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1 January 2001 Fault detection and feature analysis in interferometric fringe patterns by the application of wavelet filters in convolution processors
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1.

Introduction

In the field of nondestructive material testing several optical techniques, e.g., shearography, classical and holographic interferometry, are applied. New rewritable memory materials like photorefractive crystals or bacteriorhodopsine and digital holography lead to the ability to record hologram interferometric fringe patterns at video frame rates. These image data should be preprocessed, classified, and evaluated in real time.

Besides strictly digital image processing, hybrid optoelectronic systems are a point of current research and development.

Fig. 6

Extraction of horizontal features of an eye-chain-class defect structure and second filter step resulting in a line-shaped pattern.

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Fig. 9

Three-dimensional sketch of the correlation output applying a Morlet filter to the input image of Fig. 6.

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In the frequency plane of our current optical system (Fig. 3) we are only able to modulate amplitude or phase of the incident light wave. Concerning the optical realization of wavelet filtering we are principally engaged in the implementation of Hermitic real-value functions. The appropriate Fourier transform of these functions is positive real valued and can be addressed as a transmission distribution to a SLM. Strictly speaking, the twisted nematic liquid crystal SLMs always show a coupled amplitude and phase modulation, but they are not selectively addressable. So, with a certain polarizer and analyzer configuration, it is possible to operate in an amplitude- or phase-mostly (optimized) mode.6 7

Fig. 3

Optoelectronic image processor.

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In the presented simulations and optical implementations we focused on three wavelet functions: Mexican-hat wavelet, Morlet wavelet and Lemarie-Battle wavelet

Eq. (1)

ΦMex(x)=(1x2)ex/2,

Eq. (2)

[FΦMex](k)=(2πk)22πex/2,

Eq. (3)

ΦMorlet(x)=2 cos(2πkox)ex/2,

Eq. (4)

[FΦMorlet](k)=2π[e2(π(kk0))2+e2(π(kk0))2].
The Morlet wavelet is derived from a cosine function attenuated by an exponential function. The Mexican-hat wavelet function can be deduced from the second derivate of the Gaussian function. The Lemarie-Battle wavelet is a multi-resolution approximation built on polynomial splines of order 2p+1 (p denotes the degree of continuous differentiability). Lemarie8 showed such approximation by the scaling function

Eq. (5)

[FΨ](k)=1knΣ2n(k)with n=2+2pand

Eq. (6)

n(k)=m=+1(k+2mπ)n,2(k)=14 sin2(k/2).
We can derive the Fourier transform of the wavelet (see Refs. 8 and 9) as

Eq. (7)

[FΦLemBat](k)=eik/2kn2n(k2+π)2n(k)2n(k2).
The transmission distributions of these three functions in the Fourier plane are presented in Fig. 2. The profiles in Fig. 1 demonstrate the different localizations in spatial and frequency domain. As the Morlet function is strongly localized in the frequency plane, it shows a wider extension in spatial domain. The Mexican-hat shows the least extension in spatial domain, whereas the Lemarie-Battle wavelet takes some midway position within this group.

Fig. 1

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Fig. 2

Amplitude of the isotropic wavelet filters in frequency domain: Mexican hat, Morlet and Lemarie-Battle.

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For spatial filtering applications they can be realized as two-dimensional isotropic or as separable filter distributions. The property of rotation symmetry leads to a radius-only dependent expression of these functions

(kk=kx2+ky2)
as two-dimensional isotropic functions.10 Also separable filters can be realized by applying a Gaussian function in the second dimension, where the bandwidth determines the extension and the strength of the separability, respectively. These separable filters allow the extraction of spatial frequencies of certain resolution and alignment.

3.

Simulations and Optical Realization

The optical correlation has been simulated in a digital 4-f system, which suits perfectly to investigate the characteristics of the single wavelet function. However, all these results were simultaneously reviewed in the optical system, on the one hand to evaluate the quality and sensitivity of the setup and on the other hand to estimate the feasibility of the single filter procedures. Concerning the classification of the fringe patterns we adopted the graduation made at the BIAS (Bremer Institut fu¨r Angewandte Strahltechnik).11

The current optical setup (Fig. 3) contains in the input plane as well as in the filter plane twisted nematic liquid crystal displays with a pixel pitch of 40 μ m at VGA resolution.12 Because there is a scaling offset between the optically realized power spectrum and the addressed filter function (at λ=633 and 240 mm focal length of the FT lens), the functions do not match pixel on pixel.13 Furthermore, due to the overlap of higher orders of diffraction in the detector plane, only a part of the input display can be used for addressing the input image. However, for investigating the filter method and realizing first experiments they are convenient, because they show a very flat surface,13 14 a high filling factor and a remarkable contrast ratio (intensity ratio 1:1000). The latter property enables us to also use these SLMs in the filter plane. Earlier SLMs did not satisfy in resolution and especially not in contrast, so we had to use static filter distributions realized as chrome masks on quartz glass.

The early aim of our image processing task is the detection of defect patterns. Furthermore, the obtained features contain the enhanced defect structure and each feature itself represents a data reduction compared to the input image. Figure 4 shows samples of such fringe patterns recorded in a real-time hologram interferometric setup and charge coupled device camera shots of the processed images. The bent fringes are caused by a structural defect within the material or, as can be seen in the second sample, compressed by an applied stress. The output image of a first filter procedure utilizing a separable wavelet shows edges of high contrast. The parameters concerning the rotation and separability indicate the alignment and extension of the bend or the boundary of the compression. Depending on the directional approach of the filter function to the spatial frequencies of the defect structure in the Fourier plane, the extracted feature can appear on different sides of the edge (Fig. 5).

Fig. 4

Feature extraction in interferometric fringe patterns: Detection of bend and compression structures.

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Fig. 5

Extended filtering procedure: The upper row shows the input image and detected bends from both directions. In the lower row the depicted separable Mexican-hat wavelet applied on the extracted features reduces the feature to a dual curve.

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A second possible filtering step depicted in Fig. 5 leads to another reduction of the image volume to a dual curve. The image data were reduced to the minimum necessary information: the position and the extent of the bend. The intensity minimum between the curves denotes the position of the bend. Here especially the Mexican-hat wavelet is suited to determine the position because of its more localized behavior in the spatial domain.

Embedded impurities below the materials surface cause circular structures (circlets) in the interferogram, categorized as “eye” and “eye chain” in Ref. 11 for single impurities or crowds of them. In order to detect those structures, again combinations of separable filters lead to an extraction and data reduction. Figure 6 depicts the extraction of horizontal components in the image and the conversion into vertical lines. A second approach implies the application of isotropic Morlet filters to correlate the circlets.

4.

Classification and Real-Time Detection

The task of the optical preprocessing consists of the reduction of the image data and the support of a classification. In the previous chapter, methods in wavelet filtering were presented describing the optimal extraction of a desired feature. If we combine the different extraction characteristics of the wavelets we can form an algorithm to distinguish between the already mentioned categories of interferograms.11 Of course, those classes were graduated due to the different defect causes within the material. So, the shape of the fringe patterns is often quite similar or they merge from one class to another. For instance, a local bend will also lead to a compression in its surroundings. In Fig. 7, a simple algorithm for classifying the fringe patterns is sketched. The most striking defect structure is the eye structure, which should be detected first by applying a Morlet wavelet. Of course, every kind of wavelet filter depicted in Fig. 7 means a set of those filters with different scaling factors. The Morlet filter correlates the eye structure to a peak, in contrast to the other classes, which show different kinds of line-shaped features.

Fig. 7

Sketch of a classification algorithm.

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Subsequently, the application of separable filters leads to distinctive features for the “compression” and the “bend” class. Both show as extracted features fringe patterns, which indicate an edge at the position of the bend or the compression transition. Varying the scaling of the applied filter, the different parts of compressed fringes can be detected showing both an edge of high contrast at the same position. For the bend class, a rotation of the filter also leads to the extraction of that side of fringe pattern according to its rotational direction. The classes “groove” and “displacement” are nearly indistinguishable by this method. In general, a displacement shows a stronger feature, because the fringes are highly distorted.

In order to apply an optoelectronic image processor for industrial quality control we implemented video sequences of fringe patterns to simulate a real-time operation. The video sequences showed an eye structure caused by a spherical embedding. Since the pressure was increased while recording the sequence, the fringes spread cause of the billowed surface and also the feature changed its shape. So, it was not necessary to employ a full set of filters (multi-resolution analysis = MRA), but one single Morlet filter frequently showed a peak. This strategy leads to a kind of inverse MRA, because here the feature scales itself instead of the filter. In Fig. 8, a set of frames represents the experimental output data, which were set in comparison to the input sequence.

Fig. 8

Filtering of real-time sequences under rising pressure. The feature, which is detected by the applied Morlet filter, alternates for a certain pressure change.

010101j.8.jpg

5.

Conclusion and Future Direction

In the presented work we have demonstrated a basic approach of optical preprocessing of interferometric fringe patterns in simulation and experimental realization. The wavelet transform has been proven as a suitable tool for extracting features, which contain information about the defect properties and support a subsequent classification process.

These studies were done using samples of interferometric fringe patterns and sequences of real-time recorded fringe patterns. The investigation concerning the classification properties should be extended, because the number of test images and sequences was limited. The values of varying physical parameters which lead to moving fringes in the video sequences will be taken into consideration. So, the characteristic change of the extracted feature can be set into proportion of the varying physical parameters (e.g., pressure, temperature).

The investigations concerning the correlation of circular structures by Morlet-wavelet filters of different spectral resolution will be extended, because the first results (Fig. 9) seem to be promising with respect to the assignment of correlation peak intensities, spectral resolution and defect attributes and position.

The results of current measurements on the SLMs to determine the full complex modulation behavior will increase the quality of the optical filtering. The experimental setup will be modified to a system using a XGA-resolution SLM in the input plane and a ferroelectric binary 256×256 display in the filter plane addressable with kHz frame rates.

Acknowledgments

The interferometric images were placed at our disposal by the BIAS. The investigations in applying wavelet functions and the further developments and analysis on the presented optical hardware are supported by the Deutsche Forschungsgemeinschaft under Contract No. El 147/1-2 and by the Bundesministerium fu¨r Bildung, Wissenschaft, Forschung und Technologie, Germany, under Grant No. 03-WE9HU1-5.

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Sven Kru¨ger received his MS degree in physics from Humboldt-University, Berlin, in 1997. Currently, he is working on his PhD thesis. He is particularly interested in the application of coherent methods in optical pattern recognition and the development of computer generated diffractive optical elements.

Gu¨nther Wernicke received his MS in 1966 in applied mechanics, a PhD in 1975 in holographic interferometry from the Rostock University, and his Dr. sc. in 1985 in mechanical engineering from the Technical University Dresden, Germany. From 1969 to 1983 he was with the Institute of Mechanics in Berlin working in the application of holographic interferometry in applied mechanics. Since 1983 he has been with Humboldt-University, and currently heads the Laboratory of Coherent Optics. His interests are the application of coherent optical methods in mechanics and biophysics. He has published a book on holographic interferometry and more than 70 papers in international journals and conference proceedings. He is a member of the SPIE.

Wolfgang Osten received his BSc from the University of Jena in 1979. From 1979 to 1984 he was a coworker of the Institue of Mechanics in Berlin. In 1983 he received his PhD degree from the University of Halle in the field of holographic interferometry. From 1984 to 1991 he was employed at The Central Institute of Cybernetics and Information Processes, Berlin, making investigations in digital image processing and computer vision. Since 1991 he has been employed at the Bremen Institute of Applied Beam Technology (BIAS) as the head of the Department of Optical 3-D Sensing. He is concerned with new concepts in industrial inspection by combining modern principles of optical metrology and image processing. He is a member of SPIE.

Nazif Demoli received his BS in 1978, his MS in 1983, and his PhD in 1989 from the University of Zagreb, all in physics. He joined the Institute of Physics, University of Zagreb, as a postgraduate scholar in 1979, where he has been full time since 1979. His main research activities are in optical and computer pattern recognition, synthetic discriminant functions, spatial light modulators, and holography. He has published more than 50 papers in international journals and conference proceedings. His main contributions to the field in the sense of introducing original systems and techniques are an extended optical correlator, quasit-phase-only matched spatial filtering, the feasibility to estimate correlation filters, and the coherent optical averaging of signals. He is a member of SPIE.

Hartmut Gruber received his MS degree in physics from Humboldt University in 1982. He is a research engineer at the Laboratory of Coherent Optics. His work includes biophysics, holographic interferometry, and optical and digital image processing.

©(2001) Society of Photo-Optical Instrumentation Engineers (SPIE)
Sven Krueger, Guenther K.G. Wernicke, Wolfgang Osten, Daniel Kayser, Nazif Demoli, and Hartmut Gruber "Fault detection and feature analysis in interferometric fringe patterns by the application of wavelet filters in convolution processors," Journal of Electronic Imaging 10(1), (1 January 2001). https://doi.org/10.1117/1.1318908
Published: 1 January 2001
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