1. 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

2. Objective
Difficulty

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

4. Consistency of Interpretation2
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

5. Image Regularities
Line parallelism
Line verticality

6. 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

7. 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]

8. 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

9. Results