希爾伯特轉換增強強度傳播解得之相位影像

Abstract

最近已經有許多相關文獻提到，沿著傳播方向量到的強度分佈微分後，透過所謂的強度傳播方程式 (Transport of Intensity equation, TIE) 可以解得相位影像。此 TIE是由霍姆赫茲波動方程式 (Helmholtz wave equation) 加以推導獲得，其形式為波松方程式 (Poission equation)。因此利用 TIE這個以非干涉性基礎做為相位恢復的方法，已經被應用在很多實驗上。證實此方法是可以取代傳統透過干涉成像技術獲得相位影像之方法；因為利用干涉成像這個方法，在光學系統技術上，要求相對來說比較多。 我們架設簡單實驗以取得強度分佈微分後，透過解TIE衍生出來的3種方法，即傳統 TIE（conventional TIE, c-TIE）法，一維偏微分法（one partial derivative, OPD）和直接希爾伯特（Direct Hilbert transform） 法，來還原非週期性的蜻蜓翅膀結構與週期性的微透鏡陣列之強相位片之相位影像。另外，透過 Hilbert transform 處理後，可以發現能夠增強邊緣和圖像本身的內部構造。我們發現使用c-TIE 加上 Hilbert transform處理過後和 Direct Hilbert transform 的還原影像結果是雷同的且較佳。所以透由 Hilbert transform取代解相位梯度變化的過程，不僅讓相位還原的過程變簡單，相位圖像也會變平滑。此外，透過 OPD 還原的影像則是因為本身在 y 方向微小濾波參數的影響，會產生許多條紋雜訊的干擾。

It has been known that the full field can be calculated if the derivative of intensity profile along the propagation is measured by solving the transport of intensity equation (TIE), derived from the free space Helmholtz wave equation under the paraxial wave approximation. Using the TIE, phase recovery by non-interferometric propagation-based techniques has been employed in many experiments to replace the interferometric imaging, in which phase recovery is based on the analysis of experimental interferograms and is technically demanding. Separately using aperiodic dragonfly wings and periodic structure a micro-lens (ML) array, a strong phase plate, as our specimens, we demonstrate recovery of phase images by using conventional TIE (c-TIE), one partial derivative (OPD) and direct Hilbert transform methods. Furthermore, take the advantages of the Hilbert transform for enhancing edges and fine structures of the images, we found in Hilbert treated c-TIE as well as direct Hilbert transform methods display similar and better results. Replacing the process of solving the phase gradient by the Hilbert transform, it is not only simpler but also has smoother phase image than c-TIE and OPD alone. Due to the weak filtering with small parameter of lambda in the other dimension without derivative (says, the y-direction), the OPD images appear many streaks due to integration of noise line-by-line in y-direction.