  We consider a category-level perception problem, where one is given 3D sensor
data picturing an object of a given category (e.g. a car), and has to
reconstruct the pose and shape of the object despite intra-class variability
(i.e. different car models have different shapes). We consider an active shape
model, where -- for an object category -- we are given a library of potential
CAD models describing objects in that category, and we adopt a standard
formulation where pose and shape estimation are formulated as a non-convex
optimization. Our first contribution is to provide the first certifiably
optimal solver for pose and shape estimation. In particular, we show that
rotation estimation can be decoupled from the estimation of the object
translation and shape, and we demonstrate that (i) the optimal object rotation
can be computed via a tight (small-size) semidefinite relaxation, and (ii) the
translation and shape parameters can be computed in closed-form given the
rotation. Our second contribution is to add an outlier rejection layer to our
solver, hence making it robust to a large number of misdetections. Towards this
goal, we wrap our optimal solver in a robust estimation scheme based on
graduated non-convexity. To further enhance robustness to outliers, we also
develop the first graph-theoretic formulation to prune outliers in
category-level perception, which removes outliers via convex hull and maximum
clique computations; the resulting approach is robust to 70%-90% outliers. Our
third contribution is an extensive experimental evaluation. Besides providing
an ablation study on a simulated dataset and on the PASCAL3D+ dataset, we
combine our solver with a deep-learned keypoint detector, and show that the
resulting approach improves over the state of the art in vehicle pose
estimation in the ApolloScape datasets.
