achieved by applying compressed sensing techniques. In this work, using effective modelling of possible defects, such as air gaps between layers, we constructΒ ...
Compressed Sensing and Defect-Based Dictionaries for Characteristics Extraction in mm-Wave Non-Destructive Testing Edison Cristofani1,2, Mathias Becquaert1,2, Gokarna Pandey2, Marijke Vandewal1, Nikos Deligiannis2,4 and Johan Stiens2,3 1
CISS Department, Royal Military Academy, 30 Av. de la Renaissance, 1000 Brussels, Belgium 2 ETRO Department, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium 3 SSET-department, Imec, Kapeldreef 75, 3001 Leuven, Belgium 4 iMinds, 9050 Ghent, Belgium
AbstractβIn ultra-wideband non-destructive testing of large multilayered polymers, data collection and reduction can be achieved by applying compressed sensing techniques. In this work, using effective modelling of possible defects, such as air gaps between layers, we construct defect dictionaries and use them as support data for a signal similarity-based classifier, which will automatically extract the main characteristics of the inspected defect.
I. INTRODUCTION
N
ON-Destructive Testing (NDT) of industrial materials or parts is a fundamental aspect in maintenance to guarantee and maximize life-span of larger structures. Teο¬on, ABS or ο¬berglass can be found in critical parts which undergo strong mechanical stress such as wind turbine blades, aircraft wings, or certain gas and oil pipelines. These materials present a very high transparency at mm-wave frequencies enabling see-through testing capabilities [1]. Although testing at higher mm waves is an attractive option, parts of interest can be relatively large, thus performing any type of measurement becomes cumbersome if not impractical. This situation can be even more accentuated when inspection systems produce resolutions down to the cm or mm. A reduction in the measurement time and data collection burden is therefore a desirable solution, provided that inspection performance remains unaffected and the material characteristics are properly extracted. II. COMPRESSED SENSING AND ULTRA-WIDEBAND NDT Sensors providing millimeter scanning accuracies generate enormous amounts of data due to the high number of measurement positions required. Achieving full signal reconstruction using lower sampling rates can be decisive when it comes to guaranteeing NDT using Ultra-Wideband (UWB) sensors. The sparse sampling theorem that forms the crux of Compressed Sensing (CS) proves to be an effective technique for reducing the amount of collected data, assuming that the signal of interest has a sparse representation in a given basis or learned dictionary [2]. Let π₯ β βπ be a vector containing a complete measurement and π¦ β βπ be a partial or incomplete observation satisfying π < π. Let π¨ be a π Γ π matrix defining the relation π¦ = π¨π₯, which corresponds to an underdetermined linear system with infinite solutions. This system may only be solved by imposing a priori knowledge of unknown vector π₯ being sparse, that is, most of its elements are zero. Additionally, matrix π¨ must satisfy one of the following conditions: the Restricted Isometry Property (RIP), the null space property, and the mutual coherence property [4]. In order to solve this problem, a minimization problem can be formulated as
π₯Μ = arg minβπ₯β0 π . π‘. π¨π₯ = π¦, π₯
(1)
where βπ₯β0 is the β0 -pseudo-norm of vector π₯, which imposes sparsity. The two described sets of constraints become impractical but can be relaxed using the following considerations [5]: i. ii.
The β0 -pseudo-norm makes the minimization in (1) an NP-hard problem. The β1 -norm is used instead. Satisfying the constraints for π¨ is unrealistic for large datasets. Populating matrix π¨ with randomly generated elements obeying a Gaussian distribution allows vector recovery with high probability.
In the application hereby proposed, we consider the vector π₯ to be the fully sampled vector according to the ShannonNyquist sampling theorem [6]. In a conventional, noiseless measurement, matrix π¨ would be the identity matrix of size π Γ π with vectors π₯ and π¦ of length π. For practical reasons, π¨ cannot be a random matrix as stated in consideration ii) and random sampling is implemented by randomly selecting subsets of the identity matrix. The sparsity condition may not always be satisfied in the measurement domain and, thus, a π Γ π transformation matrix πΏ is added to the problem statement in (1) resulting in π¦ = π¨πΏπ, where π = πΏ T π₯ is defined as the transformed and sparsified representation of π₯. We solve the proposed problem by applying CS algorithms based on orthogonal matching pursuit [7] or convex relaxation [8]. UWB sensors can beneο¬t from CS principles in two ways: Firstly, lower sampling rates produce less data to be stored or transferred, and thus a simpler, less hardware-demanding detection chain is required [3], [9]. Secondly, CS is of special interest when using UWB stepped-frequency systems for NDT, for which discrete frequencies within the operation bandwidth are measured multiple times to stabilize the signal and improve Signal-to-Noise Ratio (SNR). Therefore, reducing measured frequencies thanks to CS implies reducing measurement times as well. III. SCENE DESCRIPTION AND EXPERIMENTAL SETUP Air gaps produced by defective gluing during the production process are common artifacts in multilayered materials and can affect structural integrity at a larger scale. In order to understand the possibilities of applying mm-wave inspection and CS, two 20-mm (π€1 = π€2 = 20 ππ) ABS pieces (refractive indices π1 = π2 β 1.7 [10], see Fig. 1) are produced to simulate controlled air gaps π ranging from 1 mm to 15 mm and including the no-defect case, or zero-thickness air gap. These configurations are measured with a mm-wave vector network analyzer operating a in the 40 β 75 πΊπ»π§ range and a frequency step of 5.83 ππ»π§.
Random DS factor
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sensor
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0.7 0.4
0.6 0.5
0.6
0.4 0.3
0.8 1 -10
In-depth dimension
Fig. 1: Schema of the proposed measurement setup. Two materials π1 and π2 are placed in parallel at a certain distance (air gap).
0.8
0.2 0.1 0
10 SNR (dB)
20
30
Fig. 2: Successful reconstruction rates of two-layer measurements over 50 iterations. π·π varies between 0.05 and 1. Operation point for typical SNR yields a minimum DS factor of 0.2.
The in-depth resolution in free space is defined as ππ β2π΅ β 4 ππ, where ππ is the speed of light in free space and π΅ is the systemβs bandwidth.
Tests on real measurements show that sparse data recovery using CS theory allows extraction of the thickness and location of air gaps within the explored materials by reconstructing the media interfaces. To achieve such results, we use a Fourier sparsifying matrix πΏ with the purpose to increase sparsity since the data recorded directly from the sensor is not sparse enough. Fig. 2 shows actual results for successful air gap reconstruction (1 perfect match; 0 no match) for a range of random downsampling factors and SNR. The downsampling factor is defined as π·π = π/π, where π is the number of randomly measured samples. The no operation region can be clearly identified in the left-hand-side of the figure, due to low SNR and/or low π·π. For a typical measurement under controlled conditions, SNR may vary between 15 and 25 dB. Thanks to applying CS principles, our approach enables a downsampling factor as low as 0.2 without lessening the overall sensor performance. Fig. 3 depicts a successful reconstruction of a scene presenting a 9-mm air gap under typical SNR (20 ππ΅) and keeping only 20 % (π·π = 0.2) of the original samples. Although the amplitudes of the media interfaces show a slight mismatch, the locations are perfectly determined. Thus, by choosing the adequate parameters, our CS-based approach enables faster exploitation of the see-through capabilities of UWB sensors in the mm-wave range while preserving system performances. Regarding future work, automatic air gap determination thanks to robust classification via the β1 minimization [11] will be possible by using either real observations or a simulator to model and generate ideal observations of air gaps that will populate a dictionary of possible defects in (1). This technique, which operates like a similarity-based classifier, can be then used during the sparse data reconstruction process to recover not only the original sensed signal data but also to determine which of the elements in the dictionary provides the highest likelihood. Combining this approach with different sparsifying bases and dictionary learning techniques could reduce even more the amount of required data for perfect measurement reconstructions.
Arbitrary amplitude
IV. RESULTS AND FUTURE WORK
1
Original signal Reconstructed signal
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In-depth dimension
Fig. 3: Reconstruction of a 9-mm air gap using a two-layer configuration in ABS and typical SNR and π·π = 0.2. Interfaces are perfectly located.
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