit in matlab (click here)

Voodoo camera tracker (click here)

All Together Video

Click here

- feature detection

- feature matching (epipolar geometry)

- structure from motion

- stereo reconstruction

- triangulation

- texture mapping

Documents

CSCE 643 Computer Vision: Structure from Motion Jinxiang Chai