The amount of data, especially images, being transferred and stored is growing dramatically, necessitating the design of effective methods to solve the problem of digital image authentication, particularly for important digital images whose security must be preserved. Image authentication is the application of image science to determine if a particular image is an accurate representation of the original data based on a defined criterion. Optical information security technology has been studied and developed for various practical applications for protecting secret data during storage and transmission. One of the optical methods that has been applied to image authentication is double random phase-encoding (DRPE) [1-14]. Since the introduction of the DRPE method, a survey of this method has been considered in many studies. In Ref. [13] Liu mentioned the DRPE spread-space spread-spectrum watermarking (DRPE SS-SS) technique is robust to scaling, and to JPEG compression distortion. Also, it is robust to spatial cropping and both low and high pass filtering. According to Ref. [15], analyzing the resistance of the encryption scheme under some of the commonly known attacks reported in the literature can demonstrate the validity of our proposed method. Although conventional DRPE is robust against brute force attack, it is still fragile to a few specific attacks. It has been determined that DRPE is vulnerable to impulse attacks [16, 17]. To overcome this issue integrating photon-counting and DRPE has been introduced [18-24]. Photon-counting imaging (PCI) technique generates distributions with far fewer photons than conventional imaging, which may permit bandwidth reduction since it generates sparse encrypted data. In addition, the output image from the system does not resemble its input image and cannot be visually distinguished from its counterpart, which can safeguard DRPE-PCI from unauthorized attacks and improve its security to the desired level [20, 21].

At a basic level, DRPE transforms an input image into phase and amplitude objects. The storage space or transmission time required for these objects makes the DRPE method unsuitable for practical applications. Therefore, developing efficient compression schemes is essential for speeding up transmission time and decreasing storage size of DRPE results. For compression methods, the Joint Photography Experts Group (JPEG) has proposed many successful standards. The JPEG proposed many popular compression techniques for imaging applications. These techniques are used in applications ranging from the internet to digital photography and show good performance for the storage of many images in storage media elements [25-35].

In this paper, we propose efficient compression schemes to reduce the size of the encrypted image data from the fusion of the DRPE algorithm, the virtual PCI technique, and two main compression algorithms. The compression method is applied to the sparse encrypted data of the PCI technique. The proposed method has several advantages. First, the complex images yielded by the proposed authentication procedures cannot be visually recognized because they contain photon-limited encrypted data, obtained by combining the DRPE and PCI methods. Second, by applying the proposed compression methods, the size of the encrypted images is reduced significantly without affecting authentication results. Finally, the photon-limited input phase image can be authenticated using a nonlinear cross-correlation metric.

Before describing the proposed approach, we first briefly review the basic techniques that we have used for efficient compression of DRPE data for image authentication.

DRPE allows the encoding of a primary image into complex stationary white noise using two random phase masks [1]. The random phase masks for the spatial and frequency domains, φ₁(x, y) and φ₂(μ, ν), respectively, are statistically independent and uniformly distributed over [0, 2π]. The encryption process of the DRPE algorithm is described by the following equation [1]:

where ℑ and ℑ⁻¹ represent a Fourier transform and inverse Fourier transform, respectively. The decryption process for DRPE is the reverse of the encryption process shown in Eq. (1).

Photon Counting Imaging (PCI) is a special class of optical imaging techniques that was designed for low-light conditions or night vision imaging systems. PCI can also be computationally simulated by changing a limited number of photons based on the expected number of incident photons in the entire scene. In the virtual PCI scheme, the probability of counting photons (p_j) at an arbitrary pixel (x_j) in an image can be modeled as a Poisson distribution, as follows:

where λ_j is the Poisson parameter that is computed by λ_j = N_pf_j, N_p is the expected number of incident photons, and f_j is the normalized irradiance at pixel (x_j) such that , with M being the total number of pixels in the image. In the proposed compression scheme, PCI is virtually applied to the amplitude portion of the encrypted complex amplitude image f_c(x, y) obtained from DRPE [19, 20].

As mentioned earlier, the output of the DRPE technique is very large. Thus, efficient compression techniques are crucial for storage and transmission purposes. Because the JPEG and JP2K compression techniques have superior performance compared to existing standards [27-29], these two techniques are explained briefly in this section.

2.3.1. JPEG

JPEG is a well-known standardized image compression technique. It has been widely used for several purposes, including reduction of the size of image files, storage of full-color information, and automatic error recovery [25]. This technique is the most popular compression technique and supports lossy coding. Lossy compression reduces the original image size by removing non-vital information. In the baseline mode, the image is divided into 8 × 8 blocks and each block is transformed using the discrete cosine transform (DCT). The DCT is typically applied to reduce spatial redundancy to achieve good compression performance. The transformed blocks are quantized using a uniform scalar quantization, zig-zag scanned, and eventually, entropy coded using Huffman coding. The quantization step size for each of the 64 DCT coefficients is specified in a quantization table, which remains the same for all blocks [25].

2.3.2. JPEG 2000 (JP2K)

JP2K is based on a discrete wavelet transform (DWT), scalar quantization, context modeling, arithmetic coding, and post-compression rate allocation. Preprocessing steps include tile component partitioning, DC shifting, and component transformation. During preprocessing, each slice of the original image block is partitioned into one or more disjoint rectangular regions called tiles. In terms of coding, these tile components are independent. DC shifting converts the input unsigned sample values of the image tile components into signed sample values with zero point symmetry. Thus, the relationships between the image tile components are decreased through component transformation. JP2K uses a DWT for transformation as its core coding technology. A DWT can support the transmission of multi-resolution images by using an image multi-resolution representation and decrease the correlation between pixels in the full frame to reduce the blocking effect in the codec process. After the entire image is transformed, the resulting coefficients are quantized and different levels of image quality are acquired based on the minimal precision required. The quantizer assigns different quantization levels for different sub-bands by using a scalar quantization method with a dead zone. The final step of encoding is called entropy encoding. The entropy encoder divides the wavelet sub-band into code blocks. DWT coefficients are then organized into binary bit planes. The entropy encoder uses context modeling and bit-plane arithmetic coding to encode the binary bit planes [26-33].

Figure 1 presents the proposed compression procedure for DRPE-PCI image authentication. By following the algorithm in Fig. 1, the input image is encrypted and then transferred to the image authentication-verification part. In addition, the output of each step is demonstrated in the table that is linked to each related section, which shows how pixel values of the input image are affected during the execution of the proposed encryption process. In the proposed method, a quantization method based on the PCI technique is applied to the amplitude (labeled as a(x, y)) portion of the encrypted image, f_c(x, y) obtained from DRPE. The virtual PCI technique changes the pixel values in a(x, y) to zero or very small values. The phase portion p(x, y) of DRPE contains a real value in the range [−π, π]. Thus, p(x, y) is converted into quantized values through a uniform quantization method similar to that used in [19]. The number of bits used for the uniform quantization process determines the quantized integer range. In this study, two-bit and four-bit quantization processes are considered. To construct a phase image, a binary mask of the photon limited amplitude portion can be directly multiplied by the quantized phase values. The entry for each binary mask is unity if the corresponding photon limited amplitude portion is a nonzero value; otherwise, the entry is zero. Finally, the JPEG and JP2K compression techniques are applied to the new phase image obtained by using the binary mask and uniform quantization.

Although the amplitude portions a(x, y) of the input image is not sent during the transmission, the new phase image can still be verified in the authentication step. It should be mentioned that authentication verification rejects false input images when their peak-to-correlation energy (PCE) is smaller than 0.1. The authentication system or receiver decompresses the compressed phase images for the performance evaluation of the proposed compression schemes. Because the decrypted images from the proposed procedure are not visually recognizable, it is necessary to adopt a comparison scheme to authenticate the retrieved images. In this study, nonlinear cross-correlation (NCC) is used to compare the reference image to the input image, which has a different photon-limited amplitude mask from that of the reference image (see Fig. 1) [19, 20].

Our proposed image authentication schemes efficiently compress the DRPE-encrypted images, store the compressed images in the authentication system, and utilize the decompressed images for private user image verification. It should be noted that it is likely a bad idea to directly store original images as a reference in a system for verifying personal user images because these images would be prime targets for attackers or malicious system administrators. Therefore, another advantage of the proposed approach is that, even if attackers or malicious administrators invade the system, they cannot obtain the original user images because our method utilizes efficiently compressed DRPE-encrypted images, rather than storing raw images (or original images) for user image verification.

The following simulations were performed on a PC with a 32-bit Windows 7 Enterprise OS, Intel(R) Core(TM) i5-2500K 3.30 GHz processor, 4 GB of RAM. The test images (256 × 256 pixels) shown in Figs. 2 and 3 are used to evaluate the proposed image authentication schemes. In the following experiments, two pair of images are used “Peppers”, “Cameraman” and “Baboon”, “Mona-Lisa”. Two input images are selected to be verified. For the true input image, “Peppers, “Baboon” are selected, and for the false input image, “Cameraman” and “Mona-Lisa” are considered. All simulations were implemented in MATLAB 2016. There is a full implementation of JP2K available in a low-level C API under a BSD 2-clause license (Version 2.1.0). We downloaded and used this source code with some minor changes based on our data types and the experiments we wished to perform.

In the following simulation, the lossy compression of JPEG and JP2K are tested on phase images from the DRPE-PCI scheme with various N_p and quantization bit sizes (n_k = 2 and n_k = 4 bits). We attempt to obtain the highest possible compression ratio without losing authentication efficiency. The compression ratio (CR) in this paper is defined as

Figures 2(b) and 3(b) show the phase portions (p(x, y)) of the encrypted Peppers image (Fig. 2(a)) and Baboon image (Fig. 3(a)) from DRPE, respectively. The phase image of Peppers and Baboon, as it has been explained previously, are shown in Figs. 2(c) and 3(c), respectively. Figures 2(d) and 3(d) show the phase image after applying compression and decompression using JP2K. Based on the properties of PCI, it is difficult to reconstruct a reference image after applying the PCI algorithm. Therefore, PCE is used for authentication between the reference image and input image, with its value calculated as follows:

where M and N are the image sizes along x and y axes, respectively, and ncc(x, y) is

where the parameter k defines the strength of the applied nonlinearity and determines the performance features of the processor.

Figures 4(a), 4(b), and 4(c) plot the PCE of non-compressed images and compressed phase images using the JPEG and JP2K methods, respectively. It has been demonstrated that PCE increases with an increase in the expected number of photons for the true class, particularly when N_p > 10⁵. As expected, PCE values are near unity when the parameter k is equal to 0.1. It can be seen in Figs. 4(d), 4(e), and 4(f) that PCE is less than 0.1 in false input images, even when increasing the total number of photons. It can be readily seen in Fig. 4 that PCE values have nearly the same trends in non-compressed and compressed images. In this case, the quantization bit size for the phase value is set to 2 bits. In addition, Fig. 5 demonstrates the results of Baboon (true input image) and Mona-Lisa (false input image) image in non-compressed images and compressed phase images using the JPEG and JP2K methods similarly to Fig. 4.

According to Figs. 6(a), 6(b) and Figs. 7(a), 7(b) (the results of 4-bit quantization), when increasing the total number of photons (N_p > = 10⁵), the PCEs of the compressed phase images using the JPEG and JP2K methods increase similar to Figs. 4(b), 4(c) and Figs. 5(b), 5(c), respectively. Additionally, one can see from Figs. 6(c), 6(d) and Figs. 7(c), 7(d) that both JPEG and JP2K yield similar PCE results in cases with a false input image. Furthermore, PCE in JP2K is marginally larger than the same value in JPEG in the case of true class input. It specifies that JP2K have less information lost comparing with the JPEG method.

Figures 8(a) and 8(b) show the PCE changes when applying JPEG and JP2K with various CR and N_p values when n_k = 2. It is observed that by increasing N_p, PCE is also increased in both graphs. In addition, the authentication strategy rejects false input images because PCE is far below 0.1, as shown in Figs. 8(c) and 8(d). It is worth noting that throughout the experiments, the random phase values (keys) used for encryption of the input images are assumed to be equal to those used for the reference images. Otherwise, the PCE value falls dramatically. Figure 9 shows the PCE changes when applying JPEG and JP2K with various CR in Baboon and Mona-Lisa images. The results is similar to that shown in Fig. 8.

Additionally, we tested the true input image with a photon-limited amplitude mask equal to that of the reference image in two cases of non-compressed (Figs. 10(a) and 11(a)) and compressed ones (Figs. 10(b) and 11(b)). As has been expected, PCE values are around 1 when the parameter k value is equal to 0. The trend in all figures is similar and shows that applying compression does not degrade the phase image significantly.

We are also interested in measuring the size of phase images compared to original gray-scale images (without applying any image encryption or compression techniques) to determine the compression gain (we call it true compression gain or TCG) of the proposed technique. The true compression gain can be obtained by

Figure 12 shows PCE, CR, and TCG values for n_k = 1, 2, and 4 in black, blue, and red lines, respectively. TCG values are indicated by the labels on the graph. The test image was Peppers. Various CR values were evaluated. The y-axis of the graph shows PCE changes and the x-axis represents various CR values (calculated using Eq.3). We can see that 4-bit quantization has the highest PCE values, but the lowest TCG values due to the usage of 4 bits.

It can be observed that by using the compression methods the PCE is marginally similar to the non-compressed scheme. Indeed, we can claim that JP2k marginally outperforms JPEG if PCE is the main concern. Also, it worth mentioning that the proposed authentication system not only resists against unauthorized attacks but also has had the influence of decreasing the size of the encrypted image by combining compression method and PCI.

We aimed to provide a comparative evaluation and assessment of JPEG and JP2K compression algorithm performance on the quantized phase images from DRPE-PCI using various numbers of photons from the perspective of authentication efficiency. The proposed technique considers not only quantized phase values but also a binary mask of photon-counted amplitude values. This study demonstrated that as far as lossy compression is concerned, JP2K seems to perform reasonably well in terms of its ability to efficiently handle various CRs. The results for the JP2K method showed that phase images can be compressed several times while still allowing images to be verified using a nonlinear cross-correlation technique.