Smooth Approximation of L_infinity-Norm for Multi-view Geometry

Recently the $L_\infty$-norm optimization has been introduced to multi-view geometry to achieve global optimality. It is solved through solving a sequence of SOCP (second order cone programming) feasibility problems which needs sophisticated solvers and time consuming. This paper presents an efficient smooth approximation of $L_\infty$-norm optimization in multi-view geometry using log-sum-exp functions. We have proven that the proposed approximation is pseudo-convex with the property of uniform convergence. This allows us to solve the problem using gradient based algorithms such as gradient descent to overcome the non-differentiable property of $L_\infty$ norm. Experiments on both synthetic and real image sequence have shown that the proposed algorithm achieves high precision and also significantly speeds up the implementation.