Simultaneous compression and encryption of closely resembling images: application to video sequences and polarimetric images

M. Aldossari; A. Alfalou; C. Brosseau

doi:10.1364/OE.22.022349

1. Introduction

As the amount of information to be transmitted become larger and faster, data compression is becoming a critical challenge in, e.g. video imaging. The objective of image compression is to reduce irrelevance and redundancy of the image data in order to be able to store or transmit data in an efficient form [1–7]. Image compression can be classified as lossy or lossless. Lossless compression is preferred for archival purposes and is often used for medical imaging. Lossy compression methods, especially when used at low bit rates, introduce compression artefacts. Lossy methods are especially suitable for natural images such as photographs in applications where minor (sometimes imperceptible) loss of fidelity is acceptable to achieve a substantial reduction in bit rate. In this work, we focus on lossy data compression. In an attempt to cope with the problem, several studies have been conducted in the past in academic institutions and industry, especially for telecommunications, access control, biometrics, and security systems. But only in the last few years quantitative techniques have been developed for simultaneous compression and encryption of images [1–11], and we have made contributions to this development [8–11]. These techniques open the way to a fuller control of the reconstruction of target images.

In the following, we shall focus our optical image processing analysis on the 4f system [12]. The justification for such an approach follows from the fact that image spectra can be manipulated between the reading and formation steps of the process. Using such an approach, the correlation between images has most recently been studied in numerical simulations and experimental observations for face recognition applications [13,14]. Very recently, Alfalou and Brosseau [1] pointed out that the 4f system gives a consistent way to compress and encrypt an image with a specific filter. On the one hand, redundant information can be suppressed. On the other hand, changing the distribution of the frequencies in images can dramatically change the representation of the data and allows us to render them useless for a hacker [1]. The problem one encounters is that the proposed methods have either good performances in terms of compression or encryption of static images [1–8]. In general, these methods are useless if one wishes to process video sequences and realize simultaneously compression and encryption.

Simultaneous compression and optical encryption schemes have been scarcely proposed and experimentally demonstrated. Most of the efforts have been restricted to encryption. For example, Abuturab and associates [14,15] suggested different approaches for encrypting color images based on the gyrator transform. Rajput and associates [16] used polarization for encrypting images in the Fresnel domain. Hence, the full taxonomy of the methods for simultaneously compress and encrypt multiple images which closely resemble like those of a video sequence still does not exist. For some problems, see Refs [17–24]. In an earlier study [23], we reported on a simultaneous compression and encryption method based on the use of the discrete cosine transform (DCT). The 2D DCT gives an image of the intensity of low frequency to high frequency information, but the low frequency information is in the top-left corner and the high frequency information is in the bottom-right corner. This makes the information easier to manipulate. Mosso and associates [24] introduced a method to perform encryption of video images which combines standard double random phase encoding [2] and a specific fusion in the output plane. This method shows good performances but does not optimize compression, i.e. the fusion is done without taking into account the possible spectra overlap in the output plane. More specifically, this method requires large encoding bit sizes which has for effect to increase significantly the amount of data to be transmitted and/or stored. In addition, the role played by the size of the encrypted image in double random phase-amplitude optical encryption has attracted much interest [25–28]. For example, the authors in [28] proposed a method via phase compression to enhance double random-phase encoding security. In their method, only a compressed phase distribution is available in the CCD plane, and the amplitude component is not available or requested for optical decryption. Using a nonlinear correlation algorithm for authenticating the decrypted image high security can be achieved for this scheme.

In this study, we shall use different fusion methods [8–11] which are motivated by their application to face recognition in video sequences. Our primary objective is to develop a technique to operate simultaneously compression and encryption. We point out that our algorithm is rapid and accurate. Our results distinguish well among existing state-of-the-art by the ability of the proposed method to achieve simultaneously compression and encryption. The robustness and performances of our analysis were tested against experimental data, i.e. static images, video sequences, and polarimetric images. In each case, the images show close resemblance. It is further important to observe at the outset that video sequences were treated as multiple images without taking care of temporal redundancy, motion detection, or binary encoding [29,30], which are outside the scope of this study.

This article is structured as follows. In the succeeding section, we begin by presenting some important results on a simple method of compression and encryption using a spectral data fusion technique. In section 3 we analyze the performances of the suggested optimization and its impact on the compression rate and quality of the reconstructed images. The most important point of this work stems from the data in sections 7 and 9. In section 7, this approach is specialized to the analysis of video sequences. As an additional outcome of this approach, we study the double level encryption issue in section 9. We provide examples of decompression and decryption video, demonstrating the validity of the proposed technique. It will also be shown below that our analysis offers good performances for polarimetric images. In Section 10, we offer some conclusions.

2. Multiple-image optical compression and encryption (MIOCE) method

We have recalled that lossy compression consists of a transformation to reduce the amount of data needed to represent an image. In general, compression leads to degraded quality of the reconstructed image. As systems for performing real time optical comparisons (e. g. using an optical correlator which permits to compare a sampled image to a wide variety of reference images) have appeared optical compression is becoming a critical challenge [1–11]. Among those methods, those dealing with the phase of the Fourier transform (FT) of the image are notable. Figure 1 shows a much simplified schematics of the scheme [8]. We take advantage of the fact that the useful spectrum of an image does not spread over the entire Fourier plane (Fig. 1(a)). The rightmost panel of Fig. 1 shows that the spectral amplitude is localized in specific areas of the Fourier plane. This property can be used to realize data fusion with other images. For a proper understanding of this scheme it is important to be able to choose a criterion allowing us to select relevant data for each image. This arises because when an inappropriate fusion criterion is used (Fig. 1(b)), the reconstructed images have generally poor quality. This is due to the overlapping of the different spectra [8,9].

Fig. 1 (a) A spectrum of a typical image, (b) Schematic of typical spectral fusion.

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It has been proposed [8] that this can be overcome by shifting the spectra before the fusion is realized (Fig. 2(a)). We first estimate the spectrum width of each target image. For that, we used an adapted criterion based on the root-mean-square (RMS). Commonly accepted band-pass root-mean-square (RMS) criterion to determine the spectrum’ width is expressed in our case as

Δ_{I} = n \sqrt{\int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} (u^{2} + v^{2}) {| S_{I} (u, v) |}^{2} d u d v},

where the index I denotes a target image,

S_{I} (u, v)

is its spectrum,

(u, v)

are the spectral coordinates, and n is a parameter characterizing the minimal width of the spectrum. Each spectrum is then shifted in order to minimize their overlap. Next, the fusion criterion detailed in section 3 is carried out (Fig. 2(b)).

Fig. 2 (a) Schematic illustration of the MIOCE (multiple-image optical compression and encryption) method, (b) An example showing four shifted and merged spectra.

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3. Spectral segmentation

Our explicit fusion strategy in the MIOCE method begins by partitioning the spectral plane into a large number of small regions (Fig. 2(a)). To each of these regions is assigned information from the spectrum of the target images. Figure 3 shows the key aspects of this assignment based on local energy. A simple example dealing with two target images (A and B), each with N pixels can be useful. Our analysis begins by calculating the spectrum of these two images. Then, for each pixel of the Fourier plane, we compare the (relative) spectral energy of the image A in pixel (i,j), i.e. X^A_ij = $E_{i j}^{A} / \sum_{i = 1}^{N} \sum_{j = 1}^{N} E_{i j}^{A}$ ,with its counterpart for image B X^B_ij = $E_{i j}^{B} / \sum_{i = 1}^{N} \sum_{j = 1}^{N} E_{i j}^{B}$ . The decision about pixel assignment to one of the two spectra is taken by comparing the (relative) spectral energies of A and B. This spectral segmentation has for aim to optimize the band-pass RMS of the Fourier plane which contains the fusion of the two spectra of the target images. Segmentation’s rule can be expressed as:

Fig. 3 Segmentation criterion and spectral assignment.

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{\begin{matrix} P i x e l (i, j) = S p e c t r u m_A (i, j) i f X_{i j}^{A} \geq X_{i j}^{B} \\ P i x e l (i, j) = S p e c t r u m_B (i, j) i f X_{i j}^{A} ≺ X_{i j}^{B} \end{matrix}

The success of this approach provides strong support for the contention that a single spectral plane provides all pertinent information for reconstructing the target images A and B (Fig. 3).

4. Image reconstruction with the segmentation criterion and without spectral shifting

One detrimental features of this segmentation protocol (proposed in [8]) is the occurrence of isolated pixels, i.e. an isolated pixel represents a pixel of a spectrum of image A which is surrounded by pixels of the spectrum of other images. This issue can be critical since this paper deals with sets of reference images bearing a strong resemblance (video sequence). A common descriptor of the quality of the reconstructed images is the mean square error (MSE) which characterizes the differences between the target and reconstructed images

M S E = \frac{1}{N^{2}} {\sum_{i = 1}^{N} \sum_{j = 1}^{N} | I_{d} (i, j) - I (i, j) |}^{2}

where N is the number of pixels of the target image (here, set to 256), I_d is the decompressed image, and I is the target image. Preliminary results are shown in Table 1 for 4 selected images of a video sequence showing a person walking (column 2 of Table 1). The first column of Table 1 denotes the number in the sequence of numbered images. It is noticeable how poorly the reconstructed images (without spectral shifting, column 3 of Table 1) are. We attribute this low performance to the overlap between the different spectra, as is also evident from the large MSE values in this particular case. Similar degradations are observed when the number of input target images is increased.

Table 1. Reconstructed images obtained by making use of our fusion criterion. Columns 3 and 4 correspond respectively to without and with spectral shifting.

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5. Influence of spectral shifting on the quality of the reconstructed images

The same analysis was performed with spectral shifting in order to avoid overlap as is shown in Fig. 4, where X and X’ denote the spectral width (Eq. (1)) of the spectrum of each target images considered in this specific example.

Fig. 4 Synoptic diagram of the MIOCE method optimized by making use of the shifted (RMS) criterion.

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A few words are in order about our methodology. The results of using this spectral shifting to the above example are shown in Table (1-column 4). This technique achieves good performances if we compare visually with the results of column 3. This is also consistent with the smaller MSE values. Many tests have shown that this very effective technique has great impact on both on the image detail and compression rate. These tests also demonstrated that careful manipulation of the center of the spectrum image should be studied in order to provide a good trade-off between image quality and compression rate. The choice of shifting configuration influences the compression ratio. As an example, we have presented tow possibilities for this shifting. To find the good shifting configuration, for a given application, we propose the optimization presented on section 6.

6. Optimization of the MIOCE method and performance analysis as function of compression rate and quality of reconstructed images

6.1 Spectral shift

The above outlined method showed that the shifting operation in the Fourier plane is a crucial step. As an illustration of the manner in which the shifting of the center of three spectrum images can affect the reconstructed and decompression images (i.e. the MSE values), four situations are compared in Fig. 5.Note that in each case the spectral shift is larger than the spectral width calculated from Eq. (1).

Fig. 5 Influence of the spectrum’s position on the quality of reconstructed images.

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It is revealing to see that the smallest MSE values are seen in Fig. 5(a), illustrating a spectral shift along the main diagonal. To this arrangement corresponds the smallest overlap between the three spectra. In contrast, the larger values of the MSE observed for the situations illustrated in Fig. 5(d) are consistent with the large overlap between the spectra.

6.2 Influence of the spectral shift of multiple target images on the quality of reconstructed images

The problem we address in this subsection is how the spectral shift d (Fig. 6) between the different spectra should be chosen? Or stated in other words, how d is chosen to get a good trade-off between reconstructed image detail and compression rate (Fig. 6).

Fig. 6 Filter fabrication in the Fourier plane.

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Figure 6 presents a given spectral plan $(N \times N)$ pixels. In this figure, we consider 4 images and a given shift d (in pixels). Then, the filter’ size in the Fourier plane t is chosen by keeping only the pertinent information for reconstructing output images. The expression of t is

t = 2 Δ I + x = 2 Δ I + \sqrt{2} d

where

Δ I

is obtained from Eq. (1). Consider the influence of the spectral shift d in the range [4, 64] pixels on the image quality. Figure 7(a) shows the four shifted spectra (

d = 4

pixels) and segmented by making use of the energy criterion (Eq. (2)). Figure 7(b) shows the corresponding filter and Fig. 7(c) shows the results of the filtering operation. Figures 7(d)-7(f) show similar results for d = 64.

Fig. 7 Spectrum corresponding to four merged, shifted, and filtered target images.

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In Fig. 7, we show that the spectral arrangement displayed in Fig. 6 presents the spectral plane with minimal width, but with poor quality of the reconstructed images (d = 4 pixels) since there is a strong overlap between the spectra. Increasing d to 64 pixels has for effect to obtain a significantly better quality of images. In Fig. 8, we plot the MSE as a function of d (in pixels).

Fig. 8 Influence of the spectral shift on the reconstructed image quality: MSE as a function of the number of shifting pixels.

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Figure 8 shows that the MSE decreases strongly up to d = 16 pixels, and then slowly decreases. It is worrying how poorly the quality of the reconstructed images is improved using this technique for d>16 pixels. To avoid the need for search for an appropriate d every time, we set d = 16 pixels in the following and use the remaining band-pass of the Fourier plane for merging more target images. Hence, compression rate T_c is increased while the images are reconstructed correctly.

6.3 Compression rate

The compression rate T_c is calculated as a function of the number of shifting pixels between spectra from

T_{c} = (1 - \frac{256 \times 256 \times P r i \times B i t d}{256 \times 256 \times n \times B i t c}) \times 100

where Pri is set to 2 (the spectrum is complex), Bitd is the number of bits used for encoding the compressed spectrum (set to 16), n is the number of target images, and Bitc is the number of bits used for encoding the target images (set to 8 for the gray level images considered). Hence

T_{c} = (1 - \frac{4}{n}) \times 100

Figure 9(a) summarizes the compression rate T_c and MSE as a function of the number of shifting pixels. As shown in Fig. 9(a), for the full range of parameters examined we consistently found that the T_c and MSE curves intersect at $d \approx 16$ pixels, consistent with our previous observation.

Fig. 9 (a) Compression rate and MSE as a function of the number of shifting pixels, (b) reconstructed image with a spectral shift set to 16 bits.

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7. Adaptation to the compression of video images: Fusion and conjugate symmetry

To demonstrate the efficiency of the optimized-MIOCE method, we employ it to images of a scene with moving objects and varying illumination. Because one of our primary applications is the analysis of video sequence, we would like to process a large number of images. For that purpose, the use of symmetry of the spectrum allows us to eliminate half of the spectrum plane. Before the shifting and fusion operations are achieved, we begin by fabricating a binary filter with two blocks such as that shown in Fig. 10.The first block has pixels equal to 1 while the other has pixels set to 0. Then, the images spectra are multiplied two by two with this filter in the manner shown in Fig. 10. This approach has the advantage of keeping only 50% of each spectrum; the other 50% can be found by the conjugate symmetry which a basic property of the FT.

Fig. 10 Synoptic diagram of the compression scheme par conjugate symmetry.

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In the following, we consider a video sequence with 26 images (Fig. 11). These 26 images are then grouped two by two following the rule I_n and I_{n +
13} as shown in Fig. 11.

Fig. 11 Two by two grouping of a video sequence with 26 images.

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The spectral shifting and the segmentation operations are then applied. This compression technique allows has to group 26 images in a single Fourier plane (Fig. 12).

Fig. 12 Synoptic diagram of our compression technique adapted to video sequences.

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8. Decompression and reconstruction of video images

Decompression is realized by applying the reverse of the compression operation. The 13 spectra are separated by multiplication with using a specific carrier signal. Only half of these 13 spectra are selected. The other half is reconstructed by making use of the conjugate symmetry. Finally, the inverse FT is realized to obtain the reconstructed image. We now apply our strategy of compression and encryption to special cases of video sequences. As a proof of principle, let us first consider a video sequence 1 (26 images). In Fig. 13 we present the spectrum resulting from the merging of 26 images of a video sequence.

Fig. 13 An example dealing with the merging of 26 compressed spectra.

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In Fig. 14, the corresponding MSE is illustrated as a function of the number of target images. We show in Fig. 14(a) that the MSE increases (i.e. the quality of the reconstructed images decreases) as this number is increased. This is consistent with the fact that the large the number of target images is the small the band-pass of each spectral plane is. Additionally, several plateaus can be observed. These plateaus are related to the spectra localization after the spectral shifting. We begin by positioning the spectra as far as possible from image number 1 (upper right corner of Fig. 13). Close to this image the MSE values increase.

Fig. 14 (a) MSE as a function of the number of target images, (b) Examples of input and reconstructed images.

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The fact that the MSE values are very small (even for the case of 26 images) demonstrates the good performances of our compression technique. This technique achieves good quality of the reconstructed images as shown by the comparison between the image in Fig. 14(b), i.e. one of the 26 images of the sequence, and the reconstructed image at the output of the system (Fig. 14(c)). We now consider the second example dealing 26 images of a second video sequence.

Table 2 summarizes the compression rate T_c as a function of the number of images (for 26 images, MSE = 3.9 $\times 10^{- 3}$ ).

Table 2. Simulation results for the second video sequence (15sec.avi).

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We first consider the case of five target images to be compressed. Taking a quantitative example from Table 2, if one considers 26 images, then the compression rate achieves its largest value of 84.6%, i.e. with only 25% of the information contained in each image. Our algorithm is effective to reconstruct the sequence with minimal MSE (3.5 $\times 10^{- 3}$ ). Our technique results in a good reconstructed image quality and a good compression rate is achieved.

The third example deals with video containing 26 images of a tank moving at a constant speed (15sec.avi [32]). We present the compressed and merged spectrum to transmit and/or to store. Next, the compressed and encrypted spectrum is shown. The right part shows the reconstructed images obtained from our compression and decompression technique. The results of using our method to this example are shown in Table 3. The reconstructed images have good quality and the MSE values are small. This is clear indication that our method is robust and easy to implement even for moving targets. See the video results d-15sec.avi [32] for details.

Table 3. Compression results with tank video (See d_15sec.avi for details in [32]).

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9. Encryption

In this section, our objective is to provide an effective encryption protocol to encrypt of the merged and compressed spectrum (e. g. Figure 15). It is of interest to see how the confidentiality of data in video sequences is preserved. For that, we used the principle of spectral segmentation (described in section 3) to change the spectral distribution in the Fourier domain. Once the spectral plane merging the spectra of the different target images is obtained (Fig. 15), it is multiplied by a random mask (amplitude and phase). The latter should be fabricated for the considered application and it should allow us to hide the amplitudes as well as the phases of the merged spectra. It is worth emphasizing that the use of a simple phase mask in the spectral plane is insufficient to hide all information since the spectral amplitude remains unaffected like in the double random phase scheme. Additionally, one cannot apply phase masks to the target images since they will impact the segmentation operation described in section 3. As an alternative we suggest to fabricate an encryption mask allowing us to encrypt both amplitude and phase in the spectral plane. To illustrate this encryption approach we consider the case of two target images (Fig. 15)

Fig. 15 Example of target spectrum resulting from the merging of two target images.

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The fabrication of the encryption mask is presented in Fig. 16. First, we consider two key images (for m target images, m key images should be considered). After applying the FT to these key images the spectral segmentation scheme developed in section 3 is applied. Hence, two key spectra are obtained. In Fig. 16, we illustrate the selected (blue and red) areas for each spectrum. Next, two random amplitude and phase masks are fabricated by considering the maximum values (rand ∈[0, m_max]) of the two key spectra (m_max: denotes the maximum value of the spectrum’s amplitude according to the considered spectrum). Next the two key spectra are merged to obtain the amplitude and phase encryption mask. Finally, this mask is multiplied by the target spectrum shown in Fig. 15.

Fig. 16 Illustrating the fabrication of the encryption mask.

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Figure 17 shows clearly that this is not an effective encryption technique. Several regions can still be detected which may be used by a hacker to look for useful information. This arises because of the dynamics of the frequencies contained in the spectrum, i.e. the value of max-min. For the illustrative example shown in Fig. 17, an encryption mask with random values ranging between min and max is fabricated.

Fig. 17 Target spectrum encrypted with a random mask.

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9.1 Optimization of the encryption key

To overcome this problem, we proceed as follows. We divide the encryption mask in three regions corresponding to high, intermediate, and low frequencies, which are processed separately (Fig. 18). For each region a random mask is fabricated by determining the min and max values for each region.

Fig. 18 Division of the encryption mask in three regions (M1, M2, and M3).

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In a first step, we calculate the maximum value of the amplitude spectrum m_max in the weak encryption area (shown in Fig. 17). Then we divide the spectral plane into three parts (Fig. 18): M1 correspond to frequencies with amplitude $> m_{\max}$ , M2 correspond to frequencies with amplitude in the range $] \frac{m_{\max}}{4}, m_{\max} [$ , and M3 corresponds to frequencies with amplitude $\leq \frac{m_{\max}}{4}$ . These three components are used to fabricate three random masks S1_cryp, S2_cryp, and S3_cryp (see Fig. 19).

Fig. 19 Fabrication of the encrypted mask: C = S_1cryp + S_2cryp + S_3cryp.

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The three random masks satisfy

\begin{matrix} S_{1 c r y p} = r a n d [\min (M 1), \max (M 1)] \\ S_{2 c r y p} = r a n d [\min (M 2), \max (M 2)] \\ S_{3 c r y p} = r a n d [\min (M 3), \max (M 3)] \end{matrix}

where the function rand (a,b) generates a random number between a and b. Encryption of the target spectra is realized using the key C = S_1cryp + S_2cryp + S_3cryp (Fig. 20(a)). The encrypted spectrum with these three masks is displayed in Fig. 20(b). This produces improvement in encryption quality for each test realized.

Fig. 20 (a) Target spectrum merging two images, (b) encrypted spectrum with key C = S_1cryp + S_2cryp + S_3cryp

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9.2 Second encryption level

A second encryption key, based on the double random-phase encryption scheme, is considered to improve the security level of our method. The spectrum obtained at the first encryption level (Fig. 20(b)) $C = S_{1 c r y p} + S_{2 c r u p} + S_{3 c r y p}$ is multiplied by the random phase $ϕ_{R a n}^{1}$ (with the constraint that $ϕ_{E n c r y p}^{} + ϕ_{R a n}^{1} \neq 0$ )

C^{'} = (S_{1 E n c r y p} + S_{2 E n c r y p} + S_{3 E n c r y p}) \exp (i ϕ_{R a n}^{1}) = S_{E n c r y p} \exp [i (ϕ_{E n c r y p}^{} + ϕ_{R a n}^{1})] .

Figure 21 illustrates the second level encryption scheme. The input of our system is the first-level encrypted complex plane (Fig. 20(b)). Then the real and imaginary parts of the spectrum are independently multiplied by random phase keys. After a FT we obtain the encryption image that is multiplied by the second encryption random phase key

ϕ_{R a n}^{2}

. Then,

C^{''} = F T (C^{'}) \exp (i ϕ_{R a n}^{2}) = F T {S_{c r y p} \exp [i (ϕ_{E n c r y p}^{} + ϕ_{R a n}^{1})]} \exp (i ϕ_{R a n}^{2}) .

Finally, after a second FT, the two-level encrypted image in the output plane is obtained

Fig. 21 Illustrating the second level of encryption.

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T_{t w o_l e v e l} = F T 〈 F T {S_{E n c r y p} \exp [i (ϕ_{E n c r y p}^{} + ϕ_{R a n}^{1})]} \times \exp (i ϕ_{R a n}^{2}) 〉 .

Once the encrypted spectrum is obtained, the reverse encryption operation (decryption) is done by suppressing the two random phases, then the inverse FT is realized to find the images used for fabricating the encryption keys, and eventually find the target images. The results of using this compression/encryption–decryption/decompression algorithm are shown in Table 4.

Table 4. Simulation results with two target images.

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We now briefly comment these results. These observations, taken together, provide strong support that our method has good performances in terms of compression and encryption. To validate our approach, we use the sequence video containing images of a tank moving at a constant speed (see supplemental material T-15seco.avi for details: see ref [32], and Fig. 22.This video sequence shows the target images (Fig. 22(a)), the multiplexed spectrum (Fig. 22(b)), the two-level encrypted spectrum (Fig. 22(c)), and the reconstructed images (Fig. 22(d)).

Fig. 22 Compression and encryption results with tank video: Example (See d_15sec.avi [32] for details).

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9.3 Compression and encryption of polarimetric images

We now discuss the analysis of polarimetric images by our method. Figure 23(a) illustrates the optical setup described in detail in [27]. The light from a He-Ne (632 nm) laser source passes through a rotating diffuser, a spatial filter composed of a convergent lens and a pinhole, and the collimating lens L₃. Two polarizers (Pol) and two quarterwave plates (QWP) are used to select and analyze polarization states. The light is detected on a CCD camera thanks to the lens L₄. The sample (Fig. 23(b)) used in this study is a Smiley (cork) nearby plastic (top right) and lead (bottom right) spheres. Our experiments were performed in air. Four images can be obtained depending on the generated (linear or circular) and analyzed (crossed or parallel) polarized states [31]. These four images were encrypted and compressed, and then reconstructed after transmission. Finally, the DOPL (degree of linear polarization) and DOPC (degree of circular polarization) images were calculated [31].

Fig. 23 (a) Scheme of the experimental setup for obtaining polarimetric images ( $θ = 10 °$ ). POL: polarizer; QWP: quarterwave plate; L: lens; (b) The sample considered (see text).

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Figure 24(a) shows the four images: (a-1) obtained with circularly polarized light and crossed polarizers, (a-2) obtained with circularly polarized light and parallel polarizers, (a-3) obtained with linearly polarized light and crossed polarizers, and (a-4) obtained with linearly polarized light and parallel polarizers. Figure 24 (b) shows the image after compression and encryption following our approach. In Fig. 24 (c), we show the reconstructed images and their corresponding MSE values: (c-1) obtained with circularly polarized light and crossed polarizers, (c-2) obtained with circularly polarized light and parallel polarizers, (c-3) obtained with linearly polarized light and crossed polarizers, and (c-4) obtained with linearly polarized light and parallel polarizers. These images show the good performances of our method since the MSE values are very small. The DOP images are shown in Fig. 24(d). The good quality of the DOP is also remarkable, i.e. the low- and high-frequency spectral information are preserved although the four images closely resemble.

Fig. 24 Experimental results of simultaneous compression and encryption of polarimetric images.

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10. Summary and conclusions

We have introduced a new and versatile method to compress and encrypt closely resembling images. The novelty of this scheme is that not only both operations are realized simultaneously but also it enables to deals with video sequences and polarimetric images. Our technique overcomes the bottleneck associated with fusion information in the Fourier plane. It relies on an optimization of the spatial-frequency spectrum of the image using a specific segmentation criterion. That is an energy criterion allowing us to merge several image spectra into one spectrum which contains pertinent information required for reconstructing the target images. This procedure allows us to include all pertinent information required for reconstructing the multiple images in a spectrum of size N $\times$ N pixels. The optimization of the MIOCE method requires realizing three operations: (1) a shifting, (2) a segmentation (i.e. assignment of a specific spectral area to a target spectrum), and (3) a filtering. As mentioned above, this method allows us to increase significantly the number of images to be merged in the specrum of size N $\times$ N pixels. In practice, this permits to reduce significantly the amount of information to send or to store. In addition, we presented a robust encryption scheme to overcome the constraint of the double random phase-amplitude optical encryption method.The performance of this multiple-image optical compression and encryption method is verified by analyzing several video sequences and polarimetric images.

Direct testing with numerical simulations and experiments shows the efficiency of this simultaneous compression and encryption method of closely resembling images. This work might be extended in several directions, including (a) the experimental implementation of this numerical scheme, (b) the extension to noisy situations will be considered in future work, and (c) the investigation of the protocol’s immunity to attacks.

Acknowledgment

This work is partly supported by the Royal Saudi Navy-RSNF, Ministry of Defense. The authors thank Dr. I. Leonard.

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N°	Output images without spectral shifting	Output images with spectral shifting
1	MSE = 0.0294	MSE = 0.0013
2	MSE = 0.2982	MSE = 0.0012
3	MSE = 0.3547	MSE = 0.0012
4	MSE = 0.3068	MSE = 0.0015

Number of target images	Compression rate T_c
5	20%
6	34%
7	43%
8	50%
9	56%
10	60%
11	64%
12	67%
13	70%
14	71.5%
15	73.3%
16	75%
17	76.5
18	78.8
19	79%
20	80%
21	81%
22	81.8%
23	82.6%
24	83.3%
25	84%
26	84.6%

	Target image Size : 256 256 pixels	Transmitted spectrum Size: 256 256 pixels	Reconstructed image Size: 256 256 pixels
Example of d-15sec
		MSE= 0.0018

N°	Output images without spectral shifting	Output images with spectral shifting
1	MSE = 0.0294	MSE = 0.0013
2	MSE = 0.2982	MSE = 0.0012
3	MSE = 0.3547	MSE = 0.0012
4	MSE = 0.3068	MSE = 0.0015

Number of target images	Compression rate T_c
5	20%
6	34%
7	43%
8	50%
9	56%
10	60%
11	64%
12	67%
13	70%
14	71.5%
15	73.3%
16	75%
17	76.5
18	78.8
19	79%
20	80%
21	81%
22	81.8%
23	82.6%
24	83.3%
25	84%
26	84.6%

Simultaneous compression and encryption of closely resembling images: application to video sequences and polarimetric images

Abstract

1. Introduction

2. Multiple-image optical compression and encryption (MIOCE) method

3. Spectral segmentation

4. Image reconstruction with the segmentation criterion and without spectral shifting

5. Influence of spectral shifting on the quality of the reconstructed images

6. Optimization of the MIOCE method and performance analysis as function of compression rate and quality of reconstructed images

6.1 Spectral shift

6.2 Influence of the spectral shift of multiple target images on the quality of reconstructed images

6.3 Compression rate

7. Adaptation to the compression of video images: Fusion and conjugate symmetry

8. Decompression and reconstruction of video images

9. Encryption

9.1 Optimization of the encryption key

9.2 Second encryption level

9.3 Compression and encryption of polarimetric images

10. Summary and conclusions

Acknowledgment

References and links

Cited By

Figures (24)

Tables (4)

Equations (10)

Optics Express

Target images
Encrypted spectrum
Output reconstructed images