Shape reconstruction for deflectometry based on chebyshev polynomials and iteratively reweighted least squares regression

Abstract

Deflectometry is a technique for measuring the slope data of specular surfaces, and shape reconstruction is the final process based on the measured slopes. Modal methods reconstruct surfaces with expansion polynomials. The coefficients of each polynomial mode are calculated by linear equations composed of the gradient of the polynomials and the measured slope data. Conventional approaches use ordinary least squares to solve the linear equations. However, the equations are overdetermined, and the random outliers will decrease the reconstruction accuracy. The Chebyshev polynomials are suitable for discrete slope data and can be utilized to reconstruct the surface shape in deflectometry. Hence, this paper uses 2D Chebyshev polynomials as the gradient polynomial basis set. An iteratively reweighted least squares algorithm, which iteratively calculates an additional scale factor for each data point, is applied to accomplish robust linear regression. The experiments with both synthetic and measured data prove that the proposed method is robust against noise and has higher reconstruction accuracy for shape reconstruction.