隨機離散分數信號轉換之研究

A Research on Random Discrete Fractional Signal Transforms

郭博炫(Bo-Xuan Guo)

指導教授 ： 許文良

中原大學/工學院/電機工程研究所/碩士(2011年)

https://doi.org/10.6840/cycu201100495

摘要

本論文提出了多種離散信號轉換的隨機離散分數轉換矩陣，分別為隨機離散分數餘弦與正弦轉換、隨機離散分數哈達馬轉換矩陣以及隨機離散分數哈特利轉換矩陣。由於第一、四、五及八類的離散餘弦與正弦轉換矩陣、離散哈達馬轉換矩陣及離散哈特利轉換矩陣皆具週期矩陣的性質，故我提出了兩個公式作為其交替矩陣以進一步得到上述各類信號轉換的隨機單位正交特徵向量基底。再來使用隨機多參數向量作為隨機離散分數信號轉換的特徵值矩陣次方，使得特徵向量與特徵值都有隨機的性質。我並提出降低隨機離散分數傅利葉轉換與隨機離散分數哈特利轉換計算量的方法，該方法係以隨機離散分數餘弦與正弦轉換來計算隨機離散分數傅利葉轉換與隨機離散分數哈特利轉換，可降低它們一半的計算量。
各類隨機離散分數信號轉換都可應用於影像加密，分為直接對影像矩陣轉換得到一行與列的振幅皆為隨機分佈的加密影像，及改善雙重隨機相位影像加解密系統，以提高系統的複雜度與安全性兩種。

關鍵字

影像加密 ； 週期矩陣 ； 隨機特徵向量 ； 隨機離散分數哈特利轉換 ； 隨機離散分數哈達馬轉換 ； 隨機離散分數餘弦與正弦轉換 ； 多參數轉換

並列摘要

This thesis proposes random discrete fractional versions of various important discrete signal transforms, including random discrete fractional cosine and sine transforms, random discrete fractional Hadamard transform and random discrete fractional Hartley transform. Because discrete cosine and sine transform of types I, IV, V and VIII, discrete Hadamard transform as well as discrete Hartley transform all have the property of periodic matrices, we propose two general formulas as commuting matrices of these discrete signal transforms such that we can construct their random orthonormal eigenvector bases. With the random eigenvectors, we can then define random discrete fractional signal transforms whose eigenvectors and eigenvalues are all random after taking random eigenvalue powers. Besides, we propose alternative methods to compute random discrete fractional Fourier and Hartley transforms by random discrete fractional cosine and sine transforms in order that half computations of both random discrete fractional Fourier and Hartely transforms can be reduced.
All kinds of random discrete fractional signal transforms can be applied to two image encryption schemes. The first scheme is to apply a random transform to the input image matrix directly and get an encrypted image whose amplitudes for rows and columns are all random. In the second scheme, we apply random transforms for improving the existing double random phase image encryption system to enhance its complexity and security.

並列關鍵字

random discrete fractional cosine and sine transform ； periodic matrix ； image encryption ； random discrete fractional Hartley transform ； multiple-parameter transform ； random eigenvector ； random discrete fractional Hadamard transform

參考文獻

[1] V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl., vol. 25, pp. 241–265, 1980.

[2] L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Processing, vol. 42, pp. 3084–3091, Nov. 1994.

[3] H. M. Ozaktas, M. A. Kutay, and D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” in Advances in Imaging Electronics and Physics. ew York: Academic, 1999, ch. 4.

[4] A. C. McBride and F. H. Kerr, “On Namias's fractional Fourier transform,” IMA J. Appl. Math., vol. 39, pp. 159–175, 1987.

[5] S. C. Pei and M. H. Yeh, “Improved discrete fractional fourier transform,” Opt. Lett., vol. 22, pp. 1047–1049, July 1997.