1982 the-visual-inference-of-shape-computation-from-local-feature
This dissertation develops a rigorous computational framework for recovering three‑dimensional surface shape from a single image using local image cues such as shading, contours, and illumination direction. Central to the theory are analytic derivations that (1) identify the image direction of maximal second‑derivative intensity as the surface tilt, (2) relate the normalized image Laplacian (∇²I/I) to the principal curvatures along the tilt and its orthogonal direction, and (3) propose a maximum‑likelihood estimator for surface slant under a uniform curvature prior. The work also examines how these estimators behave on three imaging modalities—standard positive images, scanning‑electron‑microscope (EM) images, and photographic negatives—showing that simple intensity normalization can mitigate some modality‑specific errors. Detailed mathematical proofs cover coordinate‑rotation transformations of second‑order image derivatives, the convexity/concavity ambiguity inherent in tilt estimation, and the derivation of the slant estimator formula zN = σκ² (|∇²I/I| – σ_κ²)⁻². Empirical validation uses synthetic sphere intensity profiles to illustrate the estimator’s cup‑shaped response for positive images, its inverse‑square slant scaling for EM images, and the blow‑up behavior for negatives. The dissertation situates these contributions within a broader literature on shape‑from‑shading, edge detection, and visual neuroscience, providing a foundation for both psychophysical modeling and practical computer‑vision algorithms.
Topics
Tilt Direction as the Axis of Maximum Second‑Derivative
Derivation showing that the image direction where the second‑derivative of intensity is maximal coincides with surface tilt, expressed τ = ½ arctan(2I_xy/(I_xx‑I_yy)). Includes the coordinate‑rotation transformation of second‑order derivatives (I_xx, I_yy, I_xy) under a rotation by τ and discusses the inherent ±π convexity/concavity ambiguity that prevents distinguishing convex from concave patches from a single shading cue.