On differential photometric reconstruction for unknown, isotropic BRDFs

Area of Science:

• Computer Vision
• Computer Graphics
• Computational Geometry

Background:

• Photometric surface reconstruction aims to recover 3D shape from 2D images.
• Existing methods often require known lighting conditions or simplified surface reflectance models (BRDF).
• Image derivatives offer rich information about surface geometry.

Purpose of the Study:

• To develop a comprehensive theory for photometric surface reconstruction with general, unknown isotropic Bidirectional Reflectance Distribution Functions (BRDFs).
• To identify precise conditions and priors for full geometric reconstruction.
• To establish new invariants relating image derivatives to surface geometry.

Main Methods:

• Exploiting the linearity of chain rule differentiation to derive photometric invariants.
• Analyzing shape-from-shading and photometric stereo under unknown isotropic BRDFs.
• Proving theoretical results on surface determination up to specific geometric contours.

Main Results:

• Discovered photometric invariants linking image derivatives to surface geometry, independent of BRDF form.
• Showed shape-from-shading reconstruction is possible up to gradient magnitude isocontours.
• Demonstrated that two derivative measurements suffice for photometric stereo recovery.
• Introduced 'photometric flow' determining surface up to gradient magnitude and depth isocontours.
• Proved a single surface normal specification fully determines depth from these isocontours.

Conclusions:

• The derived photometric invariants provide a fundamental basis for surface reconstruction with unknown BRDFs.
• The theory offers precise topological classes for surface determination.
• Practical algorithms are proposed, extending theoretical results to real-world scenarios.