Information Authentication of Three-Dimensional Photon Counting Double Random Phase Encryption Using Nonlinear Maximum Average Correlation Height Filter

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  • ABSTRACT

    In this paper, we propose a nonlinear maximum average correlation height (MACH) filter for information authentication of photon counting double random phase encryption (DRPE). To enhance the security of DRPE, photon counting imaging can be applied because of its sparseness. However, under severely photon-starved conditions, information authentication of DRPE may not be implemented successfully. To visualize the photon counting DRPE, a three-dimensional imaging technique such as integral imaging can be used. In addition, a nonlinear MACH filter can be utilized for helping the information authentication. Therefore, in this paper, we use integral imaging and nonlinear MACH filter to implement the information authentication of photon counting DRPE. To verify our method, we implement optical experiments and computer simulation.


  • KEYWORD

    Integral imaging , Photon counting imaging , Double random phase encryption (DRPE) , Nonlinear maximum average correlation height (MACH) filter

  • I. INTRODUCTION

    Recently, data encryption techniques have been widely researched [1-11]. Double random phase encryption (DRPE) that is one optical encryption technique has many advantages; easy implementation and high encryption speed [5]. DRPE uses two independent random phase masks, which follow the statistical distribution such as Uniform distribution with support [0, 1] to encrypt the primary data. For decryption, it uses one random phase mask used in the encryption stage as the key information. Thus, if this key information is revealed, the security is destroyed. To enhance the security level of DRPE, a photon counting imaging technique has been applied [6, 7] that follows a Poisson distribution. Due to its sparseness for image sensing, attackers cannot recognize the decrypted data successfully. However, under severely photon-starved conditions, the primary data cannot be decrypted with the key information correctly. Therefore, to visualize and authenticate the primary information, three-dimensional (3D) imaging techniques such as integral imaging [12] and correlation filters [13, 14] can be utilized. In integral imaging, multiple images with different perspectives can be used for 3D reconstruction. In this paper, we use integral imaging and nonlinear maximum average correlation height (MACH) filter to obtain more accuracy of information authentication for optical encryption compared with a nonlinear correlation filter.

    The paper is organized as follows. We present 3D photon counting DRPE concept and information authentication using nonlinear MACH filter in Section II. Then, we show the experimental results for proving our proposed method in Section III. Finally, we conclude our method with a summary in Section IV.

    II. INFORMATION AUTHENTICATION OF 3D PHOTON COUNTING DRPE USING NONLINEAR MACH FILTER

    In photon counting DRPE, the decrypted data may not be authenticated well. To improve the information authentication, a 3D integral imaging technique can be applied because it uses multiple data to reconstruct 3D data as shown in Fig. 1. Therefore, this technique can be applied to DRPE as shown in Fig. 2. In this paper, for simple computation, we use one-dimensional notation only. The encryption process of DRPE can be described by the following:

    image

    where Sm(x) is the mth primary data of integral imaging, ns(x) and nf(μ) are random phase masks in spatial and spatial frequency domains, 𝔍 and 𝔍-1 are Fourier transform and inverse Fourier transform, and Sem(x) is the mth encrypted data, respectively.

    Now, we have the encrypted data for the primary data. To enhance the security level, we can apply the photon counting imaging technique to the amplitude component of the encrypted data. In general, the photon counting imaging technique can be utilized for detecting the information of objects in low light level environments. Its characteristic seems to be sparse detection. Thus, using sparse detection of the photon counting imaging technique, we can enhance the security level of optical encryption. Photon counting detection can be modeled by statistical distribution such as Poisson distribution since events of photons occur rarely in unit time and space [15]. Figure 3 illustrates the concept of photon counting detection using a statistical distribution. To extract photons from the original scene, Poisson random generation with the expected number of photons Np can be used as shown in Fig. 3. In this paper, we apply computational photon counting detection to DRPE to enhance the security level. Therefore, photon-limited optical encryption can be implemented by the following:

    image

    where Np is the expected number of photons (extracted number of photons from the image) and Sphm(x) is the mth encrypted data for photon counting DRPE, respectively. Also, conventional decryption process can be done by the following:

    image

    where Sdm(x) is the mth decrypted data for conventional DRPE. Using the same manner, photon counting encrypted data can be decrypted by the following:

    image

    where Sph,dm(x) is the mth decrypted data for photon counting DRPE and ϕem(x) is the phase of the mth encrypted data, respectively. Using multiple photon counting decrypted data and computational reconstruction technique [7], 3D data can be obtained by the following:

    image
    image

    where s(x, z) is the 3D reconstructed data at reconstruction depth z, O(x, z) is the superposition matrix for 3D reconstruction, Δx is the shifted pixels for 3D reconstruction, M is the index of images for integral imaging, Nx is the total number of pixels for each image, p is the pitch between image sensors, f is the focal length of the image sensor, and cx is the size of the image sensor, respectively.

    To authenticate the information, we need one particular filter design algorithm known as Maximum Average Correlation Height (MACH) filter [14], which has been found to be robust over a broad range of conditions such as variations in the object’s scale, its orientation, background clutter and noise. This filter uses many reconstructed 3D images at different reconstruction depths as the training data as shown in Fig. 4. In the Fourier domain, the MACH filter is given by the following:

    image

    where M(μ), D(μ), S(μ), and C(μ) are an average, power spectrum, and spectral variance of the training images and power spectrum of additive noise, α , β , and γ are parameters of MACH filter for the sharpness of correlation peak, the stability of the peak value in the presence of distortions, and the response to additive noise, respectively. To improve the performance of MACH filter, we introduce the kth-law nonlinear filter as the following:

    image

    R(x) is the correlation results with nonlinear MACH filter, k is the nonlinear coefficient with real number range (0, 1), respectively. Optimum k can be chosen by a performance metric for the correlation results such as peak sidelobe ratio (PSR). In addition, nonlinear MACH filter can help to improve the information authentication since the target object has 3D information. In other words, the MACH filter has many training data via various reconstruction depths and nonlinearity can improve the performance of correlation process, thus the nonlinear MACH filter is suitable for information authentication of 3D photon-limited DRPE.

    III. EXPERIMENTAL RESULTS

    Using Eq. (1) with primary image as shown in Fig. 5(a), the conventional encrypted image can be generated as shown in Fig. 5(b). The resolution of primary image is 1350(H)×1350(V) pixels. To obtain multiple 2D images with different perspectives, synthetic aperture integral imaging (SAII) is used. The camera array consists of 10(H)×10(V) cameras. The focal length of the camera is 50 mm. The distance between cameras is 2 mm. Perfectly decrypted image that is the same as Fig. 5(a) can be obtained using Eq. (3). Figure 5(c) shows a photon counting encrypted image using Eq. (2). We can notice that the pixel sparse exists in the image by adjusting Np, because the total number of pixels for each image is 1,822,500 and the number of the extracted photons is about 1000. In other words, since the sparsity can enhance the security level of optical encryption, we set Np=1000. Using Eq. (4), photon counting decrypted image can be achieved as shown in Fig. 5(d). Here, the decrypted image is not well recognized due to lack of photons. Figure 6 shows elemental images for true and false classes by SAII. True class has three objects with different depths. In addition, false class has three objects with different depths. Figure 7 illustrates 3D reconstruction results of true and false classes at various depths using Eqs. (5) and (6) with Np=100000. As shown in Fig. 7, 3D objects can be revealed at the correct depths. However, because of severely photon-starved conditions, 3D objects cannot be recognized well by human eyes. Therefore, using Eqs. (7) and (8), information authentication can be implemented as shown in Fig. 8. Since conventional MACH filter with k = 1 is a linear correlation filter, the result is not perfectly sharp. On the other hand, the result using the MACH filter with 0 < k < 1 is sharper than the conventional one. It means that nonlinearity of the MACH filter can enhance the correlation result. We choose the optimum nonlinear coefficient k = 0.4 since the performance of the correlation such as peak sidelobe ratio (PSR) at z=340 mm is the best among others as shown in Fig. 9. As shown in Fig. 8 and 9, in case of false class, enough value of both correlation peak and PSR cannot be achieved. Therefore, we show that our proposed method can enhance the ability of information authentication for optical encryption by this experiment.

    IV. CONCLUSION

    We have presented information authentication of 3D photon counting DRPE using a nonlinear MACH filter. In conventional DRPE, since the number of photons is sufficient for recognition, when attackers know the key information, the encrypted data can be decrypted easily. To avoid this attack, photon-limited DRPE can be utilized but the decrypted data may not be recognized well due to lack of photons. Thus, for information authentication, a nonlinear correlation filter may be used to enhance the recognition rate. However, in severely photon-starved conditions, nonlinear correlation filter may not obtain the better results. In addition, for 3D encrypted data, this correlation filter cannot recognize the information well because it is for 2D data. On the other hand, MACH filter can have the better performance for 3D data because it uses much training data with different depth information. In addition, to improve the information authentication, a nonlinear MACH filter can be used. Therefore, adjusting parameters of MACH filter and nonlinear coefficient k, the performance of correlation can be enhanced.

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  • [FIG. 1.] Concept of computational integral imaging reconstruction.
    Concept of computational integral imaging reconstruction.
  • [FIG. 2.] Double random phase encryption scheme. (a) Encryption and (b) decryption.
    Double random phase encryption scheme. (a) Encryption and (b) decryption.
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  • [FIG. 3.] Mathematical model of photon counting detection.
    Mathematical model of photon counting detection.
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  • [FIG. 4.] Training images for MACH filter. (a) z = 290 mm, (b) z = 340 mm, (c) z = 400 mm.
    Training images for MACH filter. (a) z = 290 mm, (b) z = 340 mm, (c) z = 400 mm.
  • [FIG. 5.] Photon counting DRPE results. (a) Primary image, (b) encrypted image, (c) photon counting encrypted image, and (d) photon counting decrypted image.
    Photon counting DRPE results. (a) Primary image, (b) encrypted image, (c) photon counting encrypted image, and (d) photon counting decrypted image.
  • [FIG. 6.] Elemental images by SAII for (a) true class and (b) false class.
    Elemental images by SAII for (a) true class and (b) false class.
  • [FIG. 7.] 3D reconstruction results for photon counting DRPE of true class at (a) z = 290 mm, (b) z = 350 mm, (c) z = 400 mm and false class at (d) z = 290 mm, (e) z = 350 mm, and (f) z = 400 mm.
    3D reconstruction results for photon counting DRPE of true class at (a) z = 290 mm, (b) z = 350 mm, (c) z = 400 mm and false class at (d) z = 290 mm, (e) z = 350 mm, and (f) z = 400 mm.
  • [FIG. 8.] Correlation results for true class using (a) MACH filter, (b) nonlinear MACH filter with k = 0.4 and false class using (c) MACH filter, (d) nonlinear MACH filter with k = 0.4.
    Correlation results for true class using (a) MACH filter, (b) nonlinear MACH filter with k = 0.4 and false class using (c) MACH filter, (d) nonlinear MACH filter with k = 0.4.
  • [FIG. 9.] Peak sidelobe ratio (PSR) results of nonlinear MACH filter at z = 340 mm via the number of photons for different nonlinearities. (a) true class and (b) false class.
    Peak sidelobe ratio (PSR) results of nonlinear MACH filter at z = 340 mm via the number of photons for different nonlinearities. (a) true class and (b) false class.