A Theory of Shape Reconstruction
from Heat Conduction and Shading

Shape from shading using a single image of a Lambertian surface is inherently ambiguous. When the light source direction is known, the surface normal estimation has a cone-ambiguity, which worsens when the source is unknown. Recently, shape from heat conduction has emerged as an approach that leverages heat transport equations to estimate the Shape Laplacian operator, an intrinsic measure of shape. However, deriving surface normals from the Laplacian operator encounters a local binary convex/concave ambiguity. We introduce a novel theory to resolve these local shape ambiguities (excluding a few degeneracies) without relying on priors like smoothness, by combining the cues from shading and heat conduction. Our method ensures the mathematical constraints of both shading and the Laplacian are satisfied simultaneously, even with an unknown light source. We validate our theory through simulations of complex shapes and analyze its performance in the presence of noise, as well as on a noisy single thermal video of real-world objects with complex shapes and material properties, including varying albedo.