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### From Digital Photos to Textured 3D Models on Internet

### Part I

### Part II

### Part V

Gang Xu

Ritsumeikan University &

3D Media Co., Ltd.

Applications

- Cultural heritage modeling
- Car & ship modeling
- 3D Survey
- Plant Modeling
- Game
- Virtual Reality
- Web design

Outline of This Tutorial

- How can 3D be recovered from 2D?
- Mathematical Preparations: Solving Linear Equations and Non-linear Optimization
- Camera Model and Projection
- Epipolar Geometry Between Images
- Linear Algorithm for 2-View SFM
- Bundle Adjustment (non-linear optimization)

General Strategy

- Step 1: Find a linear or closed-form solution
- Step 2: Non-linear optimization
- A linear or closed-form solution via many intermediate representations usually includes large errors. But this solution can serve as the initial guess in the non-linear optimization.

Mathematical Preparation

Solving Linear Equations (2)

- When n=2
- When n>2, the inverse of A does not exist.

Solving Linear Homogeneous Equations

- When c=0, we cannot use

x

Define

Minimize

Under ||x||=1

Solution: Eigen vector associated with the smallest eigen value

of

An Example (1)

- Fitting a line to a 2D point set
- Plane equation
- Minimize

Non-Linear Optimization (1)

- Minimize non-linear function

Non-Linear Optimization (2)

- High-dimensional space
- No closed-form solution
- Multiple local minima
- Iterative procedure, starting from an initial guess, assuming that f is differentiable everywhere.

An Example: Bundle Adjustment

Bundle Adjustment: minimizing difference between observations and back-projections

3 Common Used Algorithms

- Gradient Method (1st-order derivative)
- Newton Method (2nd-orde derivative)
- Levenberg-Marquartd Method = Gradient+Newton, used in 3DM Modeler, 3DM Stitcher and 3DM Calibrator

Camera and Projection

Projection Matrix

Homogeneous Coordinates

Principal point

x

(u0,v0)

v

y

Image Coordinate SystemsIntrinsic parameters

Principal point approximately at

the image center

A: Intrinsic matrix

3D Coordinate Transform

Z’

X’

Y’

Object coordinate

system

Rotation R

Translation t

Z

X

Y

Camera coordinate system

What Is Unknown?

- Intrinsic Parameters: focal length f
- Extrinsic Parameters: Rotation matrix R, translation vector t

Rotation Representations

r/||r||

||r||

- 3×3 Rotation MatrixR
- Satisfying
- There are only 3 degrees of freedom.
- A rotation can be represented by a 3-vector r, with ||r|| representing the rotation amount, and r/||r|| representing the rotation axis.
- Others: Euler Angles, Roll-Pitch-Yaw, Quarternion

Essential Matrix

E: Essential Matrix

Fundamental Matrix

F: Fundamental matrix

Degenerate Cases Lead to Homography

- Case 1: t=0 (no translation, only rotation. Panorama- 3DM Stitcher)
- Case 2: planar scene (3DM Calibrator)
- In both cases: the image can be transformed into another by a homography.

Determining E, F and H Matricesfrom Point Matches

- E, F and H can only be determined up to scale.
- F (and E) can be linearly determined from 8 or more point matches.
- H can be linearly determined from 4 or more point matches.

Determining Translation from Essential Matrix

- Assuming that the intrinsic matrix A is available
- First compute Essential matrix
- Determining t (up to a scale) from
- There are two solutions t and -t

Determining Rotation

R

- We can consider the problem as determining R from 3 pairs of vectors before and after the rotation

Determining Rotation from n Pairs of Vectors

R

- Given (pi,pi’), i=1,…,n (n>=2),
- Define and minimize
- There is a linear solution using SVD.

…

…

F’

R,t

Determining 3D coordinates- Given R, t, x and x’, determine unknown scales s and s’ such that the two lines “meet” in the 3D space.

Bundle Adjustment

Coordinate System

- All space points and camera positions & camera poses are defined in a common coordinate system.

Match Matrix

- Match Matrix w(i,j)=1, if i-th point appears in j-th image; otherwise w(i,j)=0

j

1 2 3 …

i

0

1

1

1

1

2

.

.

1

1

0

Parameters to Determine

- The parameters to determine are
- (Xi,Yi,Zi), i=1,…,n. Totally 3n
- (tj,rj,fj), j=1,…,m. Totally 7m
- Altogether 3n+7m

Initial Guess

- LM method needs good initial guesses.
- They can be obtained using the previous linear algorithm.
- Needs to transform all data into the common coordinate system.

References

- Olivier Faugeras: 3D Computer Vision: A Geometry Viewpoint, MIT Press, 1993
- Gang Xu and Zhengyou Zhang: Epipolar Geometry in Stereo, Motion and Object Recognition, Kluwer, 1996
- Richard Hartley and Andrew Zisserman: Multiple View Geometry, Cambridge Univ Press, 2000

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