Continuous Ratio Optimization via Convex Relaxation with Applications to Multiview 3D Reconstruction

We introduce a convex relaxation framework to optimally minimize continuous surface ratios. The key idea is to minimize the continuous surface ratio by solving a sequence of convex optimization problems. We show that such minimal ratios are superior to traditionally, used minimal surface formulations in that they do not suffer from a shrinking bias and no longer require the choice of a regularity parameter The absence of a shrinking bias in the minimal ratio model is proven analytically. Furthermore we demonstrate that continuous ratio optimization can be applied to derive a new algorithm for reconstructing three-dimensional silhouette-consistent objects from multiple views. Experimental results confirm that our approach allows to accurately reconstruct deep concavities even without the specification of tuning parameters.