應用於三維微波成像中之重建過疏採樣的插值方法

在許多高頻的微波成像應用當中，奈奎斯特取樣定理所規範的取樣頻率常常造成系統製作上的困難，或者是量測時大量的時間成本。然而過少的取樣資料根據奈奎斯特取樣定理則無法正確還原，而降低微波成像的影像品質。為了解決此問題，本論文主要透過兩種不同的插值方法來進行過疏採樣的函數重建。值得一提的是，不同於壓縮感知探討隨機且不等距取樣間距的情況，本論文探討等間距取樣的情況，以符合多數成像系統中獲取資料的實際情形。以本實驗室之三維微波成像全像術演算法作為基礎，本論文首先針對開路電壓函數的數學行為進行分析，並以其連續平緩的性質為做為出發點，利用薄板樣條插值法分別對於開路電壓的振幅以及相位進行重建。其中相位會因其週期性的不連續而造成插值結果有誤差。因此在進行相位的插值重建之前，須透過相位展開來消除相位資料中的不連續。本論文利用一個品質導向的二維相位展開作為預處理的方式，消除相位取樣資料中的不連續後，再利用插值來還原出完整的相位分布。如此一來，即便在取樣區間不符合奈奎斯特取樣定理的條件，也可以利用插值來還原出清晰的物體影像。薄板樣條插值有著將函數平滑化的性質，也因此無法準確地插值重建變化速度較快或梯度較大的資料。為了解決這個問題，本論文利用有理型薄板樣條插值，透過將插值函數表示成有理數的方式，更精確的插值出完整開路電壓分布中變化較為快速的部分。在決定有理型薄板樣條插值的函數形式時，我們利用最小彎曲能的條件，制定出一個最小特徵值問題，並用該問題的解來決定插值函數所使用的基底函數及其係數，最後再對離群值進行偵測，刪除後再重新評估出合理的估測值，如此一來，即可還原出開路電壓的完整分布。為了驗證所提出插值方法的效用，本論文利用電磁數值模擬得到的散射場資料來進行成像，並且與現有的其中一種常見壓縮感知演算法―SPGL1演算法進行比較。結果顯示所解算出的物體影像品質都能透過本論文的插值方法獲得顯著改善，並且優於SPGL1演算法解算出的物體影像品質，然而薄板樣條插值受限於相位展開本身對於過疏採樣的限制，因此無法順利在極度過疏採樣的情況下進行成像，而有理化薄板樣條插值則沒有這項問題，因此潛在的應用範圍更為廣泛。除此之外，本論文亦利用隨機雜訊進行測試，而大部分結果也驗證了這些插值方法對測量誤差的穩健度。未來期待能將此插值方法用於實際測量得到的散射場資料上。關鍵字: 相位展開、薄板樣條插值、三維微波成像術。

關鍵字

相位展開；薄板樣條插值；三維微波成像術

並列摘要

In many microwave imaging applications, the sampling criterion posed by the Nyquist sampling theorem often leads to infeasibility in system design or excessively long data-acquisition time. However, according to the Nyquist theorem, one is unable to reconstruct from undersampled data, which leads to a degradation in image quality. To tackle this issue, interpolation methods are proposed to reconstruct the full data from undersampled data. It is noteworthy that, unlike compressed sensing where random undersampling is considered, the uniformly undersampled, or equidistant undersampling, is concerned since they are closer to most imaging system designs or measurement setups. Starting with a review of the formulation based on the open-circuit voltage previously proposed in our lab [1], the function behavior of open-circuit voltage is briefly discussed. The smoothness properties of the magnitude and phase of open-circuit voltage are exploited for function reconstruction using the thin-plate spline interpolation. However, the 2π-ambiguities between the original phases and their principle argument angles incur an angular discontinuity of 2π wherever the original phases surpass an interval of 2π. This contradicts the smoothness assumption posed by the thin-plate spline interpolation. Hence, a quality-guided two-dimensional phase unwrapping algorithm is adopted to serve as a pre-processing step in phase interpolation to resolve the 2π-discontinuities and recover the original smooth phase distribution. After recovering the magnitude and phase using the thin-plate spline interpolation, the full data of the magnitude and phase of the open-circuit voltage can be recovered. As a result, a clear image of the scatterer can still be obtained even with undersampled data. Nevertheless, the smoothness property of the thin-plate spline interpolation method brings some disadvantages. First of all, the steep-gradient parts or discontinuities in the full data, if any, will be lost, which contain important information about the boundaries of the scatterer. Second, it makes the interpolation method vulnerable to fluctuating sampled data. To solve this issue, the rational form of the thin-plate spline interpolation is applied to accommodate the rapid variations in the sampled data. In the determination of coefficients in the rational form, the smallest bending criteria for rational bases are used, and a corresponding smallest eigenvalue problem is formulated and solved to give the coefficients in the rational form. Finally, a remedy procedure for outliers is performed to replace singularities with reasonably estimated values. As a result, the full data can be reconstructed, and the rapid variations in the full data can be approximated more precisely. To verify the interpolation methods proposed in this work, images of several scatterer shapes and arrangements with various sampling distances are reconstructed from the full-wave simulated scattering data. These interpolation methods are also compared to a common compressed sensing algorithm called the SPGL1 algorithm. All the results show significant improvements in the qualities of the reconstructed images with these interpolation methods, which are also better than those with the SPGL1 algorithm. However, the TPS method suffers from the restriction of the unwrapping process in extremely undersampled cases. On the contrary, the rational TPS method is free from unwrapping and hence possesses more potential in wider imaging applications. In addition, this work also examines the imaging robustness of these interpolation methods. Most of the results show reasonable robustness with respect to random noises. It can be expected that these interpolation methods can also bring benefits to the reconstruction of undersampled data in real measurement setups. Keywords―phase unwrapping, thin-plate spline interpolation, three-dimensional microwave imaging.

並列關鍵字

phase unwrapping ； thin-plate spline interpolation ； three-dimensional microwave imaging

參考文獻

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