As Photometric Stereo is a means of measuring the gradient field of a surface, an essential step in the measurement of a surface structure is the reconstruction of a surface from its measured gradient field. Given that the surface normals are subject to noise, straightforward integration does not provide an adequate reconstruction of the surface. In fact, if the noise in the gradient can be considered to be Gaussian, the optimal reconstruction based on maximum likelihood principles is obtained by the method of least-squares. However, since the reconstruction of a surface from its gradient is an inverse problem, it is usually necessary to introduce some form of regularization of the solution. This paper describes and demonstrates the functionality of a library of MATLAB functions for the regularized reconstruction of a surface from its measured gradient field. The library of functions, entitled “Surface Reconstruction from Gradient Fields: grad2Surf Version 1.0” is available at the MATLAB file-exchange<p> </p> http://www.mathworks.com/matlabcentral/fileexchange/authors/321598 <p> </p>
The toolbox is the culmination of a number of papers on the least-squares reconstruction of a surface from its measured gradient field, regularized solutions to the problem, and real-time implementations of the algorithms.<sup>1-4</sup>
This paper presents a machine vision system for the control of dual-shield Tunnel Boring Machines. The system consists of a camera with ultra bright LED illumination and a target system consisting of multiple retro-reflectors. The camera mounted on the gripper shield measures the relative position and orientation of the target which is mounted on the cutting shield. In this manner the position of the cutting shield relative to the gripper shield is determined. Morphological operators are used to detect the retro-reflectors in the image and a covariance optimized circle fit is used to determine the center point of each reflector. A graph matching algorithm is used to ensure a robust matching of the constellation of the observed target with the ideal target geometry.
This paper describes the physical setup and mathematical modelling of a device for the measurement of structural deformations over large scales, e.g., a mining shaft. Image processing techniques are used to determine the deformation by measuring the position of a target relative to a reference laser beam. A particular novelty is the incorporation of electro-active glass; the polymer dispersion liquid crystal shutters enable the simultaneous calibration of any number of consecutive measurement units without manual intervention, i.e., the process is fully automatic. It is necessary to compensate for optical distortion if high accuracy is to be achieved in a compact hardware design where lenses with short focal lengths are used. Wide-angle lenses exhibit significant distortion, which are typically characterized using Zernike polynomials. Radial distortion models assume that the lens is rotationally symmetric; such models are insufficient in the application at hand. This paper presents a new coordinate mapping procedure based on a tensor product of discrete orthogonal polynomials. Both lens distortion and the projection are compensated by a single linear transformation. Once calibrated, to acquire the measurement data, it is necessary to localize a single laser spot in the image. For this purpose, complete interpolation and rectification of the image is not required; hence, we have developed a new hierarchical approach based on a quad-tree subdivision.
Cross-validation tests verify the validity, demonstrating that the proposed method accurately models both the optical distortion as well as the projection. The achievable accuracy is <i>e</i> ≤ ±0.01 [mm] in a field of view of 150 [mm] x 150 [mm] at a distance of the laser source of 120 [m]. Finally, a Kolmogorov Smirnov test shows that the error distribution in localizing a laser spot is Gaussian. Consequently, due to the linearity of the proposed method, this also applies for the algorithm's output. Therefore, first-order covariance propagation provides an accurate estimate of the measurement uncertainty, which is essential for any measurement device.
This paper presents a new approach to optical material stress analysis, which eliminates the need to apply
a random dot pattern to the surface of the sample being tested. A multi-resolution hierarchical sub-division
is implemented, with a consistent polynomial decimation applied at each layer of the tree. The degree of
decimation must be selected depending on the nature of the structure of the surface of the sample being At each
layer the individual patches are registered using a modified normalized phase correlation, whereby the Fourier
basis functions are projected onto the orthogonal complement of a low degree Gram polynomial basis. This
reduces the effect of the Gibbs error on the local registration. The registration positions are then subjected to
a regularization via an entropy weighted tensor-polynomial approximation. The Gibbs polynomial basis is used
for the tensor product, since they are orthonormal and model the continuous deformation associated with an
elastic deformation. The stability of the proposed method is demonstrated in real measurements and the results
with and without the application of the random pattern are compared.
This paper presents a new approach to non-rigid elastic registration. The method is applied to hyper spectral imaging
data for the automatic quality control of decorative foils which are subject to deformation during lamination. A new image
decimation procedure based on Savitzky-Golay smoothing is presented and applied in a multiresolution pyramid. Modified
Fourier basis functions implemented by projection onto the orthogonal complement of a truncated Gram polynomial basis
are presented. The modified functions are used to compute spectra whereby the Gibbs error associated with local gradients
in the image are reduced. The paper also presents the first direct linear solution to weighted tensor product polynomial
approximation. This method is used to regularize the patch coordinates, the solution is equivalent to a Galerkin type
solution to a partial differential equations. The new solution is applied to published standard data set and to data acquired
in a production environment. The speed of the new solution justifies explicit reference: the present solution implemented
in MATLAB requires approximatly 1.3s to register an image of size 800 ×× 500 pixels. This is approximately a factor 10
to 100 faster than previously published results for the same data set.
This paper presents a new approach to non-rigid registration. A hierarchical subdivision approach is applied, with
local normalized phase correlation for patch registration. The major improvement is achieved by implementing a
suitable decimation at each level. The decimation is implemented via a Gram polynomial basis. Both global and
local polynomial approximation are considered and compared with the use of a Fourier basis. The issue of Gibbs
error in polynomial decimation is examined. It is shown that the Gram basis is superior when applied to signals
with strong gradient, i.e., a gradient which generates a significant Gibbs error with a Fourier basis. A bivariate
Gram polynomial tensor product approximation is used to implement regularization. It is demonstrated that
the new method performs well on both synthetic and real image data. The procedure requires approximately
1.3 sec. to register an image with 800 × 500 pixels.
We extend the theory of polynomial moments by proving their spectral behavior with respect to Gaussian noise. This opens the door to doing computations on the signal-to-noise ratios of polynomial filters and with this, the comparability to classical filters is made possible. The compactness of the information in the polynomial and Fourier spectra can be compared to determine which solution will give the best performance and numerical efficiency. A general formalism for filtering with orthogonal basis functions is proposed. The frequency response of the polynomials is determined by analyzing the projection onto the basis functions. This reveals the tendency of polynomials to oscillate at the boundaries of the support; the resonant frequency of this oscillation can be determined. The new theory is applied to the extraction of 3-D embossed digits from cluttered surfaces. A three-component surface model is used consisting of a global component, corresponding to the surface; a Gaussian noise component, and local anomalies corresponding to the digits. The extraction of the geometric information associated with the digits is a preprocessing step for digit recognition. It is shown that the discrete polynomial basis functions are better suited than Fourier basis functions to fulfill this task.
A thorough analysis of discrete polynomial moments and their suitability for application to geometric surface inspection is presented. A new approach is taken to the analysis based on matrix algebra, revealing some formerly unknown fundamental properties. It is proven that there is one and only one unitary polynomial basis that is complete, i.e., the polynomial basis for a Chebychev system. Furthermore, it is proven that the errors in the computation of moments are almost exclusively associated with the application of the recurrence relationship, and it is shown that QR decomposition can be used to eliminate the systematic propagation of errors. It is also shown that QR decomposition produces a truly orthogonal basis set despite the presence of stochastic errors. Fourier analysis is applied to the polynomial bases to determine the spectral distribution of the numerical errors. The new unitary basis offers almost perfect numerical behavior, enabling the modeling of larger images with higher-degree polynomials for the first time. The application of a unitary polynomial basis eliminates the need to compute pseudo-inverses. This improvement in numerical efficiency enables real-time modeling of surfaces in industrial surface inspection. Two applications in industrial quality control via artificial vision are demonstrated.
A new method for fitting implicit curves to scattered data is proposed. The method is based on orthogonal matrix projections and singular value decomposition. The incremental aspect of the algorithm deals with each order of data individually in an incrementing manner, whereby a matrix approximation procedure is applied at each level. This determines the fit quality at each step, and hence provides co-linearity detection of each polynomial order. The best implicit polynomial fit of minimal order is provided, which essentially combines object identification and classification with object fitting.