Post on 22-Feb-2016
description
Reconnaissance d’objetset vision artificielle
Jean Ponce (ponce@di.ens.fr)http://www.di.ens.fr/~ponce
Equipe-projet WILLOWENS/INRIA/CNRS UMR 8548Laboratoire d’Informatique
Ecole Normale Supérieure, Paris
Outline
• Motivation: Making mosaics• Perspetive and weak perspective• Coordinates changes• Intrinsic and extrensic parameters• Affine registration• Projective registration
Feature-based alignment outline
Feature-based alignment outline
Extract features
Feature-based alignment outline
Extract featuresCompute putative matches
Feature-based alignment outline
Extract featuresCompute putative matchesLoop:
• Hypothesize transformation T (small group of putative matches that are related by T)
Feature-based alignment outline
Extract featuresCompute putative matchesLoop:
• Hypothesize transformation T (small group of putative matches that are related by T)
• Verify transformation (search for other matches consistent with T)
Feature-based alignment outline
Extract featuresCompute putative matchesLoop:
• Hypothesize transformation T (small group of putative matches that are related by T)
• Verify transformation (search for other matches consistent with T)
2D transformation models
Similarity(translation, scale, rotation)
Affine
Projective(homography)
Why these transformations ???
Pinhole perspective equation
zyfy
zxfx
''
''NOTE: z is always negative..
Affine models: Weak perspective projection
0
'where''
zfm
myymxx
is the magnification.
When the scene relief is small compared its distance from theCamera, m can be taken constant: weak perspective projection.
Affine models: Orthographic projection
yyxx
'' When the camera is at a
(roughly constant) distancefrom the scene, take m=1.
Analytical camera geometry
Coordinate Changes: Pure Translations
OBP = OBOA + OAP , BP = AP + BOA
Coordinate Changes: Pure Rotations
BABABA
BABABA
BABABABAR
kkkjkijkjjjiikijii
.........
TB
A
TB
A
TB
A
kji
ABA
BA
B kji
Coordinate Changes: Rotations about
the z Axis
1000cossin0sincos
RBA
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-handedorthonormal coordinate system.
Coordinate changes: pure rotations
PRP
zyx
zyx
OP
ABA
B
B
B
B
BBBA
A
A
AAA
kjikji
Coordinate Changes: Rigid Transformations
ABAB
AB OPRP
11111P
TOPRPORP A
BA
ABAB
AA
TA
BBA
B
0
Pinhole perspective equation
zyfy
zxfx
''
''NOTE: z is always negative..
The intrinsic parameters of a camera
Normalized imagecoordinates
Physical image coordinates
Units:k,l : pixel/mf : ma,b : pixel
The intrinsic parameters of a camera
Calibration matrix
The perspectiveprojection equation
The extrinsic parameters of a camera
Perspective projections induce projective transformations between planes
Weak-perspective projection
Paraperspective projection
Affine cameras
Orthographic projection
Parallel projection
More affine cameras
Weak-perspective projection model
r(p and P are in homogeneous coordinates)
p = A P + b (neither p nor P is in hom. coordinates)
p = M P (P is in homogeneous coordinates)
Affine projections induce affine transformations from planes onto their images.
Affine transformationsAn affine transformation maps a parallelogram ontoanother parallelogram
11001''
22221
11211
vu
baabaa
vu
Fitting an affine transformationAssume we know the correspondences, how do we get
the transformation?
2
1
43
21
tt
yx
mmmm
yx
i
i
i
i
i
i
ii
ii
yx
ttmmmm
yxyx
2
1
4
3
2
1
10000100
),( ii yx ),( ii yx
Fitting an affine transformation
Linear system with six unknownsEach match gives us two linearly independent
equations: need at least three to solve for the transformation parameters
i
i
ii
ii
yx
ttmmmm
yxyx
2
1
4
3
2
1
10000100
Beyond affine transformationsWhat is the transformation between two views of a
planar surface?
What is the transformation between images from two cameras that share the same center?
Perspective projections induce projective transformations between planes
Beyond affine transformationsHomography: plane projective transformation
(transformation taking a quad to another arbitrary quad)
Fitting a homographyRecall: homogenenous coordinates
Converting to homogenenousimage coordinates
Converting from homogenenousimage coordinates
Fitting a homographyRecall: homogenenous coordinates
Equation for homography:
Converting to homogenenousimage coordinates
Converting from homogenenousimage coordinates
11 333231
232221
131211
yx
hhhhhhhhh
yx
Fitting a homographyEquation for homography:
iT
T
T
ii xhhh
xHx
3
2
1
11 333231
232221
131211
i
i
i
i
yx
hhhhhhhhh
yx
0 ii xHx
iT
iiT
i
iT
iiT
iT
iT
i
ii
yxx
y
xhxhxhxhxhxh
xHx
12
31
23
00
00
3
2
1
hhh
xxxxxx
TTii
Tii
Tii
TTi
Tii
Ti
T
xyxy
3 equations, only 2 linearly independent
9 entries, 8 degrees of freedom(scale is arbitrary)
Direct linear transform
H has 8 degrees of freedom (9 parameters, but scale is arbitrary)
One match gives us two linearly independent equationsFour matches needed for a minimal solution (null space
of 8x9 matrix)More than four: homogeneous least squares
0
00
00
3
2
1111
111
hhh
xxxx
xxxx
Tnn
TTn
Tnn
Tn
T
TTT
TTT
xy
xy
0hA
Application: Panorama stitching
Images courtesy of A. Zisserman.