Image fusion using wavelet transform and its application to asymmetric cryptosystem and hiding

Isha Mehra; Naveen K. Nishchal

doi:10.1364/OE.22.005474

1. Introduction

Over the past few decades, securing information using optical methods has been an attractive research area because of inherent multidimensional nature and parallel processing capability [1]. Most of the optical schemes for information security reported in literature deal with binary or gray-scale images. Even the color images are encrypted and decrypted by a monochromatic light source and hence they do not preserve their original color information. Also, color image contribute to a higher level of security than binary and gray-value images. Zhang and Karim [2] for the first time reported a method using double random phase encoding (DRPE) for securing color images. Securing, a high resolution color image is a challenging task and is of growing interest in the optical information processing field [3–6].

In the literature, usually for recording a digital hologram, a single wavelength is used but, if multiple wavelengths are used, then a high quality color object can be reconstructed. Javidi et al. [7] recorded two holograms using red and green wavelengths and then fused all reconstructed images to obtain a high quality colored image. Further, this idea was extended employing multiple wavelengths [8]. Fusion techniques play an important role in all such multi-wavelength phenomenon [9–14] and provide better quality fused image for interpreting the image data. Alfalou and Brosseau [15] discussed several optical compression and encryption methods. They reported the relation between compression and encryption. Further, they also proposed an algorithm to compress and encrypt simultaneously multiple images [16].

The conventional DRPE technique [1] in which two random phase masks (RPMs) are used as the security keys to encrypt an input image, belongs to the class of symmetric cryptosystem. In such cryptosystems, the keys used for encryption are identical to the decryption keys. Hence, linearity makes such cryptosystems insecure. To overcome this issue, asymmetric cryptosystem has been proposed [17–21]. Asymmetric cryptosystem due to its nonlinear behavior and asymmetric nature of keys makes the encryption system more strong.

Apart from encryption, image hiding is an attractive way to encode information into white noise or some other image. Kishk and Javidi [22] proposed a method to hide a double phase-encoded image in the spatial domain. Takai and Mifune [23] proposed a method to embed the watermarking image by digital holography. Several optical techniques for image hiding and watermarking employing various domains have been reported [23–30].

Wavelet transform (WT) is also one of the signal processing tools, which is used for analysis of optical and digital signals [31]. It has good local optimization features as well as the multi-resolution analysis features, which makes it suitable for optical information processing [31–33]. Dang and Chau [33] used WT for image encryption. They presented a scheme after combining discrete WT for image compression and block cipher data encryption standard for image encryption. Martin et al. [34] proposed efficient encryption scheme by using a stream cipher to encrypt only the significant bits for individual coefficients using color set partitioning in hierarchical trees (C-SPIHT) algorithm. Recently, Bao et al. [35] proposed an image encryption scheme in wavelet domain where input image is encrypted to some other image instead of a noisy distribution.

In this paper, we propose color image fusion technique using WT for security and image hiding applications. The different components of a color image corresponding to different wavelengths are fused together using discrete WT for obtaining a better quality color image. The individual fused color components are encrypted using amplitude- and phase-truncation approach in Fresnel transform domain. Also, the encrypted components are transformed into different cover images in order to result disguising information of input image to an attacker. Asymmetric keys, Fresnel propagation parameters, weighing factor, and three cover images provide enhanced security. The simulation results demonstrate that data fusion is an effective technique, which can also be used for multispectral image encryption. Apart from multispectral image encryption, the proposed scheme can be further extended for simultaneous compression and encryption of multiple images [15,16].

2. Fusion technique for high quality colored image

In this section, color image components corresponding to different wavelengths are decomposed into wavelet coefficients using discrete WT [8]. In a one-dimensional (1-D) discrete WT, an input function f(x) is decomposed into coefficients with starting scale m_o as:

W_{ϕ} (m_{0}, k) = (\frac{1}{\sqrt{L^{'}}}) \sum_{k} f (x) ϕ_{m_{0}, k}

W_{ψ} (m, k) = (\frac{1}{\sqrt{L^{'}}}) \sum_{k} f (x) ϕ_{m, k}

Here, L' scaling parameter of wavelet. W_φ(m_o,k) and W_ψ(m,k) are called as approximation and detailed coefficients respectively. In a two-dimensional (2D) discrete WT, a 1-D discrete WT is first performed on the rows and on the columns of data. This results in one set of approximation coefficients W_φ(m,r,s) and three sets of detailed coefficients W ^η_ψ(m,r,s) where η = {H,V,D} represents horizontal, vertical and diagonal components.

Let f(x,y) be a 2D color image to be encrypted. Here, f_r(x,y), f_g(x,y), and f_b(x,y) represent their red, green, and blue components, respectively. Suppose, the recovered images corresponding to wavelengths λ_r and λ_r + δλ_r are given as f'_1r(x,y) and f'_2r(x,y) respectively. Similarly, the recovered images corresponding to wavelengths λ_g and λ_g + δλ_g are given as f'_1g(x,y) and f'_2g(x,y) respectively. Similarly, the recovered images corresponding to wavelengths λ_b and λ_b + δλ_b are given as f'_1b(x,y) and f'_2b(x,y) respectively. Now, discrete WT is performed for fusion in order to reconstruct better quality color image [8].

W_{FLLr} = AVG (W_{LLr1}, W_{LLr2})

W_{FHLr} = AVG (W_{HLr1}, W_{HLr2})

W_{FLHr} = AVG (W_{LHr1}, W_{LHr2})

W_{FHHr} = AVG (W_{HHr1}, W_{HHr2})

The combination of all four components is given as

g'_{r} (ψ, ϕ) = {W_{FLLr}, W_{FHLr}, W_{FLHr}, W_{FHHr}}

Similarly for green and blue components, the combinations of coefficients are

g'_{g} (ψ, ϕ) = {W_{FLLg}, W_{FHLg}, W_{FLHg}, W_{FHHg}}

g'_{b} (ψ, ϕ) = {W_{FLLb}, W_{FHLb}, W_{FLHb}, W_{FHHb}}

Hence, individual fused components of color image are obtained using WT in order to retrieve better quality of a color image.

3. Encryption of fused color components using phase-truncation approach

In order to encrypt all fused color components, amplitude- and phase truncation based asymmetric encryption scheme has been carried out. The principle of encryption has been depicted in Fig. 1(a) for red component only. Fused red component, g'_r(ψ,Φ) is bonded with an RPM, R₁(ψ,Φ) and its Fresnel transformation (FrT) is obtained at some distance z₁. The obtained spectrum is phase-truncated (PT) and we get, H_r (u,v).

H_{r} (u, v) = P T {F r T_{z_{1}} [{g^{'}}_{r} (ψ, ϕ) \times R_{1} (ψ, ϕ)]}

Here, R₁(ψ,φ) is the random white sequence uniformly distributed in the interval [0,2π]. For successful retrieval of the red component, the decryption key; k_r(u,v) is generated after amplitude-truncation (AT), as defined in the following Equation,

k_{r} (u, v) = A T {F r T_{z_{1}} [{g^{'}}_{r} (ψ, ϕ) \times R_{1} (ψ, ϕ)]}

Similarly, fused green and blue components are encrypted using different RPMs and corresponding PT operated and AT operated values are obtained.

H_{g} (u, v) = P T {F r T_{z_{2}} [{g^{'}}_{g} (ψ, ϕ) \times R_{2} (ψ, ϕ)]}

For successful retrieval of the green component, the decryption key; k_g(u,v) is generated.

k_{g} (u, v) = A T {F r T_{z_{2}} [{g^{'}}_{g} (ψ, φ) \times R_{2} (ψ, φ)]}

H_{b} (u, v) = P T {F r T_{z_{3}}^{} [{g^{'}}_{b} (ψ, φ) \times R_{3} (ψ, φ)]}

For successful retrieval of the blue component, the decryption key; k_b(u,v) is generated.

k_{b} (u, v) = A T {F r T_{z_{3}}^{} [{g^{'}}_{b} (ψ, φ) \times R_{3} (ψ, φ)]}

Thus, individual fused components are encrypted using amplitude-and phase truncation approach and the decryption keys corresponding to red, green and blue components are generated.

Fig. 1 (a) Block diagram for asymmetric cryptosystem; (b) DWT based image encryption; and (c) decryption process.

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4. Converting each encrypted color component into different visible cover image

The obtained three different color encrypted images are separately converted to three different cover images to give the disguising information of input image to the attacker. The flowchart is given in Fig. 1(b). A different cover image, m_r(ξ,η) having its size four times larger than the existing encrypted image is considered. The discrete WT is applied to the cover image. Suppose the wavelet coefficients after discrete WT operation are obtained as {W_LLcr, W_HLcr, W_LHrc, W_HHrc}, so to convert the encrypted red component image into cover image, HH sub-band of the cover image is replaced with encrypted red component with the weighing factor α_r. Here α_r varies from 1 to 100. Higher the value of α_r, better will be the quality of the reconstructed cover image.

Finally, the inverse discrete WT (IDWT) operation will yield the cover image corresponding to red component.

m'_{r} (ξ, η) = IDWT (W_{LLcr}, W_{HLcr}, W_{LHcr}, H_{r} / α_{r})

Similarly green and blue encrypted images are converted to corresponding cover images i.e. m'_g (ξ,η) and m'_b (ξ,η), respectively.

For image decryption, use of correct keys and same wavelet are necessary as shown in Fig. 1(c). We used Haar wavelet in this study. All the three cover images are first decomposed into four wavelet sub-bands using Haar wavelet. After being extracted from the HH sub-band, the encrypted images are then applied with the inverse process of the image encryption to reconstruct the original image color components as follows:

For decryption of red color component, the key obtained k_r(u,v), is bonded with the extracted cipher-text, H_r(u,v), and its inverse Fresnel transformation is obtained for distance z₁. The obtained spectrum is phase-truncated, as given by f'_r(x,y).

{g^{'}}_{r} (ψ, ϕ) = P T {F r T_{- z_{2}} [H_{r} (u, v) \times k_{r} (u, v)]}

{g^{'}}_{g} (ψ, ϕ) = P T {F r T_{- z_{2}} [H_{g} (u, v) \times k_{g} (u, v)]}

{g^{'}}_{b} (ψ, ϕ) = P T {F r T_{- z_{2}} [H_{b} (u, v) \times k_{b} (u, v)]}

Then, IDWT is applied to all three components separately to have a good quality of red component of reconstructed image back.

{f^{'}}_{r} (x, y) = I D W T ({g^{'}}_{r} (ψ, ϕ))

{f^{'}}_{g} (x, y) = I D W T ({g^{'}}_{g} (ψ, ϕ))

{f^{'}}_{b} (x, y) = I D W T ({g^{'}}_{b} (ψ, ϕ))

Finally, better quality colored image is given as

f (x, y) = {f^{'}}_{r} (x, y) \otimes {f^{'}}_{g} (x, y) \otimes {f^{'}}_{b} (x, y)

Thus, use of correct asymmetric keys and same wavelet i.e. Haar wavelet, decrypts a better quality color image.

5. Simulation results

The simulation study was carried out using MATLAB 2010(b). Figure 2(a) shows the input color image of size 256 ☓ 256 pixels. Figures 2(b) and 2(c) represent the red components corresponding to λ_r = 632.8 nm and λ_r + δλ_r = 631.8 nm, respectively. Figures 2(d) and 2(e) represent the green components corresponding to λ_g = 532 nm and λ_g + δλ_g = 531 nm, respectively. Figures 2(f) and 2(g) represent the blue components corresponding to λ_b = 475 nm and λ_b + δλ_b = 476 nm, respectively.

Fig. 2 (a) Input color of size 256 ☓ 256 pixels; (b) red component of colored image corresponding to λ_r = 632.8nm; (c) red component of colored image corresponding to λ_r + δλ_r = 631.8nm; (d) green component of colored image corresponding to λ_g = 532nm; (e) green component of colored image corresponding to λ_g + δλ_g = 531nm; (f) blue component of colored image corresponding to λ_b = 475nm; and (g) blue component of colored image corresponding to λ_b + δλ_b = 476nm.

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Discrete WT has been carried out for fusion purpose in order to reconstruct better quality color image. Figures 3(a)-3(d) show all four coefficients of discrete WT of reconstructed red component corresponding to wavelength λ_r. Figures 3(e)-3(h) show all four coefficients of discrete WT of reconstructed red component corresponding to wavelength λ_r + δλ_r.

Fig. 3 (a)-(d) Four coefficients of DWT of reconstructed red component corresponding to wavelength, λ_r, (e)-(h) all four coefficients of DWT of reconstructed red component corresponding to wavelength λ_r + δλ_r.

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All the three fused color components are shown in Figs. 4(a)-4(c). After fusion of wavelet coefficients, all three color components in WT domain will undergo encryption process through amplitude- and phase truncation approach. Figure 5(a) shows asymmetric (decryption) key, k_r(u,v) corresponding to red component. Figure 5(b) shows encrypted image, H_r(u,v) corresponding to red component. Figure 5(c) shows asymmetric (decryption) key, k_g(u,v) corresponding to green component. Figure 5(d) shows encrypted image, H_g(u,v) corresponding to green component. Figure 5(e) shows asymmetric (decryption) key, k_b(u,v) corresponding to blue component. Figure 5(f) shows encrypted image, H_b(u,v) corresponding to blue component.

Fig. 4 (a)-(c) All three fused color components.

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Fig. 5 (a) Decryption key, k_r(u,v) corresponding to red component; (b) corresponding encrypted image, H_r(u,v); (c) decryption key, k_g(u,v) corresponding to green component; (d) corresponding encrypted image, H_g(u,v); (e) decryption key, k_b(u,v) corresponding to blue component; (f) corresponding encrypted image, H_b(u,v).

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The three different colored encrypted images, H_r(u,v), H_g(u,v), and H_b(u,v) are separately converted to three different cover images to give the disguising information of input image to the attacker. The three cover images of size 512 ☓ 512 pixels for all three components have been shown in Figs. 6(a)-6(c). After discrete WT operation of input cover images and replacement of higher frequency component with encrypted images, corresponding results has been shown in Figs. 6(d)-6(f). Thus, The three different colored encrypted images, H_r(u,v), H_g(u,v), and H_b(u,v) are separately converted to three different cover images to give the disguising appearance of input image. The final cover images obtained after inverse discrete WT operation has been shown in Figs. 6(g)-6(i). Finally, use of correct asymmetric keys and Haar wavelet decrypts a better quality color image as shown in Fig. 6(j).

Fig. 6 (a)-(c) The three cover images,; (d)-(f) fusion for all three components; (g)-(i) the final cover images obtained after IDWT operation; (j) retrieved color image; and (k) specific attack result, mean square error (MSE) versus number of iterations.

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To check the security of the proposed scheme, we applied the special attack [18] to the proposed color image encryption scheme. To start with the specific attack algorithm, the exact real valued encrypted image is required. But, this encrypted image is further hidden into cover images using parameters α, type (Haar) and level of wavelets. These parameters are unknown to the attacker. So, the proposed scheme resists specific attack. The mean square error (MSE) values calculated between input color image and retrieved color image has been plotted against the number of iterations as shown in Fig. 6(k). In this case, we computed the MSE for 100 iterations. It can be observed that the MSE value never converged to zero. Also asymmetric keys, fractional orders, and encryption keys corresponding to three primary color components enhance the security key space in the proposed scheme. Also, Bao et al. [35] have claimed that hiding through such schemes withstand the brute force attack. Hence, it can be inferred that the proposed scheme resists both specific and brute force attack.

To study the effectiveness of the proposed security scheme, we calculated peak signal to noise ratio (PSNR) [30]. The value of PSNR for the retrieved, color image is 41.5606. With the calculated PSNR value, we infer that when the correct keys are used the PSNR value is extremely high.

6. Conclusion

We proposed color image fusion using wavelet transform for security applications that employs asymmetric encryption scheme and image hiding. Discrete Haar wavelet transform has been used for color image fusion. Further, each fused color components are encrypted using amplitude- and phase-truncation approach. Then, the individual color components are transformed into different cover image or fake image in order to mislead the attacker. Asymmetric keys, Fresnel propagation parameters, weighing factor, and three cover images constitutes the security parameters in the proposed scheme. Apart from this, lens less scheme provides error free environment to the proposed security scheme. The simulation results demonstrate that data fusion is an effective technique, which can be used for multispectral image encryption.

Acknowledgment

The authors acknowledge the funding from the Council of Scientific and Industrial Research, Government of India, under Grant No. 03/(1183)/10/EMR-II.

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Image fusion using wavelet transform and its application to asymmetric cryptosystem and hiding

Abstract

1. Introduction

2. Fusion technique for high quality colored image

3. Encryption of fused color components using phase-truncation approach

4. Converting each encrypted color component into different visible cover image

5. Simulation results

6. Conclusion

Acknowledgment

References and links

Cited By

Figures (6)

Equations (23)

Optics Express