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**Multiple View Geometry**

Marc Pollefeys COMP 256

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**Last class Gaussian pyramid Laplacian pyramid Gabor Fourier filters**

transform

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**Histograms : co-occurrence matrix**

Not last class … Histograms : co-occurrence matrix

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**Texture synthesis [Zalesny & Van Gool 2000]**

2 analysis iterations 6 analysis iterations 9 analysis iterations

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**View-dependent texture synthesis [Zalesny & Van Gool 2000]**

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**Efros & Leung ’99 p Assuming Markov property, compute P(p|N(p))**

non-parametric sampling Synthesizing a pixel Input image Assuming Markov property, compute P(p|N(p)) Building explicit probability tables infeasible Instead, let’s search the input image for all similar neighborhoods — that’s our histogram for p To synthesize p, just pick one match at random

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**Efros & Leung ’99 extended**

p B Idea: unit of synthesis = block Exactly the same but now we want P(B|N(B)) Much faster: synthesize all pixels in a block at once Not the same as multi-scale! Synthesizing a block non-parametric sampling Input image Observation: neighbor pixels are highly correlated

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**constrained by overlap**

block Input texture B1 B2 Neighboring blocks constrained by overlap B1 B2 Minimal error boundary cut B1 B2 Random placement of blocks

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**Minimal error boundary**

overlapping blocks vertical boundary _ = 2 overlap error min. error boundary

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**Why do we see more flowers in the distance?**

[Leung & Malik CVPR97] Perpendicular textures

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Shape-from-texture

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**Tentative class schedule**

Jan 16/18 - Introduction Jan 23/25 Cameras Radiometry Jan 30/Feb1 Sources & Shadows Color Feb 6/8 Linear filters & edges Texture Feb 13/15 Multi-View Geometry Stereo Feb 20/22 Optical flow Project proposals Feb27/Mar1 Affine SfM Projective SfM Mar 6/8 Camera Calibration Silhouettes and Photoconsistency Mar 13/15 Springbreak Mar 20/22 Segmentation Fitting Mar 27/29 Prob. Segmentation Project Update Apr 3/5 Tracking Apr 10/12 Object Recognition Apr 17/19 Range data Apr 24/26 Final project

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**THE GEOMETRY OF MULTIPLE VIEWS**

Epipolar Geometry The Essential Matrix The Fundamental Matrix The Trifocal Tensor The Quadrifocal Tensor Reading: Chapter 10.

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Epipolar Geometry Epipolar Plane Baseline Epipoles Epipolar Lines

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**Potential matches for p have to lie on the corresponding**

Epipolar Constraint Potential matches for p have to lie on the corresponding epipolar line l’. Potential matches for p’ have to lie on the corresponding epipolar line l.

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**Epipolar Constraint: Calibrated Case**

Essential Matrix (Longuet-Higgins, 1981)

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**E p’ is the epipolar line associated with p’.**

Properties of the Essential Matrix T E p’ is the epipolar line associated with p’. ETp is the epipolar line associated with p. E e’=0 and ETe=0. E is singular. E has two equal non-zero singular values (Huang and Faugeras, 1989). T

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**Epipolar Constraint: Small Motions**

To First-Order: Pure translation: Focus of Expansion

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**Epipolar Constraint: Uncalibrated Case**

Fundamental Matrix (Faugeras and Luong, 1992)

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**Properties of the Fundamental Matrix**

F p’ is the epipolar line associated with p’. FT p is the epipolar line associated with p. F e’=0 and FT e=0. F is singular. T T

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**The Eight-Point Algorithm (Longuet-Higgins, 1981)**

|F| =1. Minimize: under the constraint 2

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**Non-Linear Least-Squares Approach**

(Luong et al., 1993) Minimize with respect to the coefficients of F , using an appropriate rank-2 parameterization.

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**Problem with eight-point algorithm**

linear least-squares: unit norm vector F yielding smallest residual What happens when there is noise?

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**The Normalized Eight-Point Algorithm (Hartley, 1995)**

Center the image data at the origin, and scale it so the mean squared distance between the origin and the data points is 2 pixels: q = T p , q’ = T’ p’. Use the eight-point algorithm to compute F from the points q and q’ . Enforce the rank-2 constraint. Output T F T’. i i i i i i T

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**Epipolar geometry example**

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**courtesy of Andrew Zisserman**

Example: converging cameras courtesy of Andrew Zisserman

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**Example: motion parallel with image plane**

(simple for stereo rectification) courtesy of Andrew Zisserman

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**courtesy of Andrew Zisserman**

Example: forward motion e’ e courtesy of Andrew Zisserman

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**courtesy of Andrew Zisserman**

Fundamental matrix for pure translation auto-epipolar courtesy of Andrew Zisserman

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**courtesy of Andrew Zisserman**

Fundamental matrix for pure translation courtesy of Andrew Zisserman

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**Trinocular Epipolar Constraints**

These constraints are not independent!

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**Trinocular Epipolar Constraints: Transfer**

Given p and p , p can be computed as the solution of linear equations. 1 2 3

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**Trinocular Epipolar Constraints: Transfer**

problem for epipolar transfer in trifocal plane! There must be more to trifocal geometry… image from Hartley and Zisserman

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Trifocal Constraints

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**Trifocal Constraints Calibrated Case All 3x3 minors must be zero!**

Trifocal Tensor

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Trifocal Constraints Uncalibrated Case Trifocal Tensor

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**Trifocal Constraints: 3 Points**

Pick any two lines l and l through p and p . 2 3 2 3 Do it again. T( p , p , p )=0 1 2 3

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**For any matching epipolar lines, l G l = 0.**

Properties of the Trifocal Tensor For any matching epipolar lines, l G l = 0. The matrices G are singular. They satisfy 8 independent constraints in the uncalibrated case (Faugeras and Mourrain, 1995). T i 2 1 3 i 1 Estimating the Trifocal Tensor Ignore the non-linear constraints and use linear least-squares Impose the constraints a posteriori.

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**For any matching epipolar lines, l G l = 0.**

2 1 3 The backprojections of the two lines do not define a line!

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**courtesy of Andrew Zisserman**

Trifocal Tensor Example 108 putative matches 18 outliers (26 samples) 88 inliers 95 final inliers (0.43) (0.23) (0.19) courtesy of Andrew Zisserman

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**Trifocal Tensor Example**

additional line matches images courtesy of Andrew Zisserman

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**Transfer: trifocal transfer**

(using tensor notation) doesn’t work if l’=epipolar line image courtesy of Hartley and Zisserman

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**Image warping using T(1,2,N)**

(Avidan and Shashua `97)

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**Multiple Views (Faugeras and Mourrain, 1995)**

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Two Views Epipolar Constraint

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Three Views Trifocal Constraint

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Four Views Quadrifocal Constraint (Triggs, 1995)

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**Geometrically, the four rays must intersect in P..**

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Quadrifocal Tensor and Lines

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**Quadrifocal tensor determinant is multilinear**

thus linear in coefficients of lines ! There must exist a tensor with 81 coefficients containing all possible combination of x,y,w coefficients for all 4 images: the quadrifocal tensor

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**Scale-Restraint Condition from Photogrammetry**

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**from perspective to omnidirectional cameras**

3 constraints allow to reconstruct 3D point perspective camera (2 constraints / feature) more constraints also tell something about cameras l=(y,-x) (x,y) (0,0) multilinear constraints known as epipolar, trifocal and quadrifocal constraints radial camera (uncalibrated) (1 constraints / feature)

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**Radial quadrifocal tensor**

(x,y) Radial quadrifocal tensor Linearly compute radial quadrifocal tensor Qijkl from 15 pts in 4 views Reconstruct 3D scene and use it for calibration (2x2x2x2 tensor) Not easy for real data, hard to avoid degenerate cases (e.g. 3 optical axes intersect in single point). However, degenerate case leads to simpler 3 view algorithm for pure rotation Radial trifocal tensor Tijk from 7 points in 3 views Reconstruct 2D panorama and use it for calibration (2x2x2 tensor)

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**Non-parametric distortion calibration**

(Thirthala and Pollefeys, ICCV’05) Models fish-eye lenses, cata-dioptric systems, etc. angle normalized radius

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**Non-parametric distortion calibration**

(Thirthala and Pollefeys, ICCV’05) Models fish-eye lenses, cata-dioptric systems, etc. 90o angle normalized radius

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**Next class: Stereo (x´,y´)=(x+D(x,y),y) F&P Chapter 11 image I´(x´,y´)**

Disparity map D(x,y) image I´(x´,y´) (x´,y´)=(x+D(x,y),y) F&P Chapter 11

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