近年來，影像設備不斷地在進步，相機可拍攝的彩色影像畫素數量變得愈來愈多，然而，隨著單一像素愈做愈小，影像感測器的輸出訊號對光學雜訊的敏感度也愈來愈大，因此濾除雜訊成為影像處理最重要的議題之一，如何使用差分進化演算法設計一個有效適用於相機RAW影像的最佳雜訊抑制演算法，是本論文的第一個重點。
真實世界的最佳化往往存在著無法避免的不確定性，全域最佳解的性能對參數的微小變化很可能是非常敏感的，因此，真正要找的解是一個具有高容錯率且高穩定性的最穩解，而不是傳統的全域最佳解。本論文對於最穩解的定義是在不確定性影響最嚴厲情況下的最佳解，它可由一個制限最小最大最佳化問題所描述，而如何使用差分進化演算法有效地找出最穩解，是本論文的第二個重點。
針對以上兩個重點，本論文提出四個重要的議題進行探討。第一，提出基於特徵向量的交配運算子解決差分進化演算法性能對座標軸系統旋轉敏感的問題。第二，提出成功母體選擇法解決差分進化演算法在高維度多峰目標函數中容易停止收斂的問題。第三，探討實際相機取得的RAW影像的雜訊模型及雜訊抑制方法，並使用差分進化演算法設計一個基於模糊方塊配對之相機RAW影像雜訊去除演算法。最後，探討並提出制限最小最大最佳化問題的定義及定理，並提出一種限制啟動型的差分進化演算法解決制限最小最大最佳化問題。
差分進化演算法是解決連續空間全域最佳化問題最有效的方法之一，然而差分進化演算法的性能對於座標軸系統的旋轉是非常敏感的，尤其是在高條件數的目標函數下，差分進化演算法的性能更是明顯變差。因此本論文提出基於特徵向量的交配運算子來增強差分進化演算法的性能，透過候選解的共變異矩陣的特徵向量，使原本不具旋轉不變性的交配運算子具有旋轉不變性，解決了差分進化演算法敏感於座標軸系統旋轉的問題。並且提出的性能提升框架可以套用於任一種交配運算子上，只須極小的修改就可以運作。實驗結果證實提出的基於特徵向量之交配運算子在不可切割單峰函數上明顯地提升了差分進化演算法的性能，而在標準測試函數上，其整體平均性能亦有顯著提升。
其次，差分進化演算法的性能在高維度多峰複雜函數容易發生停滯問題，在問題發生時，差分進化演算法無法將解收斂到一個點上，也無法再找到更好的解。針對此問題，本論文提出成功母體選擇法來提升差分進化演算法的性能，在演化過程中儲存最近更新的解作為替代的母體，給予連續多次未更新的解一個重新選擇的機會，使母體的選擇來源不只是候選解本身，還多了替代的母體可供選擇。提出的成功母體選擇法可以套用於任何一種差分進化演算法，並且幫助差分進化演算法脫離停滯的狀態。實驗結果也顯示提出的演算法加快了差分進化演算法的收斂速度，也增快了候選解的更新頻率，更重要的是，提出的演算法幾乎在所有標準測試函數上都顯著地提升了差分進化演算法的性能，而整體平均性能更是有明顯的提升。
在應用的部分，將針對相機RAW影像雜訊去除進行討論。近年來，許多影像濾波器的研究假設影像雜訊是獨立且相同的高斯分佈，但事實上，光學雜訊的變異數和訊號強度經常呈現正相關性，而且影像感測器的排列設計也會影響雜訊的變異數。因此，本論文針對未處理過的RAW影像，提出基於模糊方塊配對之影像雜訊去除演算法將影像雜訊去除，提出的模糊方塊配對法尋找相似的方塊進行加權平均運算，此加權數由一個模糊邏輯系統給定，並利用變化穩定轉換法將雜訊變異數穩定化，使提出的方法在明亮與陰暗的區域都有優良的雜訊去除性能，最後使用本論文改良的差分進化演算法進一步提升模糊方塊配對法的性能。實驗結果顯示提出的演算法可有效地提升影像去雜訊的性能，其濾波結果在主觀視覺與客觀數據上皆優於兩個文獻上嶄新的影像去雜訊演算法。
最後，真實世界往往含有不確定性，由模擬實驗設計的最佳系統不一定適用於變化劇烈的真實世界，為了解決不確定性的問題，在設計階段就得考慮最穩解，而最穩解的尋找可以描述為一個制限最小最大最佳化問題。本論文提出制限最小最大最佳化問題的定義與定理，證明不論是最小最大化問題還是最大最小化問題，都可用最小最大化演算法解決，基於此定理，本論文提出一種限制啟動型差分進化演算法尋找最穩解，提出的演算法主要是透過直接尋找最靠近限制式邊界的解，大幅提升搜尋最穩解的效率。實驗結果顯示，在誤差小於1E–6的成功率評比下，提出的演算法在五個數值測試基準上皆表現出100%的成功率。

Recently, with the rapid advance of imaging devices, there are more and more pixels in a color image taken by a camera. However, the sensitivity of image sensor output signals to photon noise has become greater with the smaller size of each pixel. Therefore, denoising becomes one of the most important issue in image processing. How to design an optimal denoising algorithm by differential evolution for camera raw images is the first application in this dissertation.
Uncertainties are often occurred and unable to avoid in real-world optimization scenarios. The global optimum might be very sensitive to small variations in parameters. Therefore, a preferable solution is probably not the global optimum, but one with high tolerance and robustness to uncertainties. In this dissertation, the most robust solution is defined under the best possible worst-case performance, which can be described by a constrained min-max optimization problem. How to use differential evolution to effectively find the most robust solution is the second application in this dissertation.
Four topics are studied in this dissertation. First, an eigenvector-based crossover operator is proposed to solve the problem of differential evolution of which the performance is sensitive to the rotation of coordinate systems. Second, a successful-parent-selecting framework is proposed to solve the stagnation problem of differential evolution in high-dimensional multimodal objective functions. Third, with the analysis of the noise model and denoising method for raw images acquired by a digital camera, the differential evolution is used to design a fuzzy block-matching based denoising algorithm for camera raw image restoration. Finally, the definition and theorem of the constrained min-max optimization problem are specified and proposed, and a constraint-activated differential evolution is proposed to solve the constrained min-max optimization problem.
In the first topic, differential evolution is one of the most effective method to solve global optimization problems in continuous search domain. However, the performance of differential evolution is sensitive to the rotation of coordinate systems especially when the objective function is with high-conditioning. The performance of differential evolution may decrease dramatically. Therefore, an eigenvector-based crossover operator is proposed to enhance the performance of differential evolution. The eigenvector information of the covariance matrix of the solutions is utilized to make the crossover operator become rotationally invariant, and solves the sensitivity problem on the rotation of coordinate systems. The proposed operator can be applied to any crossover strategy with minimal changes. The experimental results show that the proposed eigenvector-based crossover significantly enhances the performance of differential evolution on non-separable unimodal functions. In standard test functions, the proposed operator also significantly improves the overall performance of differential evolution.
The stagnation problem of differential evolution is discussed in the second topic. When stagnation is happening, differential evolution cannot converge solutions to a fixed point, and the algorithm cannot find any better solutions. To solve this problem, a successful-parent-selecting framework is proposed to improve the performance of differential evolution. During evolution, recently updated solutions are stored into an alternative set of parents, which provides a solution, which is continuously not updated for more than unacceptable times, an alternative selection of parents. The proposed successful-parent-selecting framework can be applied to any differential evolution, and effectively helps the algorithm escaping the situation of stagnation. The simulation results show that the proposed framework accelerates the convergence speed of differential evolution, and increases the update rate of solutions. In addition, the proposed framework significantly improves the performance of differential evolution in terms of Wilcoxon rank-sum test on almost all standard test functions. The overall performance is also increased significantly.
Image denoising in practical cases is discussed in the third topic. Recently, many studies on image denoising assume image noise is a sequence of independent and identical distributed random variables which follow a Gaussian distribution. However, in fact, the variance of photon noise depends on the magnitude of signal. In addition, the arrangement design of image sensors also affects the noise variance. Therefore, an image denoising algorithm should be designed for an RAW image. In this dissertation, a fuzzy block-matching based image denoising algorithm is proposed to remove noise from an RAW image. The proposed block-matching finds similar blocks by the use of a fuzzy logic system. Then, these similar blocks are averaged with the weightings which are determined by the fuzzy logic system. A variance stabilization transform is used to stabilize the noise variance, and thus make the proposed method suitable to eliminate noise for both of bright and dark regions. Finally, the proposed differential evolution is used to further improve the performance of the proposed denoising algorithm. The experimental results show that the proposed denoising algorithm effectively improves the performance of image denoising. Furthermore, the average performance of the proposed method is better than those of two state-of-the-art image denoising algorithms in subjective and objective measure.
Finally, an optimal system which is designed from simulated environment might not be optimal in real world with uncertainties. To deal with the uncertainties, the most robust solution should be considered in design level, and it can be described as a constrained min-max optimization problem. To provide theoretical understanding of this problem, the desired solution is specified in the proposed definition. Based on the definition, a theorem is proposed to prove that a min-max algorithm can be used to solve a max-min problem without any algorithmic changes. Based on the theorem, a constraint-activated differential evolution is proposed to solve the constrained min-max optimization problem. The proposed constraint activation directly finds a solution which can best activate constraints to improve efficiency on finding a robust solution. The simulation results show that the proposed method attains 100% success rate on all of the five numerical benchmarks in terms of 1E–6 solution error.

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