二次拉普拉斯算子正規化求最佳解之理論分析與計算方法

Abstract

在空間變化性的影像處理技法中，輔助資訊的準確猜測在各種影像強化的應用中佔了很重要的地位。藉由使用者的導引或是演算法的估計得到了輔助資訊粗略估計之後，接下來一個共同的方案便是再將輔助資訊準確對齊影像內容。這樣子精製的核心步驟中很常使用二次拉普拉斯算子正規化，因為它具有閉合型式解。然而，在對其產生的大型線性系統去求解的過程中，常常對現有的硬體運算系統中造成了很大的負擔，尤其是在運算時間或是記憶體的使用量上，因此發展了許多有效率的但不失準確度的計算演算法。另外，現在仍舊有許多致命但未知的難題造成無效的精製結果，這些難題一直使精製的方法無法實際應用。對自然影像的處理應用中，輔助資訊的精製品質也是個主要考量，而這點也影響要如何適當應用二次拉普拉斯算子正規化。 在這篇論文當中，我們先分析了二次拉普拉斯算子正規化的幾何意義，這當中使用它線性系統的代數性質去證明出一個會導致無效精製結果的因素。除此之外，我們也分析了二次拉普拉斯算子正規化的頻率性質，藉此可以幫助我們將它去類比傳統慣用的線性濾波器的性質，進而將它與一些常用的非線性抹平濾波器關聯起來。有了這些分析之後，我們以精製品質以及有效性的角度，比較了一些現存的輔助資訊精製方法，並藉此讓我們以頻率性質角度更適當的使用二次拉普拉斯算子正規化得到更好的精製品質。 相對應的，我們提出了一個最佳化求解的方法，這個方法可以套用現存的快速非線性濾波器以很有效率的方式去趨近求解，而且我們也套用了它的頻率性質，去證明了我們這個方法的紮實性。這個最佳求解的方法，也被證明了它可以在運算時間上表現更好，而無損於精製的有效性以及品質。我們也將它與現有的最佳求解法關聯，得澄清出它們還未被證實的風險問題。除此之外，這樣子的關聯也得而讓我們使用簡單而有效的傳統數學技巧去再進一步加速我們的最佳化求解法。因此，我們提出的方法的確是對二次拉普拉斯算子正規化精製演算法相當的有效且有效率。 考量到精製的品質，我們也提出了一個去雜訊演算法，以解決拉普拉斯算子使用時會放大雜訊影響的問題。這個去雜訊的演算法在去雜訊之餘也可以完美的保留了原本影像內容，基於一個有效的策略，而這策略是建於一個顏色在影像的局部只會散開在一條線的假設上。除此之外，我們適當的利用了導引濾波器去實現我們的去雜訊策略，當中完美的借用了積分影像技法。因此，這個去雜訊演算法可以有效且有效率的去處理影像，並進而使輔助資訊的精製可以更抵抗雜訊造成的不良影響。 總結而論，考量輔助資訊的精製品質以及有效性，我們對二次拉普拉斯算子正規化提供了綜合性的分析，也對它的最佳化求解提供了一個有效且有效率的方法。這些研究發展，可以在空間變化性的影像處理應用上，對它的品質、運算效率、以及有效性，幫助給我們一個全面性進化的思考參考。

In spatially varying image processing, the precise inference of auxiliary information dominates various image enhancement applications. Given the rough auxiliary information provided by users or inference algorithms, a common scenario is to refine it with respect to the image content. Quadratic Laplacian regularization is generally used as the refinement framework because of the availability of closed-form solutions. However, solving the resultant large linear system imposes a great burden on commodity computing hardware systems in the form of computational time and memory consumption, so efficient computing algorithms without losing precision are being developed. Additionally, there have been critical but unclear issues that result in ineffective refinement result, which have made the refinement applications impractical. For the natural image processing, the quality of auxiliary information refinement is also a dominant consideration, which affects how to apply quadratic Laplacian regularization appropriately. In this dissertation, we first analyze the geometric nature of the quadratic Laplacian regularization associated with the algebraic property of the corresponding linear system, which proves the unclear factors that lead to ineffective refinement results. In addition, we analyze the spectral property of the quadratic Laplacian regularizer, which helps us to make an analogy to the traditional filter design principle of LTI system and thereby relate the quadratic Laplacian regularization with the common nonlinear smoothing filters. Based on these theoretical analyses, we compare the existing auxiliary information refinement frameworks in terms of refinement quality and robustness of results, allowing us to apply quadratic Laplacian regularization appropriately to obtain good refinement quality by considering spectral property. Correspondingly, we propose an optimization scheme that is capable of approaching the solution in an efficient manner using existing fast local filters, and we perform spectral analysis to validate the robustness of this method. This optimization scheme is proved to outperform the existing optimization methods in total computation time without sacrificing effectiveness and refinement quality. We also connect our optimization scheme to some existing optimization methods, proving their potential issues that have not been clarified. In addition, this connection can also help us to accelerate our optimization scheme using the classical techniques with only simple and efficient efforts. Therefore, the proposed computation scheme provides an efficient and effective method for refinement process using quadratic Laplacian regularization. Considering the quality of refinement, we also propose a denoising method for suppressing the noise effect that is amplified by the classic Laplacian adoption. This denoising method can perfectly preserve the content while eliminating noise, based on a useful denoising strategy built on the local color line assumption. In addition, the guided filter is utilized appropriately to achieve this denoising strategy, cooperating with the integral image technique perfectly. Therefore, this denoising method can perform effectively and efficiently, and the refinement process is demonstrated to be more robust against noise by using our denoising method. In conclusion, we provide comprehensive analyses on optimizing the quadratic Laplacian regularization for considering the refinement quality and effectiveness and a methodology about performing optimization for evaluating refinement results efficiently and robustly. These developments can help the applications of spatially varying natural image processing to improve their quality, efficiency, and robustness with thorough considerations.