Layer Extraction with a Bayesian Model of Shapes

Layer Extraction with a Bayesian Model of Shapes 1 2 2 Philip H. S. Torr Anthony Dick Roberto Cipolla 1 - Microsoft Research, Cambridge 2 - Department of Engineering, University of Cambridge, UK Likelihood Evaluation The Problem • Project model into each image • Accurate recovery of depth for all points in an image • Assume image intensities subject to Gaussian noise • Resolve ambiguities using prior knowledge of scene Individual Shapes Model AIC BIC Occam • Individual shape priors are uniform over a closed region Rectangle 2. 5829 2. 5850 2. 5870 • Hyperpriors express belief about relation of shapes Arch 2. 5632 2. 5657 2. 5678 • Eg Hyperprior expressing that shapes may occur in rows: Bev. Rect. 2. 5821 2. 5842 2. 5872 Bev. Arch 2. 5634 2. 5662 2. 5691 Model AIC BIC Occam Rectangle 8. 2213 8. 2536 8. 2704 Arch 8. 2183 8. 2453 8. 2789 Bev. Rect. 8. 1153 8. 1524 8. 1750 Bev. Arch 8. 1582 8. 1785 Formulation of Priors • Obtain dense 3 D surface and texture from few images Samples from a hyperprior for rows of shapes Evidence evaluation likelihood Occam factor is the maximum likelihood parameter vector for model Mi Untextured & textured views of 3 D model 8. 1155 AIC, BIC and Occam Factor evidence for each shape model, plus best fit shape Multiple Shapes is the covariance matrix at • Occam factor approximates volume of prior probability space Shape models occupied by maximum likelihood peak • Select one of a family of shapes: M 1: Rectangle: c = 0, r = 0 M 2 : Arch: r = 0 M 3: Bev. Rect: c =0 M 4: Bev Arch: no constraints c b Results Rows of layers are automatically detected (x, y) Occam factors for one and two variables d (x, y) r a Frontal View Comparison with other evidence metrics: a • AIC: • BIC: Overhead View • Occam factor: Bayesian Model Selection • Bayes’ Rule, expressing probability of a model: Multiple Planes model prior Initialisation From parallax-based correspondence algorithm constant • Model selection depends mainly on the evidence: Automatic layer extraction Conclusion data term (likelihood) Where • Use of parameterised shapes simplifies structure recovery prior are the parameters of model i • Bayesian framework provides principled way of utilising Initial reconstruction Initial Layers prior shape models