Comparison of two-dimensional integration methods for shape reconstruction from gradient data

Highlights

• Finite difference least squares integration methods are reviewed.

• Transformed-based integration methods are compared with their differences revealed.

• The “best integrators” from each family are selected for the finial comparison.

• Merits and drawbacks of each method are disclosed clearly.

Abstract

As a requisite and key step in some gradient-based measurement techniques, the reconstruction of the shape, more generally the scalar potential, from the measured gradient data has been studied for many years. In this work, three types of two-dimensional integration methods are compared under various conditions. The merits and drawbacks of each integration method are consequently revealed to provide suggestions in selection of a proper integration method for a particular application.

Introduction

In metrology, some physical quantities from measurement may not be our desired objective directly, but they may have certain relationship with our objective. The measurements are therefore still very useful and can be employed to get our desired quantities. Many optical metrology techniques belong to this indirect measurement. For instance, wavefront measurement techniques (Hartmann-based wavefront sensing [1], [2], [3], lateral shearing interferometry [4], [5], [6], etc.) reconstruct the wavefront from the slopes measured by optical sensors. Moreover, the technique of shape from shading [7], [8], [9] estimates the surface profile by integrating the calculated gradient data. Similarly, three-dimensional shape measurement for specular surfaces, e.g. phase measuring deflectometry [10], [11], [12], [13], [14], [15], [16], [17], [18], integrates gradient data from metrology to get the surface shape as shown in Fig. 1. All these techniques above only measure the derivatives of the wanted quantity. In order to achieve our final goal, a two-dimensional (2D) integration procedure is necessary to reconstruct the shape from the measured derivatives.

Due to its wide application, 2D integration methods are investigated by many researchers and there are lots of articles in literatures [7], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29]. The 2D integration problem can be considered as solving a Poisson equation with Neumann boundary conditions [30]. Research on 2D integration methods can be found since 1970s for wavefront reconstruction [19], [20], [22], [30]. Generally, finite difference approaches were employed in those methods to connect the measured slope and desired shape, and least squares estimations are made for shape reconstruction. Fourier transform has been introduced into 2D integration in 1980s [7], [31]. By applying the properties of Fourier transform, the integration operation can be implemented easily and quickly due to the well-known Fast Fourier Transform (FFT) algorithm. In 2004, Li et al. [23] compared the finite difference based least squares integration method and Fourier transform integration method with showing the finite difference based least squares integration method has higher accuracy at that time. By considering the boundary conditions, Talmi and Ribak [24] pointed out the best solution of gradient integration could be expressed in a Fourier cosine series, not the periodic Fourier series. The cosine transform integration method is therefore proposed with providing the integration result at half-integer positions in 2006. Coming to 2008, Ettl et al. [25] introduced an integration method by employing the radial basis functions which is flexible and robust. In 2012, Bon et al. [26] proposed a boundary-artifact-free Fourier integration method by simply padding slope matrices with accordingly flipped and positive or negative slope values. By noticing the accuracy of the traditional finite difference based least squares integration method is limited by its biquadratic shape assumption, Huang and Asundi [27], [28] proposed an iterative compensation approach to obtain more accurate integration results. Recently, Li et al. [29] improved the finite difference based least squares integration method by applying higher-order numerical differentiation formats.

In this work, three families of integration methods are selected into our comparison, since they have been widely used in numerous applications. Several 2D integration methods are chosen as the representative of their corresponding families to make a comparison in both reconstruction accuracy and processing speed. These integration methods are invented in different application fields, and developed along their own paths. It is very interesting to know their merits and drawbacks after the improvement in recent years. Their “abilities” and “tempers” are revealed in order to help the selection of a proper 2D integration method for a specific application.