臺灣博碩士論文加值系統

在分數信號轉換的研究中，幾乎沒有能進行快速運算的分數信號轉換。因此本論文基於克羅內克積的特性，提出了多個能夠進行快速運算的分數信號轉換。我們首先討論目前分數信號轉換領域中對於降低計算量的研究成果：離散分數HOT轉換(Hirschman optimal transform)，再來我們利用類似的方法以多參數離散分數傅立葉轉換等分數信號轉換與單位矩陣進行克羅內克積運算，提出更廣義化之低運算量分數轉換。最後我們根據克羅內克積的特性，進而以離散分數信號轉換與對角矩陣進行克羅內克積運算，得到新的轉換。以上所提出之轉換，皆能降低分數信號轉換所要求的計算量，因此我們將其運用於計算量龐大的影像加密中，能夠達到明顯的效果。
分數信號轉換在影像加密的應用上，我們可用雙重隨機相位加解密系統對影像加密。因此我們將本論文所提出之低運算量分數信號轉換運用於雙重隨機相位加解密系統中，並分析系統的安全性。最後我們設法找出一種轉換，能夠在低運算量的情況下，成功對影像加解密，並且能夠進行高安全性的影像加密，抵擋非法解密。

In the research of fractional signal transforms, there are no existing fast versions of fractional transforms. Therefore, this thesis, based on the properties of Kronecker product proposes several fractional transforms which require low computation complexity. We first discuss the existing research in the low computation fractional transform: discrete fractional HOT(Hirschman optimal transform). Then, similarly, we use the multiple-parameter discrete fractional Fourier transform and other fractional signal transforms to implement Kronecker products with the identity matrix to get the generalized fractional transforms with low computational complexity. Finally, according to the properties of Kronecker product, we generalize the diagonal matrices from the identity matrix, and use the Kronecker product of the fractional signal transforms and the diagonal matrices to define new fractional transforms. Due to the fact that the fractional transforms we propose above require low computation complexity, so the efficiency is obvious when we apply them to the high computation complexity process, for example, image encryption.
In application, we usually apply the double random phase encoding for image encryption. So we employ the proposed low computation fractional transforms to double random phase encoding, and then we analyze the security of the encoding system. Finally, we try to find out one kind of low computation fractional transform that can be applied successfully to image encryption and decryption, and has high security to resist the illegal decryption.

目錄
摘要………………………………………………I
Abstract……………………………………………………………III
誌謝…………………………………………………………………V
目錄…………………………………………………………………VI
圖目錄…………………………………………………………….VIII
表目錄……………………………………………………………X
第一章 緒論………………………………………………………1
1.1前言………………………………………………………1
1.2研究動機與目的…………………………………………2
第二章 離散分數信號轉換………………………………………3
2.1 連續分數傅利葉轉換之時頻域特性……………………3
2.2離散分數傅利葉轉換……………………………………8
2.3多參數離散分數傅利葉轉換……………………………19
2.4隨機離散分數傅利葉轉換………………………………22
2.5離散分數隨機轉換………………………………………28
第三章 克羅內克積與離散分數HOT轉換……………………31
3.1克羅內克積………………………………………………31
3.2離散HOT轉換……………………………………………34
3.3離散分數HOT轉換………………………………………39
3.4離散分數HOT之推廣轉換矩陣…………………………42
第四章 離散分數信號轉換應用於影像加密與敏感度分析……………………………………………………………………48
4.1離散分數信號轉換與離散分數HOT轉換………………48
4.2離散分數HOT推廣轉換之敏感度分析…………………59
第五章 結論……………………………………………………74
參考文獻……………………………………………………………75

圖目錄
圖2.1-1 時頻平面與轉換軸之關係.............................................5
圖2.2-1 矩陣的四個特徵向量，依零交點數分別對應0~4階的赫米特-高斯函數.....................................................................14
圖2.2-2 四個不同次冪的DFRFT對方波轉換的波形.................18
圖2.4-1 DFT交替矩陣之隨機特徵向量.....................................25
圖2.4-2 方波經RDFRFT轉換之振福與相角.............................27
圖2.5-1 方波經DFRNT轉換後之振福與相角...........................30
圖4.1-1 雙重隨機相位影像加密系統.......................................49
圖4.1-2 雙重隨機相位影像解密系統.......................................49
圖4.1-3 分數信號轉換之雙重隨機相位影像加密系統…...........50
圖4.1-4 分數信號轉換之雙重隨機相位影像解密系統…...........50
圖4.1-5 DFRFT運用於雙重隨機相位加解密系統之模擬結果....51
圖4.1-6 DFRFT之敏感度..........................................................52
圖4.1-7 DFRFT解密結果圖，次冪誤差 ..........................53
圖4.1-8 DFRFT與MPDFRFT之敏感度.....................................54
圖4.1-9 DFRFT、MPDFRFT與RDFRFT之敏感度......................55
圖4.1-10 DFRFT、MPDFRFT與DFRNT之敏感度......................56
圖4.1-11 DFRFT與DFRHOT之敏感度......................................57
圖4.1-12 DFRFT、 、DFRHOT以及 敏感度圖..................58
圖4.2-1 DFRFT、DFRHOT與MPDFRHOT之敏感度圖...............60
圖4.2-2 DFRFT、DFRHOT與RDFRHOT之敏感度圖..................61
圖4.2-3 DFRFT、DFRHOT與 之敏感度圖....................61
圖4.2-4 DFRHOT與 之敏感度圖...............................62
圖4.2-5 將能量平衡之雙重隨機相位加密系統.........................65
圖4.2-6 將能量平衡之雙重隨機相位解密系統.........................65
圖4.2-7 DFRFT、DFRHOT與 之敏感度.....................66
圖4.2-8 之原始影像、加密影像以及錯誤解密影.......67
圖4.2-9 之敏感度..............................................69
圖4.2-10原始影像及 之加解密影像....................70
圖4.2-11 之加解密影像......................................72

表目錄
表2.2-1 DFRFT特徵值分配規則………………………………………16
表2.2-2 DFRFT DFT特徵值分配規則………………………………17

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