Affine structure from motion

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Affine structure from motion

• Marc Pollefeys
• COMP 256

Some slides and illustrations from J. Ponce, A.
Zisserman, R. Hartley, Luc Van Gool,
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Last time Optical Flow
Ixu
Ix
u
Ixu- It
It
Aperture problem

• two solutions
• - regularize (smoothness prior)
• constant over window
• (i.e. Lucas-Kanade)

Coarse-to-fine, parametric models, etc
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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 Springbreak
Mar 20/22 Segmentation Fitting
Mar 27/29 Prob. Segmentation Project Update
Apr 3/5 Tracking Tracking
Apr 10/12 Object Recognition Object Recognition
Apr 17/19 Range data Range data
Apr 24/26 Final project Final project
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AFFINE STRUCTURE FROM MOTION

• The Affine Structure from Motion Problem
• Elements of Affine Geometry
• Affine Structure from Motion from two Views
• A Geometric Approach
• Affine Epipolar Geometry
• An Algebraic Approach
• Affine Structure from Motion from Multiple Views
• From Affine to Euclidean Images
• Structure from motion of multiple and deforming
  object

Reading Chapter 12.
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Affine Structure from Motion
Reprinted with permission from Affine Structure
from Motion, by J.J. (Koenderink and A.J.Van
Doorn, Journal of the Optical Society of America
A, 8377-385 (1990). ? 1990 Optical Society of
America.

• Given m pictures of n points, can we recover
• the three-dimensional configuration of these
  points?
• the camera configurations?

(structure) (motion)
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Orthographic Projection
Parallel Projection
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Weak-Perspective Projection
Paraperspective Projection
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The Affine Structure-from-Motion Problem
Given m images of n fixed points P we can write
j
2mn equations in 8m3n unknowns
Overconstrained problem, that can be solved using
(non-linear) least squares!
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The Affine Ambiguity of Affine SFM
When the intrinsic and extrinsic parameters are
unknown
So are M and P where
i
j
and
Q is an affine transformation.
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Affine Spaces (Semi-Formal) Definition
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2
Example R as an Affine Space
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In General
The notation
is justified by the fact that choosing some
origin O in X allows us to identify the point P
with the vector OP.
Warning Pu and Q-P are defined independently
of O!!
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Barycentric Combinations

• Can we add points? RPQ

NO!

• But, when

we can define

• Note

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Affine Subspaces
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Affine Coordinates

• Coordinate system for U

• Coordinate system for YOU

• Affine coordinates

• Coordinate system for Y

• Barycentric
• coordinates

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When do m1 points define a p-dimensional
subspace Y of an n-dimensional affine space X
equipped with some coordinate frame basis?
Rank ( D ) p1, where
Writing that all minors of size (p2)x(p2) of D
are equal to zero gives the equations of Y.
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Affine Transformations

• Bijections from X to Y that
• map m-dimensional subspaces of X onto
  m-dimensional
• subspaces of Y
• map parallel subspaces onto parallel subspaces
  and
• preserve affine (or barycentric) coordinates.

• Bijections from X to Y that
• map lines of X onto lines of Y and
• preserve the ratios of signed lengths of
• line segments.

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In E they are combinations of rigid
transformations, non-uniform scalings and shears.
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Affine Transformations II

• Given two affine spaces X and Y of dimension m,
  and two
• coordinate frames (A) and (B) for these spaces,
  there exists
• a unique affine transformation mapping (A) onto
  (B).

• Given an affine transformation from X to Y, one
  can always write

• When coordinate frames have been chosen for X
  and Y,
• this translates into

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Affine projections induce affine transformations
from planes onto their images.
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Affine Shape
Two point sets S and S in some affine space X
are affinely equivalent when there exists an
affine transformation y X X such that X
y ( X ).
Affine structure from motion affine shape
recovery.
recovery of the corresponding motion
equivalence classes.
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Geometric affine scene reconstruction from two
images (Koenderink and Van Doorn, 1991).
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Affine Structure from Motion
Reprinted with permission from Affine Structure
from Motion, by J.J. (Koenderink and A.J.Van
Doorn, Journal of the Optical Society of America
A, 8377-385 (1990). ? 1990 Optical Society of
America.
(Koenderink and Van Doorn, 1991)
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The Affine Epipolar Constraint
Note the epipolar lines are parallel.
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Affine Epipolar Geometry
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The Affine Fundamental Matrix
where
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An Affine Trick..
Algebraic Scene Reconstruction
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The Affine Structure of Affine Images
Suppose we observe a scene with m fixed cameras..
The set of all images of a fixed scene is a 3D
affine space!
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has rank 4!
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From Affine to Vectorial Structure
Idea pick one of the points (or their center of
mass) as the origin.
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What if we could factorize D? (Tomasi and
Kanade, 1992)
Affine SFM is solved!
Singular Value Decomposition
We can take
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From uncalibrated to calibrated cameras
Weak-perspective camera
Calibrated camera
Problem what is Q ?
Note Absolute scale cannot be recovered. The
Euclidean shape (defined up to an arbitrary
similitude) is recovered.
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Reconstruction Results (Tomasi and Kanade, 1992)
Reprinted from Factoring Image Sequences into
Shape and Motion, by C. Tomasi and T. Kanade,
Proc. IEEE Workshop on Visual Motion (1991). ?
1991 IEEE.
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More examples
Tomasi Kanade92, Poelman Kanade94
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More examples
Tomasi Kanade92, Poelman Kanade94
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More examples
Tomasi Kanade92, Poelman Kanade94
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Further Factorization work

• Factorization with uncertainty
• Factorization for indep. moving objects (now)
• Factorization for articulated objects (now)
• Factorization for dynamic objects (now)
• Perspective factorization (next week)
• Factorization with outliers and missing pts.

(Irani Anandan, IJCV02)
(Costeira and Kanade 94)
(Yan and Pollefeys 05)
(Bregler et al. 2000, Brand 2001)
(Sturm Triggs 1996, )
(Jacobs 97 (affine), Martinek Pajdla01
Aanaes02 (perspective))
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Structure from motion of multiple moving objects
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Structure from motion of multiple moving objects
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Shape interaction matrix

• Shape interaction matrix for articulated objects
  looses block diagonal structure

Costeira and Kanades approach is not usable for
articulated bodies (assumes independent motions)
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Articulated motion subspaces
Motion subspaces for articulated bodies intersect
(Yan and Pollefeys, CVPR05) (Tresadern and Reid,
CVPR05)
Joint (1D intersection)
(jointorigin)
(rank8-1)
Hinge (2D intersection)
(hingez-axis)
(rank8-2)
Exploit rank constraint to obtain better estimate
Also for non-rigid parts if
(Yan Pollefeys, 06?)
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Results

• Toy truck
• Segmentation
• Intersection

• Student
• Segmentation
• Intersection

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Articulated shape and motion factorization
(Yan and Pollefeys, 2006?)

• Automated kinematic chain building for
  articulated non-rigid obj.
• Estimate principal angles between subspaces
• Compute affinities based on principal angles
• Compute minimum spanning tree

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Structure from motion of deforming objects
(Bregler et al 00 Brand 01)

• Extend factorization approaches to deal with
  dynamic shapes

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Representing dynamic shapes
(fig. M.Brand)
represent dynamic shape as varying linear
combination of basis shapes
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Projecting dynamic shapes
(figs. M.Brand)
Rewrite
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Dynamic image sequences
One image
(figs. M.Brand)
Multiple images
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Dynamic SfM factorization?
Problem find J so that M has proper structure
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Dynamic SfM factorization
(Bregler et al 00)
Assumption SVD preserves order and orientation
of basis shape components
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Results
(Bregler et al 00)
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Dynamic SfM factorization
(Brand 01)
constraints to be satisfied for M
constraints to be satisfied for M, use to compute
J
hard!
(different methods are possible, not so simple
and also not optimal)
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Non-rigid 3D subspace flow
(Brand 01)

• Same is also possible using optical flow in stead
  of features, also takes uncertainty into account

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Results
(Brand 01)
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Results
(Brand 01)
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Results
(Bregler et al 01)
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Next class Projective structure from motion
Reading Chapter 13