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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion 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

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Examples of Superquadrics [1]

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Examples of Superquadrics Wireframes of Superquadrics [1]

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Definition of Superquadrics Figures from Superquadrics [3]

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion 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. [9]

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Classification of Superquadrics [2,9] Classification of Superquadrics a) Superellipsoids. b) Superhyperboloids of one piece. c) Superhyperboloids of two pieces. d) Supertoroids. a) b) c) d) [9]

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Definition of Spherical Products Example. For two 2D curves: circle and parabola, the spherical product is 3D paraboloid. For two 2x1 vectors [a b] and [c d] the spherical product is 3x1 vector for which:

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Spherical Products [9] 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.

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Superellipsoids The equation of Superellipse is Spherical Products for Superellipsoids and in parametric form: Superellipsoids can be obtained by a spherical products of a pair of such superellipses: The implicit equation is: - are parameters of shape squareness; - parameters of Superellipsoid sizes. where

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Vector r(η,ω) sweeps out a closed surface in space when η,ω change in the given intervals: η ω x y z r(η,ω) Creation of Superellipsoids in spherical coordinates η,ω – independent parameters (latitude and longitude angles) of vector r(η,ω) expressed in spherical coordinates. a1a1 a2a2 a3a3 Superellipsoids

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion x,y – independent parameters (Cartesian coordinates of SQ) are used to obtain z. Use the implicit equation in Cartesian coordinates, considering f(x,y,z) = 1 -a 1 x a 1 -a 2 y a 2 05/04/201111/20 x x y z y z z=NaN y=a 2 x=a 1 Creation of Superellipsoids in Cartesian coordinates 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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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Warning: complex numbers in SQ equation 1. If ε 1 or ε 2 < 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, 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 ε 1. f(x,y,z) ε1 = 1 12/2 0 05/04/2011

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Examples of Superellipsoids [4] 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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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion What ε 1 and ε 2 mean? ε 1 = 0.1 ε 1 = 1ε 1 = 2 ε 2 = 0.1 ε 2 = 1 ε 2 = 2 [9] 05/04/201114/20 ε 1 and ε 2 – are parameters of shape squareness.

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Superellipsoids shapes varying from ε 1, ε 2 05/04/201115/20 ε 1 = 3 ε 2 = 1 ε 1 = 1 ε 2 = 3 ε 1 = 3 ε 2 = 3

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion What a 1, a 2 and a 3 mean? 05/04/201116/20 a 1, a 2 and a 3 – are parameters of parallelepipeds semi-sides. The parameters of shape squareness for parallelepiped are: ε 1 = ε 1 = 0.1 If parallelepiped has dimensions: 20 x 30 x 10 cm, it means that a 1 = 10, a 2 = 15 and a 3 = 5.

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Superhyperboloids of one piece 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: - are parameters of shape squareness; - parameters of Superellipsoid sizes. where

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Superhyperboloids of two pieces 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: - are parameters of shape squareness; - parameters of Superellipsoid sizes. where

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Supertoroids Spherical Products for Supertoroids Supertoroids can be obtained by a spherical products of the following surface vectors: The implicit equation is: - are parameters of shape squareness; - parameters of Superellipsoid sizes. where

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Classification of Superquadrics a) Superellipsoids. a)b) c)d) b) Superhyperboloids of one piece. c) Superhyperboloids of two piece. d) Supertoroids.

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Rotation and translation of SQ Elevation Azimuth x y z zWzW xWxW yWyW T – transformation matrix. n – amounts of points in SQ surface. SQ – coordinates of points of SQ surface. P w – coordinates of points of rotated SQ surface. x W, y W, z W – world system of coordinates (with center in viewpoint). 05/04/201121/20

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Elevation Azimuth x y z zWzW xWxW yWyW Rotation and translation of SQ pxpx az,el,px,py,pz,x,y,z – are given; x W,y W,z W – should be found. TSQPwPw 05/04/201122/20

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion 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. 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. 2. Scene reconstruction and recognition. Rendering. In order to fit a superquadric to a surface region, 11 parameters must be determined: three extent parameters (a 1, a 2, a 3 ), two shape parameters (ε 1 and ε 2 ), three translation parameters, and three rotation parameters. Applications with Superquadrics

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Reconstruction of complex object [6] Applications with SQ Reconstruction of complex object [8] Reconstruction of complex object [7] 05/04/201124/2 0

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Reconstruction of multiple objects [9] Applications with SQ Reconstruction of multiple objects [9] 05/04/201125/2 0

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M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Links 1.A. Skowronski, J. Feldman. Superquadrics. cs557 Project, McGill University. 2.Barr A.H. Superquadrics and Angle-Preserving Transformations. IEEE Computer Graphics and Applications, 1, Chevalier L., etc. Segmentation and superquadric modeling of 3D objects. Journal of WSCG, V.11 (1), ISSN 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): , 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, Bhabhrawala T., Krovi V., Mendel F. and Govindaraju V. Extended Superquadrics. // Technical Report. New York, pages. 9.Jaklic Ales, Leonardis Ales, Solina Franc. Segmentation and Recovery of Superquadrics. // Computational imaging and vision 20, Kluwer, Dordrecht, Grazie per attenzione!!

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