Robotica e Sensor Fusion per i Sistemi Meccatronici

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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

About Superquadrics The term of Superquadrics was defined by Alan Barr in [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. [9] About Superquadrics

b) c) a) Superellipsoids. b) Superhyperboloids of one piece. c) Superhyperboloids of two pieces. d) Supertoroids. d) [9] Classification of Superquadrics [2,9]

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

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(ω) horizontally and a half circle m(η) vertically. Spherical Products [9]

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

Creation of Superellipsoids in spherical coordinates
x y z r(η,ω) Vector r(η,ω) sweeps out a closed surface in space when η,ω change in the given intervals: a1 a2 a3 η ω η,ω – independent parameters (latitude and longitude angles) of vector r(η,ω) expressed in spherical coordinates. Superellipsoids

Creation of Superellipsoids in Cartesian coordinates
x y z Use the implicit equation in Cartesian coordinates, considering x’ -a1 ≤ x ≤ a1 f(x,y,z) = 1 -a2 ≤ y ≤ a2 z’ z=NaN y=a2 x=a1 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. 05/04/2011

Warning: complex numbers in SQ equation
1. If ε1 or ε2 < 1 and cos or sin of angles ω or η < 0, then vector r(η,ω) 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 < 1 and ε1 > 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 ε1. f(x,y,z)ε1 = 1 05/04/2011

Examples of Superellipsoids
Superellipsoids can model spheres, cylinders, parallelepipeds and shapes in between. Modeling capabilities can be enhanced by tapering, bending and making cavities. [4] Examples of Superellipsoids

What ε1 and ε2 mean? ε1 and ε2 – are parameters of shape squareness.
ε1 = 0.1 ε1 = 1 ε1 = 2 ε2 = 0.1 ε2 = 1 ε2 = 2 [9] 05/04/2011 14/20

Superellipsoids shapes varying from ε1, ε2
ε1 = 3 ε2 = 1 ε1 = 1 ε2 = 3 ε1 = 3 ε2 = 3 05/04/2011 15/20

What a1, a2 and a3 mean? a1, a2 and a3 – are parameters of parallelepiped’s 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: ε1 = ε1 = 0.1 05/04/2011 16/20

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

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

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

a) Superellipsoids. b) Superhyperboloids of one piece. a) b) c) Superhyperboloids of two piece. d) Supertoroids. c) d) Classification of Superquadrics

Rotation and translation of SQ
Elevation Azimuth x y z zW xW yW 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). 05/04/2011 21/20

Rotation and translation of SQ
Elevation Azimuth x y z zW xW yW px T SQ Pw az,el,px,py,pz,x,y,z – are given; xW,yW,zW – should be found. 05/04/2011 22/20

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 (ε1 and ε2), 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

Applications with SQ 05/04/2011 24/20
Reconstruction of complex object [6] Reconstruction of complex object [8] Reconstruction of complex object [7] 05/04/2011 24/20

Applications with SQ 05/04/2011 25/20
Reconstruction of multiple objects [9] Reconstruction of multiple objects [9] 05/04/2011 25/20