A Framework for Inverse Scattering

Abstract

Scattering materials are ubiquitous: from our skin and food, to every objects such as wax and soap, to industrial materials such as coatings and soft tissues. Common to all these materials is the complex way in which they interact with light. Their appearance is the result of photons that penetrate the material surface, and perform random walks inside the material before emerging towards a camera. Inverse scattering is, then, the problem of inverting this light transport process, in order to infer scattering parameters from images of a material.

We approach inverse scattering as an appearance matching problem: given a set of measurements (images) of a material, we search for the scattering parameters which, when used to computationally render new images, minimize the difference with the captured ones. In full generality, this is a very challenging optimization problem, due to the high-dimensional search space and the non-linear dependence of images on scattering parameters. We present several contributions for making this optimization problem tractable.

First, we present the results of a large-scale study of human perception of scattering material (translucent) appearance. Our study identifies a two-dimensional embedding of the physical scattering parameters in a perceptually-meaningful appearance space. Through our analysis of this space, we find uniform parameterizations of its two axes by analytical expressions of moments of the phase function, and provide an intuitive characterization of the visual effects that can be achieved at different parts of it. Our findings highlight the important role phase function and mid-order scattering can have in controlling translucent appearance, motivating the development of inverse scattering algorithms that can handle these effects.

Second, we introduce a computational framework for efficiently solving inverse scattering appearance matching problems. Our framework is based on a combination of operator theory, stochastic gradient descent, Monte Carlo rendering, and material dictionary representations. It allows inverting the light transport process in a broad range of scattering materials, without having to rely on common approximations such as single scattering and diffusion. Additionally, it accommodates rich, high-dimensional material representations, enabling us to accurately measure parameters such as the scattering phase function shape, without having to rely on restrictive low-parameter models. To evaluate this framework experimentally, we create an acquisition setup that images thin material slabs under narrow-beam illumination from multiple lighting and viewing directions. Using measurements from this setup, we recover parameters of homogeneous (spatially-uniform) scattering materials, including arbitrary phase function shapes.

Third, we generalize our computational framework to address the heterogeneous inverse scattering problem, where the material parameters vary from point to point inside the volume. To this end, we make our algorithm applicable to measurements where photon contributions are decomposed based on criteria such as the distance they travelled inside the material, or their point of origin on the source illuminating the material. Additionally, we use path-space formulations of light transport, to allow our stochastic optimization framework to scale up to hundreds of thousands of scattering parameter unknowns. Through simulated experiments, we find that these extensions allow our algorithm to recover all spatially-varying scattering parameters for different types of scattering materials.

Fourth and finally, we present a computational imaging system that allows capturing the decomposed measurements used by our heterogeneous inverse scattering algorithms. Our system is based on interferometric techniques, and specifically on the optical coherence tomography framework. Our use of interferometry allows us to capture these decompositions at micron-scale resolutions, two to three orders of magnitude larger than previously possible. Such resolutions are necessary when collecting measurements for inverse scattering applications. We describe how to construct and optimize an optical assembly for this technique, and we build a prototype to measure and visualize scattering materials, and other optical phenomena.