H. Lipson and M. Shpitalni CAD, 1996 Optimization-Based Reconstruction of a 3D Object From a Single Freehand Line Drawing H. Lipson and M. Shpitalni CAD, 1996

Objective Difficulty

Inflated into 3D by Optimization Given: 2D edge-vertex graph Setting up “potential energies” so that the equilibrium Z values satisfy the best configuration

Consistency of Interpretation 2 x: the value to be checked a: nominal value b: reasonable deviation : 7° of angular difference as a threshold for determining parallelism Only things that are pretty close need to be considered the same

Image Regularities Line parallelism Line verticality

Other Image Regularities How to model this? Other Image Regularities Isometry Segment length ratios about the same Skew face orthogonality Skew face symmetry Face perpendicularity Adjacent faces must be perpendicular Line orthogonality Line pairs in a junction (except the collinear ones) are perpendicular in 3D

Optimization-based Reconstruction Input: 2D edge-vertex graph Variable: vector Z containing the z coordinates of the vertices Compliance function F(Z): summing up the contributions of the regularity terms with a global balancing vector W F(Z) = WT[a]

Optimization Algorithms Tried Brent minimization 1D algorithm, cyclic application (per vertex) Conjugate gradient Genetic algorithm Metaheuristic method to avoid local minima Brent seems to work best

Results