WO2023040365A1 - 基于多尺度压缩感知和马尔科夫模型的图像加密方法 - Google Patents
基于多尺度压缩感知和马尔科夫模型的图像加密方法 Download PDFInfo
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
- H04L9/001—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols using chaotic signals
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- G06F—ELECTRIC DIGITAL DATA PROCESSING
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- G06F17/10—Complex mathematical operations
- G06F17/16—Matrix or vector computation, e.g. matrix-matrix or matrix-vector multiplication, matrix factorization
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- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F18/00—Pattern recognition
- G06F18/20—Analysing
- G06F18/29—Graphical models, e.g. Bayesian networks
- G06F18/295—Markov models or related models, e.g. semi-Markov models; Markov random fields; Networks embedding Markov models
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Definitions
- the invention relates to the technical field of image encryption, in particular to an image encryption method based on multi-scale compressed sensing and a Markov model.
- the present invention proposes an image encryption method based on multi-scale compressed sensing and Markov model, which expands the application range of compressed sensing theory in the field of image encryption, and uses the Markov model to design A new pixel scrambling method is proposed, and finally the ciphertext image with better encryption effect and the decrypted image with better decryption effect are obtained.
- Image encryption method based on multi-scale compressed sensing and Markov model including:
- the sub-sampling rate of the original coefficient matrix at each level is obtained by using the multi-scale block compressive sensing theory
- the plaintext image is subjected to a three-level discrete wavelet transform to obtain the original coefficient matrix of each level, and the original coefficient matrix of each level is divided into blocks to construct a new coefficient matrix of each level;
- the low-frequency coefficient matrix Retain the low-frequency coefficient matrix, combine the measured non-low-frequency coefficient matrices at all levels, construct the matrix T, and perform microprocessing on the matrix T to obtain the test matrix NT, and generate the row-state transition probability matrix and column-state respectively according to the test matrix NT Transition probability matrix;
- One of the chaotic sequences is used for independent diffusion of the overall matrix elements, and the other chaotic sequences are used for global diffusion of the overall matrix elements to obtain the final ciphertext image.
- parameters and initial values of the one-dimensional chaotic map are generated according to the plaintext image information, specifically:
- L 0 and L 1 are respectively the number of 0s and the number of 1s in the hash value K of the plaintext image, and l 0 and l 1 are respectively the length of the longest continuous 0 sequence and the length of the longest continuous 1 sequence.
- r end is the right endpoint of the chaotic interval, r 0 is a certain value of the chaotic interval.
- t 1 -t 4 is the external key
- r start is the left endpoint of the chaotic interval
- r end is the right endpoint of the chaotic interval
- r 1 and r 2 is a certain value in the chaotic interval.
- scrambling is performed on corresponding coefficient matrices at all levels, and the scrambling method is specifically: taking the low-frequency coefficient A3 of the third-level wavelet decomposition as an example, assuming that matrix A3 is an m ⁇ n matrix, Expand A 3 into a sequence of length m ⁇ n, i is the index of the i-th element in the sequence, and the scrambling method is:
- a 3 ′ is the scrambled sequence of A 3
- L 0 and L 1 are the number of 0s and 1s in the hash value K of the plaintext image respectively, l 0 and l 1 are the length of the longest continuous 0 sequence and the length of the longest continuous 1 sequence respectively, and NT is The matrix to be tested after processing T.
- the non-low frequency coefficient matrices of all levels are first merged to obtain the overall matrix, specifically:
- the first group of sorted matrices is denoted as Y 1
- the internal three block matrices are respectively denoted as Y 11 , Y 12 , Y 13 , and the arrangement order is determined by Lind 1 '
- the second group of sorted matrices is denoted as Y 2
- the internal The three block matrices are respectively marked as Y 21 , Y 22 , and Y 23 , and the arrangement order is determined by Lind 2 '
- the third group of sorted matrices is marked as Y 3
- the three internal block matrices are respectively marked as Y 31 , Y 32 , Y 33 , the arrangement order is determined by Lind 3 '.
- sub-matrix is inserted into different positions of the overall matrix, specifically:
- T' (Y 1 ′ Y 2 ′ Y 3 ′ )
- Y 1 ' is the matrix after the first combination, and the arrangement order is determined by Hind 1 ';
- Y 2 ' is the matrix after the second combination, and the arrangement order is determined by Hind 2 ';
- Y 3 ' is the third combination After the matrix, the arrangement order is determined by Hind 3 '.
- U, S, VT are three sub-matrices obtained by SVD decomposition of the low-frequency coefficient matrix.
- T' is the combined matrix of Y 1 ', Y 2 ', and Y 3 '.
- Max is the maximum value of matrix T'
- Min is the minimum value of T'
- d 1 is the largest integer less than or equal to the average value of elements in T'
- d 2 is the smallest integer greater than or equal to the average value of Max and Min
- d 1 ' is the remainder of d 1 divided by 10
- d 2 ' is the remainder of d 2 divided by 10
- d 12 is the larger value of d 1 ' and d 2 '.
- V' is a generated and processed chaotic sequence
- w 1 to w 4 are used as control parameters for subsequent shift operations, and fix is a rounding-to-zero function.
- m 1 ' is the largest prime factor of the matrix row dimension m 1
- n 1 ' is the largest prime factor of the matrix column dimension n 1 .
- the ciphertext image obtained by the present invention has higher information entropy, and it is difficult to obtain the relevant information of the original image. At the same time, it has better plaintext sensitivity and key sensitivity, and can resist various attack. Compared with the decrypted image generated by the existing scheme, the decrypted image obtained by the present invention is of higher quality, better visual effect and more complete original image information can be obtained.
- transition probability matrix in the Markov model is introduced to scramble the image, the state space is defined according to the characteristics of the image pixel values in the encryption process, and the state transition probability matrix is constructed based on the distribution of pixel values, so that the encryption process It has good randomness, so it is difficult to be predicted.
- the plaintext image and chaotic sequence information are used in the entire encryption process, so that the encryption scheme has better plaintext sensitivity and can effectively resist chosen plaintext attacks and known plaintext attacks.
- Fig. 1 is a frame diagram of the encryption process of the present invention
- Fig. 2 is a schematic diagram of three-level wavelet decomposition in the present invention.
- wavelet decomposition is performed on the image to obtain the original coefficient matrices of each level, and then the coefficient matrices of each level are scrambled, and then the corresponding coefficient matrices are measured with multiple measurement matrices, and then the combined state transition probability matrix is used to measure the The measured value matrix is scrambled twice, and finally diffused to obtain the final ciphertext image.
- the decryption process is the reverse process of the encryption process.
- TSL mapping, TLT mapping, and Hybrid mapping are used to construct the measurement matrix, the index sequence of the first scrambling, and the index value of the coefficient matrix merging rule.
- the ISEL map is used to generate control parameters for the second scrambling, diffusion sequences for independent diffusion and global diffusion.
- the construction method of the state transition probability matrix used in the present invention is as follows:
- a 4 ⁇ 4 matrix f is initialized, and the matrix f is used to record the frequency of state transitions.
- the matrix f is used to record the frequency of state transitions. For each column vector of the matrix to be measured, first determine which type of number the first element belongs to, that is, determine the row coordinates of the state transition matrix. Then determine which type of number the next element belongs to, that is, determine the column coordinates of the state transition matrix. Add 1 to the value of the position in the matrix f corresponding to the coordinate point.
- the updated matrix f is obtained as shown in Table 1. Calculate the sum of each row of the above matrix f, and divide each element in a row by the sum of the corresponding row to obtain the corresponding probability. After all the probabilities are calculated, the final row-state transition probability matrix is obtained as shown in Table 2.
- the generation method of the column-state transition probability matrix is similar to that of the row-state transition probability matrix, except that the column vector of the matrix to be tested is not taken each time, but the row vector.
- the pixel scrambling method based on the state transition probability matrix used in the present invention is as follows:
- Row coordinates indicate selection of odd/even column (row) vectors and direction of movement up/down (left/right). If the row coordinate is an odd number, select an odd column (row), and if the row coordinate is an even number, select an even column (row); if the row coordinate is a positive number, the selected element will move up (to the left); if the row coordinate is a negative number, the selected element will move to the left. Move down (right).
- the row of the selected number corresponds to a positive odd number, move all the odd columns (rows) of the matrix up (to the left); if the row of the selected number corresponds to a negative odd number, move all the odd columns (rows) of the matrix down (right) move; if the row of the selected number corresponds to a positive even number, move all the even columns (rows) of the matrix up (left); if the row of the selected number corresponds to a negative even number, then move all the even columns (rows) of the matrix to Move down (right).
- the column coordinates indicate the shift number of the cyclic shift, and the specific value is generated by the chaotic map.
- the external key set in the instance can be:
- the chaos maps used are as described above.
- Step 1 Generate the parameters and initial values of the one-dimensional chaotic map according to the plaintext image information.
- the hash value K of the plaintext image is generated by using the SHA256 function, and K is converted into a binary number to generate 32 sets of binary numbers k 1 , k 2 . . . , k 32 in groups of 8 bits.
- the parameters and initial values of the one-dimensional chaotic map are generated according to the following formulas.
- r 0 , r 1 , r 2 are the parameter values of Hybrid mapping, TLT mapping, and TST mapping respectively.
- a, b, c are the parameter values of ISEL mapping.
- Step 2 According to the total target sampling rate of the plaintext image, the sub-sampling rate of the original coefficient matrix of each level is obtained by using the multi-scale block compressive sensing theory.
- Step 3 Substituting the parameters and initial values of the one-dimensional chaotic map into the corresponding chaotic system to generate a chaotic sequence, converting the chaotic sequence into a matrix form, and obtaining the corresponding orthogonal basis matrix; extracting according to the sub-sampling rate Some elements of the orthogonal base matrix are used as a measurement matrix;
- the parameter r 2 and the initial value y 00 of the chaotic system generated in step 1 are substituted into the TST mapping iteration t+n 1 ⁇ n 1 times, and r 1 and x 00 are substituted into the TLT mapping iteration t+n 2 ⁇ n 2 times , r 0 and x 0 are substituted into the Hybrid mapping iteration t+n 3 ⁇ n 3 times, where n 1 , n 2 , and n 3 are determined by the given block size in advance.
- the initial state value of the next iteration is perturbed by multiplying the sine value of the initial state by a small coefficient, discarding the first t elements to obtain the corresponding chaotic sequence X 1 , X 2 , X 3 .
- X 1 is a sequence of length n 1 ⁇ n 1
- X 2 is a sequence of length n 2 ⁇ n 2
- X 3 is a sequence of length n 3 ⁇ n 3 .
- the generated chaotic sequence is further processed.
- X 1 ′′ is an n 1 ⁇ n 1 matrix
- X 2 ′′ is an n 2 ⁇ n 2 matrix
- X 3 ” is an n 3 ⁇ n 3 matrix.
- the corresponding orthogonal basis matrices ⁇ 1 , ⁇ 2 , ⁇ 3 of X 1 ′′, X 2 ′′ and X 3 ′′ are used as the redundant measurement matrix.
- Step 4 the plaintext image is subjected to three-level discrete wavelet transform to obtain the original coefficient matrix of each level, and the original coefficient matrix of each level is divided into blocks to construct a new coefficient matrix of each level;
- Step 5 use the chaotic sequence to generate an index sequence, and scramble the corresponding coefficient matrices at all levels;
- the value of the element represents the block size.
- the third-level wavelet decomposition coefficient matrix H 3 , V 3 , D 3 is divided into blocks of block_size 3 ⁇ block_size 3
- the second-level wavelet decomposition matrix H 2 , V 2 , D 2 is divided into blocks of block_size 2 ⁇ block_size 2
- the wavelet decomposition coefficient matrices H 1 , V 1 , and D 1 are divided into blocks of block_size 1 ⁇ block_size 1
- each block matrix is expanded into a column vector after being divided into blocks.
- a 3 is an m ⁇ n matrix, expand A 3 into a sequence of length m ⁇ n, and i is the index of the i-th element in the sequence.
- a 3 ' is the sequence after A 3 is scrambled.
- Step 6 using the measurement matrix to measure the coefficient matrices at all levels
- the coefficient matrix A 3 ' remains unchanged, and the coefficient matrix H 3 ', V 3 ', D 3 ' is measured with ⁇ 3 ', to obtain the measured value matrix H 3 ", V 3 ", D 3 ";
- Use ⁇ 2 ' to measure the coefficient matrix H 2 ', V 2 ', D 2 ', and obtain the measured value matrix H 2 ", V 2 ", D 2 ";
- use ⁇ 1 ' to measure the coefficient matrix H 1 ', V 1 ', D 1 ' to measure, and obtain the measured value matrix H 1 ", V 1 ", D 1 ".
- Step 7 Keep the low-frequency coefficient matrix, combine the measured non-low-frequency coefficient matrices at all levels, construct the matrix T, and perform microprocessing on the matrix T to obtain the matrix NT to be tested, and generate row-state transition probability matrices and Column - state transition probability matrix;
- step 1 In order to enhance the randomness, according to the information l 0 , l 1 , L 0 , L 1 obtained in step 1, do microprocessing on the test matrix T to obtain the matrix NT.
- Step 8 Perform SVD decomposition on the low-frequency coefficient matrix to obtain a sub-matrix, and quantize the elements in the sub-matrix and non-low-frequency coefficient matrices at all levels to a preset interval;
- Step 9 Generate an index value according to the information of the chaotic sequence and determine the merging rule through the index value. According to the merging rule, first merge the non-low frequency coefficient matrices at all levels to obtain the overall matrix, and then insert the sub-matrix into the overall matrix different positions of
- H 1 "', H 2 "', H 3 "' are divided into one group, which is recorded as the first group.
- D 1 "', D 2 “', D 3 "' are divided into one group and recorded as the third group.
- the index value Lind 1 ' of the first group is determined by the last element of X 1 ', X 2 ', X 3 '
- the index value Lind 2 ' of the second group is determined by the X 1 ', X 2 ', X 3 '
- the second-to-last element is determined
- the index value Lind 3 ' of the third group is determined by the third-to-last element of X 1 ', X 2 ', and X 3 '.
- the three index values are mapped to the interval [1, 6], indicating that there are 6 possible orderings for each group.
- the first sorted matrix is denoted as Y 1
- the three internal block matrices are denoted as Y 11 , Y 12 , Y 13 respectively
- the second sorted matrix is denoted as Y 2
- the internal three block matrices are respectively denoted as are Y 21 , Y 22 , Y 23
- the third sorted matrix is marked as Y 3
- the three internal block matrices are respectively marked as Y 31 , Y 32 , and Y 33 .
- the index value Hind1' of the first group is determined by the first element of X 1 ', X 2 ', X 3 '
- the index value Hind 2 ' of the second group is determined by the first element of X 1 ', X 2 ', X 3 '
- the second element is determined
- the index value Hind 3 ' of the third group is determined by the third element of X 1 ', X 2 ', X 3 '.
- the three index values are mapped to the [1, 4] interval, indicating that the three matrices of U, S, and VT have four insertion positions respectively, that is, the top of each matrix group, and the position between two adjacent block matrices in the group. Between, the bottom of the matrix group. The result is shown below.
- the first combined matrix is marked as Y 1 ′
- the second combined matrix is marked as Y 2 ′
- the third combined matrix is marked as Y 3 ′.
- T' (Y 1 'Y 2 'Y 3 ')
- Step 10 adjusting the dimensions of the overall matrix and acquiring element information of the overall matrix to generate control parameters for secondary scrambling;
- Max is the maximum value of matrix T'
- Min is the minimum value of T'
- d 1 is the largest integer less than or equal to the average value of elements in T'
- d 2 is the smallest integer greater than or equal to the average value of Max and Min
- d 1 ' is the remainder of d 1 divided by 10
- d 2 ' is the remainder of d 2 divided by 10
- d 12 is the larger value of d 1 ' and d 2 '.
- Step 11 scrambling the combined overall matrix according to the row-state transition probability matrix and the column-state transition probability matrix, and setting corresponding flag bits;
- round is a rounding function.
- control parameters of the scrambling process are generated according to V 1 , V 2 , V 3 , and V 4 , and the calculation process is as follows.
- w 1 to w 4 are used as control parameters for subsequent shift operations, and fix is a rounding-to-zero function.
- m 1 ' is the largest prime factor of the matrix row dimension m 1
- n 1 ' is the largest prime factor of the matrix column dimension n 1 .
- Step 12 Use one of the chaotic sequences to perform independent diffusion of the overall matrix elements, and use the remaining chaotic sequences to perform global diffusion of the overall matrix elements to obtain the final ciphertext image;
- the column vector is scrambled. If the column of the selected number corresponds to a positive odd number, the shifting digit of the matrix element is w 1 ; if the column of the selected number corresponds to a negative odd number, the shifting digit of the matrix element is w 3 ; if the column of the selected number corresponds to a positive even number, Then the shifting digit of the matrix element is w 2 ; if the column of the selected number corresponds to a negative even number, the shifting digit of the matrix element is w 4 . Note that the matrix after scrambling is T". Set the state transition flag bit matrix, the initial value is 0. If the element at the corresponding position has been shifted, the flag bit at the corresponding position will change from 0 to 1.
- the row vector is scrambled, and the rules are consistent with the above steps.
- the matrix after scrambling is T"'. Also set the initial flag bit matrix, and if a shift operation occurs at the corresponding position, the flag bit at the corresponding position will change from 0 to 1.
- the interval of V" is The sequence V 1 ' is obtained by sampling d 1 ', and the sequence V 2 ' is obtained by sampling V" with an interval of d 2 '.
- the position of the last element of the sequence V 2 ' is used as the starting position, and the sequence V" is backward Take continuous m 1 ⁇ n 1 data and record it as sequence V 0 .
- T" as the matrix A.
- V 1 ' performs a forward addition modulo operation on matrix A to obtain matrix B, and then uses the previous element of each element in B to perform a circular left shift operation to obtain B'; use V 2 ' to perform matrix B' Perform backward addition modulo operation to obtain matrix C, and then use the last element of each element in C to perform circular left shift operation to obtain C'.
- the final ciphertext image is obtained.
- the present invention proposes an image encryption method based on multi-scale compressed sensing and a Markov model. According to the difference in information carried by the low-frequency coefficients and high-frequency coefficients of the image, different sampling rates are set for the low-frequency coefficients and high-frequency coefficients of the image, which can effectively Improve reconstruction quality of decrypted images.
- the image is first scrambled within the coefficient matrix, then the image is scrambled between the coefficient matrices, and finally the encryption process is completed by independent diffusion and global diffusion.
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Abstract
本发明公开了一种基于多尺度压缩感知和马尔科夫模型的图像加密方法,根据图像的低频系数与高频系数携带信息的不同,对图像的低频系数与高频系数设置不同的采样率,能够有效地提高解密图像的重构质量。另外,通过结合混沌系统与马尔科夫模型,用先对图像进行系数矩阵内置乱,后对图像进行系数矩阵间置乱,最后进行独立扩散与全局扩散的策略完成加密的过程。相比于已有方案生成的密文图像信息熵更高,难以获取原始图像的相关信息,同时有较好的明文敏感性和密钥敏感性,可以抵抗各种攻击。经过本发明获得的解密图像,相比于已有方案生成的解密图像质量更高,可以取得更好的视觉效果以及更完整的原图像信息。
Description
本申请要求于2021年09月18日提交中国专利局、申请号为202111111707.6、发明名称为“基于多尺度压缩感知和马尔科夫模型的图像加密方法”的中国专利申请的优先权,其全部内容通过引用结合在本申请中。
本发明涉及图像加密技术领域,特别是涉及基于多尺度压缩感知和马尔科夫模型的图像加密方法。
随着社会生产力和互联网技术的不断发展,数字图像信息作为互联网信息的一种重要载体,在互联网信息的交互中起到了非常重要的作用。在涉及到个人隐私、商业秘密、国防机密、国家安全等诸多领域,数字图像信息的安全性要求也越来越高,因此数字图像加密技术也越来越重要。目前常用的数字图像加密技术已较为成熟,但其中大部分理论核心仍主要是由国外提出。纵观目前现有的图像加密方案,存在着诸如:容易获取原始图像的相关信息,明文敏感性和密钥敏感性差、解密图像的重构质量以及加密效果不够好等弱点。故此,综合改进并有所创新的提出一套更为完善且安全性更强的数字图像加密方法,是非常迫切与必要的需求。
发明内容
针对现有技术存在上述弱点,本发明提出了基于多尺度压缩感知和马尔科夫模型的图像加密方法,该方法扩大了压缩感知理论在图像加密领域中的应用范围,并且借助马尔科夫模型设计了新的像素置乱方法,最终得到了加密效果较好的密文图像以及解密效果较好的解密图像。
为实现上述目的,本发明的技术方案如下:
基于多尺度压缩感知和马尔科夫模型的图像加密方法,包括:
根据明文图像信息生成一维混沌映射的参数及初始值;
通过所述明文图像的总目标采样率,利用多尺度分块压缩感知理论得到原各级系数矩阵的子采样率;
将所述一维混沌映射的参数及初始值代入相应的混沌系统生成混沌序列,把所述混沌序列转换为矩阵形式,并获取相应的正交基矩阵;根据所述子采样率提取所述正交基矩阵的部分元素作为测量矩阵;
所述明文图像进行三级离散小波变换,得到原各级系数矩阵,对所述原各级系数矩阵分块,构造新的各级系数矩阵;
利用混沌序列生成索引序列,对相应的各级系数矩阵作块进行置乱;
用所述测量矩阵对各级系数矩阵进行测量;
保留低频系数矩阵,合并测量后的非低频各级系数矩阵,构造矩阵T,对矩阵T作微处理,得到待测矩阵NT,根据待测矩阵NT分别生成行-状态转移概率矩阵和列-状态转移概率矩阵;
对所述低频系数矩阵作SVD分解得到子矩阵,将所述子矩阵以及非低频各级系数矩阵中元素量化到预设区间;
根据所述混沌序列的信息生成索引值并通过所述索引值确定合并规则,根据合并规则先合并所述非低频各级系数矩阵得到整体矩阵,再将子矩阵插入到所述整体矩阵的不同位置;
调整所述整体矩阵维度以及获取所述整体矩阵元素信息,生成二次置乱的控制参数;
根据所述行-状态转移概率矩阵和列-状态转移概率矩阵对合并后的整体矩阵进行置乱,同时设置相应的标志位;
用其中一个混沌序列进行整体矩阵元素的独立扩散,用其余混沌序列进行整体矩阵元素的全局扩散,得到最终密文图像。
进一步的,根据明文图像信息生成一维混沌映射的参数及初始值,具体为:
利用SHA256函数生成明文图像的hash值K,将K转化成二进制数后按每8位一组生成32组二进制数k
1,k
2...,k
32;
其中,L
0,L
1分别为明文图像hash值K中0的个数,1的个数,l
0,l
1分别为最长连续0序列的长度,最长连续1序列的长度。r
end为混沌区间右端点,r
0为混沌区间的某个值。
其中,k
i,i=1,2,…12为K的第i组二进制数,t
1-t
4为外部密钥,r
start为混沌区间左端点,r
end为混沌区间右端点,r
1和r
2为混沌区间的某个值。
其中,k
i,i=13,14,…24为K的第i组二进制数,t
5-t
8为外部密钥,r
start为混沌区间左端点,r
end为混沌区间右端点,a,b,c分别为相应混沌区间的某个值。
根据上式构造Z
1和Z
2两个矩阵,然后求两个矩阵的克罗内克积得到Z
3。其中,k
i,i=25,14,…32为K的第i组二进制数,t
1-t
8为外部密钥。
进一步的,对相应的各级系数矩阵作块进行置乱,所述置乱方法具体为:以第三级小波分解的低频系数A
3为例,设矩阵A
3是一个m×n的矩阵,将A
3展开为一个长度为m×n的序列,i为序列中第i个元素的索引,其置乱方式为:
A
3′(Ind
3(m×n-i+1))=A
3(Ind
3(i))
A
3′为A
3置乱后的序列,Ind
3为对第三级小波分解系数矩阵进行置乱的索引序列;同理得到其余置乱后的系数矩阵H
i',V
i',D
i',i=1,2,3。
进一步的,对矩阵T作微处理,得到待测矩阵NT的具体方式为:
其中,H
i”,V
i”,D
i”,i=1,2,3为将H
i',V
i',D
i',i=1,2,3压缩后的系数矩阵,T为合并后的矩阵。
其中,L
0,L
1分别为明文图像hash值K中0的个数,1的个数,l
0,l
1分别为最长连续0序列的长度,最长连续1序列的长度,NT为对T处理后的待测矩阵。
进一步的,根据合并规则先合并所述非低频各级系数矩阵得到整体矩阵,具体为:
其中,Lind
i,i=1,2,3是混沌序列X
i',i=1,2,3生成的索引值,n
i×n
i,i=1,2,3是X
i',i=1,2,3的长度,Lind
i',i=1,2,3是Lind
i,i=1,2,3映射到区间[1,6]上的结果。
其中,H
i”',V
i”',D
i”',i=1,2,3是对H
i”,V
i”,D
i”,i=1,2,3量化后的结果;第一组排序后的矩阵记为Y
1,内部三个块矩阵分别记为Y
11,Y
12,Y
13,排列顺序由Lind
1'决定;第二组排序后的矩阵记为Y
2,内部三个块矩阵分别记为Y
21,Y
22,Y
23,排列顺序由Lind
2'决定;第三组排序后的矩阵记为Y
3,内部三个块矩阵分别记为Y
31,Y
32,Y
33,排列顺序由Lind
3'决定。
进一步的,将子矩阵插入到所述整体矩阵的不同位置,具体为:
其中,Hind
i,i=1,2,3是混沌序列X
i',i=1,2,3生成的索引值,Hind
i',i=1,2,3是Hind
i,i=1,2,3映射到区间[1,4]上的结果。
T′=(Y
1′ Y
2′ Y
3’)
其中,Y
1'为第一组合并后的矩阵,排列顺序由Hind
1'决定;Y
2'为第二组合并后的矩阵,排列顺序由Hind
2'决定;Y
3'为第三组合并后的矩阵,排列顺序由Hind
3'决定。U,S,VT为对低频系数矩阵进行SVD分解得到的三个子矩阵。T'为Y
1',Y
2',Y
3'合并后的矩阵。
进一步的,调整所述整体矩阵维度以及获取所述整体矩阵元素信息,具体方式为:
其中,Max为矩阵T'的最大值,Min为T'的最小值,d
1为小于或等于T'中元素平均值的最大整数,d
2为大于或等于Max和Min平均值的最小整数,d
1'为d
1除以10的余数,d
2'为d
2除以10的余数,d
12为d
1'和d
2'中的较大值。
更进一步的,生成二次置乱的控制参数,具体方式为:
其中,V'为生成并处理的混沌序列,V
i,i=1,2,3,4为由V'生成的子序列。
其中,w
1~w
4作为控制参数用于后续移位操作,fix为向零取整函数。其中m
1'是矩阵行维数m
1的最大质因子,n
1'是矩阵列维数n
1的最大质因子。
经过本发明获得的密文图像,相比于已有方案生成的密文图像信息熵更高,难以获取原始图像的相关信息,同时有较好的明文敏感性和密钥敏感性,可以抵抗各种攻击。经过本发明获得的解密图像,相比于已有方案生成的解密图像质量更高,可以取得更好的视觉效果以及更完整的原图像 信息。
本发明与已有的方法,在以下方面存在优势:
1.引入了马尔科夫模型中转移概率矩阵的概念对图像进行置乱操作,根据加密过程中图像像素值的特征定义状态空间,以像素值的分布为依据构造状态转移概率矩阵,使得加密过程具有较好的随机性,从而很难被预测。
2.将明文图像和混沌序列的信息用于整个加密过程中,使得加密方案有较好的明文敏感性,可以有效抵抗选择明文攻击和已知明文攻击。
3.引入了多尺度分块压缩感知理论,更合理地设置图像的采样率,使得解密图像的重构质量有较大提高。
说明书附图
下面结合附图对本发明作进一步说明:
图1为本发明加密过程框架图;
图2为本发明三级小波分解示意图。
下面结合本发明实施例中的附图,对本发明实施例中技术方案进行详细的描述,显然,所描述的实施例仅是本发明一部分实施例,而不是全部的实施例;基于本发明中的实施例,本领域普通技术人员在没有做出创造性劳动前提下所获得的所有其他实施例都属于本发明保护的范围。
首先对图像进行小波分解得到原各级系数矩阵,其次分别对每级系数矩阵作一次置乱,接着用多个测量矩阵分别对相应的系数矩阵进行测量,之后借助状态转移概率矩阵对合并后的测量值矩阵进行二次置乱,最后进行扩散得到最终的密文图像。解密过程即为加密过程的逆过程。
本发明用到的混沌映射有四个,分别是TSL映射,TLT映射、Hybrid映射和ISEL映射。
TSL映射,TLT映射、Hybrid映射用于构造测量矩阵,第一次置乱的索引序列,系数矩阵合并规则的索引值。ISEL映射用于生成第二次置乱的控制参数,独立扩散和全局扩散的扩散序列。
本发明用到的状态转移概率矩阵的构造方法如下所示:
首先构造状态空间,将数分为四类,定义如下:
(1)如果一个数的整数部分既是正数又是奇数,则该数为正奇数。
(2)如果一个数的整数部分既是负数又是奇数,则该数为负奇数。
(3)如果一个数的整数部分既是正数又是偶数,则该数为正偶数。
(4)如果一个数的整数部分既是负数又是偶数,则该数为负偶数。
之后,初始化一个4×4的矩阵f,矩阵f用来记录状态转移的频数。对于待测矩阵的每一列向量,先判断第一个元素属于哪一类数,即确定了状态转移矩阵的行坐标。之后确定下一个元素属于哪一类数,即确定了状态转移矩阵的列坐标。该坐标点所对应矩阵f中的位置的数值加1。依此类推,得到更新后的矩阵f如表1所示,计算上述矩阵f每一行的和,对于一行内的每个元素,除以对应行的和,得到相应的概率。所有概率计算出来之后,就得到了最终的行-状态转移概率矩阵如表2所示。列-状态转移概率矩阵与行-状态转移概率矩阵的生成方式相似,只是每次取的不是待测矩阵的列向量,而是行向量。
表1
表2
本发明用到的基于状态转移概率矩阵的像素置乱方法如下所示:
对于行-状态转移概率矩阵(列-状态转移概率矩阵),找到矩阵中概率值大于0.25的值所处的位置。行坐标指示选择奇数/偶数列(行)向量以及移动方向上/下(左/右)。如果行坐标是奇数就选择奇数列(行),行坐标是偶数就选择偶数列(行);行坐标是正数被选中的元素就向上(左)移动,行坐标是负数被选中的元素就向下(右)移动。具体来说,如果选中的数所在行对应正奇数,则将矩阵所有奇数列(行)向上(左)移动;如果选中的数所在行对应负奇数,则将矩阵所有奇数列(行)向下(右)移动;如果选中的数所在行对应正偶数,则将矩阵所有偶数列(行)向上(左)移动;如果选中的数所在行对应负偶数,则将矩阵所有偶数列(行)向下(右)移动。列坐标指示循环移位的移位数,具体数值由混沌映射生成。
实施例1
如图1所示,本发明的实施例是基于本发明技术方案进行实施的,给出了详细的实施方式和具体的操作过程,但本发明的保护范围不限于下述实施例。实例中设置外部密钥可以为:
t
1=0.11,t
2=0.22,t
3=0.33,t
4=0.44,t
5=2.723,t
6=0.618,t
7=3.141,t
8=4.6692,t=600,设置决定矩阵分块大小的数组block_size为[32,16,8],即block_size
1=32,block_size
2=16,block_size
3=8,设置采样率为0.25。所用混沌映射如上所述。
步骤1:根据明文图像信息生成一维混沌映射的参数及初始值。
示例性的,利用SHA256函数生成明文图像的hash值K,将K转化成二进制数后按每8位一组生成32组二进制数k
1,k
2...,k
32。统计明文hash值K中0的个数L
0,1的个数L
1,最长连续0序列的长度l
0与最长连续1序列的长度l
1。按如下各式生成一维混沌映射的参数及初始值。
其中,k
i,i=1,2,...32为K的第i组二进制数,r
0,r
1,r
2分别是Hybrid映射,TLT映射,TST映射的参数值。
其中,a,b,c是ISEL映射的参数值。
根据上式构造Z
1和Z
2两个矩阵,然后求两个矩阵的克罗内克积得到Z
3。
步骤2:通过所述明文图像的总目标采样率,利用多尺度分块压缩感知理论得到原各级系数矩阵的子采样率。
步骤3:将所述一维混沌映射的参数及初始值代入相应的混沌系统生成混沌序列,把所述混沌序列转换为矩阵形式,并获取相应的正交基矩阵;根据所述子采样率提取所述正交基矩阵的部分元素作为测量矩阵;
示例性的,将步骤1生成的混沌系统的参数r
2及初始值y
00代入TST映射迭代t+n
1×n
1次,r
1和x
00代入TLT映射迭代t+n
2×n
2次,r
0和x
0代入Hybrid映射迭代t+n
3×n
3次,其中,n
1,n
2,n
3由事先给定的分块大小确定。每迭代2000次,用初始状态的正弦值乘以微小的系数来扰动下一次迭代的初始状态值,舍弃前t个元素得到相应的混沌序列X
1,X
2,X
3。其中,X
1是长度为n
1×n
1的序列,X
2是一个长度为n
2×n
2的序列,X
3是一个长度为n
3×n
3的序列。为了加强混沌序列的随机性,对生成的混沌序列作进一步处理。即将序列乘以干扰系数α,之后加上干扰系数β,得到序列X
1',X
2',X
3',此处,取α=-2,β=1。将处理后的混沌序列X
1',X
2',X
3'转化为矩阵形式得到X
1”,X
2”,X
3”。其中,X
1”是一个n
1×n
1的矩阵,X
2”是一个n
2×n
2的矩阵,X
3”是一个n
3×n
3的矩阵。之后,将X
1”,X
2”,X
3”相应的正交基矩阵Φ
1,Φ
2,Φ
3作为冗余测量矩阵。将冗余测量矩阵的行维数n
i,i=1,2,3乘以相应的采样率得到新的行维数m
i,i=1,2,3,然后分别提取冗余测量矩阵的前m
i行作为正式的测量矩阵Φ
1',Φ
2',Φ
3'。
步骤4:所述明文图像进行三级离散小波变换,得到原各级系数矩阵,对所述原各级系数矩阵分块,构造新的各级系数矩阵;
示例性的,如图2所示,构造新的各级系数矩阵为一个低频系数矩阵A
3和九个水平、垂直、对角方向的高频系数矩阵H
i,V
i,D
i,i=1,2,3。
步骤5:利用混沌序列生成索引序列,对相应的各级系数矩阵作块进行置乱;
示例性的,给定长度为3的数组block_size,元素的数值表示分块大小。第三级小波分解系数矩阵H
3,V
3,D
3分成的block_size
3×block_size
3块,第二级小波分解矩阵H
2,V
2,D
2分成block_size
2×block_size
2的块, 第一级小波分解系数矩阵H
1,V
1,D
1分成block_size
1×block_size
1的块,分块之后将每个块矩阵展开为列向量。最后依次合并相应的列向量就构成了九个新的系数矩阵NH
3,NV
3,ND
3,NH
2,NV
2,ND
2,NH
1,NV
1,ND
1。
对步骤3生成的三个混沌序列X
1',X
2',X
3'按升序排序得到相应的索引序列Ind
1,Ind
2,Ind
3。用Ind
3对第三级小波分解系数矩阵A
3,NH
3,NV
3,ND
3进行置乱得到A
3',H
3',V
3',D
3',Ind
2对第二级小波分解系数矩阵NH
2,NV
2,ND
2进行置乱得到H
2',V
2',D
2',Ind
1对第一级小波分解系数矩阵NH
1,NV
1,ND
1进行置乱得到H
1',V
1',D
1'。以A
3为例,设矩阵A
3是一个m×n的矩阵,将A
3展开为一个长度为m×n的序列,i为序列中第i个元素的索引。其中,A
3'为A
3置乱后的序列。
A
3′(Ind
3(m×n-i+1))=A
3(Ind
3(i))
步骤6:用所述测量矩阵对各级系数矩阵进行测量;
示例性的,系数矩阵A
3'保持不变,用Φ
3'对系数矩阵H
3',V
3',D
3'进行测量,得到测量值矩阵H
3”,V
3”,D
3”;用Φ
2'对系数矩阵H
2',V
2',D
2'进行测量,得到测量值矩阵H
2”,V
2”,D
2”;用Φ
1'对系数矩阵H
1',V
1',D
1'进行测量,得到测量值矩阵H
1”,V
1”,D
1”。
步骤7:保留低频系数矩阵,合并测量后的非低频各级系数矩阵,构造矩阵T,对矩阵T作微处理,得到待测矩阵NT,根据待测矩阵NT分别生成行-状态转移概率矩阵和列-状态转移概率矩阵;
为了增强随机性,根据步骤1中求得的信息l
0,l
1,L
0,L
1对待测矩阵T作微处理,得到矩阵NT。
结合矩阵NT的元素信息构造行-状态转移概率矩阵和列-状态转移概率矩阵。
步骤8:对所述低频系数矩阵作SVD分解得到子矩阵,将所述子矩阵以及非低频各级系数矩阵中元素量化到预设区间;
示例性的,分别求出各级小波分解后H
i”,V
i”,D
i”,i=1,2,3九个系数矩阵的最大值与最小值,并将系数矩阵量化到[0,255]的区间上。对H
i”,V
i”,D
i”,i=1,2,3九个系数矩阵的量化完成后,就得到了H
i”',V
i”',D
i”',i=1,2,3九个相对应的系数矩阵。对低频矩阵A
3'进行SVD分解,得到u,s,vt三个相同维度的子矩阵,然后分别求出三个子矩阵的最大值与最小值,将u,s,vt三个子矩阵量化到区间[0,255]上,最终得到相对应的矩阵U,S,VT。
步骤9:根据所述混沌序列的信息生成索引值并通过所述索引值确定合并规则,根据合并规则先合并所述非低频各级系数矩阵得到整体矩阵,再将子矩阵插入到所述整体矩阵的不同位置;
示例性的,同方向系数合并为一组:H
1”',H
2”',H
3”'分在一个组,记为第一组。V
1”',V
2”',V
3”'分在一个组,记为第二组。D
1”',D
2”',D
3”'分在一个组,记为第三组。利用步骤3生成的三个混沌序列X
1',X
2',X
3'求索引值。
即第一组的索引值Lind
1'由X
1',X
2',X
3'的最后一个元素决定,第二组的索引值Lind
2'由X
1',X
2',X
3'的倒数第二个元素决定,第三组的索引值Lind
3'由X
1',X
2',X
3'的倒数第三个元素决定。同时将三个索引值映射到[1,6]区间上,表示每个组存在6种可能的排列顺序。
其中,第一组排序后的矩阵记为Y
1,内部三个块矩阵分别记为Y
11,Y
12,Y
13;第二组排序后的矩阵记为Y
2,内部三个块矩阵分别记为Y
21,Y
22,Y
23;第三组排序后的矩阵记为Y
3,内部三个块矩阵分别记为Y
31,Y
32,Y
33。
将量化后的矩阵U,S,VT分别插入到第一轮合并后的矩阵中。即矩阵U插到第一组Y
1中,矩阵S插到第二组Y
2中,矩阵VT插到第三组Y
3中。利用步骤3生成的三个混沌序列X
1',X
2',X
3'求索引值。
即第一组的索引值Hind1'由X
1',X
2',X
3'的第一个元素决定,第二组的索引值Hind
2'由X
1',X
2',X
3'的第二个元素决定,第三组的索引值Hind
3'由X
1',X
2',X
3'的第三个元素决定。同时将三个索引值映射到[1,4]区间上,表示U,S,VT三个矩阵分别有四个插入的位置,即每个矩阵组的顶部,组内两个相邻块矩阵之间,矩阵组的底部。结果如下所示。
其中,第一组合并后的矩阵记为Y
1',将第二组合并后的矩阵记为Y
2',将第三组合并后的矩阵记为Y
3'。将三个矩阵Y
1',Y
2',Y
3'合并,记合并后的矩阵为T'。
T'=(Y
1'Y
2'Y
3')
步骤10,调整所述整体矩阵维度以及获取所述整体矩阵元素信息,生成二次置乱的控制参数;
示例性的,假设合并后的矩阵维度为m×n,则寻找m×n的两个因子m
1和n
1,使得m
1×n
1=m×n,同时|m
1-n
1|最小,然后将矩阵维度调整为m
1×n
1。之后获取矩阵信息如下所示。
其中,Max为矩阵T'的最大值,Min为T'的最小值,d
1为小于或等于T'中元素平均值的最大整数,d
2为大于或等于Max和Min平均值的最小整数,d
1'为d
1除以10的余数,d
2'为d
2除以10的余数,d
12为d
1'和d
2'中的较大值。
步骤11:根据所述行-状态转移概率矩阵和列-状态转移概率矩阵对合并后的整体矩阵进行置乱,同时设置相应的标志位;
示例性的,将步骤1生成的参数a,b,c和初始值v
0带入ISEL映射,共迭代t+(d
12+1)×m
1×n
1次,每迭代2000次,用初始状态的正弦值乘以微小的系数来扰动下一次迭代的初始状态值。再舍弃前t个数,最后生成一个混沌序列V。为增强随机性,对序列V作进一步处理。即将序列乘以干扰系数α,之后加上干扰系数β,得到序列V',此处,取α=-4,β=1。之后根据混沌序列V'生成四个子序列V
1,V
2,V
3,V
4。
其中,round为四舍五入函数。
根据V
1,V
2,V
3,V
4生成置乱过程的控制参数,计算过程如下所示。
其中,w
1~w
4作为控制参数用于后续移位操作,fix为向零取整函数。其中m
1'是矩阵行维数m
1的最大质因子,n
1'是矩阵列维数n
1的最大质因子。
步骤12:用其中一个混沌序列进行整体矩阵元素的独立扩散,用其余混沌序列进行整体矩阵元素的全局扩散,得到最终密文图像;
示例性的,根据步骤7产生的行-状态转移概率矩阵,对列向量进行置乱操作。如果选中的数所在列对应正奇数,则矩阵元素移动位数为w
1;如果选中的数所在列对应负奇数,则矩阵元素移动位数为w
3;如果选中的数所在列对应正偶数,则矩阵元素移动位数为w
2;如果选中的数所在列对应负偶数,则矩阵元素移动位数为w
4。记置乱后的矩阵为T”。设置状态转移标志位矩阵,初始值为0。如果对应位置的元素发生了移位操作,则相应位置的标志位由0变成1。
根据步骤7产生的列-状态转移概率矩阵,对行向量进行置乱操作, 规则同上述步骤一致。记置乱后的矩阵为T”'。同样设置初始的标志位矩阵,如果对应位置发生了移位操作,则相应位置的标志位由0变成1。
将步骤11生成的混沌序列V'量化到[0,255]区间上,得到序列V”。将步骤10中获取的d
1'和d
2'作为混沌序列的采样间隔,对V”进行间隔为d
1'的采样得到序列V
1',对V”进行间隔为d
2'的采样得到序列V
2'。以序列V
2'的最后一个元素的位置作为起始位置,由序列V”向后取连续的m
1×n
1个数据,记为序列V
0。将图像矩阵T”'与序列V
0作异或运算得到矩阵T””。将T””记为矩阵A。
所示序列V
1'对矩阵A作前向加取模运算,得到矩阵B,之后利用B中每个元素的前一个元素作循环左移位运算得到B';利用V
2'对矩阵B'作后向加取模运算,得到矩阵C,之后利用C中每个元素的后一个元素作循环左移位运算得到C'。至此,最终的密文图像就得到了。
本发明提出基于多尺度压缩感知和马尔科夫模型的图像加密方法,根据图像的低频系数与高频系数携带信息的不同,对图像的低频系数与高频系数设置不同的采样率,能够有效地提高解密图像的重构质量。另外,通过结合混沌系统与马尔科夫模型,用先对图像进行系数矩阵内置乱,后对图像进行系数矩阵间置乱,最后进行独立扩散与全局扩散的策略完成加密的过程。
上面结合附图对本发明的实施方式作了详细说明,但是本发明并不限于上述实施方式,在所属技术领域普通技术人员所具备的知识范围内,还可以在不脱离本发明宗旨的前提下做出各种变化。
Claims (8)
- 基于多尺度压缩感知和马尔科夫模型的图像加密方法,其特征在于,包括:根据明文图像信息生成一维混沌映射的参数及初始值;通过所述明文图像的总目标采样率,利用多尺度分块压缩感知理论得到原各级系数矩阵的子采样率;将所述一维混沌映射的参数及初始值代入相应的混沌系统生成混沌序列,把所述混沌序列转换为矩阵形式,并获取相应的正交基矩阵;根据所述子采样率提取所述正交基矩阵的部分元素作为测量矩阵;所述明文图像进行三级离散小波变换,得到原各级系数矩阵,对所述原各级系数矩阵分块,构造新的各级系数矩阵;利用混沌序列生成索引序列,对相应的各级系数矩阵作块进行置乱;用所述测量矩阵对各级系数矩阵进行测量;保留低频系数矩阵,合并测量后的非低频各级系数矩阵,构造矩阵T,对矩阵T作微处理,得到待测矩阵NT,根据待测矩阵NT分别生成行-状态转移概率矩阵和列-状态转移概率矩阵;对所述低频系数矩阵作SVD分解得到子矩阵,将所述子矩阵以及非低频各级系数矩阵中元素量化到预设区间;根据所述混沌序列的信息生成索引值并通过所述索引值确定合并规则,根据合并规则先合并所述非低频各级系数矩阵得到整体矩阵,再将子矩阵插入到所述整体矩阵的不同位置;调整所述整体矩阵维度以及获取所述整体矩阵元素信息,生成二次置乱的控制参数;根据所述行-状态转移概率矩阵和列-状态转移概率矩阵对合并后的整体矩阵进行置乱,同时设置相应的标志位;用其中一个混沌序列进行整体矩阵元素的独立扩散,用其余混沌序列进行整体矩阵元素的全局扩散,得到最终密文图像。
- 根据权利要求1所述基于多尺度压缩感知和马尔科夫模型的图像加密方法,其特征在于,根据明文图像信息生成一维混沌映射的参数及初始值,具体方法为:其中,L 0,L 1分别为明文图像hash值K中0的个数,1的个数,l 0,l 1分别为最长连续0序列的长度,最长连续1序列的长度;r end为混沌区间右端点,r 0为混沌区间的某个值;其中,k i,i=1,2,…12为K的第i组二进制数,t 1-t 4为外部密钥,r start为混沌区间左端点,r end为混沌区间右端点,r 1和r 2为混沌区间的某个值;其中,k i,i=13,14,…24为K的第i组二进制数,t 5-t 8为外部密钥,r start为混沌区间左端点,r end为混沌区间右端点,a,b,c分别为相应混沌区间的某个值;其中,k i,i=25,14,…32为K的第i组二进制数,t 1-t 8为外部密钥;Z 1,Z 2为过渡矩阵;
- 根据权利要求1所述基于多尺度压缩感知和马尔科夫模型的图像加密方法,其特征在于,对相应的各级系数矩阵作块进行置乱,所述置乱方法具体为:设矩阵A 3是一个m×n的矩阵,将矩阵A 3展开为一个长度为m×n的序列,i为序列中第i个元素的索引,其置乱方式为:A 3′(Ind 3(m×n-i+1))=A 3(Ind 3(i))A 3'为A 3置乱后的序列,Ind 3为对第三级小波分解系数矩阵进行置乱的索引序列;同理得到其余置乱后的系数矩阵H i',V i',D i',i=1,2,3。
- 根据权利要求1所述基于多尺度压缩感知和马尔科夫模型的图像加密方法,其特征在于,根据合并规则先合并所述非低频各级系数矩阵得到整体矩阵,具体为:其中,Lind i,i=1,2,3是混沌序列X i',i=1,2,3生成的索引值,n i×n i,i=1,2,3是X i',i=1,2,3的长度,Lind i',i=1,2,3是Lind i,i=1,2,3映射到区间[1,6]上的结果;其中,H i”',V i”',D i”',i=1,2,3是对H i”,V i”,D i”,i=1,2,3量化到[0,255]后的结果;第一组排序后的矩阵记为Y 1,内部三个块矩阵分别记为Y 11,Y 12,Y 13,排列顺序由Lind 1'决定;第二组排序后的矩阵记为Y 2,内部三个块矩阵分别记为Y 21,Y 22,Y 23,排列顺序由Lind 2'决定;第三组排序后的矩阵记为Y 3,内部三个块矩阵分别记为Y 31,Y 32,Y 33,排列顺序由Lind 3'决定。
- 根据权利要求1所述基于多尺度压缩感知和马尔科夫模型的图像加密方法,其特征在于,将子矩阵插入到所述整体矩阵的不同位置,具体为:其中,Hind i,i=1,2,3是混沌序列X i',i=1,2,3生成的索引值,Hind i',i=1,2,3是Hind i,i=1,2,3映射到区间[1,4]上的结果;T′=(Y 1′ Y 2′ Y 3′)其中,Y 1'为第一组合并后的矩阵,排列顺序由Hind 1'决定;Y 2'为第二组合并后的矩阵,排列顺序由Hind 2'决定;Y 3'为第三组合并后的矩阵,排列顺序由Hind 3'决定;T'为Y 1',Y 2',Y 3'合并后的矩阵;U,S,VT为对低频各级系数矩阵进行SVD分解得到的三个子矩阵。
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SHI YUANDI, HU YINAN, WANG BIN: "Image Encryption Scheme Based on Multiscale Block Compressed Sensing and Markov Model", ENTROPY, vol. 23, no. 10, pages 1297, XP093049715, DOI: 10.3390/e23101297 * |
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