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This article presents a mathematical framework to simultaneously tackle theproblems of 3D reconstruction, pose estimation and object classification, froma single 2D image. In sharp contrast with state of the art methods that relyprimarily on 2D information and solve each of these three problems separatelyor iteratively, we propose a mathematical framework that incorporates prior"knowledge" about the 3D shapes of different object classes and solves theseproblems jointly and simultaneously, using a hypothesize-and-bound (H&B)algorithm. In the proposed H&B algorithm one hypothesis is defined for eachpossible pair [object class, object pose], and the algorithm selects thehypothesis H that maximizes a function L(H) encoding how well each hypothesis"explains" the input image. To find this maximum efficiently, the function L(H)is not evaluated exactly for each hypothesis H, but rather upper and lowerbounds for it are computed at a much lower cost. In order to obtain bounds forL(H) that are tight yet inexpensive to compute, we extend the theory of shapesdescribed in [14] to handle projections of shapes. This extension allows us todefine a probabilistic relationship between the prior knowledge given in 3D andthe 2D input image. This relationship is derived from first principles and isproven to be the only relationship having the properties that we intuitivelyexpect from a "projection." In addition to the efficiency and optimalitycharacteristics of H&B algorithms, the proposed framework has the desirableproperty of integrating information in the 2D image with information in the 3Dprior to estimate the optimal reconstruction. While this article focusesprimarily on the problem mentioned above, we believe that the theory presentedherein has multiple other potential applications.

Hypothesize and Bound: A Computational Focus of Attention Mechanism for  Simultaneous 3D Shape Reconstruction, Pose Estimation and Classification from  a Single 2D Image