We present a generalized method for reconstructing the shape of an object from measured gradient data. A certain class of optical sensors does not measure the shape of an object but rather its local slope. These sensors display several advantages, including high information efficiency, sensitivity, and robustness. For many applications, however, it is necessary to acquire the shape, which must be calculated from the slopes by numerical integration. Existing integration techniques show drawbacks that render them unusable in many cases. Our method is based on an approximation employing radial basis functions. It can be applied to irregularly sampled, noisy, and incomplete data, and it reconstructs surfaces both locally and globally with high accuracy.

Shape reconstruction from gradient data.

This paper presents a method that allows a conventional dual-camera structured light system to directly acquire the three-dimensional shape of the whole surface of an object with high dynamic range of surface reflectivity. To reduce the degradation in area-based correlation caused by specular highlights and diffused darkness, we first disregard these highly specular and dark pixels. Then, to solve this problem and further obtain unmatched area data, this binocular vision system was also used as two camera-projector monocular systems operated from different viewing angles at the same time to fill in missing data of the binocular reconstruction. This method involves producing measurable images by integrating such techniques as multiple exposures and high dynamic range imaging to ensure the capture of high-quality phase of each point. An image-segmentation technique was also introduced to distinguish which monocular system is suitable to reconstruct a certain lost point accurately. Our experiments demonstrate that these techniques extended the measurable areas on the high dynamic range of surface reflectivity such as specular objects or scenes with high contrast to the whole projector-illuminated field.

3D shape measurement of objects with high dynamic range of surface reflectivity.

Least-squares (LS)-based integration computes the function values by solving a set of integration equations (IEs) in LS sense, and is widely used in wavefront reconstruction and other fields where the measured data forms a slope. It is considered that the applications of IEs with smaller truncation errors (TEs) will improve the reconstruction accuracy. This paper proposes a general method based on the Taylor theorem to derive all kinds of IEs, and finds that an IE with a smaller TE has a higher-order TE. Three specific IEs with higher-order TEs in the Southwell geometry are deduced using this method, and three LS-based integration algorithms corresponding to these three IEs are formulated. A series of simulations demonstrate the validity of applying IEs with higher-order TEs in improving reconstruction accuracy. In addition, the IEs with higher-order TEs in the Hudgin and Fried geometries are also deduced using the proposed method, and the performances of these IEs in wavefront reconstruction are presented.

Improving wavefront reconstruction accuracy by using integration equations with higher-order truncation errors in the Southwell geometry

The fast development in the fields of integrated circuits, photovoltaics, the automobile industry, advanced manufacturing, and astronomy have led to the importance and necessity of quickly and accurately obtaining three-dimensional (3D) shape data of specular surfaces for quality control and function evaluation. Owing to the advantages of a large dynamic range, non-contact operation, full-field and fast acquisition, high accuracy, and automatic data processing, phase-measuring deflectometry (PMD, also called fringe reflection profilometry) has been widely studied and applied in many fields. Phase information coded in the reflected fringe patterns relates to the local slope and height of the measured specular objects. The 3D shape is obtained by integrating the local gradient data or directly calculating the depth data from the phase information. We present a review of the relevant techniques regarding classical PMD. The improved PMD technique is then used to measure specular objects having discontinuous and/or isolated surfaces. Some influential factors on the measured results are presented. The challenges and future research directions are discussed to further advance PMD techniques. Finally, the application fields of PMD are briefly introduced.

Three-Dimensional Shape Measurements of Specular Objects Using Phase-Measuring Deflectometry.

Specular surface measurement by using least squares light tracking technique

Methods of numerical integration of sampled data are compared in terms of their frequency responses and resolving power. Compared, theoretically and by numerical experiments, are trapezoidal, Simpson, Simpson-3/8 methods, method based on cubic spline data interpolation and Discrete Fourier Transform (DFT) based method. Boundary effects associated with DFT- based and spline-based methods are investigated and an improved Discrete Cosine Transform based method is suggested and shown to be superior to all other methods both in terms of approximation to the ideal continuous integrator and of the level of the boundary effects.

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Frequency responses and resolving power of numerical integration of sampled data.

In some measurement techniques the profile, f(x), of a function should be obtained from the data on measured slope f(?)(x) by integration. The slope is measured in a given set of points, and from these data we should obtain the profile with the highest possible accuracy. Most frequently, the integration is carried out by numerical integration methods [Press et al., Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1987)] that assume different kinds of polynomial approximation of data between sampling points. We propose the integration of the function in the Fourier domain, by which the most-accurate interpolation is automatically carried out. Analysis of the integration methods in the Fourier domain permits us to easily study and compare the methods' behavior.

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Integration in the Fourier domain for restoration of a function from its slope: comparison of four methods

A compact scanning deflectometer is presented for the fast topography measurement of semiconductor wafers. The technique, however, is equally well suited for any flat or slightly curved specular reflective surface. The measurement principle is based on the 2D measurement of the local slope vector by means of a narrow Laser beam scanning rapidly across the sample surface. The fast linear scanning is combined with sample rotation to measure the complete surface of circular samples. There is no physical contact to the measured surface. The topography of the sample is derived from the slope data by a novel 2D integration method, which is robust with respect to noise in the slope signals. We present the full-size topography of unpatterned and patterned wafers of different polishing quality.

A fast optical scanning deflectometer for measuring the topography of large silicon wafers

The objective of profilometry is to obtain the topography of a surface. Some methods are based on the measurement of the slope of the test surface. Then, by integration the profile of a surface can be determined. The slope is measured in a given set of points and from these data it is necessary to obtain the profile with the highest possible accuracy. Most frequently, the integration is carried out by numerical integration methods that assume different kinds of polynomial approximation of data between sampling points. We propose the integration of the function by means of processing in the Fourier domain. The analysis of the different integration methods in the Fourier domain enables us to easily study and compare their performance.

Frequency response of five integration methods to obtain the profile from its slope

We propose a new method for evaluating and correcting aberrations in a Vander Lugt correlator. The technique is achieved with liquid-crystal displays of the correlator and allows the task to be performed in situ. We present the theory on which the method is based and the experimental results that we obtained by applying it in a convergent correlator.

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Alfonso Moreno

Papers

Frequency responses and resolving power of numerical integration of sampled data.

Integration in the Fourier domain for restoration of a function from its slope: comparison of four methods

A fast optical scanning deflectometer for measuring the topography of large silicon wafers

Frequency response of five integration methods to obtain the profile from its slope

Evaluation and correction of aberrations in an optical correlator by phase-shifting interferometry.