Regularized Reconstruction of a Surface from its Measured Gradient Field

Abstract

This paper presents several new algorithms for the regularized reconstruction of a surface from its measured gradient field. By taking a matrix-algebraic approach, we establish general framework for the regularized reconstruction problem based on the Sylvester Matrix Equation. Specifically, Spectral Regularization via Generalized Fourier Series (e.g., Discrete Cosine Functions, Gram Polynomials, Haar Functions, etc.), Tikhonov regularization, Constrained Regularization by imposing boundary conditions, and regularization via Weighted least squares can all be solved expediently in the context of the Sylvester Equation framework. State-of-the-art solutions to this problem are based on sparse matrix methods, which are no better than $$\mathcal {O}\!\left( n^6\right) $$ algorithms for an $$m\times n$$ surface with $$m \sim n$$ . In contrast, the newly proposed methods are based on the global least squares cost function and are all $$\mathcal {O}\!\left( n^3\right) $$ algorithms. In fact, the new algorithms have the same computational complexity as an SVD of the same size. The new algorithms are several orders of magnitude faster than the state-of-the-art; we therefore present, for the first time, Monte-Carlo simulations demonstrating the statistical behaviour of the algorithms when subject to various forms of noise. We establish methods that yield the lower bound of their respective cost functions, and therefore represent the “Gold-Standard” benchmark solutions for the various forms of noise. The new methods are the first algorithms for regularized reconstruction on the order of megapixels, which is essential to methods such as Photometric Stereo.