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superquadrics towards learning primitive based representations humans develop a common sense understanding of the physical behaviour of the world within the first year of their life for example we are able to identify 3d objects in a scene and infer their geometric and physical properties even when only parts of these objects are visible it has long been hypothesized that the human visual system processes the vast amount of raw visual input into compact parsimonious representations where complex objects are decomposed into a small number of atomic elements primitives that can each be represented using low dimensional descriptions in the early days of computer vision researches explored various shape primitives that could potentially mimic the human s perception such as 3d polyhedrals generalized cylinders geons and superquadrics however it proved very difficult to extract such representations from images due to the lack of computational resources and training data at that time recently shape primitives have been revisited in the context of deep learning primitive based representations provide an interpretable alternative towards traditional shape extraction methods that do not take into consideration the constituent parts of the target object in cvpr 2019 pdf code bib in cvpr 2020 pdf code bib inproceedings paschalidou2019cvpr title superquadrics revisited learning 3d shape parsing beyond cuboids author paschalidou despoina and ulusoy ali osman and geiger andreas booktitle proceedings ieee conf on computer vision and pattern recognition cvpr month jun year 2019 inproceedings paschalidou2020cvpr title learning unsupervised hierarhical part decomposition of 3d objects from a single rgb image author paschalidou despoina and luc van gool and geiger andreas booktitle proceedings ieee conf on computer vision and pattern recognition cvpr month jun year 2020 why superquadrics inspired by the nature of the human s cognitive system that perceives an object as a decomposition of parts researchers have proposed to represent objects as a set of atomic elements which we refer to as primitives examples for such primitives include 3d polyhedral shapes generalized cylinders and geons for decomposing 3d objects into a set of parts in 1986 pentland introduced a parametric version of generalized cylinders based on deformable superquadrics he proposed a system able to represent the scene structure using multiple superquadrics superquadrics are a parametric family of surfaces that can describe cubes cylinders spheres ellipsoids etc in a single continuous parameter space they are fully described using 11 parameters 6 for the pose 3 for the size and 2 for the shape one of the most important features of superquadric surfaces is their interchangeable implicit and explicit function definition the explicit superquadric function defines the surface vector mathbf r that can be used for sampling points on the superquadric surface begin equation mathbf r eta omega begin bmatrix alpha_ 1 cos epsilon_ 1 eta cos epsilon_ 2 omega alpha_ 2 cos epsilon_ 1 eta sin epsilon_ 2 omega alpha_ 3 sin epsilon_ 1 eta end bmatrix quad begin aligned pi 2 leq eta leq pi 2 pi leq omega leq pi end aligned label eq parametric_eq_supp end equation where mathbf alpha alpha_ 1 alpha_ 2 alpha_ 3 determine the size and mathbf epsilon epsilon_ 1 epsilon_ 2 determine the global shape of the superquadric below we visualize the shape of superquadrics for different values of epsilon_ 1 and epsilon_ 2 in addition to the shape parameters each superquadric is associated with a rigid body transformation this transformation is represented by a translation vector mathbf t t_ x t_ y t_ z and a quaternion mathbf q q_ 0 q_ 1 q_ 2 q_ 3 that determines the coordinate system transformation from world coordinates to local primitive centric coordinates superquadrics shape vocabulary their ability to model various shapes with few parameters makes them a natural choice for geometric primitives the implicit function can be used to decide the relative position of any 3d point w r t the superquadric surface in particular for any point mathbf x in mathbb r 3 we can determine whether it lies inside or outside a superquadric using its implicit surface function which is commonly referred to as the inside outside function begin equation f mathbf x lambda left left frac x alpha_ 1 right frac 2 epsilon_ 2 left frac y alpha_ 2 right frac 2 epsilon_ 2 right frac epsilon_ 2 epsilon_ 1 left frac z alpha_ 3 right frac 2 epsilon_ 1 label eq implicit_sq end equation if f mathbf x lambda 1 0 mathbf x lies on the surface of the superquadric if f mathbf x lambda 1 0 the point lies outside the superquadric we provide an easy to use script that can be used for visualizing the superquadric surface given a set of parameters code is available here references peter elias and lawrence g roberts machine perception of three dimensional solids phd thesis massachusetts institute of technology 1963 i binford visual perception by computer in ieee conference of systems and control 1971 alan h barr superquadrics and angle preserving transformations ieee computer graphics and applications cga 1981 irving biederman recognition by components a theory of human image understanding psychological review 94 2 115 1987 ales jaklic ales leonardis and franc solina segmentation and recovery of superquadrics volume 20 of computational imaging and vision springer 2000