  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 $\mathcal{O}!(n^6)$ algorithms
for an $m\times n$ surface. In contrast, the newly proposed methods are based
on the global least squares cost function and are all $\mathcal{O}!(n^3)$
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.
