Development of spatial and temporal phase evaluation techniques in digital holography 5

CHAPTER FIVE RESULTS AND DISCUSSION CHAPTER FIVE RESULTS AND DISCUSSION 5.1 Spatial phase evaluation analysis 5.1.1 Wrapped phase map extraction As mentioned in Section 3.1.1, the wrapped phase map can be determined by the conventional digital phase subtraction method, and can also be obtained by using the proposed complex phasor method With the proposed method, direct phase manipulation is effectively avoided, so better results for the extracted phase map can be expected as shown in the following descriptions Figure 5.1(a) shows an unfiltered wrapped phase map between the initial and deformed states Figure 5.1(b) shows the central row (along the horizontal direction) of Fig 5.1(a) As can be seen in Fig 5.1(b), the wrapped phase map is noisy Figure 5.2(a) shows a wrapped phase map obtained by the proposed complex phasor method with an average filter Figure 5.2(b) shows the central row of Fig 5.2(a) Figure 5.3(a) shows a wrapped phase map obtained by the complex phasor method with a median filter Figure 5.3(b) shows the central row of Fig 5.3(a) It can be seen from Figs 5.15.3 that the phase map extracted by complex phasor method is much smoother, and complex phasor method with an average filter performs better than that with a median filter in the suppression of speckle noise Only the central part of the phase difference map calculated by complex phasor method is investigated Phase maps [as shown in Figs 5.1(a), 5.2(a) and 5.3(a)] correspond to the deformation of the plate, and a 113 CHAPTER FIVE RESULTS AND DISCUSSION comparison between conventional methods and the proposed method in a determination of higher-order displacement derivatives is presented in Section 5.1.2 (a) Rad -1 -2 -3 -4 50 100 150 Pixel 200 250 300 (b) Figure 5.1 (a) An unfiltered wrapped phase map between the initial and deformed states; (b) the central row of Fig 5.1(a) 114 CHAPTER FIVE RESULTS AND DISCUSSION (a) Rad -1 -2 -3 -4 50 100 150 Pixel 200 250 300 (b) Figure 5.2 (a) A wrapped phase map obtained by the proposed complex phasor method with an average filter; (b) the central row of Fig 5.2(a) 115 CHAPTER FIVE RESULTS AND DISCUSSION (a) Rad -1 -2 -3 -4 50 100 150 Pixel 200 250 300 (b) Figure 5.3 (a) A wrapped phase map obtained by the proposed complex phasor method with a median filter; (b) the central row of Fig 5.3(a) 116 CHAPTER FIVE RESULTS AND DISCUSSION 5.1.2 Determination of displacement derivative 5.1.2.1 First-order displacement derivative As described in Section 3.1.2.1, after the complex amplitude corresponding to the deformation distribution is determined, the first-order displacement derivative can be further obtained by using the proposed method Figure 5.4(a) shows a wrapped phase map corresponding to ∂w ∂ξ ' without any filtering Figure 5.4(b) shows a wrapped phase map corresponding to ∂w ∂ξ ' by filtering Fig 5.4(a) using a direct phase manipulation with an average filter It can be seen from Fig 5.4(b) that the blur increases Figures 5.4(c) and 5.4(d) show two wrapped phase maps corresponding to ∂w ∂ξ ' filtered by sine/cosine transformation with average and median filters, respectively A 3× filtering window in the algorithms is employed In the filtering methods, the iterative cycle is The results shown in Figs 5.4(c) and 5.4(d) are obtained by processing the wrapped phase map in Fig 5.4(a) with the sine/cosine transformation method It is seen from Figs 5.4(c) and 5.4(d) that in this study, conventional sine/cosine transformation method does not work well Figure 5.5(a) shows a wrapped phase map corresponding to ∂w ∂ξ ' obtained by complex phasor method with a median filter It can be seen from Fig 5.5(a) that the results are not satisfactory Figure 5.5(b) shows a wrapped phase map corresponding to ∂w ∂ξ ' obtained by complex phasor method with an average filter A shearing value of 56 pixels is used in Figs 5.4, 5.5(a) and 5.5(b) It can be seen that the phase map obtained by the proposed method with an average filter shows a higher quality than previously proposed methods (Schnars and Jüptner, 1994b; Liu, 2003) In the conventional sine/cosine transformation, each pixel is given an equal weight with fixed amplitude without considering the reliability of the pixel In the complex phasor 117 CHAPTER FIVE RESULTS AND DISCUSSION method with an average filter, direct phase manipulation is avoided, so better results are clearly observed [see Fig 5.5(b)] In the filtering methods, the iterative cycle is also If the iterative cycle increases slightly, the results can be improved using the sine/cosine transformation method but are still not as good as those using the proposed complex phasor method with an average filter However, the iterative cycle is kept relatively small to ensure that dense fringes not become blurred Main advantages of the proposed method in the determination of displacement derivatives are: (1) Sign ambiguity can be overcome but without the need to discriminate the cases as in the conventional digital phase subtraction method (Kreis, 2005); (2) real and imaginary parts of complex values Γ(m, n, 2)Γ* (m, n,1) and A' (m, n) A* (m, n) are respectively processed, and direct phase manipulation is avoided; (3) a filter (such as an average filter) can be defined easily; (4) a larger and more controllable range of the sensitivity is achieved In the proposed complex phasor method, an averaged point is dominated by pixels with relatively large amplitudes A pixel with the largest amplitude will have the highest influence on the results However, a complex phasor can not be lined up in a definite way, and low-pass filters such as a median filter are not suitable as real amplitudes vary within the filtering window (Ströbel, 1996; Ghiglia and Pritt, 1998) The appropriate selection of a window size in the average filter is also important, and the proposed method is more suitable for the wrapped phase maps with low-density fringes When the phase maps are dense, iterative cycles should be relatively small in order not to smear out dense fringes As the two-dimensional short-time Fourier transform is further applied, a wrapped phase map corresponding to ∂w ∂ξ ' is obtained as shown in Fig 5.5(c) 118 CHAPTER FIVE RESULTS AND DISCUSSION Since noise usually has small coefficients, coefficients lower than the preset threshold can be fully eliminated In the proposed two-dimensional short-time Fourier transform method, the preset threshold is 3, and the values of σ ξ ' and σ η ' are set at 10 After the filtered phase map is obtained, an unwrapping algorithm can be carried out in order to correct the 2π jumps The unwrapped phase map obtained can correspond to the first-order displacement derivative [according to Eq (3.18)] Figure 5.5(d) shows an unwrapped phase distribution for Fig 5.5(c) with a quality-guided (phase derivative variance) unwrapping algorithm (Ghiglia and Pritt, 1998) It can be seen that two eyes in the conventional shearography is obtained in digital holography, and the first-order displacement derivative can be calculated by Eq (3.18) (a) (b) (c) (d) Figure 5.4 Wrapped phase maps corresponding to ∂w ∂ξ ' (a) obtained by complex phasor method without any filtering; (b) after directly filtering with an average filter; (c) after sine/cosine transformation with an average filter; (d) after sine/cosine transformation with a median filter 119 CHAPTER FIVE RESULTS AND DISCUSSION (a) (b) (c) (d) Figure 5.5 (a) A wrapped phase map corresponding to ∂w ∂ξ ' obtained by complex phasor method with a median filter; (b) a wrapped phase map corresponding to ∂w ∂ξ ' obtained by complex phasor method with an average filter; (c) a wrapped phase map obtained by processing the phase map in Fig.5.5(b) with short-time Fourier transform algorithm; (d) a continuous phase distribution The effectiveness of the proposed two-dimensional short-time Fourier transform is due to the following reasons (Mallat, 1999): (1) The coherence between the kernel and the wrapped phase map results in relatively large coefficients; (2) the redundancy of short-time Fourier transform algorithm can make it more robust to the thresholding operation To show the advantages of the proposed method, the central rows of wrapped phase maps are extracted Figures 5.6(a)-5.6(d) show the central rows of the wrapped phase maps in Figs 5.4(a), 5.4(c), 5.5(b) and 5.5(c), respectively It can be seen from Figs 5.6(a)-5.6(d) that better results are obtained by using the proposed method 120 CHAPTER FIVE RESULTS AND DISCUSSION 2 1 Rad Rad 0 -1 -1 -2 -2 -3 -3 -4 50 100 150 Pixel 200 -4 250 50 100 200 250 200 250 (b) (a) 3 2 1 Rad Rad 150 Pixel 0 -1 -1 -2 -2 -3 -3 -4 50 100 150 Pixel 200 250 (c) Figure 5.6 -4 50 100 150 Pixel (d) The central rows of wrapped phase maps in (a) Fig 5.4(a), (b) Fig 5.4(c), (c) Fig 5.5(b) and (d) Fig 5.5(c) 5.1.2.2 Second-order displacement derivative As mentioned in Section 3.1.2.2, after the complex amplitude corresponding to firstorder displacement derivative is obtained, the second-order displacement derivatives can be further determined by using the proposed method Figure 5.7(a) shows an extracted wrapped phase map obtained by the proposed complex phasor method An average filter with × pixels window is used A 3D plot of the continuous phase map that corresponds to the displacement of the test plate is shown in Fig 5.7(b) 121 CHAPTER FIVE RESULTS AND DISCUSSION 50 Pixel 100 150 200 250 50 100 150 Pixel (a) 200 250 (b) Figure 5.7 (a) A wrapped phase map obtained by the proposed method; (b) a 3D plot of the continuous phase map for the deformation of the test plate After the complex amplitude is obtained by using the proposed complex phasor method, shifting operations are carried out Figure 5.8(a) shows an extracted wrapped phase map corresponding to the curvature along the ξ ' direction without using a filter It can be seen from Fig 5.8(a) that the wrapped phase map is noisy Since the phase map contains 2π jumps, a phase unwrapping algorithm is required to determine a continuous phase map However, when the wrapped phase map is noisy, the unwrapping operation is usually unsuccessful Hence, two strategies are usually adopted to solve this problem One is to improve the filtering algorithm, while another is to develop a more powerful unwrapping algorithm In this case study, the first strategy is considered, and a filtering algorithm is developed to remove the noise Figure 5.8(b) shows a wrapped phase map corresponding to the curvature ∂ w ∂ξ '2 using the proposed complex phasor method with an average filter After the phase map [Fig 5.8(b)] is obtained, conventional sine/cosine transformation is used to further remove the noise Figure 5.8(c) shows a filtered wrapped phase map corresponding to the curvature along the ξ ' direction using sine/cosine transformation with a × average filter Iterative cycle is set as 19 It can be seen 122 CHAPTER FIVE RESULTS AND DISCUSSION 5.5 Optical image encryption 5.5.1 Arnold transform and interference As mentioned in Section 3.5.1, the approach based on Arnold transform and interference can be applied to encrypt a color image with three-wavelength channels An original color image of Mandrill with 512 × 512 pixels is shown in Fig 5.38(a), and pixel size in the image plane is 10 µm In this case study, the original color image (standard test object) is downloaded from the website (http://sipi.usc.edu/database), and this image is first normalized before image encryption Figure 5.38(b) shows an image after Arnold transform operation as the number of iterations for red, green and blue channels is 1, and 1, respectively Figure 5.38(c) shows an image after Arnold transform operation as the respective iterations are 60, 70 and 80, while in Fig 5.38(d) the respective iterations are 205, 215 and 225 The resultant correlation coefficient (CC) values in Fig 5.38(b) for red, green and blue channels are (-0.083, 0.040, 0.071), while the corresponding CC values in Figs 5.38(c) and 5.38(d) are (8.97 × 10 − , 6.68 × 10 − , - 9.64 × 10 − ) and (- 3.37 × 10 − , -0.0029, 0.0023) It is shown in Figs 5.38(b)-5.38(d) that the input image is highly blurred after the Arnold transform, and no information about the original input image can be observed Figures 5.39(a)-5.39(f) show the retrieved phase-only masks M1 and M2 for the red channel, M3 and M4 for the green channel and M5 and M6 for the blue channel, respectively The Arnold scrambling period is 384, and the number of iterations in Arnold transform for red, green and blue channels is set as 80, 90 and 100, respectively It can be seen in Figs 5.39(a)-5.39(f) that the retrieved phase masks are randomly distributed, and the original color input image is fully hidden Since a color image usually contains three channels, a digital format with three channels is used 161 CHAPTER FIVE RESULTS AND DISCUSSION (a) (c) (b) (d) Figure 5.38 (a) An original color image of Mandrill; the images after Arnold transform using different number of iterations of (b) red-1, green-1 and blue-1; (c) red-60, green-70 and blue-80; (d) red-205, green-215 and blue-225 Figure 5.39(g) shows a decrypted image using incorrect phase masks M1 and M2 but with other correct security parameters, such as distances, wavelengths, phase masks M3-M6 and the iteration number in Arnold transform Figure 5.39(h) shows a decrypted image using incorrect phase masks M1-M6 but with other correct security parameters The average CC values in Fig 5.39(h) for the red, green and blue channels are - 3.03 × 10 − , - 4.24 × 10 − and 1.54 × 10 − , respectively It is shown in Figs 5.39(g) and 5.39(h) that the real color input image and some information on the input image are lost as incorrect security keys are used Figure 5.39(i) shows a decrypted image using all correct security parameters with CC ≈ 1, which means that the original color input image is perfectly retrieved and the influence of the threshold operation in the retrieval of random phase masks can be ignored 162 CHAPTER FIVE RESULTS AND DISCUSSION (a) (b) (c) (d) (e) (f) (g) (h) (i) Figure 5.39 Retrieved phase-only masks (a) M1 and (b) M2 for red channel, (c) M3 and (d) M4 for green channel, (e) M5 and (f) M6 for blue channel; decrypted images with (g) wrong phase masks M1 and M2, (h) wrong phase masks M1-M6 and (i) all correct security keys 163 CHAPTER FIVE RESULTS AND DISCUSSION To evaluate the influence of iteration number on the decrypted image, a relationship between CC values and iteration errors (for red channel) is shown in Fig 5.40(a) The relationships in the green and blue channels are shown in Figs 5.40(b) and 5.40(c) It can be seen in Figs 5.40(a)-5.40(c) that a small iteration error of results in the CC value close to zero Figure 5.40(d) shows a decrypted image without using the inverse Arnold transform during image decryption The resultant average CC values in Fig 5.40(d) for red, green and blue channels are - 2.12 × 10 − , 7.60 × 10 − and 3.21 × 10 − , respectively It can be seen in Fig 5.40(d) that no information on the original input image can be observed Figures 5.40(e)-5.40(g) show the decrypted images with an iteration number error of at the red, green, or blue channel but with other correct security keys, respectively Figure 5.40(h) shows a decrypted image with the iteration errors of for all three channels but with other correct parameters The CC values in Fig 5.40(h) for red, green and blue channels are -0.083, 0.040 and -0.071, respectively It can be seen again that the encrypted images can not be correctly decrypted using the incorrect parameters, and actual color information is also lost Although the extracted random phase masks (M1-M6) render image decryption difficult for the attackers, decryption difficulty is further increased without the information on the usage of Arnold transform The parameters, such as the distances and light source wavelengths, can also be considered security keys for image decryption Moreover, Arnold transform of every pixel within a small region of each channel using a random number of iterations followed by Arnold transform of every pixel within a relatively large region of each channel can be further investigated, which might significantly improve the security level of the designed optical system 164 CHAPTER FIVE RESULTS AND DISCUSSION Red channel Green channel 0.8 0.8 CC value 1.2 CC value 1.2 0.6 0.4 0.6 0.4 0.2 0.2 0 -0.2 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 10 12 14 16 18 20 Iteration error -0.2 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 10 12 14 16 18 20 Iteration error (a) (b) Blue channel 1.2 CC value 0.8 0.6 0.4 0.2 -0.2 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 10 12 14 16 18 20 Iteration error (c) (d) (e) (f) (g) (h) Figure 5.40 Relationships between CC values and iteration errors only for (a) red channel, (b) green channel, (c) blue channel; (d) a decrypted image without using the inverse Arnold transform operation during image decryption; decrypted images with iteration error of (e) at red channel, (f) at green channel, (g) at blue channel, and (h) at all three channels 165 CHAPTER FIVE RESULTS AND DISCUSSION 5.5.2 Fractional Fourier transform and interference As mentioned in Section 3.5.2, optical color image encryption can also be realized by using interference and virtual optics (i.e., fractional Fourier transform) Two methods are proposed, i.e., three-wavelengths and an indexed image An original color image ( 512 × 512 pixels) with different types of peppers is shown in Fig 5.41(a), and in the image plane a pixel size of 10 µm is used As a light source with a single wavelength ( λ1 , or λ2 , or λ3 ) is used, decrypted images are obtained as shown in Figs 5.41(b)5.41(d) It can be seen that using a monochromatic light the real color information of the original input image is lost Phase masks M1-M6 are obtained with the proposed method, and Figs 5.42(a) and 5.42(b) show two retrieved phase-only masks M1 and M2 (at red channel) before fractional Fourier transform operation It can be seen in Figs 5.42(a) and 5.42(b) that the retrieved phase-only masks are distributed randomly In addition, random-phase masks M3-M6 (green and blue channels) can also be retrieved by using the similar approach The obvious advantage of the interference approach is that the phase retrieval algorithm can be carried out without any iteration (Zhang and Wang, 2008) Since a color image contains three channels, a digital format with three channels is also used to demonstrate the retrieved random phase masks Figure 5.42(c) shows a decrypted image using all correct security parameters The CC values for three channels in Fig 5.42(c) are Figure 5.42(d) shows a decrypted image using incorrect phase masks M1-M6 but with all other correct security parameters, such as function orders in fractional Fourier transform, distances and wavelengths The average CC values for red, green and blue channels are 5.95 × 10 − , − 1.70 × 10 − and − 5.93 × 10 − , respectively It can be seen that no any information on the original image is obtained 166 CHAPTER FIVE RESULTS AND DISCUSSION (a) (c) (b) (d) Figure 5.41 (a) An original color image of Peppers; decrypted images with (b) a single wavelength λ1 , (c) a single wavelength λ2 , and (d) a single wavelength λ3 (a) (b) (c) (d) Figure 5.42 Retrieved phase-only masks (a) M1 and (b) M2 at the red channel; decrypted images with (c) all correct security keys and (d) incorrect masks M1-M6 167 CHAPTER FIVE RESULTS AND DISCUSSION To evaluate the sensitivity of the function order in 2D fractional Fourier transform, the relationships between the CC value and a function order error are investigated In a channel, there are two phase masks, and each phase mask requires two function orders in the 2D fractional Fourier transform Figure 5.43(a) shows that a relatively small function order error would render the CC values at red channel as low as zero Similarly, the effect of a function order error for green and blue channels is shown in Figs 5.43(b) and 5.43(c), respectively Figure 5.43(d) shows a decrypted image with only one function order error of 0.03 at red channel but with all other correct security parameters Figure 5.43(e) shows a decrypted image with only one function order error of 0.03 at red channel and another function order error of 0.03 at green channel Figure 5.43(f) shows a decrypted image with only one function order error of 0.03 at green channel and another function order error of 0.03 at blue channel It is seen in Figs 5.43(d)-5.43(f) that a real color image can not be obtained as incorrect parameters are used As one function order error of 0.03 occurs in each channel, the decrypted image is totally random and no any information is observed Figure 5.43(g) shows a decrypted image with errors of nm for wavelengths λ1 , λ2 and λ3 at both wave paths but with all other correct security keys The CC values for red, green and blue channels in Fig 5.43(g) are 0.16, 0.38 and 0.33, respectively Figure 5.43(h) shows a decrypted image using distance errors of 10 mm at both wave paths for all three channels but with all other correct security keys The CC values for red, green and blue channels in Fig 5.43(h) are 0.14, 0.33 and 0.28, respectively It is illustrated again that the real color information is lost when incorrect security keys are used Although the small errors in the security keys with wavelength and distance not make the decrypted image totally random, a threshold for the CC values can be preset to authenticate the receiver 168 CHAPTER FIVE RESULTS AND DISCUSSION 0.8 0.6 0.6 CC value 0.8 CC value 0.4 0.4 0.2 0.2 0 -0.2 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 FRFT order error (a) -0.2 -0.4 -0.35-0.3 -0.25-0.2 -0.15-0.1 -0.05 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 FRFT order error (b) CC value 0.8 0.6 0.4 0.2 -0.2 -0.4 -0.35-0.3 -0.25-0.2 -0.15-0.1 -0.05 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 FRFT order error (c) (d) (e) (f) (g) (h) Figure 5.43 Relationships between CC values and one function order error at (a) red channel, (b) green channel, (c) blue channel; decrypted images with (d) one incorrect function order at red channel, (e) two incorrect function orders, (f) two other incorrect function orders, (g) incorrect wavelengths, (h) incorrect distances 169 CHAPTER FIVE RESULTS AND DISCUSSION The tolerance to the loss of encrypted data and the occlusions of certain percentage of pixels from the encrypted complex masks (after the fractional Fourier transform operation) before image decryption have also been investigated Figures 5.44(a) and 5.44(b) show the decrypted images with 6.25 % and 50 % occlusions of all encrypted complex masks The CC values in Figs 5.44(a) and 5.44(b) for red, green and blue channels are (0.16, 0.29, 0.18) and (0.07, 0.03, 0.03), respectively It is illustrated that the quality of the decrypted image degrades with an increase of occlusion percentage The influence of noise on the decrypted image is also investigated, and it is found that the decrypted image is sensitive to noise For instance, when Gaussian noise (mean zero and variance 0.02) is added to all encrypted complex masks, the CC values of a decrypted image at red, green and blue channels are -0.01, -0.02 and -0.01, respectively The small CC values might be due to the relatively small values of the retrieved phase masks (before fractional Fourier transform) and fractional Fourier transform method (a) (b) Figure 5.44 Decrypted images with an occlusion of (a) 6.25 %; and (b) 50 % on all encrypted complex masks As mentioned in Section 3.5.2, optical color image encryption can also be realized by using an indexed image method A color image as shown in Fig 5.41(a) is also studied, and the pixel size in the image plane is 10 µm Function orders in fractional Fourier transform for the retrieved phase masks M7 and M8 are (0.42, 0.44) 170 CHAPTER FIVE RESULTS AND DISCUSSION and (0.46, 0.48), respectively Figures 5.45(a) and 5.45(b) show an image matrix ( 512 × 512 pixels) and a color map ( 256 × pixels), respectively To implement the interference method, the image matrix is divided by a constant value of 1000 before the image encryption operation Figures 5.45(c) and 5.45(d) show two retrieved phase masks M7 and M8 before the implementation of fractional Fourier transform It is shown that the retrieved phase masks are distributed randomly and no any information on the original input image can be observed (a) (b) (c) (d) Figure 5.45 (a) An image matrix; (b) a color map; (c) a retrieved phase mask M7; (d) a retrieved phase mask M8 Figure 5.46(a) shows a decrypted image using incorrect phase masks M7 and M8 but with all other correct security keys Average CC value for the decrypted image in Fig 5.46(a) is 0.0025 It is shown that the original color image can not be observed Figures 5.46(b)-5.46(d) show the decrypted images with only one function order error of 0.03 for 2D fractional Fourier transform (in the phase mask M7), a wavelength error of nm at both wave paths, and a recording distance error of 10 mm 171 CHAPTER FIVE RESULTS AND DISCUSSION at both wave paths, respectively It is shown that a real color image can not be obtained The influence of the occlusion and noise has also been studied, when the indexed image method is applied It is also found that the decrypted images are shown to be sensitive to noise and occlusion operations In addition, when only one of the correct phase masks is used, the silhouette of the original image can still be observed using the interference method Hence, the proposed pre- or post- processing methods can effectively enhance the system security, and a preset CC value can also be used as a threshold to verify the authentication of the receiver Figure 5.46(e) shows a decrypted image using all correct security keys, and the resultant CC value is (a) (b) (c) (d) (e) Figure 5.46 Decrypted images with (a) incorrect phase masks M7 and M8; (b) only one incorrect function order in the phase mask M7; (c) incorrect wavelengths; (d) incorrect distances; (e) all correct security keys 172 CHAPTER FIVE RESULTS AND DISCUSSION 5.5.3 Phase-shifting technique and a bit-plane separation As mentioned in Section 3.5.3, a method based on a bit-plane separation and phaseshifting digital holography can be applied to encrypt an image and enhance the security level Figures 5.47(a)-5.47(h) show bit planes of the original image (‘Lena’ image) In practice, the 7th bit plane plays the most important role in the image encryption As bit planes are obtained, each bit-plane image can be processed based on double random-phase masks and fractional Fourier transform After complex amplitude for each bit plane is retrieved by using phase-shifting digital holographic technique in the hologram plane, the Arnold transform approach is further applied to process the real and imaginary parts of the retrieved complex amplitude The number of iterations in the Arnold transform for the real and imaginary parts is 60 and 120, which are the same for all bit planes For the sake of brevity, only encrypted images for the 0th and 7th bit planes are shown in Figs 5.48(a)-5.48(d) It can be seen that the original input image is fully encrypted, and no information about the input image can be observed (a) (b) (c) (d) (e) (f) (g) (h) Figure 5.47 A separation of the original image: (a) 0th; (b) 1st; (c) 2nd; (d) 3rd; (e) 4th; (f) 5th; (g) 6th; and (h) 7th bit planes 173 CHAPTER FIVE (a) RESULTS AND DISCUSSION (b) (c) (d) Figure 5.48 After encryption operations: (a) real and (b) imaginary parts for 0th bit plane; (c) real and (d) imaginary parts for 7th bit plane For correct decryption, precise security keys must be used Figure 5.49(a) shows a decrypted image with a wrong iteration number of in the Arnold transform for the real part (CC=0.151), while Fig 5.49(b) shows a decrypted image with a wrong iteration number of in Arnold transform for the imaginary part (CC=0.153) Figure 5.49(c) shows a decrypted image with a wrong iteration number of in Arnold transform simultaneously for the real and imaginary parts (CC = -0.004) Note that the error of iteration number happens in all bit planes It is seen in Figs 5.49(a)-5.49(c) that a small wrong iteration number in Arnold transform can result in the incorrectly decrypted image Figure 5.49(d) shows a decrypted image using a function order error of 0.04 in fractional Fourier transform method for b1 and b2 (CC=0.047), while Fig 5.49(e) shows a decrypted image using a function order error of 0.04 in fractional Fourier transform method for a1 and a2 (CC=0.39) Figure 5.49(f) shows a decrypted image using function order errors of 0.04 in fractional Fourier transform method for b1, b2, a1 and a2 simultaneously (CC=0.049) It is seen in Figs 5.49(d)-5.49(f) that a small function order error can result in an incorrectly decrypted image Figure 5.49(g) shows a decrypted image using a wrong phase mask M2 (CC=0.037) Figure 5.49(h) shows a decrypted image with all correct security keys, and the CC value is To enhance the security of this optical cryptographic system, different function orders in 174 CHAPTER FIVE RESULTS AND DISCUSSION fractional Fourier transform and different number of iterations in the Arnold transform can be set for the different bit planes It is worth noting that the 7th bit plane plays the most important role during the decryption In addition, a CC value can also be preset as a threshold to further make an authentication of the receivers (a) (b) (c) (d) (e) (f) (g) (h) Figure 5.49 Decrypted images with (a) a wrong iteration number of in Arnold transform for real part; (b) a wrong iteration number of in Arnold transform for imaginary part; (c) a wrong iteration number of in Arnold transform for real and imaginary parts simultaneously; (d) a function order error of 0.04 in fractional Fourier transform for b1 and b2; (e) a function order error of 0.04 in fractional Fourier transform for a1 and a2; (f) a function order error of 0.04 in fractional Fourier transform for b1, b2, a1 and a2; (g) a wrong phase mask M2; and (h) all correct security keys 175 ... -51 -52 -53 -54 10 20 30 40 Pixel (b) -52 .7 -52 .8 -52 .9 Phase (Rad) -53 -53 .1 -53 .2 -53 .3 -53 .4 -53 .5 -53 .6 -53 .7 10 20 30 40 Pixel (c) Figure 5. 25 Phase variation along section A-A by (a) digital. .. in Fig 5. 8(d) 50 100 100 Pixel Pixel 50 150 150 200 200 250 250 50 100 Pixel 150 200 50 100 Pixel 150 200 (b) (a) 50 Pixel 100 150 200 250 50 100 Pixel (c) 150 200 (d) Figure 5. 8 (a) A phase map... 50 100 100 Pixel Pixel 50 150 150 200 200 50 100 150 Pixel 50 200 200 (b) (a) 50 100 100 Pixel 50 Pixel 100 150 Pixel 150 150 200 200 50 100 150 Pixel 200 (c) 50 100 150 Pixel 200 (d) Pixel 50