geometric computing for cybernetics eduardo bayro corrochanobernstei/web5/moima2016-end.pdf ·...
TRANSCRIPT
![Page 1: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/1.jpg)
Geometric Computing for Cybernetics
CINVESTAV
Center of Research and Advanced Studies, Guadalajara
Jalisco, Mexico
Department of Electrical Engineering and Computer Science
GEOVIS Laboratory
MOIMA June.23..2016, Hanover
Eduardo Bayro Corrochano [email protected]
http://www.gdl.cinvestav.mx/edb
![Page 2: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/2.jpg)
Motivation
Mathematical Framework
Robot Manipulators and Mobile Robots
Humanoids Mechanisms
Perception
Medical Robotics
Low and Middle Level Perception
Conclusions
Content
![Page 3: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/3.jpg)
Motivation: building perception action systems
What kind of
mathematical system?
Conformal geometric algebra
Modern unifying language for the
design of algorithms for PAC systems
Quaternions
Matrix algebra
Vector calculus Trigonometry
Development of
PAC systems
Tensor calculus
Symbolic computing
A sortiment of
mathematical tools
![Page 4: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/4.jpg)
Motivation: shall we go for higher dimensions ?
3D
nD
mD
Real Time ?
Do we gain something by
using other kind of geometric
representations?
Are we sofisticating the
approch for sake for
mathematical elegancy ?
![Page 5: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/5.jpg)
Geometric Algebra
![Page 6: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/6.jpg)
Quaternions, o a new system
of imaginaries in algebra
Clifford algebra: fussion
of Grassman and
Hamilton ideas
William Kingdon Clifford William Rowan Hamilton
Die lineale Ausdehnungslehre
ein neuer Zweig der
Mathematik
Hermann Günther Grassmann
Evolution of Algebras
![Page 7: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/7.jpg)
Conformal Geometric
Algebras
Lie algebras/groups
Directed distance
Projective Geometry
Versors
Homogeneous model
Null Cone
Horosphere (Wachter Gauss`
student 1792-1817)
Conformal Geometric Algebra
Incidence Algebra
Hongbo. Li, David Hestenes and Alyn Rockwood. "Generalized Homogeneous coordinates for
computational geometry".
G. Somer, editor,Geometric Computing with Clifford Algebras. Springer, Chapter I, pp. 1-45, 2001.
Math. Framework
![Page 8: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/8.jpg)
Geometric Product of two vectors
The geometric product of two vectors a y b is written as ab and can be
expressed as the sum of a scalar and a wedge products:
babaab
Clifford Geometric Algebra
scalar bivector
trivector
n-vector
ncba ...
Math. Framework
![Page 9: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/9.jpg)
Geometric Algebra Gp,q,r
In general a geometric algebra Gp,q,r is a linear space of dimension 2n, where
n=p+q+r.
Geometric Product
Let ei and ej be two orthonormal basis vectors of the vector space.
Then the geometric product for these vectors is defined as:
Clifford Geometric Algebra
jieeeee
ee
ijjiij
ji
for
n1,...,qpjifor0
qp1,...,pjifor1
p1,...,jifor1
bivector
Math. Framework
![Page 10: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/10.jpg)
Example the geometric algebra G4,1,0 has 25=32 elements.
12
4
2
3
2
2
2
1 eeee
ee
ee
Note
EII
eeE
eeeI
ec
e
5
4
54
321
:
12
5 e
Conformal Geometric Algebra for 3D space
2)ba(2
1BA metric recovery !!
02
2
21
x
eexxx oee
1 5 10 10 5 1
Scalar Vectors Bivectors Trivectors Tetravectors Pentavectors or
Pseudoscalar
3D 5D
Math. Framework
![Page 11: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/11.jpg)
Math. Framework
p
r X1
X2
X3
X4
0
2
21
sx
eps
c
c r 4321* xxxxS
0* Sxc
cISS *
d n
exxx 321
* den
cI *
Conformal Entities
Sphe
re
Pla
ne
![Page 12: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/12.jpg)
S2 S1
z
321
* xxxz 21 ssz
exxl 21
*
1
2
21 l
cIZZ *
cIll *
Lin
e
Cir
cle
Conformal Entities Math. Framework
Sphere, circle, pair of points and point
![Page 13: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/13.jpg)
Inversion and Dilation
r
eps 2
21 r
x
x’
The reflection of the point in an sphere.
1
21
'
sxsx
ees o
r
The dilation is the product of two
reflections
1
) ln(
2 1 2
2 1
2 2 1
2 1
2 1 2
2 1
'
) )( (
) )( ( ) )( ( ' '
2 1
1
r r
r r
r r
r r
r r
xD D x
e D
e e e e D
e e e e x e e e e x
E
o o
D
o o
D
o o
r
ees o
2
21
2 r
x
x’
ees o 21
1
Math. Framework
![Page 14: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/14.jpg)
Rotation in an arbitrary axis :
The Motor (moment-vector):
MMOO
RIM
RTM
s
t
ss
s
~'
12
Rotor, Translator and Motor
L
s
ltelele
s
ls
eR
eeR
TTRRL
tt
2
2222)))((())1)()(1((
1
A motor for rigid motion of geometric objects O like points, lines, planes, sphere
Translator
I
tT
21
Math. Framework
![Page 15: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/15.jpg)
Projection: sphere to affine plane
2)('
dcz
bazzMz 3R
3R
Mobius transformation result as the projection of
simple combination of motions on the sphere
Move one dimension higher and use
the Riemman sphere !!
By Douglas N. Arnold & Jonathan Rogness
Math. Framework
![Page 16: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/16.jpg)
Conformal transformation in geometric algebra
)}~
)({~
' eTexeTeKxKx aaaa
52
3
2
1)(
oeec eexxxFx
x
In conformal geometry
oaaaac eexxeTexeTeFKxKFxFxax
xxx
2'2
1´)}
~)({()
~()'(
1
1'
Versor sequence!!
Math. Framework
![Page 17: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/17.jpg)
Rotation in an arbitrary axis
Forward kinematics
L
L
ele
l
eM
eM
eM
TTRMtt
2
2
222
~
))1)()(1((
1
L
l
x
T-1
Rl
M
Mot
or
L1
L2 Ln
xp
Kin
em
atic
s ch
ain
s
Motor (dual quaternion) better operator for rigid transformation representation than a matrix representation (R, t)
MMXXtRxx~
´
![Page 18: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/18.jpg)
Differential kinematics
![Page 19: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/19.jpg)
Differential kinematics
Replacing both in previous equation
![Page 20: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/20.jpg)
Differential kinematics
since
![Page 21: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/21.jpg)
Robot Dynamics
Math. Framework
Math. Framework
![Page 22: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/22.jpg)
22
Grad, div, curl and blade-Laplacian
Laplacian
![Page 23: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/23.jpg)
Long term research goal
materials science
Neuroscience
Robotics
sensor stimuli
Brain coding
neocortex
World coordinates
Hyperspherical retina
fusion, storaging
learning
control
action
Ego coordinates
![Page 24: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/24.jpg)
Geometric Cybernetics
Development of algorithms for PAC systems
Body-sensors
calibration
RGB-d cameras
3D reconstruction
navigation
Geometric
control
Model human vision
Learning
Perception
Image Processing
Computer Vision
Grasping, object
manipulation
Humanoid
![Page 25: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/25.jpg)
Body-sensor calibration: relate the coordinate systems of
the sensors with the robot body
![Page 26: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/26.jpg)
Hand-eye calibration in CGA
10101010
BBXXXXAA tRtRtRtRXBAX
A B
X
Using motors
However, according Chen´s theorem the only what matters is the relative
motion of the involved screw lines
![Page 27: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/27.jpg)
Hand-eye calibration
The angle and pitch of the gripper (or endoscope) are equal to the angle and pitch of the camera, they remain
invariant under coordinate transformations, which is known as the Chen’s screw congruence theorem); therefore,
the problem is solved using only the screw axes lines defined by the motors
ld
ed
eM )22
sin()22
cos(
Lines can be represented using bivectors a and a`
![Page 28: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/28.jpg)
Hand-eye calibration
D
![Page 29: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/29.jpg)
The procecedure is repeated for each axis.
![Page 30: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/30.jpg)
![Page 31: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/31.jpg)
![Page 32: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/32.jpg)
![Page 33: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/33.jpg)
Calibrated robot: 3D reconstruction and controlled navigation
![Page 34: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/34.jpg)
Grasping
![Page 35: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/35.jpg)
Robot Manipulador
CINVESTAV
AdeptSix600
(6 DOF)
Barrett Hand
(4 DOF)
![Page 36: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/36.jpg)
Grasping: three contact points
Three conctact points:
the forces met at the
center of mass
Three contact
points:
Forces are parallel
Multiple conctact
points
![Page 37: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/37.jpg)
Visually guide object grasping
Segmentation
Pose estimation
Tracking
Grasping
![Page 38: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/38.jpg)
Humanoid Mechanism and Perception
![Page 39: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/39.jpg)
Long term research goal
materials science
Neuroscience
Robotics
sensor stimuli
Brain coding
neocortex
World coordinates
Cognitive Architecture
fusion, storaging
learning
control
action
Ego coordinates
Mathematical framework
Motivation
![Page 40: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/40.jpg)
Humanoid walking
45
![Page 41: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/41.jpg)
Human versus robot vision system
perception
Visual cortex
Signal flow
Use a stochastic geometric
framework for modelling biologic
concepts and develop algoriths for
intelligent mechatronic systems
Motivation
![Page 42: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/42.jpg)
,
Concepts Lie Algebra of visual perception
Hoffman L. Model of Human Vision system using Lie algebras
1960’s
More complex combinations
the first ~14 years of life
![Page 43: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/43.jpg)
Concepts Combinations of Lie operators for visual flow
![Page 44: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/44.jpg)
Binocular vision Concepts
Human visual cortex has some intrinsic manner
(subgroup of the affine group) to convert this binocular
view of the world into a ciclopean view (without losing
the 3D scene)
E.L. Schwartz, V1-V2-V3
Quasi-conformal. Mapping
(log of Mobiüs transforamtion)
Cyclopean eye
• The Superior Colliculus contains a motor map; activity in a
given region correlates with a particular a saccade-vector.
• The saccade-vectors are independent of the absolute gaze
direction
![Page 45: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/45.jpg)
Fixation, Stereopsis and Vergence
Moving an object slightly
off the horopter creates a small
retinal disparity that stimulates
stereoscopic depth perception
Relationship between the horopter,
Panum’s area and the zone of stereopsis.
Concepts
Source: Enright, 1983. Source: Bell, 1823.
![Page 46: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/46.jpg)
Listings law using rotors and translators
Using rotors
Pure rotations
Motor to Include small displacements
Parameterization of plane deformation
Applications
Listings planes of a binocualr system Tracking of moving object
![Page 47: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/47.jpg)
The computation of the motion axes is done with motors M in G4,1,0.
Body-Camera Calibration using screw lines Applications
Humanoid head
![Page 48: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/48.jpg)
Conformal Model of Humanoid robot Vision
Stereo Rig, Azimuth Plane , 3D Reconstruction
Duals of poncelet points
Applications
Spherical representation of the 3D visual space
![Page 49: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/49.jpg)
Algebra of spheres for selflocalization Applications
Modelbased Humamoid selflocalization Location hypotheses generation mechanism Rcognition
Inverse kinematics
![Page 50: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/50.jpg)
Simplex in conformal GA for 3D reconstruction
simplex
ecba
Applications
Delanauy traingulation
![Page 51: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/51.jpg)
This algorithm computes the axes of rotation based on measurements of lines taken from the real world.
![Page 52: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/52.jpg)
The procecedure is repeated for each axis.
Tilt Vergence
Left and right
Pan
![Page 53: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/53.jpg)
58
Head Control
;A A d A
Ae J A where J
L̂m
o
We obtain the orientation error dynamics as follows
So we need to use a robust control technique,
because of the unknown term .
Defining the orientation error
A de A A
Integral Sliding Modes (ISM) is a control strategy having the following
advantages.
Robustness against parameter uncertainty and external disturbances.
Robustness starting from the initial time instant.
Signal control low gain (Magnitude).
Chattering-free, pre-filtering the discontinuous control term.
dA
![Page 54: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/54.jpg)
59
Designing ISM Controller
Assuming joint velocities as control signals, we can re-define
the error dynamics as:
1 2;A A de J U A where U U U
Using U1 to control “ideal” dynamics, without unknown and disturbance
terms, and U2 to reject these terms. We define the sliding surface as:
AS e z
where z is an auxiliar variable. Then its dynamic is given by:
1 2A dS J U U A z
Defining we obtain 1Az J U
2A dS J U A
then with
1 1
2 2 ( )
A A
A
U K J e
U K J sign S
a stable error system is obtained ( 0)Ae
and with we reject the disturbance terms,
where 1 20 ; .dK K A
![Page 55: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/55.jpg)
60
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
Time (s)
X3
(ra
d)
e1 Red - e2 Blue - e3 Green
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
Time (s)
X3
(ra
d)
e1 Red - e2 Blue - e3 Green
Figure A) P I D Figure B) I S M
Comparison of controlers
![Page 56: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/56.jpg)
Estimation of Pose and 3D Structure
L R
Estimation of the orientation
and pose M=RT of the
cameras using Kalman filter
techniques
For
Oculomotorcontrol for:
vergence and tracking
Estimation von M_l=R_lT_l and M_r=R_rT_r
PC
PC
Bonus:
--independency of
errors in sensor-Body
calibration.
--3D reconstruction and
relocalization
![Page 57: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/57.jpg)
1. Egomotion and 3D reconstrucion: object searching Applications
Uisng Kalman filter techniques the rotor R and the translator T are
estimated Cameras work in a master slave fashion by fixation
Cyclopean eye
Vieht-Müller circles
and fixarion points
Locus of ZOD
Estimated rotation angles
Of left and right cameras
Hand-eye calibratio is inaccurate due uncertanties. Free coordinates egomotion estimation
helps for a better selflocalization and 3D reconstruction.
Camera was moved by
hand in a 3D trayectory
![Page 58: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/58.jpg)
1. Egomotion and 3D reconstrucion: object recognition and pose estimation Applications
Fixation point to get stereosis for better
correspondences and frature extracction
Focus of Attention
Object recognition
and pose estimation
Grasping !!
![Page 59: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/59.jpg)
Egomotion and 3D reconstrucion Applications
view starcase for climbing
![Page 60: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/60.jpg)
Egomotion and 3D reconstrucion
Applications
Views by changimg the vergence
![Page 61: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/61.jpg)
Quaternion Spike Neurons
66
![Page 62: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/62.jpg)
Quaternion Spike Neural Network
67
![Page 63: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/63.jpg)
Quaternion Spike Neurons
68
Control of a 6 DOF robot manipulator
![Page 64: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/64.jpg)
Neuronavigator
![Page 65: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/65.jpg)
Segmentation
Neural net to adjust a topological map to the object we are interested in object contour
Gradient Vector Flow input selection (for the NN) and in the adaptation process.
Geometric Algebra each neural unit (neuron) has an associated transformation, which is expressed as a motor (rotation-translation) in the GA
All the motors of the net determine the object shape (when they are applied to a specific point)
![Page 66: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/66.jpg)
Gradient Vector Flow (GVF)
Objective: spread the gradient vectors
GVF is a vector field
Minimizes
where is an image borders map
),(),(),( yxvyxuyx v
dydxffvvuu yxyx
222222 )( v
),( yxf
Big value f
0f
![Page 67: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/67.jpg)
Generalized Gradient Vector Flow (GGVF)
Change the coefficient and by the weight functions
Thus
f
efg
)( )(1)( fgfh
dydxffhfg22 )()( vv
2f
![Page 68: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/68.jpg)
Generalized Gradient Vector Flow (GGVF)
![Page 69: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/69.jpg)
Input
im age
C om pute
the vector field
G G V F
C om pute
stream lines
D eterm ine
the inputs
to the net
Training of
the net
G N G
GGVF
Inputs
Set of
versorsA pply versors
to selected
point to define
the object shape
C onform al
G eom etric
A lgebra
Segm ented
im age or 3D
object
Segmentation
![Page 70: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/70.jpg)
Algorithm (2D) to adjust the topologic map
Every certain number of iterations, insert new neurons Find the neuron ni with the highest value of rsf. If any of its
neighbors, say nj, is connected with an edge larger than cmax, then create a new neuron nnew between ni and nj with
Erase old connections between ni and nj , and create new
connections (edges) from nnew to ni and nj
Repeat previous steps
By this way, we are minimizing the quantification error
)))~
((( 2
0 XMPM
2
ji
new
TTT
2
ji
lnew
vvv
![Page 71: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/71.jpg)
Input
image
Compute
GGVF
Compute
streamlines
Determine inputs
to the net
Neural net
Apply Ti and to
extract the
object shape
Output:
Segmented
image
![Page 72: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/72.jpg)
Without GGVF With GGVF
![Page 73: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/73.jpg)
Without GGVF With GGVF
Quantification error:
1.19
Topographic error:
0.142
Quantification
error: 1.10
Topographic
error: 0.153
Quantification
error: 2.04
Topographic
error: 0.133
Quantification
error: 0.98
Topographic
error: 0.05
Quantification
error: 0.23
Topographic
error: 0.0
Quantification
error: 1.002
Topographic
error: 0.0
![Page 74: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/74.jpg)
Segmentation
Input image GGVF Streamlines
Result Inputs to the net and
initialization
![Page 75: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/75.jpg)
Segmentation
Input image GGVF Streamlines
Result Inputs to the net and
initialization
![Page 76: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/76.jpg)
Segmentation
Advantages against GGVF-Snake
Snake
![Page 77: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/77.jpg)
Segmentation
![Page 78: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/78.jpg)
Algorithm 3D to adjust the topologic map
Every certain number of iterations, insert new neurons Find the neuron ni with the highest value of rsf. If any of its neighbors,
say nj, is connected with an edge larger than cmax, then create a new neuron nnew between ni and nj with
Erase old connections between ni and nj , and create new
connections (edges) from nnew to ni and nj
Repeat previous steps
By this way, we are minimizing the quantification error
)))~
((( 2
0 XMPM
![Page 79: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/79.jpg)
Surface modeling using Neural Nets
Result with
100 neurons
Result with
170 neurons
![Page 80: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/80.jpg)
Results with volumetric data
Result with 100 neurons
Result with 200 neurons Result with 300 neurons
Inputs
![Page 81: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/81.jpg)
3D Case Models using circles / spheres
![Page 82: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/82.jpg)
Marching Spheres
Models based on spheres
0
2
0
2
0
2
)(2
1
)(2
1)(
2
1
eeccS
eeccSeeccS
iiii
iiiiiiii
ggg
j
g
mmm
j
mppp
j
p
r
rr
![Page 83: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/83.jpg)
Tumor Model based on spheres
n / d Number of spheres with each
approach
UoS – DT Proposed
3370 / 1 13480 11866
3370 / 3 8642 2602
![Page 84: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/84.jpg)
Motor Interpolation of points, lines, planes, spheres
89
![Page 85: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/85.jpg)
Motor interpolation of the pose
90
![Page 86: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/86.jpg)
Real time interpolation
91
![Page 87: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/87.jpg)
Quaternionic Phase
![Page 88: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/88.jpg)
Quaternion Wavelet Transform
QWT Is a natural extension of the real and complex wavelet transforms
Real
Complex
Quaternionic
Hypercomplex
![Page 89: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/89.jpg)
Quaternion Gabor Filters
Quaternion Gabor Filter
)/()/()2/2/( 221122
221
2
)( ycjxciyx
eeexG
Low-Pass (G)
High-Pass (H)
![Page 90: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/90.jpg)
Quaternion Gabor Wavelets with seletive orientations
15 45 75 –75 –45 –15 degrees
r
I
J
k
![Page 91: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/91.jpg)
Quaternionic Wavelet Pyramid
![Page 92: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/92.jpg)
Motion Detection
Horizontal and Vertical disparity is calculated with the formulas
With reference frequency
![Page 93: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/93.jpg)
Confidence
A Confidence Matrix is calculated for each level as
With
![Page 94: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/94.jpg)
Motion Estimation Algorithm
![Page 95: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/95.jpg)
Motion Estimation Algorithm
QWT
Original Sequence
Ph
ase
Disparity
All the levels of the
pyramid are
combined
![Page 96: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/96.jpg)
Results
![Page 97: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/97.jpg)
Results
Original Images
Confidence in 4 levels
Resulting Vectors
![Page 98: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/98.jpg)
Combination
Disparity levels are combinated
Horizontal Disparity Vertical Disparity
![Page 99: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/99.jpg)
An Experiment
Original Images
Confidence in 4 levels
Resulting Vectors
![Page 100: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/100.jpg)
Comparison
Bernard Algorithm, level 4 Using the QWT, level 4
![Page 101: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/101.jpg)
Atomic function
106
The automic function up(x) is generated by infinite
convolutions of rectangular impulses. The fucntion up(x)
has the following representation in terms of the Fourier
transform.
![Page 102: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/102.jpg)
Quaternion analytic signal
107
![Page 103: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/103.jpg)
Monogenic signal
108
![Page 104: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/104.jpg)
Hilbert and Riesz Transforms using Atomic Function
109
Riesz transform
Hilbert transfrom
![Page 105: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/105.jpg)
Atomic Function Quaternion Hilbert transform
110
![Page 106: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/106.jpg)
Atomic Function Riesz transform
111
Image Local phase orientation
![Page 107: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/107.jpg)
Multi-scale Riesz transform for symmetry feature extraction
112 Atomic filters in spatial and frequency domains at two levels of the multiresolution pyramid.
![Page 108: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/108.jpg)
113
Multi-scale Riesz transform for symmetry feature extraction
![Page 109: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/109.jpg)
Feature extraction
114
![Page 110: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/110.jpg)
Conformal Geometric Algebra Voting Scheme.
In computer vision, perceptual organization consists in obtaining meaningful
representations from row visual data.
We have reformulated the Tensor Voting algorithm [2] using representations
in terms of k-vectors of the Conformal Geometric Algebra, which allow us to
extract any kind of geometric entities or flags from images.
The essential components of this voting scheme can be summarized in two
methodologies:
◦ To represent information using CGA.
◦ To communicate information via a voting process and a clustering
technique.
115 [2] P. Mordohai and G. Medioni, Tensor voting. a perceptual organization approach to computer vision and machine learning.,
Synthesis Lectures on Image, Video, and Multimedia Processing., Morgan & Claypool Publishers, 2006.
![Page 111: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/111.jpg)
Representation of salient circles and lines.
Let p0 be a pixel, then we want to compute the perceptual structure F, that
satisfy:
In order to compute F, all tokens in the neighborhood of p0 cast a vote. Each
vote is codify as a salient geometric entity.
Definition: A salient geometric entity is a set of points together with a
function, that assigns a scalar value to each element of the set.
Using CGA G3,1 a salient line is represented by:
where pc and lc are a point, and a line, respectively, in conformal
representation.
116
![Page 112: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/112.jpg)
The geometric structure of
the vote from pi to p0 is:
The perceptual saliency is
defined by:
where:
Representation of salient circles and lines.
117
![Page 113: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/113.jpg)
Voting process for extraction of circles and lines.
118
![Page 114: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/114.jpg)
119
Voting process for extraction of circles and lines
![Page 115: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/115.jpg)
Experimental results for extraction of circles and lines
120
![Page 116: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/116.jpg)
Experimental results for extraction of circles and lines.
121
![Page 117: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/117.jpg)
And the perceptual saliency of a symmetry axis is:
Where: ◦ pjc is the nearest pixel to the point LpiL.
◦ sigma is a parameter that fix the size of the neighbourhood in which we look for point pjc.
For each pair of points: For each pair of lines:
Representation of information for detection of bilateral symmetry
122
![Page 118: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/118.jpg)
And the perceptual saliency of a symmetry axis is:
Where: ◦ pjc is the nearest pixel to the point LpiL.
◦ sigma is a parameter that fix the size of the neighbourhood in which we look for point pjc.
For each pair of intersection lines: For each pair of parallel lines
Representation of information for detection of bilateral symmetry
123
![Page 119: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/119.jpg)
124
Voting process for extraction of symmetry axis
![Page 120: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/120.jpg)
Experimental results for extraction of symmetry axis
125
![Page 121: Geometric Computing for Cybernetics Eduardo Bayro Corrochanobernstei/Web5/MOIMA2016-End.pdf · 2016-06-28 · Hand-eye calibration The angle and pitch of the gripper (or endoscope)](https://reader034.vdocuments.net/reader034/viewer/2022042201/5ea1c977defd9d4761647ed9/html5/thumbnails/121.jpg)
128
http://www.gdl.cinvestav.mx
Geometric Computing for Cybernetics
Vol I
Geometric
Algebra for
Image
Processing,
Computer
Vision and
Machine
Learning
SpringerVerlag
Vol II
Geometric
Algebra
for Robotics
Springer Verlag
Vol IIiI
Integral
Transforms
for Science and
Engineering
using
Geometric
Algebra
Springer Verlag
(In preparation)