Globally Optimal Algorithms for Stratified Autocalibration

Abstract

We present practical algorithms for stratified autocalibration with theoretical guarantees of global optimality. Given a projective reconstruction, we first upgrade it to affine by estimating the position of the plane at infinity. The plane at infinity is computed by globally minimizing a least squares formulation of the modulus constraints. In the second stage, this affine reconstruction is upgraded to a metric one by globally minimizing the infinite homography relation to compute the dual image of the absolute conic (DIAC). The positive semidefiniteness of the DIAC is explicitly enforced as part of the optimization process, rather than as a post-processing step.For each stage, we construct and minimize tight convex relaxations of the highly non-convex objective functions in a branch and bound optimization framework. We exploit the inherent problem structure to restrict the search space for the DIAC and the plane at infinity to a small, fixed number of branching dimensions, independent of the number of views. Chirality constraints are incorporated into our convex relaxations to automatically select an initial region which is guaranteed to contain the global minimum. Experimental evidence of the accuracy, speed and scalability of our algorithm is presented on synthetic and real data.

Deterministically maximizing feasible subsystem for robust model fitting with unit norm constraint

“…Our branch and bound algorithm is similar to those in many other computer vision works [13,5,19]. We should note the difference between branch and partition mentioned above.…”

“…We terminate the branch and bound iteration when the absolute gap between the current best upper bound and lower bound converges to zero. This zero-gap tolerance guarantees that our method reports the globally optimal solution, in contrast to a ε-suboptimal solution in the majority of existing works, like [5,19].…”

“…We introduce novel piecewise linear under-and overestimator for square terms. In terms of relaxation tightness, this piecewise relaxation mechanism is superior over the popular convex and concave relaxations in existing literature, such as [13,5,19].…”

“…In existing computer vision literature ( [13,5,19]), the mainstream idea is to build convex and concave relaxations for nonconvex terms. In the sense of convexity and concavity, the tightest possible relaxations are the convex and concave envelopes, the bilinear envelopes in [13,5,19] (w). Note that we omit the subscript j of x j in this whole section for simplification.…”

“…Actually, many works using the branch and bound method have appeared in the computer vision community, like [13,5,19]. Therefore, the detailed procedures are omitted here.…”