This paper presents a new algebraic framework for modeling structures and computing the reconstruction of curves from local derivatives. This is the fundamental problem behind the use of inclinometers to measure the deformation of structures. A solid theoretical background, with derivations, is provided for the solution of a real and practical measurement problem. The reconstruction problem is solved using matrix algebraic techniques applied to variational calculus. Tikhonov, spectral, and a new regularization technique based on truncated constrained basis functions are implemented. It is proved that the solution of differential equations via basis functions corresponds to computing the spectra of the bending modes with respect to the basis functions used. This insight yields a method of determining the optimal set of basis functions based on spectral compactness. A completely new approach to computing a linear differential operator which yields a global estimator is presented. The method is also extended to yield a regularizing linear differential operator. The methods are verified, and the claims made substantiated by extensive numerical simulations including Monte Carlo simulations to verify the susceptibility of the solution to noise. The methods are also applied to a real measurement problem of determining the subsidence of rail track during construction in the vicinity of the track. Simultaneous reconstruction and temporal filtering is implemented.
