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875: Recent A dvances in Geometric C omputer V ision & Recognition. Jan-Michael Frahm Fall 2011. Introductions. Class. Recent methods in computer vision Broadness of the class depends on YOU! Initial papers in class are the best papers of CVPR(2010,2011), ICCV(2011), ECCV(2010). - PowerPoint PPT Presentation
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875: Recent Advances in Geometric Computer Vision & Recognition
Jan-Michael FrahmFall 2011
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Introductions
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Class• Recent methods in computer vision• Broadness of the class depends on YOU!• Initial papers in class are the best papers of
CVPR(2010,2011), ICCV(2011), ECCV(2010)
4
Grade Requirements• Presentation of 3 papers in class
30 min talk, 10 min questions
• Papers for selection must come from: top journals: IJCV, PAMI, CVIU, IVCJ top conferences: CVPR (2010,2011), ICCV (2011),
ECCV (2010), MICCAI (2010, 2011) approval for all other venues is needed
• Final project evaluation, extension of a recent method from
the above
5
Schedule• Aug. 29th, Introduction• Aug. 31st, Geometric computer vision, first paper selection• Sept. 5th, Labor day• Sept. 7th, Geometric computer vision, Robust estimation• Sept. 12th, Optimization• Sept. 19th, Classification• Sept. 21st, 1. round of presentations starts• Oct 17th, 2. round of presentations starts• Oct 31st, definition of final projects due• Nov 7th, 3. round of presentations starts• Dec 5th and 7th, final project presentation
6
How to give a great presentation
• Structure of the talk: Motivation (motivate and explain the
problem) Overview Related work (short concise discussion) Approach Experiments Conclusion and future work
7
How to give a great presentation
• Use large enough fonts 5-6 one line bullet items on a slide
max• Keep it simple• No complex formulas in your talk• Bad Powerpoint slides• How to for presentations
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How to give a great presentation
• Abstract the material of the talk provide understanding beyond
details• Use pictures to illustrate
find pictures on the internet create a graphic (in ppt, graph tool) animate complex pictures
9
How to give a good presentation
• Avoid bad color schemes no red on blue looks awful
• Avoid using laser pointer (especially if you are nervous)
• Add pointing elements in your presentation
• Practice to stay within your time! • Don’t rush through the talk!
Projective and homogeneous points• Given: Plane in R2 embedded in P2 at coordinates w=1
viewing ray g intersects plane at v (homogeneous coordinates) all points on ray g project onto the same homogeneous point v projection of g onto is defined by scaling v=g/l = g/w
w=1
R3 ( )1
x xg v y y
w
1
1 1
xwxyv y g
w w
O
(R2)
w
x
v
y
Affine and projective transformations Affine transformation leaves infinite points at infinity
11 12 13
21 22 23
31 32 33
0 0 0 0 1 0
x
y
z
X a a a t XY a a a t YZ a a a t Z
'affineM T M
Projective transformations move infinite points into finite affine space
'projectiveM T M
11 12 13
21 22 23
31 32 33
41 42 43 441 0
xp
yp
zp
a a a tx X Xa a a ty Y Ya a a tz Z Zw w w ww
Example: Parallel lines intersect at the horizon (line of infinite points). We can see this intersection due to perspective projection!
Homogeneous coordinates
0 cbyax Ta,b,c0,0)()( kkcykbxka TT a,b,cka,b,c ~
Homogeneous representation of lines
equivalence class of vectors, any vector is representativeSet of all equivalence classes in R3(0,0,0)T forms P2
Homogeneous representation of points
0 cbyax Ta,b,cl Tyx,x on if and only if
0l 11 x,y,a,b,cx,y, T 0,1,,~1,, kyxkyx TT
The point x lies on the line l if and only if xTl=lTx=0
Homogeneous coordinatesInhomogeneous coordinates Tyx,
T321 ,, xxx but only 2DOF
Ideal points and the line at infinity
T0,,l'l ab
Intersections of parallel lines
TT and ',,l' ,,l cbacba
Example
1x 2x
Ideal points T0,, 21 xxLine at infinity T1,0,0l
l22 RP
normal direction ba,
Note that in P2 there is no distinction between ideal points and others
Pinhole Camera Model
Camera obscura
(Frankreich, 1830)
object
optical axis
image plane
principal point (u,v)
Pinhole Camera Model
pmfocal length, aspect
ratio (f, af)
Skew s
01 1
0 0 1
p p
f s um m
m K af v
aperture
Pinhole Camera Model
Selbstkalibrierung bestimmt die intrinsischen Kameraparameter
01 1
0 0 1
p p
f s um m
m K af v
Introduction to Computer Vision for Robotics
Projective Transformation
Projective Transformation maps M onto Mp in P3 space
0
01
1 0 0 00 1 0 00 0 1 00 0 01 1
p
pp p
z
x X Xy Y Y
M T MZ Z Z
W
X
YO
M
pM
Projective Transformation linearizes projection
0
projective scale factorZZ
Introduction to Computer Vision for Robotics
Projection in general pose
Rotation [R]
Projection center C M
World coordinates
Projection:m
p
0 1cam T
R CT
P
pm PM
1
0 1
T T
scene cam T
R R CT T
Introduction to Computer Vision for Robotics
Projection matrix P Camera projection matrix P combines:
inverse affine transformation Tcam-1 from general pose to origin
Perspective projection P0 to image plane at Z0 =1 affine mapping K from image to sensor coordinates
0, = T Tscenem PM P KPT K R R C
0
1 0 0 0projection: 0 1 0 0 0
0 0 1 0P I
scene pose transformation: 0 1
T T
scene T
R R CT
sensor calibration: 00 0 1
x x
y y
f s cK f c
Homography
X
Y0
M
im
jm
Homography plane to plane warping purely rotating camera
m
Self-calibration for Rotating CamerasAgapito et al.
1 1, , ,j i j i i j i jR K H K
• Rotation invariant formulation
2 1, , , , ,
T T T Tj i j i j i i j i j j j i iR R K H K K H K I
2, , ,
T T Ti i j i j i j j j iK K H K K H
Projection of the dual absolute conic into image i
Projection of the dual absolute conic into image j
, ,T
j i j iR R I
1, , ,j i j i i j i jH K R K
calibration through Choleski decomposition
Ti iK K
Removing projective distortion
333231
131211
3
1
'''
hyhxhhyhxh
xxx
333231
232221
3
2
'''
hyhxhhyhxh
xxy
131211333231' hyhxhhyhxhx 232221333231' hyhxhhyhxhy
select four points in a plane with know coordinates
(linear in hij)
(2 constraints/point, 8DOF 4 points needed)
Remark: no calibration at all necessary
Freely Moving Camera
XY
Cj
jm
M
Z
Ci
Epipolar Linie:
im
ie
m
iml
im
Computable from image correspondences
Example: motion parallel with image plane
Example: forward motion
e
e’
Introduction to Computer Vision for Robotics
The Essential Matrix E• F is the most general constraint on an image pair. If the camera
calibration matrix K is known, then more constraints are available
• Essential Matrix E
E holds the relative orientation of a calibrated camera pair. It has 5 degrees of freedom: 3 from rotation matrix Rik, 2 from direction of translation e, the epipole.
010101~~~~ mFKKmmKFmKFmm
E
TTTT
00
0][ with ][
xy
xz
yz
xx
eeeeee
eReE
Estimation of P from E• From E we can obtain a camera projection matrix pair:
E=Udiag(0,0,1)VT
• P0=[I3x3 | 03x1] and there are four choices for P1:
P1=[UWVT | +u3] or P1=[UWVT | -u3] or P1=[UWTVT | +u3] or P1=[UWTVT | -u3]
100001010
with W
four possible configurations:
only one with 3D point in front of both cameras
Kruppa Equations
, ,[ ]j i j i x i j iF K e K R
, , , ,[ ] [ ]T T
j i j j i j i x i j i i x i j iF K F K e K R e K R
, , [ ] [ ]T T T Tj i j i i x i xF K K F e K K e
1, ,[ ]j i i x i j i jF e K R K
Kruppa-equation (Faugeras et al.`92)
for constant camera calibration
Dual absolute conic limited to epipolar-geometrie