Fast parametric reciprocal-orthogonal jacket transforms
© Lee et al.; licensee Springer. 2014
Received: 14 March 2014
Accepted: 10 September 2014
Published: 29 September 2014
In this paper, we propose a new construction method for a novel class of parametric reciprocal-orthogonal jacket transform (PROJT) having parameters for a sequence length N=2r+1 that is a power of two, based on the reciprocal-orthogonal parametric (ROP) transform and block diagonal matrices. It is shown that the inverse transform of the proposed PROJT is conveniently obtained by the reciprocal of each elements of the forward matrix and transpose operation. What is more, an efficient algorithm for the computation of the PROJT has been developed with the aid of the matrix decomposition and Kronecker product. Further, the experiments show that the independent parameters in the proposed PROJT are successfully used as additional secret keys for image encryption, watermarking, and error-correcting codes.
There are a variety of discrete signal orthogonal transforms [1–3], such as discrete Fourier transform (DFT), discrete Hartley transform (DHT), Walsh-Hadamard transform (WHT), Haar transform, slant transform, and discrete cosine transform (DCT), which have various application in digital signal processing, image compressing video processing, and pattern recognition. A lot of services of these transforms are mainly due to their practical usefulness and the existence of fast and efficient algorithms for their computation. However, since each of the well-known transforms [4–9], for example DFT, DHT, WHT, DCT, etc., is fixed without any parameters, each single transform only deals with its special area of applications. In order to best match the given input signal class or the application, many parametric transforms [10–15] with matrices associating a set of parameters are presented to fit the desirable signal by choosing appropriate parameters. An advantage of parametric transforms is the possibility to implement large families of transforms with a unified software/hardware, which is efficient for every representative of the family and may be tuned to the desired transforms .
On the other hand, the DCT and the Karhunen-Loeve transform (KLT) have better compaction performance than the slant transform [10, 11]. Both the DCT and KLT have more computational complexity than the slant transform. Therefore, the need arises for slant transform improvement schemes that yield performance comparable to that of the KLT and DCT without incurring their computational complexity [10–20]. Therefore, various generalizations of the WHTs and DFTs have been attempted. Lee  proposed a center weight Hadamard transform with matrix of order 4. However, the proposed jacket transform has only three parameters at most. Recently, Bouguezel et al.  proposed a new class of reciprocal-orthogonal parametric (ROP) transforms, which have independent parameters for an input data vector of length N. Further, they showed that the inverse of the ROP transforms and matrices is easily obtained and has fast algorithm. Lee et al.  proposed a novel class of element-wise inverse jacket transforms (EIJTs) having 2N−1 parameters for an input data vector of length N=3·2 r . Moreover, Bouguezel et al.  proposed new parametric discrete Fourier transforms, Hartley transforms, and algorithms for fast computation. Ding et al.  proposed arbitrary-length Walsh-Jacket transforms. Agaian et al.  developed a class of generalized parametric Slant-Hadamard transforms with fast algorithm, whose performance is better than the classical one-transform-based model. Chen et al.  proposed a fast cocyclic jacket transform over the complex number field. Moreover, many other transforms based on complex field and finite fields were proposed [19, 20, 22].
In recent years, enormous parametric transforms corresponding to the existing determined transforms have been developed. It has been shown that parametric transforms can have more flexibility and a wider range of applications compared to its original transform. For example, the independent parameters of the fractional discrete transform are used as an additional secret key for watermarking , encryption [4, 17], error-correcting codes, etc. From this point of view, parametric transforms with matrices described in a unified form and based on a set of parameters become more and more important.
The main purpose of this paper is to propose a fast parametric reciprocal-orthogonal jacket transform (PROJT) having parameters for an input data vector N=2r+1, which is reciprocal-orthogonal and has a fast and efficient algorithm with special structure. In addition, this proposed transform has more parameters than some known proposed transforms. A lot of simulations show that the independent parameters in the PROJT are able to be used as additional secret keys for image encryption. The rest of this paper is organized as follows. In Section 2, jacket matrices and some preliminaries are recalled. In Section 3, the PROJT is proposed and developed. In Section 4, an efficient algorithm with special structure is developed for the proposed PROJT having many parameters. Examples and computer simulations are given in Section 5. We draw some conclusions and remarks in Section 6.
2 Preliminaries and notations
In this section, we introduce some definitions and notations. For an N×N matrix [J] N , the N×N associated matrix is obtained from matrix [J] N by taking the reciprocal of each entry and exchanging its row and column indices. In other words, the (k,i) entry of is equal to the reciprocal of the element in the (i,k) position in [J] N . We now recall the definition of a jacket matrix which is reciprocal-orthogonal in .
3 Proposed PROJT with many parameters
where for i=N/4,N/4+1,⋯,N/2−1.
where k,l=0,1,⋯,N/2−1. With the above notations and symbols, we are ready to propose a novel class PROJT as follows.
n=0,1,⋯,N−1, and stands for the largest integer no more than .
The PROJT matrix of order N=2r+1 have independent parameters, since there are nonzero independent parameters a00,⋯,a0,N/2−1,a1,0,⋯,aN/4−1 and independent parameters V00,⋯,V0,N,V1,0,⋯,V1,N/2−1. It is known that the more independent parameters a transform has, the more it has applications. Note that the PROJT has parameters while the ROP has parameters, which implies that the PROJT has more applications in watermarking, encryption, and error-correcting codes than the ROP. Further, it can be shown that the inverse transform of the proposed transform is easily obtained and has an efficient algorithm.
Case 2. Next, we consider that t≠n. There are four subcases.
where the last equality follows from (12).
where the last equality follows from (9).
where the second last equality follows from (15) and i=0,1,⋯,N−1. Hence we finish our proof. □
In order to understand the proposed PROJT, we give some examples to illustrate how to construct it.
4 Fast and efficient algorithm for the proposed PROJT
In this section, we analyze some properties of the proposed PROJT, which are used to present an efficient algorithm for a fast computation of the proposed PROJT by (13) and (14).
for k∈S o and 1≤k≤N/2−1.
Hence all the output points can be obtained from the output sequence of the WHT of order of N/4 of the input sequence.
where [I]2 is a 2×2 identity matrix and d i a g(V(0),…,V(N/2−1)) is the N×N block diagonal matrix whose block is 2×2 matrices.
Let be an N/2×N/2 permutation matrix whose entries (0,0),(1,N/2−1),(2,1),(3,N/2−2),(4,2),…,(2i,i),(2i+1,N/2−i−1),…,(N/2−2,N/4−1),(N/2−1,N/2−(N/4−1)−1) are 1, the other entries are 0.
Let be an N/2×N/2 permutation matrix whose entries (0,0),(1,2),(2,4),…,(i,2i),…,(N/4−1,N/2−2), (N/4,1),(N/4+1,3),…,(N/4+i,2i+1),…,(N/4+N/4−1,N/2−1) are 1, the other entries are 0.
By the mean of this decomposition of the proposed matrix, we are able to get a fast and efficient algorithm.
Moreover, let be an N/2×N/2 diagonal matrix whose diagonal entries are . Let be an N/2×N/2 diagonal matrix whose diagonal entries are a0,0,a0,N/2−1, a0,1,a0,N/2−2,…,a0,i,a0,N/2−i−1,…,a0,N/4−1,a0,N/4. Let be an N×N diagonal matrix whose diagonal entries are , . Let be an N×N diagonal matrix whose diagonal entries are V0,0,V0,1,V0,2,…,V1,N−2,V0,N−1.
5 Simulations and discussion
where j=0,1,N/4−1. Error δ can be any integer uniformly distributed in the range 1-255. Figure 3d shows the decrypted image with the minimized error δ=1, which shows that the original image is successfully protected.
Therefore, to successfully decrypt the 512×512 (N=512) image, whose elements are integers in the range 0-255, we need to know every parameter of V0,2i, V0,2i+1, V1,2i, a0,i for i=0,1,…,N/2−1 and a1,j for j=0,1,N/4−1. Intuitively, this is also true for N>512. Assume that the errors in V 0,2i′, V 0,2i+1′, V 1,2i′, a 0,i′ for i=0,1,…,N/2−1 and a 1,j′ for j=0,1,N/4−1 are uniformly distributed. Then, the probability of a successful decryption without knowing all the parameters correctly is 1/2561152.
In this paper, we have proposed a new class of PROJT of order N=2r+1 independent 3N parameters. The PROJT is based on the the proposed ROP transform , block diagonal matrices, and permutations. On one hand, the critical usefulness of the PROJT generalized the proposed ROP transform, which is a special case of the the PROJT. On the other hand, the PROJT has more parameters than the ROP transform. What is more important, some nice properties are presented, in particular, the inverse transform is fastly obtained by the reciprocal and transpose operations. With the aid of matrix decomposition and Kronecker product approach, a fast and efficient algorithm for computing the proposed PROJT is obtained. In fact, we show that the PROJT has 3N multiplications and N+N log2N addition operations. Therefore, the proposed PROJT can be employed in watermaking and encryption where the independent parameters can be used as an additional secret key.
The authors would like to thank the reviewers for their insightful suggestions and comments for improving the manuscript.
This work was supported by the MEST 2012-002521, National Research Foundation, Korea. It was also supported partly by the National Nature Science Foundation of China (11271256, 61201249) and the DHU Distinguished Young Professor Program (14D210402).
- Ahmed N, Rao KR: Orthogonal Transforms for Digital Signal Processing. Springer, New York; 1975.View ArticleMATHGoogle Scholar
- Yarlagadda RK, Hershey JE: Hadamard Matrix Analysis and Synthesis With Applications to Communications Signal/Image Processing. Kluwer Academic Publishers, Norwell, MA; 1997.View ArticleGoogle Scholar
- Horadam KJ: Hadamard Matrices and Their Applications. Princeton University Press, Princeton, NJ; 2006.MATHGoogle Scholar
- Pei S-C, Hsue W-L: The multiple-parameter discrete fractional Fourier transform. IEEE Signal Process. Lett 2006, 13(6):329-332.View ArticleGoogle Scholar
- Tseng C-C: Eigenvalues and eigenvectors of generalized DFT, generalized DHT, DCT-IV and DST-IV matrices. IEEE Signal Process 2002, 50(4):866-877. 10.1109/78.992133View ArticleMathSciNetGoogle Scholar
- Lee MH: The center weighted Hadamard transform. IEEE Trans. Circuits Syst 1989, 36(9):1247-1249. 10.1109/31.34673View ArticleGoogle Scholar
- Lee MH: A new reverse jacket transform and its fast algorithm. IEEE Trans. Circ. Syst. II 2000, 47(1):39-47. 10.1109/82.818893View ArticleMATHGoogle Scholar
- Ding JJ, Pei SC, Wu PH: Jacket Haar, Transform. Circuits and Systems (ISCAS) 2011 IEEE International Symposium 15-18 May 2011, 1520-1523.View ArticleGoogle Scholar
- Lee MH, Zhang X-D: Fast block center weighted Hadamard transform. IEEE Trans. Circuits Syst. I: Reg. Papers 2007, 54(12):2741-2745.MathSciNetView ArticleGoogle Scholar
- Agaian S, Tourshan K, Noonan JP: Parametric Slant-Hadamard transforms with applications. IEEE Signal Process. Lett 2002, 9(11):375-377.View ArticleMATHGoogle Scholar
- Agaian S, Tourshan K, Noonan JP: Generalized parametric Slant-Hadamard transforms with applications. Signal Process. Lett 2004, 84(8):1299-1306. 10.1016/j.sigpro.2004.04.009View ArticleMATHGoogle Scholar
- Bouguezel S: A reciprocal-orthogonal parametric transform and its fast algorithm. IEEE Signal Process. Lett 2012, 19(11):769-772.View ArticleGoogle Scholar
- Bouguezel S, Ahmad MO, Swamy MNS: A new class of reciprocal-orthogonal parametric transforms. IEEE Trans. Circuits Syst. I, Reg. Papers 2009, 56(4):795-804.MathSciNetView ArticleGoogle Scholar
- Bouguezel S, Ahmad MO, Swamy MNS: New parametric discrete Fourier and Hartley transforms, and algorithms for fast computation. IEEE Trans. Circuits Syst. I, Reg. Papers 2011, 58(3):562-575.MathSciNetView ArticleGoogle Scholar
- Ding JJ, Pei S-C, Wu P-H: Arbitrary-length Walsh-Jacket transforms. In Proceedings of 2011 APSIPA Annual Summit and Conference. Xi’an, China; 18-21 October 2011:1-10.Google Scholar
- Minasyan S, Astola J, Guevorkian D: An image compression scheme based on parametric, Haar-like transform. Circuits and Systems 2005. ISCAS 2005. IEEE International Symposium 23-26 May 2005, 2088-2091.Google Scholar
- Vilardy JM, Calderon JE, Torres CO, Mattos L: Digital images phase encryption using fractional Fourier transform. Electronics, Robotics and Automotive Mechanics Conference 26-29 Sept 2006, 15-18.View ArticleGoogle Scholar
- Lee MH, Zhang X-D, Song W, Xia X-G: Fast reciprocal jacket transform with many parameters. IEEE Trans. Circuits Syst. I, Reg. Papers 2012, 59(7):1472-1481.MathSciNetView ArticleGoogle Scholar
- Lee MH: A new reverse, Jacket transform based on Hadamard matrix. In IEEE International Symposium on Information Theory. Sorrento, Italy; 25-30 June 2000:471-471.Google Scholar
- Lee MH, Borissov YL: On Jacket transforms over finite fields. In IEEE International Symposium on Information Theory. Seoul, Korea; 28 June 2009-3 July 2009:2803-2807.Google Scholar
- Chen Z, Lee MH, Zeng G: Fast cocylic jacket transform. IEEE Trans. Signal Process 2008, 56(5):2143-2148.MathSciNetView ArticleGoogle Scholar
- Lee MH: Jacket Matrices: Construction and its Applications for Fast Cooperative Wireless Signal Processing. Lambert Academic Publishing, Lambert, Germany; 2012.Google Scholar
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