Introduction • Some object classes are hard to reconstruct – Lack of texture – Transparency – Reflection • Solution: shape prior – Shapes within object class similar – Local distribution of surface normals

Formulation • Baseline Method: Volumetric depth map fusion – Segmentation of a voxel space into free and occupied space: us ∈ [0, 1] • Shape prior formulation – Voxel space aligned with object of known class ix s∈ [0, 1] andi i x sP=1Labeling of a voxel space into 3 labels: free space, ground, object • Convex Energy – Unary term ∗ Computed from depth maps, local preference for solid class – Smoothness term ∗ Dependent on surface orientation, position and involved labels Overview • Locally, surface normals similar between different examples – Roof at the top of the car close to horizontal

• Local distribution of normals captured from training data • Input data regularized using trained local normal distributions • Trained anisotropic smoothness used for – free space ↔ object – ground ↔ object • ground ↔ free space generic smoothness • Label determined by smoothness Convex Energy ∈ •iρ s :≥ joint unary term at voxel s for label i •ijφ s :convex smoothness term at voxel s for labels i and j •ix s∈ [0, 1]: indicating whether label i is chosen at voxel s •ijx s−jix s3∈ [−1, 1] : represents the local surface orientation 3• ek ∈ R : k-th canonical basis vector • Optimized using primal-dual algorithm [Chambolle and Pock 2011] Unary Term • Only indicates free or occupied space

Shape Prior Training

• Training data, mesh models • Transformed into volumetric models • Per voxel s – Acquire normal directions of all training samples – Generate histogram over normal directions – Probability of normal n at s, Ps (n) given by histogram Discrete Wulff Shape • φs (·) support function of a Wulff shape Wφs [Esedoglu and Osher 2004] – Wulff shape: convex shape • Intersection of half spaces as parameterization of Wφs – n half space normal –nd sdistance of half-space boundary to origin • We have φs (n) =nd s [Esedoglu and Osher 2004] •nd s= − log (Ps (n)), determined by training data

Trained Shape Prior

Slices through the bottle shape prior: vertical, horizontal Results

Input image Depth map Vol. fusion Shape Prior Acknowledgements We gratefully acknowledge the support of the 4DVideo ERC starting grant #210806 and V-Charge grant #269916 both under the EC’s FP7/2007-2013.