An Accurate Fourier-Based Method for Three-Dimensional Reconstruction of Transparent Surfaces in the Shape-from-Polarization Method

The Fourier-based gradient field integration method can efficiently reconstruct transparent surfaces from the measured gradient data in the Shape-from-polarization method. However, for the differentiation operator having large truncation errors and lacking constraints on adjacent heights, it will increase the reconstructing error of the Fourier-based integration method. This paper presents an accurate Fourier-based integration approach to improve the reconstruction accuracy, in which a new differentiation operator is derived by limiting the truncation error and increasing heights and slopes in the operator. In order to modify the data obtained by the new operator to meet the requirements of both periodicity and size for the use of the discrete Fourier transform, we propose a method to extend the raw gradient data by first performing antisymmetric extension and then performing periodic extension. A series of simulations and experiments have been developed to verify the performance of the proposed method. By comparing the approach of this paper with other Fourier-based approaches, including Frankot-Chellappa, Southwell-FT and Simpson-FT, both of the simulation and experiment results show that the proposed Fourier-based integration method performs a higher accuracy than other approaches. In the reconstruction experiment, the reconstruction error can be reduced from 0.065 similar to 0.081 mm to 0.020 mm for the spherical surface, and from 0.060 similar to 0.12 mm to 0.016 mm for the free-form surface.