OptimizationBased Reconstruction of a 3D Object From a Single Freehand Line Drawing

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Optimization-Based Reconstruction of a 3D Object
From aSingle Freehand Line Drawing

• H. Lipson and M. Shpitalni
• CAD, 1996

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Objective
Difficulty
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Inflated into 3D by Optimization
Given 2D edge-vertex graph
Setting up potential energies so that the
equilibrium Z values satisfy the best
configuration
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Consistency of Interpretation
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
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Image Regularities
Line parallelism
Line verticality
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Other Image Regularities
How to model this?

• 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

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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) WTa

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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

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Results