Robotica e Sensor Fusion per i Sistemi Meccatronici

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@ing. unitn. it ilya. afanasyev@ing. unitn. it http: //www. mariolinodececco. altervista. org/ M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion

Object detection and localization The block diagram of recognition task: A-priori know: - Dimensions; - Features (shape, curvature, etc. ) - Superquadrics, etc. - Optimization, iterations, etc. ; - RANSAC; - Criteria of quality. Object Math. model Measurement system Data Preprocessing - Camera, laser - Kinect - Multicamera system - Point clouds - Images, video - Eliminating the background; - Changing the contrast, size, etc. Fitting Algorithm Results - Transformation matrix, angles, distance, accuracy, etc. M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion 02/04/2 2/2 1

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

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

Examples of Superellipsoids can model spheres, cylinders, parallelepipeds and shapes in between. Modeling capabilities can be enhanced by tapering, bending and making cavities. M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Examples of Superellipsoids [4]

About Superquadrics The term of Superquadrics was defined by Alan Barr in 1981 [2]. Superquadrics are a flexible family of 3 D 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] M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion About Superquadrics

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

Definition of Spherical Products For two 2 x 1 vectors [a b]’ and [c d]’ the spherical product is 3 x 1 vector for which: Example. For two 2 D curves: circle and parabola, the spherical product is 3 D paraboloid. M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Definition of Spherical Products

Spherical Products A 3 D surface can be obtained by the spherical product of two 2 D curves [2]. The spherical product is defined to operate on two 2 D curves. A unit sphere is produced by a spherical product of a circle h(ω) horizontally and a half circle m(η) vertically. M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion 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. M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Superellipsoids

Spherical Products for 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. M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Superhyperboloids of one piece

Spherical Products for 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. M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Superhyperboloids of two pieces

Spherical Products for 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. M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Supertoroids

Superquadrics in MATLAB Modelling Superquadrics in MATLAB: - Demo-function: xpquad - superquadrics plotting demonstration. - ezmesh and ezsurf functions (see, Superquadrics_visualization. m). %-------------------------------% a 1 = 4; a 2 = 2; a 3 = 1; eps 1 = 1; eps 2 = 1; figure('Name', 'Superellipsoids') ezmesh('4*sin(u)*cos(v)', '2*sin(u)*sin(v)', 'cos(u)', [0, 2*pi], 20) colormap([0 0 1]) hold on rotate 3 d on %-------------------------------- M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion 02/04/2 15/ 21

What a 1, a 2 and a 3 mean? a 1, a 2 and a 3 – are parameters of parallelepiped’s semi-sides. If parallelepiped has dimensions: 20 x 30 x 10 cm, it means that a 1 = 10, a 2 = 15 and a 3 = 5. The parameters of shape squareness for parallelepiped are: ε 1 = 0. 1 M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion 05/04/2011 16/20

What ε 1 and ε 2 mean? ε 1 and ε 2 – are parameters of shape squareness. ε 2 = 0. 1 ε 2 = 2 [9] ε 1 = 0. 1 ε 1 = 2 M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion 05/04/2011 17/20

Superellipsoids shapes varying from ε 1, ε 2 ε 1 = 3 ε 2 = 1 ε 1 = 1 ε 2 = 3 ε 1 = 3 ε 2 = 3 M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion 05/04/2011 18/20

Plotting Superellipsoids in spherical coordinates z Vector r(η, ω) sweeps out a closed surface in space when η, ω change in the given intervals: a 3 r(η, ω) η a 2 y a 1 ω x η, ω – independent parameters (latitude and longitude angles) of vector r(η, ω) expressed in spherical coordinates. M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Superellipsoids

Plotting Superellipsoids in Cartesian coordinates z Use the implicit equation in Cartesian coordinates, considering -a 1 ≤ x ≤ a 1 f(x, y, z) = 1 -a 2 ≤ y ≤ a 2 z’ y=a 2 x’ y y’ x=a 1 z=Na. N x, y – independent parameters (Cartesian coordinates of SQ) are used to obtain z. x 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. M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion 05/04/2011 20/20

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 and ε 1 > 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)ε 1 = 1 M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion 05/04/2011 21/2

Rotation and translation of SQ z z. W x. W y. W Elevation y Azimuth x 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. x. W, y. W, z. W – world system of coordinates (with center in viewpoint). M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion 05/04/2011 22/20

Rotation and translation of SQ z z. W x. W px y. W Elevation y Azimuth T SQ x az, el, px, py, pz, x, y, z – are given; x. W, y. W, z. W – should be found. M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion 05/04/2011 23/20 Pw

Applications with 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 3 D sensor data obtained by a robot. 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. 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. M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Applications with Superquadrics

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

Applications with SQ Reconstruction of complex object [6] Reconstruction of complex object [7] M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion 05/04/2011 Reconstruction of complex object [8] 26/2

Applications with SQ Reconstruction of multiple objects [9] M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion 05/04/2011 27/2

Task 1: To model Europallet in SQ Dimensions of Euro. Pallet M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion 02/04/2 28/ 21

As an example of a Pallet model à It should be created SQ model of pallet Transformation: SQc – SQr Transformation: SQc – SQl M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion 02/04/2 29/ 21

Task 2: To model objects by deformed SQ Objects like an apple, a bottle, a chair, a tin of beer M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion 02/04/2 30/ 21

Links 1. 2. 3. 4. 5. 6. 7. 8. 9. A. Skowronski, J. Feldman. Superquadrics. cs 557 Project, Mc. Gill University. http: //www. skowronski. ca/andrew/school/557/start. html Barr A. H. Superquadrics and Angle-Preserving Transformations. IEEE Computer Graphics and Applications, 1, 11 -22. 1981. Chevalier L. , etc. Segmentation and superquadric modeling of 3 D objects. Journal of WSCG, V. 11 (1), 2003. ISSN 1213 -6972. Kindlmann G. Superquadric Tensor Glyphs. EUROGRAPHICS. IEEE TCVG Symposium on Visualization (2004). Pages 8. 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. Chella A. and Pirrone R. A Neural Architecture for Segmentation and Modeling of Range Data. // 10 pages. 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. Bhabhrawala T. , Krovi V. , Mendel F. and Govindaraju V. Extended Superquadrics. // Technical Report. New York, 2007. 93 pages. Jaklic Ales, Leonardis Ales, Solina Franc. Segmentation and Recovery of Superquadrics. // Computational imaging and vision 20, Kluwer, Dordrecht, 2000. M. De Cecco - Lucidi del corso di Robotica e Sensor Fusion Grazie per attenzione!!