## Assessing the performance of a method of simultaneous compression and encryption of multiple images and its resistance against various attacks |

Optics Express, Vol. 21, Issue 7, pp. 8025-8043 (2013)

http://dx.doi.org/10.1364/OE.21.008025

Acrobat PDF (2888 KB)

### Abstract

We introduce a double optimization procedure for spectrally multiplexing multiple images. This technique is adapted from a recently proposed optical setup implementing the discrete cosine transformation (DCT). The new analysis technique is a combination of spectral fusion based on the properties of DCT, specific spectral filtering, and quantization of the remaining encoded frequencies using an optimal number of bits. Spectrally multiplexing multiple images defines a first level of encryption. A second level of encryption based on a real key image is used to reinforce encryption. A set of numerical simulations and a comparison with the well known JPEG (Joint Photographic Experts Group) image compression standard have been carried out to demonstrate the improved performances of this method. The focus here will differ from the method of simultaneous fusion, compression, and encryption of multiple images (SFCE) [Opt. Express **19**, 24023 (2011)] in the following ways. Firstly, we shall be concerned with optimizing the compression rate by adapting the size of the spectral block to each target image and decreasing the number of bits required to encode each block. This size adaptation is achieved by means of the root-mean-square (RMS) time-frequency criterion. We found that this size adaptation provides a good tradeoff between bandwidth of spectral plane and number of reconstructed output images. Secondly, the encryption rate is improved by using a real biometric key and randomly changing the rotation angle of each block before spectral fusion. By using a real-valued key image we have been able to increase the compression rate of 50% over the original SFCE method. We provide numerical examples of the effects for size, rotation, and shifting of DCT-blocks which play noteworthy roles in the optimization of the bandwidth of the spectral plane. Inspection of the results for different types of attack demonstrates the robustness of our procedure.

© 2013 OSA

## 1. Introduction

1. A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Adv. Opt. Photon. **1**(3), 589 (2009). [CrossRef]

3. P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. **20**(7), 767–769 (1995). [CrossRef] [PubMed]

4. Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express **15**(16), 10253–10265 (2007). [CrossRef] [PubMed]

5. P. Kumar, J. Joseph, and K. Singh, “Known-plaintext attack-free double random phase-amplitude optical encryption: vulnerability to impulse function attack,” J. Opt. **14**(4), 045401 (2012). [CrossRef]

1. A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Adv. Opt. Photon. **1**(3), 589 (2009). [CrossRef]

8. A. Alfalou and A. Mansour, “All optical video-image encryption enforced security level using independent component analysis,” J. Opt. A, Pure Appl. Opt. **9**, 787 (2007). [CrossRef]

6. J.-J. Huang, H.-E. Hwang, C.-Y. Chen, and C.-M. Chen, “Lensless multiple-image optical encryption based on improved phase retrieval algorithm,” Appl. Opt. **51**(13), 2388–2394 (2012). [CrossRef] [PubMed]

8. A. Alfalou and A. Mansour, “All optical video-image encryption enforced security level using independent component analysis,” J. Opt. A, Pure Appl. Opt. **9**, 787 (2007). [CrossRef]

9. Z. Liu, Q. Guo, L. Xu, M. A. Ahmad, and S. Liu, “Double image encryption by using iterative random binary encoding in gyrator domains,” Opt. Express **18**(11), 12033–12043 (2010). [CrossRef] [PubMed]

10. A. Alfalou and C. Brosseau, “Dual encryption scheme of images using polarized light,” Opt. Lett. **35**(13), 2185–2187 (2010). [CrossRef] [PubMed]

7. X. Wang and D. Zhao, “Double images encryption method with resistance against the specific attack based on an asymmetric algorithm,” Opt. Express **20**(11), 11994–12003 (2012). [CrossRef] [PubMed]

*et al.*[11

11. F. Mosso, J. F. Barrera, M. Tebaldi, N. Bolognini, and R. Torroba, “All-optical encrypted movie,” Opt. Express **19**(6), 5706–5712 (2011). [PubMed]

1. A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Adv. Opt. Photon. **1**(3), 589 (2009). [CrossRef]

11. F. Mosso, J. F. Barrera, M. Tebaldi, N. Bolognini, and R. Torroba, “All-optical encrypted movie,” Opt. Express **19**(6), 5706–5712 (2011). [PubMed]

20. A. Alfalou, C. Brosseau, N. Abdallah, and M. Jridi, “Simultaneous fusion, compression, and encryption of multiple images,” Opt. Express **19**(24), 24023–24029 (2011). [CrossRef] [PubMed]

21. M. Johnson, P. Ishwar, V. Prabhakaran, D. Schonberg, and K. Ramchandran, “On compressing encrypted data,” IEEE Trans. Signal Process. **52**(10), 2992–3006 (2004). [CrossRef]

### 1.1. Optical DCT: motivation and background

13. A. Alfalou and C. Brosseau, “Exploiting root-mean-square time-frequency structure for multiple-image optical compression and encryption,” Opt. Lett. **35**(11), 1914–1916 (2010). [CrossRef] [PubMed]

20. A. Alfalou, C. Brosseau, N. Abdallah, and M. Jridi, “Simultaneous fusion, compression, and encryption of multiple images,” Opt. Express **19**(24), 24023–24029 (2011). [CrossRef] [PubMed]

**1**(3), 589 (2009). [CrossRef]

### 1.2. Optimization of the method

20. A. Alfalou, C. Brosseau, N. Abdallah, and M. Jridi, “Simultaneous fusion, compression, and encryption of multiple images,” Opt. Express **19**(24), 24023–24029 (2011). [CrossRef] [PubMed]

13. A. Alfalou and C. Brosseau, “Exploiting root-mean-square time-frequency structure for multiple-image optical compression and encryption,” Opt. Lett. **35**(11), 1914–1916 (2010). [CrossRef] [PubMed]

**19**(24), 24023–24029 (2011). [CrossRef] [PubMed]

## 2. Simultaneous fusion, compression, and encryption of multiple images

**19**(24), 24023–24029 (2011). [CrossRef] [PubMed]

_{i}) are the target images at the input of the SFCE system. Firstly we apply separately the DCT transform separately to each of these images. Secondly, each spectrum of size (

*N*,

*N*) pixels is multiplied by a low-pass filter

*N*’,2

*N*’) pixels as shown in Fig. 1(a) (first 4 filtered DCTs). To avoid information overlap these blocks are returned and shifted. Then, quantization to each block of size (

*N’*,

*N’*) pixels is realized, as displayed in Fig. 1 (b). This permits to divide each block in several areas which are further quantized. This quantization is realized by normalizing each DCT to its maximum value. Following this quantization, we get the block shown in the right-side of Fig. 1(b). By reducing the maximum value in a given area we reduce the number of bits required to encode it.

*N*target images. The use of a first encryption-key (a specific image among those used in the input plane of SFCE), the specific fusion after rotation and the quantification constitute the first level of encryption. Then, a digital fingerprint encrypted with the classical DRP system is added to the fusion spectrum to avoid the second encryption level.

_{i}**19**(24), 24023–24029 (2011). [CrossRef] [PubMed]

*N’*,

*N’*) is insufficient to optimize the bandwidth of the spectral plane. In addition, we remark that using a unique and constant filter size for each block and image would not be optimized. It should also be noted that the number of bits for encoding frequencies is not optimized when only the maximum value of the given area is considered. In addition the encryption technique developed in [20

**19**(24), 24023–24029 (2011). [CrossRef] [PubMed]

## 3. Compression optimization

### 3.1. Adaptation of the block size according RMS criterion

*N’*,

*N’*) pixels) for each target image. Let us estimate the minimum size of the spectrum of a given target image. For that purpose we have adapted the RMS duration criterion to DCT. Following [13

13. A. Alfalou and C. Brosseau, “Exploiting root-mean-square time-frequency structure for multiple-image optical compression and encryption,” Opt. Lett. **35**(11), 1914–1916 (2010). [CrossRef] [PubMed]

*I*, and

*n*is a constant ranging from 0 and 2, e.g.

*n*ranging between 0 and 2. Table 1 shows that the reconstruction of the target image does not require the information contained in all pixels. If

*n*= 1, i.e. for a block of size 88

*n*<1 (panels (b-c-d) of Table 1). In contrast a good performance can be achieved for

*n*>1 (panels (f-g) of Table 1).

*n*from 1.5 to 2 improves the quality of the reconstructed image. Beyond

*n*= 2 there is no visible improvement of the quality of the reconstructed image. In the following

*n*is set to 2. For the purpose of later comparison of the numerically derived effect of block size on the quality of the reconstructed image and compression rate, two different metrics will be used: the mean-square error (MSE), and the peak signal-to-noise ratio (PSNR).

### 3.2. Effect of the block size on the quality of reconstructed images

*N*

_{i}target images of size

*t*

_{x}and

*t*

_{y}denote the size in pixels of each image (in this work we assume that

*N*images each of size

_{i}*N*=256 (image size equal to 256

*n*=1.5, 4 blocks of size (132

*n*. Not surprisingly, the effect of increasing the number of multiplexed images is to increase

### 3.3. Calculation of the compression rate without quantization

*T*

_{c}, we first consider the special case in the absence of quantization. The compression rate

*max(block)*and

*min (block)*denote respectively the maximal and minimal value in the blocks, and

*t*is calculated from Eq. (1).

*T*

_{c}in bits is not optimized. That is, it is smaller than

*T*

_{c}in pixels (Table 3). This is due in part to the rapidly varying spectra which necessitate a large number of encoding bits. To remedy this situation we propose to quantify the DCT blocks to obtain a fixed number of bits.

### 3.4. Fixed quantization

*m*which is required to encode the coefficients of the DCT blocks? There are several ways in which this question might be answered, but here, we first assume that

*m*=log

_{2}(max(block)) is fixed. The results showing the reconstructed images,

*T*

_{c}, and PSNR are summarized in Table 4.A notable feature of Table 4 is that obtaining a good compression rate requires a small block

*t*, i.e. a large number of images to be multiplexed. According to these numbers, the quality of the reconstructed images degrades rapidly. For example, if

*N*

_{i}is set to 16, the attained

*T*

_{c}=88% corresponding to a 10 dB loss. In addition, one sees that the values of

*T*

_{c}remain quite limited.

*T*

_{c}necessitates optimizing the quantization in order to decrease the DCT values. To attain this goal, we first address the case of fixed quantization. In [20

**19**(24), 24023–24029 (2011). [CrossRef] [PubMed]

*T*

_{c}was calculated in terms of the actual values in each area. Hence it cannot be adapted according the specificities of an application. As mentioned above we will quantify all values of every block with a fixed number of bits, i.e.

*m*. Hence, the block (

*t*,

*t*) resulting from this quantization necessitates a reduced number of bits. According this analysis, the new values of DCT with quantization are defined as followswhere

*m*is the number of bits for a given value of

*T*

_{c}. Note that in our previous work [20

**19**(24), 24023–24029 (2011). [CrossRef] [PubMed]

*N*

_{i}=2; second line). This value of

*T*

_{c}is significantly larger than the 2.5% obtained for the case of two images (Table 4). But, it should be noted that the quality of the reconstructed images remains poor. Thus, if the latter is over 8 bits this method leads to a high degradation in reconstructed image quality due to the uniform quantization.

*T*

_{c}s) indicates that a compromise between the two ways of calculating

*m*should be found. We suggest that a convenient measure of

*m*is given bywhere

*m*is the maximum number of bits used for encoding the DCT coefficients of a given block, and

### 3.5. Quantization by block optimization and normalization between 0 and max

*m*. The DCT is first divided in several areas according to a given step (Fig. 1). Next, each area is quantized (each area is represented by a specific color in Fig. 1 by considering both maximum and minimum values in this area.

*T*

_{c}with or without optimization. From left to right, the first column indicates the number of images to be multiplexed in the spectral plane. PSNR is shown in the second column. The reconstructed images without optimization [20

**19**(24), 24023–24029 (2011). [CrossRef] [PubMed]

*t,t*) is calculated with respect to the range between these two values, i.e. log

_{2}(max-min), instead of using the range from 0 to the maximum value, i.e. log

_{2}(max-0). The net effect is a decrease of

*m*for the following reason. Assume that in a given area the maximum and minimum value as are respectively set to 255 and 128. Using log

_{2}(max-0) requires 8 bits for encoding these values. Now normalizing values between 0 and max [0-128] and using log

_{2}(max-min) involves only 7 bits. Another notable feature of Table 6 is that as the PSNR is decreased the quality of the reconstructed image decreases gently while the compression rate (in bits) increases. For a given PSNR value, the compression rate is always larger when our optimization method is employed. This scheme permits a compression rate close to 97% to be obtained with good quality of reconstructed images (

*N*

_{i}=16).

### 3.6. Comparison with JPEG

*t*,

*t*). Table 7 shows a comparison of the results (gray-level images) obtained from our method with those obtained with the JPEG image compression standard for several compression rates. The data in Table 7 shows that the optimized SFCE method proved to be a very effective technique to control

*T*

_{c}. This is due to the adapted quantization and choice of the size of the DCT block. The key feature is that our technique does not suffer from the pixellization as is observed with the JPEG compression method. It is anticipated that this can be improved further by applying different geometric forms to define each DCT block as used in the Tetrolet transform [23

23. J. Krommweh, “Tetrolet transform: A new adaptive Haar wavelet algorithm for sparse image representation,” J. Vis. Commun. Image R. **21**(4), 364–374 (2010). [CrossRef]

*T*

_{c}=62% and PSNR= 32.5 dB.

## 4. Optimized encryption adapted to the SFCE method

*T*

_{c≤47%}

**,**since the encrypted images are complex valued. Hence, a second level of encryption will be added without degradation either in reconstructed image quality or in compression rate.

### 4.1 Optimization using real key image

*T*

_{c}with encryption (Table 9) and

*m*=5. The performances are good since

*T*

_{c}is decreased by only 8% compared to the case where no encryption is performed.

### 4.2 Resistance against attacks

**Attack 1**: Our first test is conducted with the cipher having no information about our technique. We assume that he or she has no knowledge of the number of target images, their rotation and quantization, the encryption keys, and the encryption protocol. To accomplish his or her goal, the cipher needs all this information which renders his or her task arduous. Indeed, within the optimized SFCE algorithm that is presented here one needs to have the fingerprint, random phase mask, quantization matrix, block sizes, their rotations and shapes to decrypt the input information. The size of the random matrix and fingerprint is

^{256×256}for the fingerprint and (10

^{15})

^{256×256}for the random matrix. The quantization matrix (for an image of size 256

**Attack 2:**The second test considers that the cipher now knows the principle of the optimized SFCE. For simplicity, a single image was used in this test. Since the cipher has access to our system, he or she can send an image corresponding to the inverse DCT (IDTC) of a matrix whole elements are all equal to 1. Matrix quantization is realized such that each value is

*m*-bit encoded, i.e. every value is multiplied by (2

^{m-1}– 1). Then, encryption block is realized. This block is the sum of the real part of the Fourier transform of the fingerprint and of the maximum value of the latter multiplied by a random matrix. The cipher can guess the encryption block in suppressing a matrix of the same dimension as the image sent but for which all the values are equal to

**Attack 3**: Finally, we consider that the cipher knows a part of the image to be sent (part

*P*=64 pixels, image size=256 pixels). In conclusion, this set of tests demonstrates the robustness of our algorithm against such kind of attacks since even with knowledge of much data the original target image cannot be decrypted.

## 5. Summary and conclusions

**35**(11), 1914–1916 (2010). [CrossRef] [PubMed]

## Acknowledgment

## References and links

1. | A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Adv. Opt. Photon. |

2. | B. Javidi, ed., |

3. | P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. |

4. | Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express |

5. | P. Kumar, J. Joseph, and K. Singh, “Known-plaintext attack-free double random phase-amplitude optical encryption: vulnerability to impulse function attack,” J. Opt. |

6. | J.-J. Huang, H.-E. Hwang, C.-Y. Chen, and C.-M. Chen, “Lensless multiple-image optical encryption based on improved phase retrieval algorithm,” Appl. Opt. |

7. | X. Wang and D. Zhao, “Double images encryption method with resistance against the specific attack based on an asymmetric algorithm,” Opt. Express |

8. | A. Alfalou and A. Mansour, “All optical video-image encryption enforced security level using independent component analysis,” J. Opt. A, Pure Appl. Opt. |

9. | Z. Liu, Q. Guo, L. Xu, M. A. Ahmad, and S. Liu, “Double image encryption by using iterative random binary encoding in gyrator domains,” Opt. Express |

10. | A. Alfalou and C. Brosseau, “Dual encryption scheme of images using polarized light,” Opt. Lett. |

11. | F. Mosso, J. F. Barrera, M. Tebaldi, N. Bolognini, and R. Torroba, “All-optical encrypted movie,” Opt. Express |

12. | M. Paturzo, P. Memmolo, L. Miccio, A. Finizio, P. Ferraro, A. Tulino, and B. Javidi, “Numerical multiplexing and demultiplexing of digital holographic information for remote reconstruction in amplitude and phase,” Opt. Lett. |

13. | A. Alfalou and C. Brosseau, “Exploiting root-mean-square time-frequency structure for multiple-image optical compression and encryption,” Opt. Lett. |

14. | E. Darakis and J. J. Soraghan, “Reconstruction domain compression of phase-shifting digital holograms,” Appl. Opt. |

15. | T. J. Naughton, J. B. McDonald, and B. Javidi, “Efficient compression of fresnel fields for Internet transmission of three-dimensional images,” Appl. Opt. |

16. | T. J. Naughton, Y. Frauel, B. Javidi, and E. Tajahuerce, “Compression of digital holograms for three-dimensional object reconstruction and recognition,” Appl. Opt. |

17. | S. L. Wijaya, M. Savvides, and B. V. K. Vijaya Kumar, “Illumination-tolerant face verification of low-bit-rate JPEG2000 wavelet images with advanced correlation filters for handheld devices,” Appl. Opt. |

18. | L. Ding, Y. Yan, Q. Xue, and G. Jin, “Wavelet packet compression for volume holographic image recognition,” Opt. Commun. |

19. | S. Yeom, A. Stern, and B. Javidi, “Compression of 3D color integral images,” Opt. Express |

20. | A. Alfalou, C. Brosseau, N. Abdallah, and M. Jridi, “Simultaneous fusion, compression, and encryption of multiple images,” Opt. Express |

21. | M. Johnson, P. Ishwar, V. Prabhakaran, D. Schonberg, and K. Ramchandran, “On compressing encrypted data,” IEEE Trans. Signal Process. |

22. | A. Alfalou, A. Mansour, M. Elbouz, and C. Brosseau, “Optical compression scheme to multiplex & simultaneously encode images”, in Optical and Digital Image Processing Fundamentals and Applications, |

23. | J. Krommweh, “Tetrolet transform: A new adaptive Haar wavelet algorithm for sparse image representation,” J. Vis. Commun. Image R. |

**OCIS Codes**

(070.0070) Fourier optics and signal processing : Fourier optics and signal processing

(100.0100) Image processing : Image processing

(200.4560) Optics in computing : Optical data processing

**ToC Category:**

Image Processing

**History**

Original Manuscript: January 8, 2013

Revised Manuscript: February 21, 2013

Manuscript Accepted: February 28, 2013

Published: March 26, 2013

**Citation**

A. Alfalou, C. Brosseau, N. Abdallah, and M. Jridi, "Assessing the performance of a method of simultaneous compression and encryption of multiple images and its resistance against various attacks," Opt. Express **21**, 8025-8043 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-7-8025

Sort: Year | Journal | Reset

### References

- A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Adv. Opt. Photon.1(3), 589 (2009). [CrossRef]
- B. Javidi, ed., Optical and Digital Techniques for Information Security (Springer Verlag, New York, 2005).
- P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett.20(7), 767–769 (1995). [CrossRef] [PubMed]
- Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express15(16), 10253–10265 (2007). [CrossRef] [PubMed]
- P. Kumar, J. Joseph, and K. Singh, “Known-plaintext attack-free double random phase-amplitude optical encryption: vulnerability to impulse function attack,” J. Opt.14(4), 045401 (2012). [CrossRef]
- J.-J. Huang, H.-E. Hwang, C.-Y. Chen, and C.-M. Chen, “Lensless multiple-image optical encryption based on improved phase retrieval algorithm,” Appl. Opt.51(13), 2388–2394 (2012). [CrossRef] [PubMed]
- X. Wang and D. Zhao, “Double images encryption method with resistance against the specific attack based on an asymmetric algorithm,” Opt. Express20(11), 11994–12003 (2012). [CrossRef] [PubMed]
- A. Alfalou and A. Mansour, “All optical video-image encryption enforced security level using independent component analysis,” J. Opt. A, Pure Appl. Opt.9, 787 (2007). [CrossRef]
- Z. Liu, Q. Guo, L. Xu, M. A. Ahmad, and S. Liu, “Double image encryption by using iterative random binary encoding in gyrator domains,” Opt. Express18(11), 12033–12043 (2010). [CrossRef] [PubMed]
- A. Alfalou and C. Brosseau, “Dual encryption scheme of images using polarized light,” Opt. Lett.35(13), 2185–2187 (2010). [CrossRef] [PubMed]
- F. Mosso, J. F. Barrera, M. Tebaldi, N. Bolognini, and R. Torroba, “All-optical encrypted movie,” Opt. Express19(6), 5706–5712 (2011). [PubMed]
- M. Paturzo, P. Memmolo, L. Miccio, A. Finizio, P. Ferraro, A. Tulino, and B. Javidi, “Numerical multiplexing and demultiplexing of digital holographic information for remote reconstruction in amplitude and phase,” Opt. Lett.33(22), 2629–2631 (2008). [CrossRef] [PubMed]
- A. Alfalou and C. Brosseau, “Exploiting root-mean-square time-frequency structure for multiple-image optical compression and encryption,” Opt. Lett.35(11), 1914–1916 (2010). [CrossRef] [PubMed]
- E. Darakis and J. J. Soraghan, “Reconstruction domain compression of phase-shifting digital holograms,” Appl. Opt.46(3), 351–356 (2007). [CrossRef] [PubMed]
- T. J. Naughton, J. B. McDonald, and B. Javidi, “Efficient compression of fresnel fields for Internet transmission of three-dimensional images,” Appl. Opt.42(23), 4758–4764 (2003). [CrossRef] [PubMed]
- T. J. Naughton, Y. Frauel, B. Javidi, and E. Tajahuerce, “Compression of digital holograms for three-dimensional object reconstruction and recognition,” Appl. Opt.41(20), 4124–4132 (2002). [CrossRef] [PubMed]
- S. L. Wijaya, M. Savvides, and B. V. K. Vijaya Kumar, “Illumination-tolerant face verification of low-bit-rate JPEG2000 wavelet images with advanced correlation filters for handheld devices,” Appl. Opt.44(5), 655–665 (2005). [CrossRef] [PubMed]
- L. Ding, Y. Yan, Q. Xue, and G. Jin, “Wavelet packet compression for volume holographic image recognition,” Opt. Commun.216(1-3), 105–113 (2003). [CrossRef]
- S. Yeom, A. Stern, and B. Javidi, “Compression of 3D color integral images,” Opt. Express12(8), 1632–1642 (2004). [CrossRef] [PubMed]
- A. Alfalou, C. Brosseau, N. Abdallah, and M. Jridi, “Simultaneous fusion, compression, and encryption of multiple images,” Opt. Express19(24), 24023–24029 (2011). [CrossRef] [PubMed]
- M. Johnson, P. Ishwar, V. Prabhakaran, D. Schonberg, and K. Ramchandran, “On compressing encrypted data,” IEEE Trans. Signal Process.52(10), 2992–3006 (2004). [CrossRef]
- A. Alfalou, A. Mansour, M. Elbouz, and C. Brosseau, “Optical compression scheme to multiplex & simultaneously encode images”, in Optical and Digital Image Processing Fundamentals and Applications, G. Cristobal (Ed.), P. Schelkens (Ed.), and H. Thienpont (Ed.), (Wiley, 2011) pp. 463–483.
- J. Krommweh, “Tetrolet transform: A new adaptive Haar wavelet algorithm for sparse image representation,” J. Vis. Commun. Image R.21(4), 364–374 (2010). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.