What's In A Patch, I: Tensors, Differential Geometry and Statistical Shading Analysis

TL;DR

This paper introduces a linear algebraic framework using tensors for shape-from-shading, analyzing how image derivatives can offer invariance to illumination changes and improve reconstruction accuracy.

Contribution

It develops a tensor-based mathematical framework and investigates the invariance properties of image derivatives under different lighting conditions.

Findings

Image derivatives show increased invariance to illumination changes.

Gradient-based shape-from-shading yields more accurate reconstructions.

Validation through computational experiments supports the theoretical claims.

Abstract

We develop a linear algebraic framework for the shape-from-shading problem, because tensors arise when scalar (e.g. image) and vector (e.g. surface normal) fields are differentiated multiple times. The work is in two parts. In this first part we investigate when image derivatives exhibit invariance to changing illumination by calculating the statistics of image derivatives under general distributions on the light source. We computationally validate the hypothesis that image orientations (derivatives) provide increased invariance to illumination by showing (for a Lambertian model) that a shape-from-shading algorithm matching gradients instead of intensities provides more accurate reconstructions when illumination is incorrectly estimated under a flatness prior.