擴展壓縮取樣應用於渦電流三維重構

Abstract

工業4.0的到來使得大型結構的品質檢測顯得愈顯重要。傳統網格掃描的檢測方式需要超過兩倍的取樣空間頻率才能達到預期的空間解析度。另一種取樣方法為壓縮取樣；利用訊號的稀疏性，比起奈奎斯特取樣需要更少的取樣數即可達到相同的解析度。然而，僅依靠訊號稀疏性的壓縮取樣在需要更詳細的深度分辨率時將會面臨困境。在本研究中，三維重構指的是重構結構內缺陷的位置和深度。關鍵在於結構中的缺陷是二元的：存在與否（存在為1，不存在為0）。因此，訊號除了稀疏性也有二元性，或者在某些情況下，值介於0和1之間以表示某些區域內的局部缺陷。 本研究著重於從線性量測中還原二元向量。解決方案有二種；其一為放寬二元約束，並採用凸優化演算法來求解。此外，文中證明演算法的收斂性。其二方法是在向量上引入伯努利先驗，並以變分法近似已知量測條件下向量的後驗機率。這些演算法可以適用於單位區間向量的還原。凸優化方法本身能夠處理單位區間向量的還原；於此同時，基於機率推論的演算法可以透過引入貝塔先驗來達成。這些演算法使用高斯隨機量測矩陣或具有共線行的量測矩陣進行測試，並在二元和單位區間向量的還原任務上展現出優於現有壓縮取樣演算法的效能。 在應用方面，使用微擾分析以線性化磁通量密度量測值和待檢測結構材料性質之間的關係。線性化的靈敏度矩陣實質上為兩個電場的內積：由線圈中的電流密度感應的電場，以及由磁感測器處的點磁流密度感應的電場。為求有效率的計算靈敏度矩陣，推導了幾種幾何形狀下電磁場的半解析解，並使用有限元素法進行驗證。 本研究將二元向量還原演算法應用於多層金屬板中缺陷的三維重構，使用數值模擬數據和實際實驗數據進行缺陷重構。結果顯示，開發的演算法在深度解析度和重構品質方面優於現有的壓縮取樣演算法。具體而言，該感測系統能夠使用間隔為4毫米的磁感測器陣列，達到重構小至2毫米的缺陷並有0.5毫米的深度解析度。 另一例為將單位區間向量還原演算法應用於金屬管的檢測。隨著感測探頭沿著金屬管移動，重構結果被重合為一。這兩個例子不但突顯了渦電流感測中高效取樣的潛力，也為應用擴展壓縮取樣至更多的物理重構問題奠定了基礎。

With the advent of Industry 4.0, ensuring the quality inspection of large structures has become increasingly crucial. Traditional raster scanning methods necessitate sampling at more than twice the spatial frequency to achieve a desired spatial resolution. Another sampling paradigm, known as compressive sampling, exploits signal sparsity to achieve the same resolution using fewer samples than Nyquist sampling would otherwise require. However, compressive sampling, which rely solely on sparsity, encounter challenges when a detailed depth resolution is necessary. In this study, 3D reconstruction refers to reconstructing both the position and depth of defects within a structure. The key is that defects in structures are binary; they are either present (represented as 1) or absent (represented as 0). Therefore, the signal of interest is not only sparse but also binary, or in some instances, have values between 0 and 1 to indicate partial defects in certain regions. This study focuses on the recovery of binary vectors from linear measurements with two approaches. One approach relaxes the binary constraint and employs convex optimization algorithms to solve the problem. Additionally, convergence of the algorithm is proved. Another approach introduces a Bernoulli prior on the vector and computes the variational approximation of the posterior probability of the vector conditioned on the measurements. These algorithms can be adapted to recover unit interval vectors. The convex method inherently handles the recovery of unit interval vectors, while the probabilistic inference-based algorithms can be modified with a beta prior to achieve this as well. These algorithms are tested using measurement matrices that are either Gaussian random or have collinear columns, and demonstrate superior performance compared to existing compressive sampling algorithms on binary vector recovery tasks. On the application side, perturbation analysis is applied to linearize the relationship between the magnetic flux density measurements and the material properties of the inspected structure. The linearized sensitivity matrix is essentially the inner product of two electric fields: one induced by electric current density in the coil, and the other induced by a point magnetic current density at the magnetic sensor. For efficient calculation of the sensitivity matrices, semi-analytical solutions are derived for the electromagnetic fields in several geometries and validated against finite element methods. Binary vector recovery algorithms are applied to provide 3D reconstructions of defects in a multilayer metal plate, using both numerically simulated and experimental data. The results show that the developed algorithms provide superior depth resolution and reconstruction quality compared to existing compressive sampling algorithms. Specifically, the sensing system is capable of reconstructing defects as small as 2~mm with a depth resolution of 0.5~mm, using a magnetic sensor array with 4~mm intervals. Another example applies unit interval vector recovery algorithms to the inspection of a metal pipe. Reconstructions are merged together as the sensing probe move along the pipe. These two examples not only highlight the potential of efficient sampling in eddy current sensing, but also establish a foundation for applying the extension of compressive sampling to a broader range of physical reconstruction problems.