Supplementary Material for Superquadrics Revisited : Learning 3 D Shape Parsing beyond Cuboids

In this supplementary document, we present a detailed derivation of the proposed analytical solution to the Chamfer loss, which avoids the need for computationally expensive reinforcement learning or iterative prediction. Moreover, we also present additional qualitative results on more complex object categories from the ShapeNet dataset [2] such as cars and motorbikes and on the SURREAL human body dataset [6]. Furthermore, we also show results on primitive prediction when using RGB images instead of 3D occupancy grids as input. Finally, we empirically demonstrate that our bi-directional Chamfer loss formulation indeed works better and results in less local minima than the original bi-directional loss formulation of Tulsiani et al. [5]. 1. Superquadrics In this work, we propose superquadrics as a shape primitive representation. Their simple parametrization in combination to their ability to represent a diverse class of shapes makes superquadrics a natural choice for geometric primitives. Moreover, their continuous parametrization is suitable for deep learning as their shape varies continuously with their parameters. Superquadrics are fully modelled using a set of 11 parameters [1]. The explicit superquadric equation defines the surface vector r r(η, ω) = α1 cos 1 η cos 2 ω α2 cos 1 η sin 2 ω α3 sin 1 η  −π/2 ≤ η ≤ π/2 −π ≤ ω ≤ π (1) where α = [α1, α2, α3] determine the size and = [ 1, 2] determine the global shape of the superquadric. Fig. 1 visualizes the shape of superquadrics for different values of 1 and 2. In addition to the shape parameters, we also associate a rigid body transformation with each superquadric. This transformation is represented by a translation vector t = [tx, ty, tz] and a quaternion q = [q0, q1, q2, q3] that determines the coordinate system transformation T (x) = R(λ)x + t(λ) from world coordinates to local primitive-centric coordinates. This transformation as well as the angles η, ω and the scale parameters α1, α2, α3 are illustrated in Fig. 2. 2. Derivation of Pointcloud-to-Primitive Loss This section provides the derivation of the pointcloud-to-primitive distance LX→P (X,P) in Eq. 11 of the main paper. For completeness, we restate our notation briefly. We represent the target point cloud as a set of 3D points X = {xi}i=1 and we approximate the continuous surface of the m primitive by a set of 3D points Ym = {y k }k=1. We further denote Tm(x) = R(λm)x + t(λm) as the mapping from world coordinates to the local coordinate system of the m primitive. The pointcloud-to-primitive distance, LX→P , measures the distance from the point cloud to the primitives to ensure that each observation is explained by at least one primitive. It can be expressed as: LX→P (X,P) = Ep(z) [∑ xi∈X min m|zm=1 ∆i ] (2) Figure 1: Superquadric Shape Space. Superquadrics are a parametric family of surfaces that can be used to describe cubes, cylinders, spheres, octahedral ellipsoids, etc. [1]. This figure visualizes superquadrics when varying the shape parameters 1 and 2, while keeping the size parameters α1, α2 and α3 constant. where ∆i denotes the minimal distance from point xi to the surface of the m’th primitive: ∆i = min k=1,. ,K ‖Tm(xi)− y k ‖2 (3) Assuming independence of the existence variables p(z) = ∏ m p(zm), we can replace the expectations in (2) with summations as follows: LX→P (X,P) = ∑