Diapositiva 1

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Robotica e Sensor Fusion per i Sistemi
Meccatronici Object Detection with Superquadrics
Prof. Mariolino De Cecco, Dr. Ilya Afanasyev,
Ing. Nicolo Biasi Department of Structural
Mechanical Engineering, University of Trento
Email mariolino.dececco_at_ing.unitn.it ilya.afanas
yev_at_ing.unitn.it http//www.mariolinodececco.alter
vista.org/
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Examples of Superquadrics
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Examples of Superquadrics
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Wireframes of Superquadrics
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Examples of Superquadrics
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Figures from Superquadrics
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Definition of Superquadrics
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About Superquadrics
The term of Superquadrics was defined by Alan
Barr in 1981 2. Superquadrics are a flexible
family of 3D parametric objects, useful for
geometric modeling. By adjusting a relatively few
number of parameters, a large variety of shapes
may be obtained. A particularly attractive
feature of superquadrics is their simple
mathematical representation.
Superquadrics are used as primitives for shape
representation and play the role of prototypical
parts and can be further deformed and glued
together into realistic looking models.
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About Superquadrics
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Classification of Superquadrics
a)
b)
c)
a) Superellipsoids. b) Superhyperboloids of one
piece. c) Superhyperboloids of two pieces. d)
Supertoroids.
d)
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Classification of Superquadrics 2,9
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Definition of Spherical Products
For two 2x1 vectors a b and c d the
spherical product is 3x1 vector for which
Example. For two 2D curves circle and parabola,
the spherical product is 3D paraboloid.
Definition of Spherical Products
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Spherical Products
A 3D surface can be obtained by the spherical
product of two 2D curves 2. The spherical
product is defined to operate on two 2D
curves.
A unit sphere is produced by a spherical product
of a circle h(?) horizontally and a half circle
m(?) vertically.
Spherical Products 9
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Spherical Products for Superellipsoids
The equation of Superellipse is
and in parametric form
Superellipsoids can be obtained by a spherical
products of a pair of such superellipses
The implicit equation is
where
- are parameters of shape squareness
- parameters of Superellipsoid sizes.
Superellipsoids
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Creation of Superellipsoids in spherical
coordinates

• Vector r(?,?) sweeps out a closed surface in
  space when ?,? change in the given intervals

?,? independent parameters (latitude and
longitude angles) of vector r(?,?) expressed
in spherical coordinates.
Superellipsoids
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Creation of Superellipsoids in Cartesian
coordinates
Use the implicit equation in Cartesian
coordinates, considering
-a1 x a1
f(x,y,z) 1
-a2 y a2
z
y
x,y independent parameters (Cartesian
coordinates of SQ) are used to obtain z.
The implicit form is important for the recovery
of Superquadrics and testing for intersections,
while the explicit form is more suitable for
scene reconstruction and rendering.
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Warning complex numbers in SQ equation

• 1. If e1 or e2 lt 1 and cos or sin of angles ? or
  ? lt 0, then vector r(?,?) has complex values. To
  escape them, it should be used signum-function of
  sin or cos and absolute values of the vector
  components.

2. Analogically if x or y lt 1 and e1 gt 1, the
function f(x,y,z) willl have the complex values
of z. To overcome it, use the f(x,y,z) in power
of exponent e1.
f(x,y,z)e1 1
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Examples of Superellipsoids
Superellipsoids can model spheres, cylinders,
parallelepipeds and shapes in between. Modeling
capabilities can be enhanced by tapering, bending
and making cavities.
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Examples of Superellipsoids
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What e1 and e2 mean?
e1 and e2 are parameters of shape squareness.
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Superellipsoids shapes varying from e1, e2
e1 3
e2 1
e1 1
e2 3
e1 3
e2 3
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What a1, a2 and a3 mean?
a1, a2 and a3 are parameters of
parallelepipeds semi-sides.
If parallelepiped has dimensions 20 x 30 x 10
cm, it means that a1 10, a2 15 and a3 5.
The parameters of shape squareness for
parallelepiped are e1 e1 0.1
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Spherical Products for Superhyperboloids of one
piece
Superhyperboloids of one piece can be obtained by
a spherical products of a hyperboloid and a
superellipse
The implicit equation is
where
- are parameters of shape squareness
- parameters of Superellipsoid sizes.
Superhyperboloids of one piece
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Spherical Products for Superhyperboloids of two
pieces
Superhyperboloids of two pieces can be obtained
by a spherical products of a pair of such
hyperboloids
The implicit equation is
where
- are parameters of shape squareness
- parameters of Superellipsoid sizes.
Superhyperboloids of two pieces
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Spherical Products for Supertoroids
Supertoroids can be obtained by a spherical
products of the following surface vectors
The implicit equation is
where
- are parameters of shape squareness
- parameters of Superellipsoid sizes.
Supertoroids
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Classification of Superquadrics
a) Superellipsoids.
b) Superhyperboloids of one piece.
a)
b)
c) Superhyperboloids of two piece.
d) Supertoroids.
c)
d)
Classification of Superquadrics
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Rotation and translation of SQ
T transformation matrix. n amounts of points
in SQ surface. SQ coordinates of points of SQ
surface. Pw coordinates of points of rotated
SQ surface. xW, yW, zW world system of
coordinates (with center in viewpoint).
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Rotation and translation of SQ
px
az,el,px,py,pz,x,y,z are given xW,yW,zW
should be found.
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Applications with Superquadrics
Superquadrics have been employed in computer
vision and robotics problems related to object
recognition. Superquadrics can be used for
1. Object recognition by fitting geometric shapes
to 3D sensor data obtained by a robot.
In order to fit a superquadric to a surface
region, 11 parameters must be determined three
extent parameters (a1, a2, a3), two shape
parameters (e1 and e2), three translation
parameters, and three rotation parameters.
2. Scene reconstruction and recognition.
Rendering.
Shape reconstruction is a low level process where
sensor data is interpreted to regenerate objects
in a scene making as few assumptions as possible
about the objects. Object recognition is a higher
level process whose goal is to abstract from the
detailed data in order to characterize objects in
a scene.
Applications with Superquadrics
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Applications with SQ
Reconstruction of complex object 6
Reconstruction of complex object 8
Reconstruction of complex object 7
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Applications with SQ
Reconstruction of multiple objects 9
Reconstruction of multiple objects 9
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Links

1. A. Skowronski, J. Feldman. Superquadrics. cs557
   Project, McGill University. http//www.skowronski.
   ca/andrew/school/557/start.html
2. Barr A.H. Superquadrics and Angle-Preserving
   Transformations. IEEE Computer Graphics and
   Applications, 1, 11-22. 1981.
3. Chevalier L., etc. Segmentation and superquadric
   modeling of 3D objects. Journal of WSCG, V.11
   (1), 2003. ISSN 1213-6972.
4. Kindlmann G. Superquadric Tensor Glyphs.
   EUROGRAPHICS. IEEE TCVG Symposium on
   Visualization (2004). Pages 8.
5. Solina F. and Bajcsy R. Recovery of parametric
   models from range images The case for
   superquadrics with global deformations. // IEEE
   Transactions on Pattern Analysis and Machine
   Intelligence PAMI-12(2)131--147, 1990.
6. Chella A. and Pirrone R. A Neural Architecture
   for Segmentation and Modeling of Range Data. //
   10 pages.
7. Leonardis A., Jaklic A., and Solina F.
   Superquadrics for Segmenting and Modeling Range
   Data. // IEEE Transactions On Pattern Analysis
   And Machine Intelligence, vol. 19, no. 11, 1997.
8. Bhabhrawala T., Krovi V., Mendel F. and
   Govindaraju V. Extended Superquadrics. //
   Technical Report. New York, 2007. 93 pages.
9. Jaklic Ales, Leonardis Ales, Solina Franc.
   Segmentation and Recovery of Superquadrics. //
   Computational imaging and vision 20, Kluwer,
   Dordrecht, 2000.

Grazie per attenzione!!