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Multi-focus image fusion and robust encryption algorithm based on compressive sensing
Optics & Laser Technology 91 (2017) 212–225
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Full length article
Multi-focus image fusion and robust encryption algorithm based on compressive sensing
MARK
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Di Xiaoa, , Lan Wanga, Tao Xianga, Yong Wangb a Key Laboratory of Dependable Service Computing in Cyber Physical Society of Ministry of Education, College of Computer Science, Chongqing University, Chongqing 400044, China b Key Laboratory of Electronic Commerce and Logistics of Chongqing, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
A R T I C L E I N F O
A BS T RAC T
Keywords: Image encryption Image fusion Compressive sensing Robust
Multi-focus image fusion schemes have been studied in recent years. However, little work has been done in multi-focus image transmission security. This paper proposes a scheme that can reduce data transmission volume and resist various attacks. First, multi-focus image fusion based on wavelet decomposition can generate complete scene images and optimize the perception of the human eye. The fused images are sparsely represented with DCT and sampled with structurally random matrix (SRM), which reduces the data volume and realizes the initial encryption. Then the obtained measurements are further encrypted to resist noise and crop attack through combining permutation and diffusion stages. At the receiver, the cipher images can be jointly decrypted and reconstructed. Simulation results demonstrate the security and robustness of the proposed scheme.
1. Introduction Nowadays, multimedia sensor networks have wide application prospects in the fields such as military affairs and environment monitor. The sensors are actually cameras that record still image or video. Due to the fact that the commonly used optical lenses suffer from a problem of limited field depth, it is often impossible to get an image that contains all relevant objects in focus and we call these images multi-focus image. Image fusion can merge the different information taken from the same scene with different focuses to obtain a new and improved image [1]. Multi-focus image fusion has been widely used in various fields, such as computer vision and remote sensing. Currently, the existing multi-focus image fusion methods are mainly based on spatial domain [2], transform domain [3] and compressive sensing (CS) [4–7]. However, these image fusion methods focus on how to integrate multi-focus images into a single visual enhanced image, and do not care about secure transmission and robustness. During image transmission over unreliable wireless channels, packets loss or malicious attack is inevitable [8]. Generally, the confidentiality is protected with encryption methods. However, conventional encryption algorithms are not suitable for image when considering their costs and traffic load [9]. Since images are often sparse or compressible while CS can enable sub-Nyquist sampling and low-energy data reduction, it is promising to utilize CS to fill in the gap.
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In the related field of image encryption based on CS, several works have been proposed. Liu et al. [8] proposed the image encryption scheme based on CS theory with robustness to packet loss. Huang et al. [10] proposed a compression-combined digital image encryption method which is robust against consecutive packet loss and malicious crop attack. Subsequently, a robust still-image transmission coding scheme based on CS is presented in [11]. A robust coder, based on CS with structurally random matrix (SRM), for encrypted images over packet transmission networks is proposed in [12]. A simultaneous image compression, fusion and encryption algorithm based on compressive sensing and chaos can be found in [13]. But in the experiment result of the method, the correlation and histograms distributions of the encrypted images are not uniform which implies the scheme is not very secure and valid in some extent. Since good fusion performance, transmission security and robustness are not always considered or valid, the aforementioned approaches are not suitable for fusion and robust encryption of multi-focus images. In this paper, a multi-focus image fusion and robust encryption scheme based on CS is presented to ensure the efficiency and security of image transmission. In this scheme, the source images are firstly fused based on wavelet decomposition and then the fused image is sparsely represented with discrete cosine transform (DCT). The obtained coefficients are measured with a structurally random measurement matrix (SRM) [14] and finally ciphertext can be obtained by
Corresponding author. E-mail address: [email protected] (D. Xiao).
http://dx.doi.org/10.1016/j.optlastec.2016.12.024 Received 26 August 2016; Received in revised form 11 November 2016; Accepted 20 December 2016 0030-3992/ © 2016 Elsevier Ltd. All rights reserved.
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DWT
Image fusion
M easurem ent
Fused image C om pressive sensing
Measurement encryption Ciphertext
key1
key2
DWT
Measurement
Fused image
Decryption
Reconstruction
Ciphertext
Fig. 1. The flowchart of the proposed main steps.
This is equivalent to find the sparsest solutions to x = φs, provided M ≥ CK log N/K , where C is a small constant. Compressive sensing theory indicates that x can be acquired by the following random measurement matrix ∅ ∈ RM×N (M ≪ N), i.e.,
encrypting the integer of measurement with the combined scrambling and diffusion stages. At the receiver, the fused image can be reconstructed by decrypting the cipher image and using a recovery algorithm. The analyses and the numerical experiments are given to illustrate its fusion performance, security and robustness. The rest of this paper is organized as follows. Section 2 is a brief review of the theory of CS and Logistic map. Section 3 proposes the image fusion based on wavelet decomposition and the robust encryption based on CS with SRM. In Section 4, the numerical experiments and analyses can be found. Finally, we conclude the paper in Section 5.
y = Φx = Φφs
(3)
where y is a sampled vector, and Φ is a M × N measurement matrix that is incoherent with the sparse basis φ , i.e., the maximum magnitude of the element in Φφ is small. Then x can be faithfully recovered from only M=O(K log N) measurements through l1-minimization:
min s 1 s. t. y = Φφs
2. Preliminaries
(4)
Eq. (4) presents an l1 -minimization problem which can be solved by orthogonal matching pursuit algorithm [17].
In this section, we show the basic background, primarily on the theory of CS and Logistic map.
2.2. Logistic map 2.1. Compressive sensing Logistic map is usually used to generate pseudo-random sequences. Logistic map is defined as:
The fundamental Shannon/Nyquist sampling theory is widely accepted as the keystone in signal acquisition and reconstruction. Nevertheless, the number of required measurements can be so large that the storage becomes unbearable and the acquisition time can be very long sometimes. Compressive sensing (CS) in [15], as an emerging and fascinating field, shows that K-sparse signals or images can be recovered exactly from M measurements far fewer than N samples what are considered being necessary in Nyquist theorem, i.e., M ≪ N . CS combines the traditional sampling and compressing process into a single non-adaptive linear measurement process by exploiting the data sparsity. Many signals acquired from the physical world are sparse or compressible in the sense that when expressed in the proper orthonormal basis (such as a standard or wavelet basis), their coefficient entries decay rapidly. A signal x ∈ RN of length N is said to be K-sparse or compressible if it can be well approximated using only K ≪ N coefficients over some sparse basis φ as follows:
x = φs
xn +1=μxn (1 − xn )
The output sequence is chaotic if one chooses μ ∈ [3. 57, 4], and μ is the control parameter. This chaos system is sensitive to the seed value x 0 and the generated sequence xn is distributed in the range of (0, 1). The initial value x 0 and control parameterμ can be combined as the secret key. 3. Proposed scheme With the background knowledge presented in the previous section, the proposed image fusion and robust encryption algorithm based on CS is now described. All images considered in this paper are gray-scale images. The proposed algorithm is realized in the following main steps: (1) image fusion based on wavelet decomposition; (2) sparse representation of the fused images; (3) SRM measure; (4) measurement encryption; (5)the receiver decrypts the ciphertext; (6)reconstruct the fused image. The flowchart of the proposed main steps is illustrated in Fig. 1.
(1)
where s is the transform coefficient vector that contains at most K significant nonzero entries. For images, typical choices of φ can consider the widely used discrete Fourier transform (DFT), discrete cosine transform (DCT), or discrete wavelet transform (DWT) matrix, etc. If φ satisfies RIP (Restricted Isometry Property), we can reconstruct signal by solving the optimization problem below [16].
min s 1 s. t. x = φs
(5)
3.1. Image fusion based on wavelet decomposition The key point in multi-focus image fusion is to decide which portions of each image are in better focus than their respective counterparts in the associated images and then combine these regions
(2)
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Fusion rule
DWT
IDWT
Fig. 2. The image fusion scheme using DWT.
If M2 (p)
to construct a well-focused image with certain fusion rules [18]. We propose an image fusion scheme based on discrete wavelet transform (DWT) [19] which involves wavelet decomposition of the source images, calculation of the wavelet coefficients, merging wavelet coefficients based on the fusion rule, and reconstruction for new fusion image, as shown in Fig. 2. The core step in image fusion based on wavelet is to merge the coefficients based on the fusion rule. In the present work, since we are focusing on multi-focus image fusion, the characteristics of the images should be considered: most information of source images is kept in the low frequency sub-band, while the high frequency sub-bands contain the detail of image. In the application, we pay more attention to better visual representation for multi-focus images, so we should preserve most relevant and complementary information with the reduced noise. The important idea of the fusion is to select the coefficients of low-frequency band by a neighborhood pixel correlation selection scheme, while the coefficients of the high-frequency bands are performed with a maximum absolute value based pixel selection scheme. The two coefficients can be combined together to reconstruct the superior image in which all the pixels can be clearly focused. Generally, an image I has its multi-scale decomposition (MSD) representation denoted as D I . Let p=(m, n, k, l) indicate the index corresponding to a particular MSD coefficient, where m and n indicate the spatial position in a given frequency band, k is the decomposition level, and l is the frequency band of the MSD representation. Therefore, DI ( p ) denotes the MSD value of the corresponding coefficient at the position (m, n) with decomposition level k and frequency band l. We propose a scheme by computing the variance in a neighborhood to select the low-frequency coefficients.
δI ( p )=
1 S×T
meanI ( p )=
S /2
T /2
∑
∑
DI (m + s, n + t , k , l ) − meanI ( p ) 2 ,
s ∈− S /2 t ∈− T /2
1 S×T
S /2
T /2
∑
∑
DI (m+s, n +t , k , l ),
s ∈− S /2 t ∈− T /2
⎧ DA ( p ), δA ( p )≥δB ( p ), Dz ( p )=⎨ ⎩ DB ( p ), δA ( p ) < δB ( p ). If M2 (p)≥T ,
1
S /2
⎧ Wmax DA ( p )+Wmin DB ( p ), δA ( p )≥δB ( p ), Dz ( p ) = ⎨ ⎩Wmin DA ( p )+Wmax DB ( p ), δA ( p ) < δB ( p ).
(10)
⎛ 1−M2 ( p ) ⎞ Wmin= 0. 5 − 0. 5 ⎜ ⎟, ⎝ 1−T ⎠
(11)
Wmax =1 − Wmin .
(12)
where T is the match threshold value and T ∈ (0. 5, 1). Then for the high-frequency wavelet coefficients, the maximum absolute value fusion rule is adopted as follows:
⎧ DA ( p ), if DA ( p ) ≥ DB ( p ) DZ ( p )=⎨ ⎩ DB ( p ), if DA ( p ) < DB ( p )
(13)
At last, the main fusion steps can be summarized as follows: Step1: Source images lA (x, y) and lB (x, y) must be registered to assure that the corresponding pixels are aligned. Step2: The wavelet transform decomposes each image into K-level wavelet planes to obtain one low-frequency portion and 3K highfrequency portions. Step3: The wavelet coefficients of the low-frequency are selected by (6) and (7) and performed fusion by (9) and (10), while the wavelet coefficients of the high-frequency are fused by (13). Step4: The fused image l f (x, y) is constructed by performing an inverse wavelet transform based on the combined transform coefficients from Step 3.
(6)
3.2. Sparse representation and Compressive sensing by SRM
(7)
In this section, we firstly divide the fused image with N × N pixels into a lot of non-overlapping blocks. Next, each image block is transformed by a sparse basis matrix where we use DCT as the sparse basis Ψ , and various DCT coefficient blocks are formed. Simultaneously, measurement matrix ΦE is deployed to sense these DCT coefficients independently within each block. The process is simply random linear projection, and can be achieved by inner product operation of the corresponding two elements between Ψ and ΦE . Here, according to the CS principle in [15], ΦE should be selected to be incoherent with Ψ. We design an efficient and appropriate measurement matrix which is called a structurally random matrix (SRM) in the literature [14]. SRM possesses the following features: optimal or near-optimal sensing
where S × T is the neighborhood size, and meanI (p) and δ I (p) denote the mean value and variance value of the coefficients centered at (m, n) in the window of S × T , respectively. We set M2 (p) as the regional variance matching degree for the source image A and source image B in S × T neighborhood and M2 (p)ϵ[0, 1]. The smaller the M2 (p) value is, the lower the regional variance matching degree for image A and B is.
M2 (p)=
(9)
T /2
2× S × T ∑s ∈− S /2 ∑t ∈− T /2 DA ( p ) − meanA ( p ) DB ( p ) − meanB ( p ) δA ( p ) + δB ( p ) (8)
The fusion scheme used for the low-frequency bands can be illustrated as follows:
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sion at the same time and the inter-block operation, as illustrated in Fig. 3. There is a measurement integer sequence seq of length M × N , as given by Eq. (17).
performance; universality; low complexity; hardware/optical implementation friendless. Generally, we have a form of Φ as follows:
Φ=
N DFR, M
(14)
where the matrix F ∈ is a block diagonal matrix, in which F represents the set of the block Hadamard matrix ΦE , D is an operator which picks up M rows of FR uniform at random, and R denotes a scrambling operator which is a binary matrix in which each row or column has exactly one and the rest are all zero. Here, inspired by the idea in the literature [20], the permutation matrix R is constructed as follows:
th
where Z(i) is the integer value of the i measurement,i ∈ [1, M × N]. Firstly, the sequence seq is partitioned into Num blocks and each block includes 64 integer values. Secondly, through the intra-block operation, the integers are scrambled and integer values are changed at the same time. Then, through the inter-block operation, all the blocks are permutated to improve the encryption effect. The pseudorandom numbers generated from Logistic map are used to change the pixel values in the blocks. Meanwhile, the position of integers is relocated according to the random number sorting. The detailed encryption algorithm is described as follows:
Step1: the random sequence p1 can be generated by Logistic map with (μ1, x1), and the control parameter μ1 and the initial value x1 are keys. Logistic map is iterated for h + N times, where h > 0 is used to improve initial value sensitivity. Step2: Initialize the N × N matrix R and set the value of each element to be zero. The re-ordered p2 is obtained by sorting p1 in an ascending order and set R(p2(m),m) to be 1, where m ∈ {1, 2, …N}.
Step1: The sequence seq is firstly divided into a set of blocks B={b1,b 2 ,…,b Num }, where each block contains 64 integer values. We need to pad the last block to make the length be a multiple of 64 and the padding is 1 or 0 if the total number of the measurement integers is not a multiple of 64. We can calculate the number of padding bits as follows:
The N × N permutation matrix R can be obtained by those steps. It should be noticed that permutation-based decryption is performed by multiplying the cipher image with the inverse permutation matrix. So, we can obtain the measurement matrix Φ . In the pseudo-random measure process, the fused image lf is firstly transformed to x in DCT domain and each column of the coefficient matrices will be measured by the constructed measurement matrix Φ , and thus the measurements y are obtained:
y = Φx =
N DFR x M
|PP| = 64 − ((M × N)mod 64), where |PP| is the length of the padding filed. Step2: Intra-block operation: for each block in set B, perform the diffusion and permutation operations simultaneously.
(15)
Iterate Logistic map (5) with (μ 2 ,x2 ) for N0 times to avoid the harmful effect of transitional procedure, where N0 is a constant, and iterate (5) with (μ 2 ,xN 0 ) to generate 64 pseudorandom numbers and sort them in ascending order. The permutated integers are obtained by the sorted index and the integer values are changed simultaneously according to
The pseudo-random measure process accomplishes the measure operation. Meanwhile, it can be viewed as compression and the first level encryption. Since these measurements are generally not finiteprecision numbers, they are quantized in [12,21] for further processing. However, this will cause distortion error and the reconstruction performance may be unacceptable to some extent. Therefore, to eliminate the distortion error, in this paper, the measurements will not be quantized and only the integers of measurements will be encrypted.
c(n) = p (n )⨁k (n )⨁c (n−1),
(18)
where p (n), k(n), c(n) and c(n-1) represent the current integer, the key stream element, the output cipher-integer and the previous cipherinteger, respectively. For the first integer, an initial value c(-1) can be set as a seed. The key stream element k(n) used for masking is calculated by Logistic map given by Eq. (5).
3.3. Measurement encryption
k(n)= mod [floor(x(n)×1015), 256].
Firstly, each measurement is divided into three parts: the positive integer Z, the decimal fraction Q and the sign bit H:
y = H × Z + Q.
(17)
seq = {Z (1), Z (2), Z (3),…Z (M × N)},
N×N
(19)
where x(n) is the current state of the chaotic map and is generated by Logistic map with (μ 2 ,x3). Different blocks have different adaptive μ 2 values in the encryption processing by Eq. (20) and Eq. (21).
(16)
To further encrypt the integer of measurement Z, inspired by the idea in the literature [22], the encryption algorithm is proposed, which includes the intra-block operation combining permutation and diffu-
⎛ j−1 ⎞ e=⎜ ⎟∈[0, 1] ⎝ Num − 1 ⎠
Fig. 3. Block diagram of the proposed image measurement encryption.
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(20)
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μ 2=
μ 2+3. 57 + 0. 43e ∈[3. 57, 4] 2
4. Experimental results and analyses (21) To test and verify the performance of the validity and security of the proposed scheme, the numerical simulation experiments are carried out. In general, the multi-focus images contain complementary content and information. Two types of multi-focus images of sizes 256×256 and 512 × 512 are used for testing and the reconstruction algorithm is GRSR in [23]. All the experiments are performed on personal computer with an Intel Core 1.8 G CPU, 2 GB memorywith a Windows 7 Ultimate 64 bits operating system. The algorithm is simulated with Matlab R2014a platform, and the secret key is randomly selected as: μ1= 3.995635698763065, μ 2 =3.997458345356721, x1= 0.2236984x2 =0.326653465125465, x3= 0.265689563447562, 55643233, x 4= 0.425653527463592 for Logistic map and block size 32 × 32 in CS. In the following experiments, we can explore the security and robustness of images over the transmission.
Logistic map is always chaotic if one chooses the initial μ 2∈[3. 57, 4], where e is the ratio between the block index and the total number of blocks. Here, the control parameter μ 2 and the two initial value x2 and x3 can serve as the keys. Step3: Inter-block operation: iterate (5) with (μ 3,x 4 ) to generate Num pseudorandom numbers and sort them in ascending order. The permutated block location is obtained by the sorted index. The initial value x 4 can serve as the key and μ 3 is a constant chosen within [3. 57, 4]. Step4: Repeat Steps 2–3 m rounds to satisfy the security requirement, and the ciphertext C′ is obtained. Step5: Recombine the ciphertext C′, the decimal fraction Q and the sign bit H by the following Eq. (22).
C = H × C′+Q.
4.1. Fusion and encryption results
(22)
In Fig. 4, the source images and fused images are as shown in the first two columns and the third column, respectively. It is obvious that the fused image is acceptable on the vision. The obtained cipher images are shown in the fourth column, while the decrypted and reconstructed images with correct keys are shown in the fifth column of Fig. 4. The PSNR of the reconstructed image is showed in Table 1 when sampling rate is 1:4. It can be seen from the decrypted results that the salient features of source images are preserved and transferred into the reconstructed image.
The encrypted measurements C are transformed to cipher image by filling random value in the last column of image, and the cipher image is transmitted to the receiver by wireless channel which may suffer from malicious attacks. 3.4. Decryption and reconstruction At the receiving side, after the receiver obtains the encrypted image, he applies joint decryption and reconstruction to recover the fused image with the decryption keys which are transmitted through security channel.
4.2. Image fusion evaluation Experiments are conducted to compare the effectiveness of the proposed fusion algorithm with three other multi-focus image fusion algorithms in [24–26] through four image quality evaluation criteria. These four metrics are: (i) Qabf metric [27] which measures the quality of the fused image (denoted as I f ) using two images as input (denoted as IA and IB , respectively), (ii) mutual information (MI) metric [28], (iii) spatial frequency (SF) metric [29], and (iv) PSNR of the fused image. For these criteria, the larger the value is, the better the fusion result is. The proposed algorithm and its counterparts belong to the same category in which image fusion is in transform domain. The proposed algorithm exploits two-level wavelet decomposition of the clock images. The objective performance comparisons are presented in Table 2, where one can see that the proposed approach always outperforms three other conventional approaches.
Step1: Through the inverse transformation of the Step5 in the encryption algorithm, the cipher C is divided into three parts: the positive integer C′, the decimal fraction D and the sign bit H. Step2: The positive integer sequence C′ is divided the same as the Step1 in the encryption algorithm. Step3: The inverse inter-block permutation for the positive integer C′ is similar to the Step3 in the encryption with the key x 4 and the constant μ 3. Step4: The inverse intra-block operation is to remove the effect of the combined permutation and diffusion on the blocks with the key μ 2 , x2 and x3. Firstly, the pixel value is obtained by Eq. (23):
p′(n) = c′(n −1)⨁k (n )⨁c′(n ),
(23)
4.3. Statistical analysis
where p′(n), k(n), c′(n) and c′(n-1) represent the decrypted integer, the key stream element, the current encrypted integer and the previous cipher-integer, respectively. The k (n ) is obtained from Eq. (19), the same as the encryption section. Secondly, the inverse permutation for the decrypted integer is similar to the Step2 in the encryption.
Confusion is a technique to ensure that the relationship among the plaintext, the ciphertext and the key is as complicated as possible. Good confusion makes it very difficult to recover the secret key even if a large number of plaintext-ciphertext pairs produced by the same encryption key are available for an attacker. We need to analyze the image histogram and pixel correlation to verify the capability of resisting statistical attacks and show the good confusion property. The histogram reveals the distribution of image pixel values. Good confusion method should ensure the cipher-image histograms of different source images have a uniform distribution to resist statistical attacks. Numerical experiments are conducted to show that the ciphertexts have similar distribution shown in Fig. 5(d) and (h), although we do not quantify the cipher image to [0,255] in order to reduce the distortion. So it is clear that the histogram does not reveal any information for the statistical analysis for attackers.
Step5: Repeat Steps3–4 (m-1) times which is the same as the Step4 in encryption. After the padding field is removed, the plaintext P is obtained. At the last round, the length of padding will be 0 when (M × N) mod 64 = 0, which means that the last block in the original message has 64 bits; while the length of padding is not 0 when (M × N) mod 64 ≠ 0 , and the length of padding is |PP|. Step6: It is the same as the Step6 of the encryption algorithm, and the measurements without distortion error are obtained. Step7: The fused image is reconstructed through l1 -minimization problem which can be solved by GRSR algorithm in [23].
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Fig. 4. Simulation results of the proposed scheme: the first row is Clock images, the second row is Pepsi images, the third row is Peppers images and the fourth row is Camera images; from the first to the fifth column are the focus-left image, focus-right image, fused image, encrypted image, and reconstructed image, respectively.
redundant in meaningful image. To evaluate the correlation property of the plain image and cipher image, the following steps are performed.
Table 1 The PSNR of the reconstructed image for different source images. images
PSNR
Clock images Pepsi images Peppers images Camera images
42.3757 40.7933 38.3611 38.3892
(1) 5000 pixels are randomly selected as samples; (2) The correlation coefficient (CC) between two adjacent pixels in horizontal, vertical and diagonal directions are calculated according to Eqs. (24)–(26).
rx, y = Table 2 The performance comparison among various image fusion approaches. Methods
MI
Qabf
SF
PSNR
Pu and Ni's approach [24] Qu et al.’s approach [25] Li and He's approach [26] Proposed approach
6.80 6.07 2.63 6.93
0.66 0.6 0.67 0.68
11.25 12.67 5.45 18.15
30.25 35.69 34.79 36.87
E (x )=
D (x )=
E {[x − E (x )][ y − E ( y)] D (x ) D ( y )
1 M 1 M
(24)
M
∑ xi i =1
(25)
M
∑ [xi −E (xi )][ yi −E ( yi)] i =1
(26)
where xi and yi are the gray-level values of the two adjacent pixel in three directions of each image and M represents the total number of the samples. E(x) and D(x) are the expectation and the variance of x, respectively.
As a general requirement for all image encryption schemes, the cipher image should be greatly different from its original form. However, the adjacent pixels usually have strong correlation and very
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Fig. 5. Two source images, cipher images and their histograms: (a) plain image1,(b)histogram of the plain image1, (c) cipher image1, (d) histogram of the cipher image1, (e) plain image2,(f) histogram of the plain image2, (g) cipher image2, (f) histogram of the cipher image2.
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Table 3 Correlation coefficients of the plain image and the cipher image. Direction
Plain image
Cipher image
Horizontal Vertical Diagonal
0.9701 0.9693 0.9452
0.0085 −0.0153 0.0025
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(c1)
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(c2)
Fig. 6. Correlation plots of two adjacent pixels: the first row is the correlation plots of the plain image in (a1) horizontal, (b1) vertical, (c1) and diagonal directions; the second row is the correlation plots of the cipher image in (a2) horizontal, (b2) vertical, (c2) and diagonal directions.
In our proposed scheme, four initial values of Logistic map and the two parameters are considered as the secret keys. To evaluate the key sensitivity of encryption phase, by introducing a slight change 10−15 to one of the keys while keeping all the others unchanged, and the encryption process is repeated. Fig. 7 shows the source image, the encrypted image with correct key, the encrypted image with incorrect key and the differential image between two cipher images. Furthermore, in a good cryptosystem, a tiny modification of the decryption key results in a totally different decrypted image. To evaluate the key sensitivity of the decryption case, the fused source images are firstly encrypted with the original keys and the cipher images are obtained. The original key is modified slightly by adding 1 to the 15th decimal place as listed in Table 4 and the cipher image is decrypted with the modified keys. Fig. 8 shows the decrypted image and it is obvious that all of the incorrectly decrypted images are completely unrecognizable and cannot reveal any perceptive information of the plaintext. Consequently, the keys are fairly sensitive to the decryption and reconstruction.
The CC values of the test plain image and its cipher image obtained by the proposed algorithm are listed in Table 3 and Fig. 6. It indicates that the CC values of adjacent pixels in the plain image and the corresponding cipher image are close to one and zero, respectively. This means that no detectable correlation exists between the original image and its corresponding cipher image. Based on the above histogram and correlation analyses, we can conclude that the proposed scheme can resist statistical analysis attack as the encryption process effectively confuses and diffuses the plain image.
4.4. Key sensitivity analysis In cryptanalysis, the attacker usually makes an alteration in the plain image and observes the corresponding change in the cipher image. A meaningful relationship between the plain image and the cipher image may appear, and this kind of attack is called differential attack. In particular, when an image is encrypted, the completely different ciphertext should be produced when slightly different keys are used to encrypt the same plaintext. Therefore, when an image is decrypted, a tiny mismatch between the encryption and decryption keys can cause the decryption failure. So the key sensitivity is a crucial feature of an effective cryptosystem.
4.5. Robustness analysis It is inevitable that the ciphertext will be affected by noise, consecutive packet loss or even losing partial information during the 219
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Fig. 7. Key sensitivity test in the encryption case: (a) fused source image; (b) cipher image with correct keys; (c) cipher image(μ1=
3.995635698763066, μ 2 =3.997458345356721,
x1= 0.223698455643233, x2= 0.265689563447562, x3= 0.326653465125465, x 4 =0.425653527463592); (d) differential image between (b) and (c); (e) cipher image(μ1= 3.995635698763065, μ 2 =3.997458345356722, x1= 0.223698455643233, x2= 0.265689563447562, x3= 0.326653465125465, x 4 =0.425653527463592); (f) differential image between (b) and (e); (f) cipher image (μ1= 3.995635698763065, μ 2 =3.997458345356721, x1=
0.223698455643234, x2=
0.265689563447562, x3= 0.326653465125465,
x 4 =0.425653527463592); (g) differential image between (b) and (f); (h) cipher image (μ1= 3.995635698763065, μ 2 =3.997458345356721, x1= 0.223698455643233, x2=
0.265689563447563, x3= 0.326653465125465, x 4 =0.425653527463592); (i) differential image between (b) and (h); (j) cipher image(μ1= 3.995635698763065,
μ 2 =3.997458345356721, x1= 0.223698455643233, x2=
0.265689563447562,x3= 0.326653465125466, x 4 =0.425653527463592); (k) differential image between (b) and (j); (l)
cipher image (μ1= 3.995635698763065, μ 2 =3.997458345356721, x1= 0.223698455643233, x2=
0.265689563447562, x3= 0.326653465125465, x 4 =0.425653527463593); (m)
differential image between (b) and (l).
Salt & pepper noise and intensity is 0.01, 0.05 and 0.10, respectively. Form the Table 5 and Fig. 10, we can see that the PSNR values of the reconstructed images decrease with the increase of noise strength and the images become fuzzier with Salt & pepper noise than Gaussian noise, which indicates our proposed scheme performs better in antiGaussian noise than Salt & pepper noise. However, from the visual sense, main information can still be recognized. The PSNR values of the
transmission channel and storage process. Therefore, we must consider the robustness of the proposed scheme. In Fig. 10, we display robustness of the proposed scheme with different intensity Gaussian noises and Salt & pepper noise added into the cipher image. Fig. 9(a)– (f) show the reconstructed images with zero-mean Gaussian noise with standard deviation of 1and intensity is 0.01, 0.05, 0.10, 0.15, 0.20 and 0.25, respectively. Fig. 9(e)–(g) show the reconstructed images with
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Table 4 Differences between cipher images produced by slightly different keys. Figures
8(b) 8(c) 8(d) 8(e) 8(f) 8(g) 8(h)
Decryption key
μ1
μ2
x1
x2
x3
x4
3.995635698763065 3.995635698763066 3.995635698763065 3.995635698763065 3.995635698763065 3.995635698763065 3.995635698763065
3.99745834535671 3.99745834535671 3.99745834535672 3.99745834535671 3.99745834535671 3.99745834535671 3.99745834535671
0.223698455643233 0.223698455643233 0.223698455643233 0.223698455643234 0.223698455643233 0.223698455643233 0.223698455643233
0.265689563447562 0.265689563447562 0.265689563447562 0.265689563447562 0.265689563447563 0.265689563447562 0.265689563447562
0.326653465125465 0.326653465125465 0.326653465125465 0.326653465125465 0.326653465125465 0.326653465125466 0.326653465125465
0.425653527463592 0.425653527463592 0.425653527463592 0.425653527463592 0.425653527463592 0.425653527463592 0.425653527463593
Fig. 8. Key sensitivity test in the decryption case: (a) encrypted image; (b) decrypted image with the original keys; (c) decrypted image (μ1= 3.995635698763066, μ 2 =3.997458345356721, x1= 0.223698455643233, x2= 0.265689563447562, x3= 0.326653465125465, x 4 =0.425653527463592); (d) decrypted image (μ1= 3.995635698763065,
μ 2 =3.997458345356722, x1= 0.223698455643233, x2= 0.265689563447562, x3= 0.326653465125465, x 4 =0.425653527463592); (e) decrypted image (μ1= 3.995635698763065,
μ 2 =3.997458345356721, x1=
0.223698455643234, x2=
μ 2 =3.997458345356721, x1= 0.223698455643233, x2=
0.265689563447562, x3= 0.326653465125465, x 4 =0.425653527463592); (f) decrypted image (μ1= 3.995635698763065,
0.265689563447563,x3= 0.326653465125465,x 4 =0.425653527463592); (g) decrypted image (μ1= 3.995635698763065,
μ 2 =3.997458345356721, x1= 0.223698455643233, x2=
0.265689563447562, x3=
0.326653465125466, x 4 =0.425653527463592); (h) decrypted image (μ1= 3.995635698763065,
μ 2 =3.997458345356721, x1= 0.223698455643233, x2=
0.265689563447562, x3=
0.326653465125465, x 4 =0.425653527463593).
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Fig. 9. The reconstructed image with different intensity in Gaussian noises:(a) 0.01, (b) 0.05,(c) 0.1,(d) 0.15, (e) 0.2, (f) 0.25. Salt and pepper noise: (g) 0.01, (h) 0.05, (i) 0.10. 45
Table 5 PSNR values of the reconstructed images for different Gaussian noise and Salt & pepper noise intensity. PSNR
9(a) 9(b) 9(c) 9(d) 9(e) 9(f) 9(g) 9(h) 9(i)
0.01 0.05 0.10 0.15 0.20 0.25 0.01 0.05 0.10
38.9336 33.6432 29.5689 28.9766 26.9484 23.8793 23.4027 16.9756 14.3429
0.25 0.2
30 25
0.15 20
Intensity
Intensity
Intensity
35
PSNR
Figures
0.3
PSNR
40
0.1
15 10
0.05 5 0
0
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
Fig. 10. The PSNR distribution of reconstructed image in different intensity noise.
channel, information block loss may affect image content. So the reconstruction performance from incomplete ciphertext is also examined. In Fig. 11, we give an example and display the robustness of the proposed scheme. During the transmission, the attacker crops the block data of 20× 20, 30× 30, 50× 50 pixels, respectively. As shown in
proposed scheme and the scheme in [30] are compared in Table 6, which shows that the proposed scheme has better performance. The results indicate that the proposed scheme could resist against the noise attack to a certain degree. When malicious crop attack happens during the transmission 222
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Table 6 PSNR values for different Gaussian noise intensity in proposed scheme compared with Ref. [30]. Intensity
Proposed scheme
Ref.[30]
0.01 0.05 0.1 0.15 0.2 0.25
38.9336 33.6432 29.5689 28.9766 26.9484 23.8793
30.5884 28.5532 25.3264 22.7359 20.5801 18.9111
Fig. 11. The reconstructed images with pixel block loss of the encrypted image, (a) 20×20 pixels loss; (b) the reconstructed image of (a); (c) 30×30 pixels loss; (d) the reconstructed image of (c) (e) 50×50 pixels loss; (f) the reconstructed image of (e).
Table 7 The PSNR of the reconstructed images with different size block loss.
Table 8 Practical feature comparison between SRM and other random sensing matrix.
Figures
block size
PSNR
10(b) 10(d) 10(f)
20×20 30×30 50×50
24.2116 21.3791 17.0550
Fig. 11(a)-(f) and Table 7, it can be seen that although the reconstruction images are distorted to some extent, the content is maintained in our result, which implies that our scheme have robustness of resisting malicious crop attack.
Features
SRMs
Completely Random Matrices
No. of measurements for exact recovery Sensing complexity Reconstruction complexity at each iteration Implementation in hardware and optics Fast computability Block-based processing
O(KlogN)
O(KlogN)
O(NlogN) O(NlogN)
O(KNlogN) O(kNlogN)
Very easy
Difficult
Yes Yes
No No
Gaussian or Bernoulli i.i.d. matrices [21]. The measurements obtained by SRM are reduced to some degree according to sampling rate and the complexity is reduced much in encryption by confusion and diffusion. The schemes in [8,10,13] perform sampling by random sensing matrices, so the memory and computation complexity are relatively higher than SRM.
4.6. Comparison analysis In the proposed scheme, a structurally random matrix (SRM) is utilized. Table 8 summarizes the practical advantages of employing a fast and efficient sensing matrix over a random sensing matrix such as 223
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Table 9 The comparison of proposed scheme with other methods. proposed method
other methods
good fusion high sensing efficiency resist block missing resist noise attack Without the quantization error high security low memory complexity low computation complexity
[31] [21] [8,10,13] [13] / [24] [13] [22,24]
Fig. 12. The designed optoelectronic hybrid setup of the proposed scheme.
Acknowledgement
In the proposed scheme, we not only use image fusion technology for multi-focus image but also consider the security and compression in transmission channel, while the schemes in [24–26] do not take this into account. We compare the proposed scheme with other methods from many aspects. As shown in Table 9, our proposed scheme has all the listed advantages, but other algorithms only have one or some of the advantages.
The work described in this paper was funded by the National Natural Science Foundation of China (Grant Nos. 61472464, 61502399, 61572089, 61672118), the Natural Science Foundation of Chongqing Science and Technology Commission (Grant Nos. cstc2013jcyjA40017, cstc2015jcyjA40039) and the Fundamental Research Funds for the Central Universities (Grant No. 106112014CDJZR185501).
4.7. Possible optical setup
References
A possible optoelectronic hybrid setup in Fig. 12 is designed. The source images are fed into the computer to accomplish the pre-processing operations include multi-focus image fusion, sparse representation, pseudo- random measure. As mentioned in [32], the DWT of an image can be obtained by carrying out a Fourier transform of this image followed by a multiplication by the WT filter. The two operations can be simultaneously implemented in coherent optics and the spatial light modulator (SLM) is used for modulating the fused image. Then the fused image is transformed with DCT. Inspired by the idea in the literature [33], we can succeed in implementing the DCT optically. The charge-coupled device (CCD) captures the transformed measurement and inputs it to the computer to accomplish the measurement encryption operation. After several iterations, the final ciphertext can be directly obtained from CCD.
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5. Conclusion In this paper, an effective multi-focus image fusion and robust encryption scheme is investigated. The fused image based on wavelet decomposition is sampled by SRM and re-encrypted by the intra-block operation and inter-block operation controlled by Logistic map. The decryption process includes the inverse intra-block and inter-block operation as well as the compressive sensing reconstruction with the GRSR algorithm. Experimental results and analyses indicate that the proposed scheme is effective, robust and secure with good fusion performance. 224
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Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Full length article
Multi-focus image fusion and robust encryption algorithm based on compressive sensing
MARK
⁎
Di Xiaoa, , Lan Wanga, Tao Xianga, Yong Wangb a Key Laboratory of Dependable Service Computing in Cyber Physical Society of Ministry of Education, College of Computer Science, Chongqing University, Chongqing 400044, China b Key Laboratory of Electronic Commerce and Logistics of Chongqing, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
A R T I C L E I N F O
A BS T RAC T
Keywords: Image encryption Image fusion Compressive sensing Robust
Multi-focus image fusion schemes have been studied in recent years. However, little work has been done in multi-focus image transmission security. This paper proposes a scheme that can reduce data transmission volume and resist various attacks. First, multi-focus image fusion based on wavelet decomposition can generate complete scene images and optimize the perception of the human eye. The fused images are sparsely represented with DCT and sampled with structurally random matrix (SRM), which reduces the data volume and realizes the initial encryption. Then the obtained measurements are further encrypted to resist noise and crop attack through combining permutation and diffusion stages. At the receiver, the cipher images can be jointly decrypted and reconstructed. Simulation results demonstrate the security and robustness of the proposed scheme.
1. Introduction Nowadays, multimedia sensor networks have wide application prospects in the fields such as military affairs and environment monitor. The sensors are actually cameras that record still image or video. Due to the fact that the commonly used optical lenses suffer from a problem of limited field depth, it is often impossible to get an image that contains all relevant objects in focus and we call these images multi-focus image. Image fusion can merge the different information taken from the same scene with different focuses to obtain a new and improved image [1]. Multi-focus image fusion has been widely used in various fields, such as computer vision and remote sensing. Currently, the existing multi-focus image fusion methods are mainly based on spatial domain [2], transform domain [3] and compressive sensing (CS) [4–7]. However, these image fusion methods focus on how to integrate multi-focus images into a single visual enhanced image, and do not care about secure transmission and robustness. During image transmission over unreliable wireless channels, packets loss or malicious attack is inevitable [8]. Generally, the confidentiality is protected with encryption methods. However, conventional encryption algorithms are not suitable for image when considering their costs and traffic load [9]. Since images are often sparse or compressible while CS can enable sub-Nyquist sampling and low-energy data reduction, it is promising to utilize CS to fill in the gap.
⁎
In the related field of image encryption based on CS, several works have been proposed. Liu et al. [8] proposed the image encryption scheme based on CS theory with robustness to packet loss. Huang et al. [10] proposed a compression-combined digital image encryption method which is robust against consecutive packet loss and malicious crop attack. Subsequently, a robust still-image transmission coding scheme based on CS is presented in [11]. A robust coder, based on CS with structurally random matrix (SRM), for encrypted images over packet transmission networks is proposed in [12]. A simultaneous image compression, fusion and encryption algorithm based on compressive sensing and chaos can be found in [13]. But in the experiment result of the method, the correlation and histograms distributions of the encrypted images are not uniform which implies the scheme is not very secure and valid in some extent. Since good fusion performance, transmission security and robustness are not always considered or valid, the aforementioned approaches are not suitable for fusion and robust encryption of multi-focus images. In this paper, a multi-focus image fusion and robust encryption scheme based on CS is presented to ensure the efficiency and security of image transmission. In this scheme, the source images are firstly fused based on wavelet decomposition and then the fused image is sparsely represented with discrete cosine transform (DCT). The obtained coefficients are measured with a structurally random measurement matrix (SRM) [14] and finally ciphertext can be obtained by
Corresponding author. E-mail address: [email protected] (D. Xiao).
http://dx.doi.org/10.1016/j.optlastec.2016.12.024 Received 26 August 2016; Received in revised form 11 November 2016; Accepted 20 December 2016 0030-3992/ © 2016 Elsevier Ltd. All rights reserved.
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DWT
Image fusion
M easurem ent
Fused image C om pressive sensing
Measurement encryption Ciphertext
key1
key2
DWT
Measurement
Fused image
Decryption
Reconstruction
Ciphertext
Fig. 1. The flowchart of the proposed main steps.
This is equivalent to find the sparsest solutions to x = φs, provided M ≥ CK log N/K , where C is a small constant. Compressive sensing theory indicates that x can be acquired by the following random measurement matrix ∅ ∈ RM×N (M ≪ N), i.e.,
encrypting the integer of measurement with the combined scrambling and diffusion stages. At the receiver, the fused image can be reconstructed by decrypting the cipher image and using a recovery algorithm. The analyses and the numerical experiments are given to illustrate its fusion performance, security and robustness. The rest of this paper is organized as follows. Section 2 is a brief review of the theory of CS and Logistic map. Section 3 proposes the image fusion based on wavelet decomposition and the robust encryption based on CS with SRM. In Section 4, the numerical experiments and analyses can be found. Finally, we conclude the paper in Section 5.
y = Φx = Φφs
(3)
where y is a sampled vector, and Φ is a M × N measurement matrix that is incoherent with the sparse basis φ , i.e., the maximum magnitude of the element in Φφ is small. Then x can be faithfully recovered from only M=O(K log N) measurements through l1-minimization:
min s 1 s. t. y = Φφs
2. Preliminaries
(4)
Eq. (4) presents an l1 -minimization problem which can be solved by orthogonal matching pursuit algorithm [17].
In this section, we show the basic background, primarily on the theory of CS and Logistic map.
2.2. Logistic map 2.1. Compressive sensing Logistic map is usually used to generate pseudo-random sequences. Logistic map is defined as:
The fundamental Shannon/Nyquist sampling theory is widely accepted as the keystone in signal acquisition and reconstruction. Nevertheless, the number of required measurements can be so large that the storage becomes unbearable and the acquisition time can be very long sometimes. Compressive sensing (CS) in [15], as an emerging and fascinating field, shows that K-sparse signals or images can be recovered exactly from M measurements far fewer than N samples what are considered being necessary in Nyquist theorem, i.e., M ≪ N . CS combines the traditional sampling and compressing process into a single non-adaptive linear measurement process by exploiting the data sparsity. Many signals acquired from the physical world are sparse or compressible in the sense that when expressed in the proper orthonormal basis (such as a standard or wavelet basis), their coefficient entries decay rapidly. A signal x ∈ RN of length N is said to be K-sparse or compressible if it can be well approximated using only K ≪ N coefficients over some sparse basis φ as follows:
x = φs
xn +1=μxn (1 − xn )
The output sequence is chaotic if one chooses μ ∈ [3. 57, 4], and μ is the control parameter. This chaos system is sensitive to the seed value x 0 and the generated sequence xn is distributed in the range of (0, 1). The initial value x 0 and control parameterμ can be combined as the secret key. 3. Proposed scheme With the background knowledge presented in the previous section, the proposed image fusion and robust encryption algorithm based on CS is now described. All images considered in this paper are gray-scale images. The proposed algorithm is realized in the following main steps: (1) image fusion based on wavelet decomposition; (2) sparse representation of the fused images; (3) SRM measure; (4) measurement encryption; (5)the receiver decrypts the ciphertext; (6)reconstruct the fused image. The flowchart of the proposed main steps is illustrated in Fig. 1.
(1)
where s is the transform coefficient vector that contains at most K significant nonzero entries. For images, typical choices of φ can consider the widely used discrete Fourier transform (DFT), discrete cosine transform (DCT), or discrete wavelet transform (DWT) matrix, etc. If φ satisfies RIP (Restricted Isometry Property), we can reconstruct signal by solving the optimization problem below [16].
min s 1 s. t. x = φs
(5)
3.1. Image fusion based on wavelet decomposition The key point in multi-focus image fusion is to decide which portions of each image are in better focus than their respective counterparts in the associated images and then combine these regions
(2)
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Fusion rule
DWT
IDWT
Fig. 2. The image fusion scheme using DWT.
If M2 (p)
to construct a well-focused image with certain fusion rules [18]. We propose an image fusion scheme based on discrete wavelet transform (DWT) [19] which involves wavelet decomposition of the source images, calculation of the wavelet coefficients, merging wavelet coefficients based on the fusion rule, and reconstruction for new fusion image, as shown in Fig. 2. The core step in image fusion based on wavelet is to merge the coefficients based on the fusion rule. In the present work, since we are focusing on multi-focus image fusion, the characteristics of the images should be considered: most information of source images is kept in the low frequency sub-band, while the high frequency sub-bands contain the detail of image. In the application, we pay more attention to better visual representation for multi-focus images, so we should preserve most relevant and complementary information with the reduced noise. The important idea of the fusion is to select the coefficients of low-frequency band by a neighborhood pixel correlation selection scheme, while the coefficients of the high-frequency bands are performed with a maximum absolute value based pixel selection scheme. The two coefficients can be combined together to reconstruct the superior image in which all the pixels can be clearly focused. Generally, an image I has its multi-scale decomposition (MSD) representation denoted as D I . Let p=(m, n, k, l) indicate the index corresponding to a particular MSD coefficient, where m and n indicate the spatial position in a given frequency band, k is the decomposition level, and l is the frequency band of the MSD representation. Therefore, DI ( p ) denotes the MSD value of the corresponding coefficient at the position (m, n) with decomposition level k and frequency band l. We propose a scheme by computing the variance in a neighborhood to select the low-frequency coefficients.
δI ( p )=
1 S×T
meanI ( p )=
S /2
T /2
∑
∑
DI (m + s, n + t , k , l ) − meanI ( p ) 2 ,
s ∈− S /2 t ∈− T /2
1 S×T
S /2
T /2
∑
∑
DI (m+s, n +t , k , l ),
s ∈− S /2 t ∈− T /2
⎧ DA ( p ), δA ( p )≥δB ( p ), Dz ( p )=⎨ ⎩ DB ( p ), δA ( p ) < δB ( p ). If M2 (p)≥T ,
1
S /2
⎧ Wmax DA ( p )+Wmin DB ( p ), δA ( p )≥δB ( p ), Dz ( p ) = ⎨ ⎩Wmin DA ( p )+Wmax DB ( p ), δA ( p ) < δB ( p ).
(10)
⎛ 1−M2 ( p ) ⎞ Wmin= 0. 5 − 0. 5 ⎜ ⎟, ⎝ 1−T ⎠
(11)
Wmax =1 − Wmin .
(12)
where T is the match threshold value and T ∈ (0. 5, 1). Then for the high-frequency wavelet coefficients, the maximum absolute value fusion rule is adopted as follows:
⎧ DA ( p ), if DA ( p ) ≥ DB ( p ) DZ ( p )=⎨ ⎩ DB ( p ), if DA ( p ) < DB ( p )
(13)
At last, the main fusion steps can be summarized as follows: Step1: Source images lA (x, y) and lB (x, y) must be registered to assure that the corresponding pixels are aligned. Step2: The wavelet transform decomposes each image into K-level wavelet planes to obtain one low-frequency portion and 3K highfrequency portions. Step3: The wavelet coefficients of the low-frequency are selected by (6) and (7) and performed fusion by (9) and (10), while the wavelet coefficients of the high-frequency are fused by (13). Step4: The fused image l f (x, y) is constructed by performing an inverse wavelet transform based on the combined transform coefficients from Step 3.
(6)
3.2. Sparse representation and Compressive sensing by SRM
(7)
In this section, we firstly divide the fused image with N × N pixels into a lot of non-overlapping blocks. Next, each image block is transformed by a sparse basis matrix where we use DCT as the sparse basis Ψ , and various DCT coefficient blocks are formed. Simultaneously, measurement matrix ΦE is deployed to sense these DCT coefficients independently within each block. The process is simply random linear projection, and can be achieved by inner product operation of the corresponding two elements between Ψ and ΦE . Here, according to the CS principle in [15], ΦE should be selected to be incoherent with Ψ. We design an efficient and appropriate measurement matrix which is called a structurally random matrix (SRM) in the literature [14]. SRM possesses the following features: optimal or near-optimal sensing
where S × T is the neighborhood size, and meanI (p) and δ I (p) denote the mean value and variance value of the coefficients centered at (m, n) in the window of S × T , respectively. We set M2 (p) as the regional variance matching degree for the source image A and source image B in S × T neighborhood and M2 (p)ϵ[0, 1]. The smaller the M2 (p) value is, the lower the regional variance matching degree for image A and B is.
M2 (p)=
(9)
T /2
2× S × T ∑s ∈− S /2 ∑t ∈− T /2 DA ( p ) − meanA ( p ) DB ( p ) − meanB ( p ) δA ( p ) + δB ( p ) (8)
The fusion scheme used for the low-frequency bands can be illustrated as follows:
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sion at the same time and the inter-block operation, as illustrated in Fig. 3. There is a measurement integer sequence seq of length M × N , as given by Eq. (17).
performance; universality; low complexity; hardware/optical implementation friendless. Generally, we have a form of Φ as follows:
Φ=
N DFR, M
(14)
where the matrix F ∈ is a block diagonal matrix, in which F represents the set of the block Hadamard matrix ΦE , D is an operator which picks up M rows of FR uniform at random, and R denotes a scrambling operator which is a binary matrix in which each row or column has exactly one and the rest are all zero. Here, inspired by the idea in the literature [20], the permutation matrix R is constructed as follows:
th
where Z(i) is the integer value of the i measurement,i ∈ [1, M × N]. Firstly, the sequence seq is partitioned into Num blocks and each block includes 64 integer values. Secondly, through the intra-block operation, the integers are scrambled and integer values are changed at the same time. Then, through the inter-block operation, all the blocks are permutated to improve the encryption effect. The pseudorandom numbers generated from Logistic map are used to change the pixel values in the blocks. Meanwhile, the position of integers is relocated according to the random number sorting. The detailed encryption algorithm is described as follows:
Step1: the random sequence p1 can be generated by Logistic map with (μ1, x1), and the control parameter μ1 and the initial value x1 are keys. Logistic map is iterated for h + N times, where h > 0 is used to improve initial value sensitivity. Step2: Initialize the N × N matrix R and set the value of each element to be zero. The re-ordered p2 is obtained by sorting p1 in an ascending order and set R(p2(m),m) to be 1, where m ∈ {1, 2, …N}.
Step1: The sequence seq is firstly divided into a set of blocks B={b1,b 2 ,…,b Num }, where each block contains 64 integer values. We need to pad the last block to make the length be a multiple of 64 and the padding is 1 or 0 if the total number of the measurement integers is not a multiple of 64. We can calculate the number of padding bits as follows:
The N × N permutation matrix R can be obtained by those steps. It should be noticed that permutation-based decryption is performed by multiplying the cipher image with the inverse permutation matrix. So, we can obtain the measurement matrix Φ . In the pseudo-random measure process, the fused image lf is firstly transformed to x in DCT domain and each column of the coefficient matrices will be measured by the constructed measurement matrix Φ , and thus the measurements y are obtained:
y = Φx =
N DFR x M
|PP| = 64 − ((M × N)mod 64), where |PP| is the length of the padding filed. Step2: Intra-block operation: for each block in set B, perform the diffusion and permutation operations simultaneously.
(15)
Iterate Logistic map (5) with (μ 2 ,x2 ) for N0 times to avoid the harmful effect of transitional procedure, where N0 is a constant, and iterate (5) with (μ 2 ,xN 0 ) to generate 64 pseudorandom numbers and sort them in ascending order. The permutated integers are obtained by the sorted index and the integer values are changed simultaneously according to
The pseudo-random measure process accomplishes the measure operation. Meanwhile, it can be viewed as compression and the first level encryption. Since these measurements are generally not finiteprecision numbers, they are quantized in [12,21] for further processing. However, this will cause distortion error and the reconstruction performance may be unacceptable to some extent. Therefore, to eliminate the distortion error, in this paper, the measurements will not be quantized and only the integers of measurements will be encrypted.
c(n) = p (n )⨁k (n )⨁c (n−1),
(18)
where p (n), k(n), c(n) and c(n-1) represent the current integer, the key stream element, the output cipher-integer and the previous cipherinteger, respectively. For the first integer, an initial value c(-1) can be set as a seed. The key stream element k(n) used for masking is calculated by Logistic map given by Eq. (5).
3.3. Measurement encryption
k(n)= mod [floor(x(n)×1015), 256].
Firstly, each measurement is divided into three parts: the positive integer Z, the decimal fraction Q and the sign bit H:
y = H × Z + Q.
(17)
seq = {Z (1), Z (2), Z (3),…Z (M × N)},
N×N
(19)
where x(n) is the current state of the chaotic map and is generated by Logistic map with (μ 2 ,x3). Different blocks have different adaptive μ 2 values in the encryption processing by Eq. (20) and Eq. (21).
(16)
To further encrypt the integer of measurement Z, inspired by the idea in the literature [22], the encryption algorithm is proposed, which includes the intra-block operation combining permutation and diffu-
⎛ j−1 ⎞ e=⎜ ⎟∈[0, 1] ⎝ Num − 1 ⎠
Fig. 3. Block diagram of the proposed image measurement encryption.
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μ 2=
μ 2+3. 57 + 0. 43e ∈[3. 57, 4] 2
4. Experimental results and analyses (21) To test and verify the performance of the validity and security of the proposed scheme, the numerical simulation experiments are carried out. In general, the multi-focus images contain complementary content and information. Two types of multi-focus images of sizes 256×256 and 512 × 512 are used for testing and the reconstruction algorithm is GRSR in [23]. All the experiments are performed on personal computer with an Intel Core 1.8 G CPU, 2 GB memorywith a Windows 7 Ultimate 64 bits operating system. The algorithm is simulated with Matlab R2014a platform, and the secret key is randomly selected as: μ1= 3.995635698763065, μ 2 =3.997458345356721, x1= 0.2236984x2 =0.326653465125465, x3= 0.265689563447562, 55643233, x 4= 0.425653527463592 for Logistic map and block size 32 × 32 in CS. In the following experiments, we can explore the security and robustness of images over the transmission.
Logistic map is always chaotic if one chooses the initial μ 2∈[3. 57, 4], where e is the ratio between the block index and the total number of blocks. Here, the control parameter μ 2 and the two initial value x2 and x3 can serve as the keys. Step3: Inter-block operation: iterate (5) with (μ 3,x 4 ) to generate Num pseudorandom numbers and sort them in ascending order. The permutated block location is obtained by the sorted index. The initial value x 4 can serve as the key and μ 3 is a constant chosen within [3. 57, 4]. Step4: Repeat Steps 2–3 m rounds to satisfy the security requirement, and the ciphertext C′ is obtained. Step5: Recombine the ciphertext C′, the decimal fraction Q and the sign bit H by the following Eq. (22).
C = H × C′+Q.
4.1. Fusion and encryption results
(22)
In Fig. 4, the source images and fused images are as shown in the first two columns and the third column, respectively. It is obvious that the fused image is acceptable on the vision. The obtained cipher images are shown in the fourth column, while the decrypted and reconstructed images with correct keys are shown in the fifth column of Fig. 4. The PSNR of the reconstructed image is showed in Table 1 when sampling rate is 1:4. It can be seen from the decrypted results that the salient features of source images are preserved and transferred into the reconstructed image.
The encrypted measurements C are transformed to cipher image by filling random value in the last column of image, and the cipher image is transmitted to the receiver by wireless channel which may suffer from malicious attacks. 3.4. Decryption and reconstruction At the receiving side, after the receiver obtains the encrypted image, he applies joint decryption and reconstruction to recover the fused image with the decryption keys which are transmitted through security channel.
4.2. Image fusion evaluation Experiments are conducted to compare the effectiveness of the proposed fusion algorithm with three other multi-focus image fusion algorithms in [24–26] through four image quality evaluation criteria. These four metrics are: (i) Qabf metric [27] which measures the quality of the fused image (denoted as I f ) using two images as input (denoted as IA and IB , respectively), (ii) mutual information (MI) metric [28], (iii) spatial frequency (SF) metric [29], and (iv) PSNR of the fused image. For these criteria, the larger the value is, the better the fusion result is. The proposed algorithm and its counterparts belong to the same category in which image fusion is in transform domain. The proposed algorithm exploits two-level wavelet decomposition of the clock images. The objective performance comparisons are presented in Table 2, where one can see that the proposed approach always outperforms three other conventional approaches.
Step1: Through the inverse transformation of the Step5 in the encryption algorithm, the cipher C is divided into three parts: the positive integer C′, the decimal fraction D and the sign bit H. Step2: The positive integer sequence C′ is divided the same as the Step1 in the encryption algorithm. Step3: The inverse inter-block permutation for the positive integer C′ is similar to the Step3 in the encryption with the key x 4 and the constant μ 3. Step4: The inverse intra-block operation is to remove the effect of the combined permutation and diffusion on the blocks with the key μ 2 , x2 and x3. Firstly, the pixel value is obtained by Eq. (23):
p′(n) = c′(n −1)⨁k (n )⨁c′(n ),
(23)
4.3. Statistical analysis
where p′(n), k(n), c′(n) and c′(n-1) represent the decrypted integer, the key stream element, the current encrypted integer and the previous cipher-integer, respectively. The k (n ) is obtained from Eq. (19), the same as the encryption section. Secondly, the inverse permutation for the decrypted integer is similar to the Step2 in the encryption.
Confusion is a technique to ensure that the relationship among the plaintext, the ciphertext and the key is as complicated as possible. Good confusion makes it very difficult to recover the secret key even if a large number of plaintext-ciphertext pairs produced by the same encryption key are available for an attacker. We need to analyze the image histogram and pixel correlation to verify the capability of resisting statistical attacks and show the good confusion property. The histogram reveals the distribution of image pixel values. Good confusion method should ensure the cipher-image histograms of different source images have a uniform distribution to resist statistical attacks. Numerical experiments are conducted to show that the ciphertexts have similar distribution shown in Fig. 5(d) and (h), although we do not quantify the cipher image to [0,255] in order to reduce the distortion. So it is clear that the histogram does not reveal any information for the statistical analysis for attackers.
Step5: Repeat Steps3–4 (m-1) times which is the same as the Step4 in encryption. After the padding field is removed, the plaintext P is obtained. At the last round, the length of padding will be 0 when (M × N) mod 64 = 0, which means that the last block in the original message has 64 bits; while the length of padding is not 0 when (M × N) mod 64 ≠ 0 , and the length of padding is |PP|. Step6: It is the same as the Step6 of the encryption algorithm, and the measurements without distortion error are obtained. Step7: The fused image is reconstructed through l1 -minimization problem which can be solved by GRSR algorithm in [23].
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Fig. 4. Simulation results of the proposed scheme: the first row is Clock images, the second row is Pepsi images, the third row is Peppers images and the fourth row is Camera images; from the first to the fifth column are the focus-left image, focus-right image, fused image, encrypted image, and reconstructed image, respectively.
redundant in meaningful image. To evaluate the correlation property of the plain image and cipher image, the following steps are performed.
Table 1 The PSNR of the reconstructed image for different source images. images
PSNR
Clock images Pepsi images Peppers images Camera images
42.3757 40.7933 38.3611 38.3892
(1) 5000 pixels are randomly selected as samples; (2) The correlation coefficient (CC) between two adjacent pixels in horizontal, vertical and diagonal directions are calculated according to Eqs. (24)–(26).
rx, y = Table 2 The performance comparison among various image fusion approaches. Methods
MI
Qabf
SF
PSNR
Pu and Ni's approach [24] Qu et al.’s approach [25] Li and He's approach [26] Proposed approach
6.80 6.07 2.63 6.93
0.66 0.6 0.67 0.68
11.25 12.67 5.45 18.15
30.25 35.69 34.79 36.87
E (x )=
D (x )=
E {[x − E (x )][ y − E ( y)] D (x ) D ( y )
1 M 1 M
(24)
M
∑ xi i =1
(25)
M
∑ [xi −E (xi )][ yi −E ( yi)] i =1
(26)
where xi and yi are the gray-level values of the two adjacent pixel in three directions of each image and M represents the total number of the samples. E(x) and D(x) are the expectation and the variance of x, respectively.
As a general requirement for all image encryption schemes, the cipher image should be greatly different from its original form. However, the adjacent pixels usually have strong correlation and very
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Fig. 5. Two source images, cipher images and their histograms: (a) plain image1,(b)histogram of the plain image1, (c) cipher image1, (d) histogram of the cipher image1, (e) plain image2,(f) histogram of the plain image2, (g) cipher image2, (f) histogram of the cipher image2.
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Table 3 Correlation coefficients of the plain image and the cipher image. Direction
Plain image
Cipher image
Horizontal Vertical Diagonal
0.9701 0.9693 0.9452
0.0085 −0.0153 0.0025
250
250
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100
50
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0
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(a1)
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0
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(a2)
200
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0
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(c1)
250
0
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(b1)
250
0
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250
(b2)
0
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(c2)
Fig. 6. Correlation plots of two adjacent pixels: the first row is the correlation plots of the plain image in (a1) horizontal, (b1) vertical, (c1) and diagonal directions; the second row is the correlation plots of the cipher image in (a2) horizontal, (b2) vertical, (c2) and diagonal directions.
In our proposed scheme, four initial values of Logistic map and the two parameters are considered as the secret keys. To evaluate the key sensitivity of encryption phase, by introducing a slight change 10−15 to one of the keys while keeping all the others unchanged, and the encryption process is repeated. Fig. 7 shows the source image, the encrypted image with correct key, the encrypted image with incorrect key and the differential image between two cipher images. Furthermore, in a good cryptosystem, a tiny modification of the decryption key results in a totally different decrypted image. To evaluate the key sensitivity of the decryption case, the fused source images are firstly encrypted with the original keys and the cipher images are obtained. The original key is modified slightly by adding 1 to the 15th decimal place as listed in Table 4 and the cipher image is decrypted with the modified keys. Fig. 8 shows the decrypted image and it is obvious that all of the incorrectly decrypted images are completely unrecognizable and cannot reveal any perceptive information of the plaintext. Consequently, the keys are fairly sensitive to the decryption and reconstruction.
The CC values of the test plain image and its cipher image obtained by the proposed algorithm are listed in Table 3 and Fig. 6. It indicates that the CC values of adjacent pixels in the plain image and the corresponding cipher image are close to one and zero, respectively. This means that no detectable correlation exists between the original image and its corresponding cipher image. Based on the above histogram and correlation analyses, we can conclude that the proposed scheme can resist statistical analysis attack as the encryption process effectively confuses and diffuses the plain image.
4.4. Key sensitivity analysis In cryptanalysis, the attacker usually makes an alteration in the plain image and observes the corresponding change in the cipher image. A meaningful relationship between the plain image and the cipher image may appear, and this kind of attack is called differential attack. In particular, when an image is encrypted, the completely different ciphertext should be produced when slightly different keys are used to encrypt the same plaintext. Therefore, when an image is decrypted, a tiny mismatch between the encryption and decryption keys can cause the decryption failure. So the key sensitivity is a crucial feature of an effective cryptosystem.
4.5. Robustness analysis It is inevitable that the ciphertext will be affected by noise, consecutive packet loss or even losing partial information during the 219
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Fig. 7. Key sensitivity test in the encryption case: (a) fused source image; (b) cipher image with correct keys; (c) cipher image(μ1=
3.995635698763066, μ 2 =3.997458345356721,
x1= 0.223698455643233, x2= 0.265689563447562, x3= 0.326653465125465, x 4 =0.425653527463592); (d) differential image between (b) and (c); (e) cipher image(μ1= 3.995635698763065, μ 2 =3.997458345356722, x1= 0.223698455643233, x2= 0.265689563447562, x3= 0.326653465125465, x 4 =0.425653527463592); (f) differential image between (b) and (e); (f) cipher image (μ1= 3.995635698763065, μ 2 =3.997458345356721, x1=
0.223698455643234, x2=
0.265689563447562, x3= 0.326653465125465,
x 4 =0.425653527463592); (g) differential image between (b) and (f); (h) cipher image (μ1= 3.995635698763065, μ 2 =3.997458345356721, x1= 0.223698455643233, x2=
0.265689563447563, x3= 0.326653465125465, x 4 =0.425653527463592); (i) differential image between (b) and (h); (j) cipher image(μ1= 3.995635698763065,
μ 2 =3.997458345356721, x1= 0.223698455643233, x2=
0.265689563447562,x3= 0.326653465125466, x 4 =0.425653527463592); (k) differential image between (b) and (j); (l)
cipher image (μ1= 3.995635698763065, μ 2 =3.997458345356721, x1= 0.223698455643233, x2=
0.265689563447562, x3= 0.326653465125465, x 4 =0.425653527463593); (m)
differential image between (b) and (l).
Salt & pepper noise and intensity is 0.01, 0.05 and 0.10, respectively. Form the Table 5 and Fig. 10, we can see that the PSNR values of the reconstructed images decrease with the increase of noise strength and the images become fuzzier with Salt & pepper noise than Gaussian noise, which indicates our proposed scheme performs better in antiGaussian noise than Salt & pepper noise. However, from the visual sense, main information can still be recognized. The PSNR values of the
transmission channel and storage process. Therefore, we must consider the robustness of the proposed scheme. In Fig. 10, we display robustness of the proposed scheme with different intensity Gaussian noises and Salt & pepper noise added into the cipher image. Fig. 9(a)– (f) show the reconstructed images with zero-mean Gaussian noise with standard deviation of 1and intensity is 0.01, 0.05, 0.10, 0.15, 0.20 and 0.25, respectively. Fig. 9(e)–(g) show the reconstructed images with
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Table 4 Differences between cipher images produced by slightly different keys. Figures
8(b) 8(c) 8(d) 8(e) 8(f) 8(g) 8(h)
Decryption key
μ1
μ2
x1
x2
x3
x4
3.995635698763065 3.995635698763066 3.995635698763065 3.995635698763065 3.995635698763065 3.995635698763065 3.995635698763065
3.99745834535671 3.99745834535671 3.99745834535672 3.99745834535671 3.99745834535671 3.99745834535671 3.99745834535671
0.223698455643233 0.223698455643233 0.223698455643233 0.223698455643234 0.223698455643233 0.223698455643233 0.223698455643233
0.265689563447562 0.265689563447562 0.265689563447562 0.265689563447562 0.265689563447563 0.265689563447562 0.265689563447562
0.326653465125465 0.326653465125465 0.326653465125465 0.326653465125465 0.326653465125465 0.326653465125466 0.326653465125465
0.425653527463592 0.425653527463592 0.425653527463592 0.425653527463592 0.425653527463592 0.425653527463592 0.425653527463593
Fig. 8. Key sensitivity test in the decryption case: (a) encrypted image; (b) decrypted image with the original keys; (c) decrypted image (μ1= 3.995635698763066, μ 2 =3.997458345356721, x1= 0.223698455643233, x2= 0.265689563447562, x3= 0.326653465125465, x 4 =0.425653527463592); (d) decrypted image (μ1= 3.995635698763065,
μ 2 =3.997458345356722, x1= 0.223698455643233, x2= 0.265689563447562, x3= 0.326653465125465, x 4 =0.425653527463592); (e) decrypted image (μ1= 3.995635698763065,
μ 2 =3.997458345356721, x1=
0.223698455643234, x2=
μ 2 =3.997458345356721, x1= 0.223698455643233, x2=
0.265689563447562, x3= 0.326653465125465, x 4 =0.425653527463592); (f) decrypted image (μ1= 3.995635698763065,
0.265689563447563,x3= 0.326653465125465,x 4 =0.425653527463592); (g) decrypted image (μ1= 3.995635698763065,
μ 2 =3.997458345356721, x1= 0.223698455643233, x2=
0.265689563447562, x3=
0.326653465125466, x 4 =0.425653527463592); (h) decrypted image (μ1= 3.995635698763065,
μ 2 =3.997458345356721, x1= 0.223698455643233, x2=
0.265689563447562, x3=
0.326653465125465, x 4 =0.425653527463593).
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Fig. 9. The reconstructed image with different intensity in Gaussian noises:(a) 0.01, (b) 0.05,(c) 0.1,(d) 0.15, (e) 0.2, (f) 0.25. Salt and pepper noise: (g) 0.01, (h) 0.05, (i) 0.10. 45
Table 5 PSNR values of the reconstructed images for different Gaussian noise and Salt & pepper noise intensity. PSNR
9(a) 9(b) 9(c) 9(d) 9(e) 9(f) 9(g) 9(h) 9(i)
0.01 0.05 0.10 0.15 0.20 0.25 0.01 0.05 0.10
38.9336 33.6432 29.5689 28.9766 26.9484 23.8793 23.4027 16.9756 14.3429
0.25 0.2
30 25
0.15 20
Intensity
Intensity
Intensity
35
PSNR
Figures
0.3
PSNR
40
0.1
15 10
0.05 5 0
0
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
Fig. 10. The PSNR distribution of reconstructed image in different intensity noise.
channel, information block loss may affect image content. So the reconstruction performance from incomplete ciphertext is also examined. In Fig. 11, we give an example and display the robustness of the proposed scheme. During the transmission, the attacker crops the block data of 20× 20, 30× 30, 50× 50 pixels, respectively. As shown in
proposed scheme and the scheme in [30] are compared in Table 6, which shows that the proposed scheme has better performance. The results indicate that the proposed scheme could resist against the noise attack to a certain degree. When malicious crop attack happens during the transmission 222
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Table 6 PSNR values for different Gaussian noise intensity in proposed scheme compared with Ref. [30]. Intensity
Proposed scheme
Ref.[30]
0.01 0.05 0.1 0.15 0.2 0.25
38.9336 33.6432 29.5689 28.9766 26.9484 23.8793
30.5884 28.5532 25.3264 22.7359 20.5801 18.9111
Fig. 11. The reconstructed images with pixel block loss of the encrypted image, (a) 20×20 pixels loss; (b) the reconstructed image of (a); (c) 30×30 pixels loss; (d) the reconstructed image of (c) (e) 50×50 pixels loss; (f) the reconstructed image of (e).
Table 7 The PSNR of the reconstructed images with different size block loss.
Table 8 Practical feature comparison between SRM and other random sensing matrix.
Figures
block size
PSNR
10(b) 10(d) 10(f)
20×20 30×30 50×50
24.2116 21.3791 17.0550
Fig. 11(a)-(f) and Table 7, it can be seen that although the reconstruction images are distorted to some extent, the content is maintained in our result, which implies that our scheme have robustness of resisting malicious crop attack.
Features
SRMs
Completely Random Matrices
No. of measurements for exact recovery Sensing complexity Reconstruction complexity at each iteration Implementation in hardware and optics Fast computability Block-based processing
O(KlogN)
O(KlogN)
O(NlogN) O(NlogN)
O(KNlogN) O(kNlogN)
Very easy
Difficult
Yes Yes
No No
Gaussian or Bernoulli i.i.d. matrices [21]. The measurements obtained by SRM are reduced to some degree according to sampling rate and the complexity is reduced much in encryption by confusion and diffusion. The schemes in [8,10,13] perform sampling by random sensing matrices, so the memory and computation complexity are relatively higher than SRM.
4.6. Comparison analysis In the proposed scheme, a structurally random matrix (SRM) is utilized. Table 8 summarizes the practical advantages of employing a fast and efficient sensing matrix over a random sensing matrix such as 223
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Table 9 The comparison of proposed scheme with other methods. proposed method
other methods
good fusion high sensing efficiency resist block missing resist noise attack Without the quantization error high security low memory complexity low computation complexity
[31] [21] [8,10,13] [13] / [24] [13] [22,24]
Fig. 12. The designed optoelectronic hybrid setup of the proposed scheme.
Acknowledgement
In the proposed scheme, we not only use image fusion technology for multi-focus image but also consider the security and compression in transmission channel, while the schemes in [24–26] do not take this into account. We compare the proposed scheme with other methods from many aspects. As shown in Table 9, our proposed scheme has all the listed advantages, but other algorithms only have one or some of the advantages.
The work described in this paper was funded by the National Natural Science Foundation of China (Grant Nos. 61472464, 61502399, 61572089, 61672118), the Natural Science Foundation of Chongqing Science and Technology Commission (Grant Nos. cstc2013jcyjA40017, cstc2015jcyjA40039) and the Fundamental Research Funds for the Central Universities (Grant No. 106112014CDJZR185501).
4.7. Possible optical setup
References
A possible optoelectronic hybrid setup in Fig. 12 is designed. The source images are fed into the computer to accomplish the pre-processing operations include multi-focus image fusion, sparse representation, pseudo- random measure. As mentioned in [32], the DWT of an image can be obtained by carrying out a Fourier transform of this image followed by a multiplication by the WT filter. The two operations can be simultaneously implemented in coherent optics and the spatial light modulator (SLM) is used for modulating the fused image. Then the fused image is transformed with DCT. Inspired by the idea in the literature [33], we can succeed in implementing the DCT optically. The charge-coupled device (CCD) captures the transformed measurement and inputs it to the computer to accomplish the measurement encryption operation. After several iterations, the final ciphertext can be directly obtained from CCD.
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