Transcript of Bases géométriques de l’informatique – Part deux Jean Ponce “Hauts du DI” Email:...
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- Bases gomtriques de linformatique Part deux Jean Ponce Hauts du
DI Email: ponce@di.ens.frponce@di.ens.fr Web:
http://www.di.ens.fr/~poncehttp://www.di.ens.fr/~ponce Planches
aprs les cours sur : http://www.di.ens.fr/~ponce/geomvis/lect1.pptx
http://www.di.ens.fr/~ponce/geomvis/lect1.pdf
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- References R. Hartley and A. Zisserman, Multiple View Geometry
in Computer Vision, Cambridge University Press, 2000. O.D.
Faugeras, Q.-T. Luong, and T. Papadopoulo, The Geometry of Multiple
Images, MIT Press, 2001. D.A. Forsyth and J. Ponce, Computer
Vision: A Modern Approach, Prentice-Hall, 2002, 2011 (2 nd
edition). J.J. Koenderink, Solid Shape, MIT Press, 1990. M. Berger,
Gomtrie, Nathan, 1992. D. Hilbert and S. Cohn-Vossen, Geometry and
the Imagination, Chelsea, 1952.
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- Images are two-dimensional patterns of brightness values. They
are formed by the projection of 3D objects.
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- How do we perceive depth?
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- Building large-scale 3D models from photographs Structure from
motionDense multiview stereo Snavely, Seitz, Szeliski, 2007
Vergauwen, Van Gool, 2006 Brown, Lowe, 2005 Schaffalitzky,
Zisserman, 2002 Furukawa, Ponce, 2007 Labatut, Pons, Keriven, 2009
Goesele, et al., 2007
http://phototour.cs.washington.edu/bundler/http://grail.cs.washington.edu/software/pmvs/
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- Google Maps Photo Tour Lucasfilm Weta Digital ( Bath &
Burke, Weta Digital, Siggraph11) PMVS
(http://www.di.ens.fr/pmvs)
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- A model built from 25,000 photos (Ubelmann, Dessales, Ponce,
2013)
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- What is a camera? x
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- What is a camera? x
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- What is a camera? x x c r y
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- x c r y c
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- x c r y x c
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- x c r y x c
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- x c r y x r y Linear family of lines x x c
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- Rank-4 (nondegenerate) families: Linear congruences Figures H.
Havlicek, VUT
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- Building a parabolic camera (Batog, Goaoc, Lavandier, Ponce,
2010)
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- Koenderink (1984) Solid shapes and their outlines
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- What does the occluding contour tell us about shape? (Marr
& Nishihara, 1978; Koenderink, 1984)
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- What does the occluding contour tell us about shape? (Marr
& Nishihara, 1978; Koenderink, 1984)
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- What does the occluding contour tell us about shape? M. Van
Hemskeerk Picasso Drer
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- The Geometry of the Gauss Map Gauss sphere Image of parabolic
curve Moving great circle Reprinted from On Computing Structural
Changes in Evolving Surfaces and their Appearance, By S. Pae and J.
Ponce, the International Journal of Computer Vision, 43(2):113-131
(2001). 2001 Kluwer Academic Publishers.
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- PLAN DU COURS: 1. Introduction gnrale 2. Camras Euclidiennes :
perspective centrale, projection parallle; paramtres intrinsques et
extrinsques; mires et talonnage Euclidien. 3. Camras affines :
lments de gomtrie affine; gomtrie multi vues, analyse du mouvement.
4. Camras projectives : lments de gomtrie projective; gomtrie multi
vues, analyse du mouvement. 5. talonnage Euclidien sans mire : la
conique absolue de Chasles et ses cousines; analyse du mouvement
Euclidien. 6. Camras purement projectives : lments de gomtrie des
droites; camras linaires gnrales . 7. Les surfaces Euclidiennes
lisses et leurs silhouettes : gomtrie diffrentielle descriptive; le
thorme de Koenderink et les graphes d'aspects.
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- Euclidean Cameras Pinhole perspective projection Orthographic
and weak-perspective models Non-standard models A detour through
sensing country Intrinsic and extrinsic parameters
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- Animal eye: a looonnng time ago. Pinhole perspective
projection: Brunelleschi, XV th Century. Camera obscura: XVI th
Century. Photographic camera: Niepce, 1816.
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- Massaccios Trinity, 1425 Pompei painting, 2000 years ago. Van
Eyk, XIV th Century Brunelleschi, 1415
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- Pinhole Perspective Equation NOTE: z is always negative..
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- Affine projection models: Weak perspective projection is the
magnification. When the scene relief is small compared its distance
from the Camera, m can be taken constant: weak perspective
projection.
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- Affine projection models: Orthographic projection When the
camera is at a (roughly constant) distance from the scene, take
m=-1.
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- Planar pinhole perspective Orthographic projection Spherical
pinhole perspective
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- Diffraction effects in pinhole cameras. Shrinking pinhole size
Use a lens!
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- Lenses Snells law n 1 sin 1 = n 2 sin 2 Descartes law
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- Thin Lenses
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- Spherical Aberration Distortion Chromatic Aberration
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- A compound lens
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- E=( /4) [ (d/z) 2 cos 4 ] L
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- Vignetting
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- Challenge: Illumination What is wrong with these pictures?
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- Photography (Niepce, La Table Servie, 1822) Milestones:
Daguerrotypes (1839) Photographic Film (Eastman, 1889) Cinema
(Lumire Brothers, 1895) Color Photography (Lumire Brothers, 1908)
Television (Baird, Farnsworth, Zworykin, 1920s) CCD Devices
(1970)
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- Image Formation: Radiometry What determines the brightness of
an image pixel? The light source(s) The surface normal The surface
properties The optics The sensor characteristics
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- Quantitative Measurements and Calibration Euclidean
Geometry
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- Euclidean Coordinate Systems
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- Planes
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- Coordinate Changes: Pure Translations O B P = O B O A + O A P,
B P = A P + B O A
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- Coordinate Changes: Pure Rotations
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- Coordinate Changes: Rotations about the z Axis
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- A rotation matrix is characterized by the following properties:
Its inverse is equal to its transpose, and its determinant is equal
to 1. Or equivalently: Its rows (or columns) form a right-handed
orthonormal coordinate system.
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- Coordinate Changes: Pure Rotations
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- Coordinate Changes: Rigid Transformations
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- Cameras and their parameters
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- Pinhole Perspective Equation
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- The Intrinsic Parameters of a Camera Normalized Image
Coordinates Physical Image Coordinates Units: k,l : pixel/m f : m :
pixel
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- The Intrinsic Parameters of a Camera Calibration Matrix The
Perspective Projection Equation
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- The Extrinsic Parameters of a Camera p M P
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- Explicit Form of the Projection Matrix Note: M is only defined
up to scale in this setting!!
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- Theorem (Faugeras, 1993)
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- Calibration Problem
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- Linear Camera Calibration
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- Linear Systems A A x xb b = = Square system: unique solution
Gaussian elimination Rectangular system ?? underconstrained:
infinity of solutions Minimize |Ax-b| 2 overconstrained: no
solution
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- How do you solve overconstrained linear equations ??
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- Homogeneous Linear Systems A A x x0 0 = = Square system: unique
solution: 0 unless Det(A)=0 Rectangular system ?? 0 is always a
solution Minimize |Ax| under the constraint |x| =1 2 2
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- How do you solve overconstrained homogeneous linear equations
?? The solution is e. 1 E(x)-E(e 1 ) = x T (U T U)x-e 1 T (U T U)e
1 = 1 1 2 + + q q 2 - 1 > 1 ( 1 2 + + q 2 -1)=0
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- Example: Line Fitting Problem: minimize with respect to
(a,b,d). Minimize E with respect to d: Minimize E with respect to
a,b: where Done !! n
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- Note: Matrix of second moments of inertia Axis of least
inertia
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- Linear Camera Calibration Linear least squares for n > 5
!
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- Once M is known, you still got to recover the intrinsic and
extrinsic parameters !!! This is a decomposition problem, not an
estimation problem. Intrinsic parameters Extrinsic parameters
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- Degenerate Point Configurations Are there other solutions
besides M ?? Coplanar points: ( )=( ) or ( ) or ( ) Points lying on
the intersection curve of two quadric surfaces = straight line +
twisted cubic Does not happen for 6 or more random points!
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- Analytical Photogrammetry Non-Linear Least-Squares Methods
Newton Gauss-Newton Levenberg-Marquardt Iterative, quadratically
convergent in favorable situations
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- Mobile Robot Localization (Devy et al., 1997)
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- (Rothganger, Sudsang, & Ponce, 2002)