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Journal ArticleDOI

Framework for gradient integration by combining radial basis functions method and least-squares method

20 Aug 2013-Applied Optics (Appl Opt)-Vol. 52, Iss: 24, pp 6016-6021

TL;DR: A framework with a combination of the radial basis functions method and the least-squares integration method is proposed to improve the integration process from gradient to shape and is accurate, automatic, easily implemented, and robust.

AbstractA framework with a combination of the radial basis functions (RBFs) method and the least-squares integration method is proposed to improve the integration process from gradient to shape. The principle of the framework is described, and the performance of the proposed method is investigated by simulation. Improvement in accuracy is verified by comparing the result with the usual RBFs-based subset-by-subset stitching method. The proposed method is accurate, automatic, easily implemented, and robust and even works with incomplete data.

Topics: Image stitching (54%), Least squares (54%), Radial basis function (51%), Reflection (physics) (50%)

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Citations
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Journal ArticleDOI
TL;DR: In this work, three types of two-dimensional integration methods are compared under various conditions to provide suggestions in selection of a proper integration method for a particular application.
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.

71 citations


Journal ArticleDOI
TL;DR: The fundamental principle and the basic concepts of PMD technique are introduced and followed by a brief overview of its key developments since it was first proposed to provide some suggestions for potential future investigations.
Abstract: As a low cost, full-field three-dimensional shape measurement technique with high dynamic range, Phase Measuring Deflectometry (PMD) has been studied and improved to be a simple and effective manner to inspect specular reflecting surfaces. In this review, the fundamental principle and the basic concepts of PMD technique are introduced and followed by a brief overview of its key developments since it was first proposed. In addition, the similarities and differences compared with other related techniques are discussed to highlight the distinguishing features of the PMD technique. Furthermore, we will address the major challenges, the existing solutions and the remaining limitations in this technique to provide some suggestions for potential future investigations.

70 citations


Journal ArticleDOI
07 Dec 2017-Sensors
TL;DR: This work presents a review of the relevant techniques regarding classical and improved phase-measuring deflectometry, and discusses the challenges and future research directions to further advance PMD techniques.
Abstract: 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.

43 citations


Cites methods from "Framework for gradient integration ..."

  • ...proposed a framework with a combination of the RBFs method and the least-squares integration method [76]....

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Journal ArticleDOI
TL;DR: Results show that the Numerical orthogonal polynomial is superior to the other three polynomials because of its high accuracy and robustness even in the case of a wavefront with incomplete data.
Abstract: Four orthogonal polynomials for reconstructing a wavefront over a square aperture based on the modal method are currently available, namely, the 2D Chebyshev polynomials, 2D Legendre polynomials, Zernike square polynomials and Numerical polynomials. They are all orthogonal over the full unit square domain. 2D Chebyshev polynomials are defined by the product of Chebyshev polynomials in x and y variables, as are 2D Legendre polynomials. Zernike square polynomials are derived by the Gram-Schmidt orthogonalization process, where the integration region across the full unit square is circumscribed outside the unit circle. Numerical polynomials are obtained by numerical calculation. The presented study is to compare these four orthogonal polynomials by theoretical analysis and numerical experiments from the aspects of reconstruction accuracy, remaining errors, and robustness. Results show that the Numerical orthogonal polynomial is superior to the other three polynomials because of its high accuracy and robustness even in the case of a wavefront with incomplete data.

27 citations


Cites background from "Framework for gradient integration ..."

  • ...Reconstruction techniques of wavefront or surface shape can be classified approximately as zonal and modal methods [2,3]....

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Journal ArticleDOI
TL;DR: A modified easy-implementation integration method based on HFLI (EI-HFLI), which can work in arbitrary domains, and can directly and conveniently handle incomplete gradient data is proposed.
Abstract: Stereo deflectometry is defined as measurement of the local slope of specular surfaces by using two CCD cameras as detectors and one LCD screen as a light source. For obtaining 3D topography, integrating the calculated slope data is needed. Currently, a high-order finite-difference-based least-squares integration (HFLI) method is used to improve the integration accuracy. However, this method cannot be easily implemented in circular domain or when gradient data are incomplete. This paper proposes a modified easy-implementation integration method based on HFLI (EI-HFLI), which can work in arbitrary domains, and can directly and conveniently handle incomplete gradient data. To carry out the proposed algorithm in a practical stereo deflectometry measurement, gradients are calculated in both CCD frames, and then are mixed together as original data to be meshed into rectangular grids format. Simulation and experiments show this modified method is feasible and can work efficiently.

25 citations


Cites methods from "Framework for gradient integration ..."

  • ...This integration method usually works together with the least-squares stitching method [7] since RBF is mainly effective at handling small size measurement data (e....

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References
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Journal ArticleDOI
TL;DR: An overview of 3-D shape measurement using various optical methods, and a focus on structured light tech- niques where various optical configurations, image acquisition technology, data postprocessing and analysis methods and advantages and limitations are presented.
Abstract: We first provide an overview of 3-D shape measurement us- ing various optical methods. Then we focus on structured light tech- niques where various optical configurations, image acquisition tech- niques, data postprocessing and analysis methods and advantages and limitations are presented. Several industrial application examples are presented. Important areas requiring further R&D are discussed. Finally, a comprehensive bibliography on 3-D shape measurement is included, although it is not intended to be exhaustive. © 2000 Society of Photo-Optical Instrumentation Engineers. (S0091-3286(00)00101-X)

1,406 citations


"Framework for gradient integration ..." refers background in this paper

  • ...Optical 3D shapemetrology is widely used for inspection purposes in industrial applications [1,2]....

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Journal ArticleDOI
Abstract: The problem of wave-front estimation from wave-front slope measurements has been examined from a least-squares curve fitting model point of view. It is shown that the slope measurement sampling geometry influences the model selection for the phase estimation. Successive over-relaxation (SOR) is employed to numerically solve the exact zonal phase estimation problem. A new zonal phase gradient model is introduced and its error propagator, which relates the mean-square wave-front error to the noisy slope measurements, has been compared with two previously used models. A technique for the rapid extraction of phase aperture functions is presented. Error propagation properties for modal estimation are evaluated and compared with zonal estimation results.

880 citations


Proceedings ArticleDOI
10 Sep 2004
Abstract: We present a new method to measure specular free-form surfaces within seconds. We call the measuring principle `Phase Measuring Deflectometry' (PMD). With a stereo based enhancement of PMD we are able to measure both the height and the slope of the surface. The basic principle is to project sinusoidal fringe patterns onto a screen located remotely from the surface under test and to observe the fringe patterns reflected via the surface. Any slope variations of the surface lead to distortions of the patterns. Using well-known phase-shift algorithms, we can precisely measure these distortions and thus calculate the surface normal in each pixel. We will deduce the method's diffraction-theoretical limits and explain how to reach them. A major challenge is the necessary calibration. We solved this task by combining various photogrammetric methods. We reach a repeatability of the local slope down to a few arc seconds and an absolute accuracy of a few arc minutes. One important field of application is the measurement of the local curvature of progressive eyeglass lenses. We will present experimental results and compare these results with the theoretical limits.

406 citations


Proceedings ArticleDOI
10 Sep 2004
Abstract: Accurate 3D shape measurement is of big importance for industrial inspection. Because of the robustness, accuracy and ease of use optical measurement techniques are gaining importance in industry. For fast 3D measurements on big surfaces fringe projection is commonly used: A projector projects fringes onto the object under investigation and the scattered light is recorded by a camera from a triangulation angle. Thus, it is possible reaching a depth resolution of about one by 10.000 of the measurement field size (e.g. 100 μm for a 1 m sized field). For non- or low scattering objects it is common to put scattering material like particle spray onto the object under investigation. Objects where this is not allowed are often regarded as problematic objects for full field non-coherent optical measurement techniques. The solution is to switch from fringe projection to fringe reflection. The fringe reflection technique needs a simple setup to evaluate a fringe pattern that is reflected from the surface under investigation. Like for fringe projection the evaluated absolute phase identifies the location of the originating fringe. This allows identifying the reflection angles on the object for every camera pixel. The results are high resolution local gradients on the object which can be integrated to get the 3D shape. The achievable depth resolution compared to fringe projection is much better and reaches to a depth resolution down to 1 nm for smooth surfaces. We have proven the ability, robustness and accuracy of the technique for various technical objects and also fluids. A parallel paper of this conference 'Evaluation Methods for Gradient Measurement Techniques' picks up further processing of the evaluated data and explains in more detail the performed calculations. This paper mainly concentrates on the fringe reflection principle, reachable resolution and possible applications.

150 citations


Journal ArticleDOI
TL;DR: A generalized method for reconstructing the shape of an object from measured gradient data based on an approximation employing radial basis functions that can be applied to irregularly sampled, noisy, and incomplete data, and it reconstructs surfaces both locally and globally with high accuracy.
Abstract: 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.

107 citations