# Variational Uncalibrated Photometric Stereo under General Lighting

## TL;DR

This paper introduces a variational method for uncalibrated photometric stereo under general lighting, jointly estimating shape, reflectance, and illumination directly from images, improving accuracy over previous methods.

## Contribution

It presents a novel variational framework that approximates Lambertian reflectance with spherical harmonics and estimates shape without normal integration, enhancing robustness and efficiency.

## Key findings

- Reduces mean angular error by 2-3 times compared to state-of-the-art.
- Validates robustness and efficiency through extensive experiments.
- Effectively estimates shape, reflectance, and lighting jointly in uncalibrated settings.

## Abstract

Photometric stereo (PS) techniques nowadays remain constrained to an ideal laboratory setup where modeling and calibration of lighting is amenable. To eliminate such restrictions, we propose an efficient principled variational approach to uncalibrated PS under general illumination. To this end, the Lambertian reflectance model is approximated through a spherical harmonic expansion, which preserves the spatial invariance of the lighting. The joint recovery of shape, reflectance and illumination is then formulated as a single variational problem. There the shape estimation is carried out directly in terms of the underlying perspective depth map, thus implicitly ensuring integrability and bypassing the need for a subsequent normal integration. To tackle the resulting nonconvex problem numerically, we undertake a two-phase procedure to initialize a balloon-like perspective depth map, followed by a "lagged" block coordinate descent scheme. The experiments validate efficiency and robustness of this approach. Across a variety of evaluations, we are able to reduce the mean angular error consistently by a factor of 2-3 compared to the state-of-the-art.