Importance Sampling via Load-Balanced Facility Location

In this paper, we consider the problem of "importance sampling" from a high dynamic range image, motivated by a computer graphics problem called image-based lighting. Image-based lighting is a method to light a scene by using real-world images as part of a 3D environment. Intuitively, the sampling problem reduces to finding representative points from the image such that they have higher density in regions of high intensity (or energy) and low density in regions of low intensity (or energy). We formulate this task as a facility location problem where the facility costs are a function of the demand served. In particular, we aim to encourage load balance amongst the facilities by using V-shaped facility costs that achieve a minimum at the "ideal" level of demand. We call this the load-balanced facility location problem, and it is a generalization of the uncapacitated facility location problem with uniform facility costs. We develop a primal-dual approximation algorithm for this problem, and analyze its approximation ratio using dual fitting and factor-revealing linear programs. We also give some experimental results from applying our algorithm to instances derived from real high dynamic range images.