Abstract Estimating scattering parameters of heterogeneous media from images is a severely under‐constrained and challenging problem. Most of the existing approaches model BSSRDF either through an analysis‐by‐synthesis approach, approximating complex path integrals, or using differentiable volume rendering techniques to account for heterogeneity. However, only a few studies have applied learning‐based methods to estimate subsurface scattering parameters, but they assume homogeneous media. Interestingly, no specific distribution is known to us that can explicitly model the heterogeneous scattering parameters in the real world. Notably, procedural noise models such as Perlin and Fractal Perlin noise have been effective in representing intricate heterogeneities of natural, organic, and inorganic surfaces. Leveraging this, we first create HeteroSynth, a synthetic dataset comprising photorealistic images of heterogeneous media whose scattering parameters are modeled using Fractal Perlin noise. Furthermore, we propose Tensorial Inverse Scattering (TensoIS), a learning‐based feed‐forward framework to estimate these Perlin‐distributed heterogeneous scattering parameters from sparse multi‐view image observations. Instead of directly predicting the 3D scattering parameter volume, TensoIS uses learnable low‐rank tensor components to represent the scattering volume. We evaluate TensoIS on unseen heterogeneous variations over shapes from the HeteroSynth test set, smoke and cloud geometries obtained from open‐source realistic volumetric simulations, and some real‐world samples to establish its effectiveness for inverse scattering. Overall, this study is an attempt to explore Perlin noise distribution, given the lack of any such well‐defined distribution in literature, to potentially model real‐world heterogeneous scattering in a feed‐forward manner. Project Page: https://yashbachwana.github.io/TensoIS/

Null Foley-Sammon Transform (NFST) has received increasing attention in the machine learning and pattern recognition literature. NFST finds a discriminative nullspace where all samples of the same class get mapped into a single point. It has a closed form solution and is free of parameters to tune. NFST has been leveraged in many areas including novelty detection, person or vehicle re-identification and achieved state-of-the-art results. Motivated from its attractive properties and its effectiveness in wide range of applications, in this paper we focus on the theoretical analysis of NFST. In previous literature, NFST was shown to exist in small sample size (SSS) case. We first prove that NFST can exist in non-SSS case also, under certain conditions. Thereby, we extend the domain of applicability of NFST to a more general case. Secondly, we perform analysis of the singular points of NFST, revealing important insights on their identities and existence. Thirdly, we show the theoretical relation between NFST of SSS data and NFST of the non-SSS data obtained by PCA. Fourthly, we show that this theoretical relation can be exploited to obtain an efficient algorithm for computing NFST on high dimensional SSS data. Finally, we perform extensive experiments to validate our theoretical analysis.