**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**30526

##### Optimal Image Representation for Linear Canonical Transform Multiplexing

**Authors:**
Navdeep Goel,
Salvador Gabarda

**Abstract:**

**Keywords:**
chirp signals,
image multiplexing,
linear canonical transform,
Image
transformation,
Polynomial
approximation

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1110750

**References:**

[1] S. Hoggar, Mathematics of Digital Images: Creation, Compression, Restoration, Recognition. New York: Cambridge University Press, 2006.

[2] G. Wallace, “The JPEG still picture compression standard,” IEEE Trans. Consum. Electron., vol. 38, no. 1, pp. xviii–xxxiv, 1992.

[3] M. Rabbani and W. P. Jones, Digital Image Compression Techniques. Bellingham, WA, USA: Society of Photo-Optical Instrumentation Engineers (SPIE), 1991.

[4] I. Sadeh, “Polynomial approximation of images,” Comput. Math. Appl., vol. 32, no. 5, pp. 99–115, 1996.

[5] M. Eden, M. Unser, and R. Leonardi, “Polynomial representation of pictures,” Signal Process., vol. 10, no. 4, pp. 385–393, 1986.

[6] H. Huangi and U. Ascher, “Faster gradient descent and the efficient recovery of images,” Vietnam J. Math., vol. 42, no. 1, pp. 115–131, 2014.

[7] Ordinary Differential Equations. Allied Publishers, 2003.

[8] Z. Omar, N. Mitianoudis, and T. Stathaki, “Two-dimensional Chebyshev polynomials for image fusion,” in Picture Coding Symposium (PCS), 2010, pp. 426–429.

[9] Al-Jarwan, I.A., and M. Zemerly, Image compression using adaptive variable degree variable segment length Chebyshev polynomials, ser. Lecture Notes in Computer Science. Springer Berlin Heidelberg, 2005, vol. 3540, pp. 1196–1207.

[10] P. Hansen, “The truncated SVD as a method for regularization,” BIT, vol. 27, no. 4, pp. 534–553, 1987.

[11] C. Liu, J. Zhou, and K. He, “Image compression based on truncated HOSVD,” in International Conference on Information Engineering and Computer Science, 2009, pp. 1–4.

[12] P. Flandrin, “Time frequency and chirps,” in Proc. SPIE , Wavelet Applications VIII, vol. 4391, 2001, pp. 161–175.

[13] S. C. Pei and J. J. Ding, “Eigenfunctions of the offset Fourier, fractional Fourier, and linear canonical transforms,” J. Opt. Soc. Am. A, vol. 20, no. 03, pp. 522–532, 2003.

[14] Q. Xiang and K. Qin, “Convolution, correlation, and sampling theorems for the offset linear canonical transform,” Signal Image Video Process., vol. 08, no. 03, pp. 433–442, 2014.

[15] K. B. Wolf, Integral Transforms in Science and Engineering. New York: Plenum Press, 2013.

[16] S. C. Pei and J. J. Ding, “Eigenfunctions of Fourier and fractional Fourier transforms with complex offsets and parameters,” IEEE Trans. Circuits Syst. I, vol. 54, no. 07, pp. 1599–1611, 2007.

[17] H. M. Ozaktas, B. Barshan, D. Mendlovic, and L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relationship to chirp and wavelet transforms,” J. Opt. Soc. Am. A, vol. 11, no. 02, pp. 547–559, 1994.

[18] R. Driggers, Encyclopedia of Optical Engineering: Las - Pho. Dekker, 2003, vol. 2.

[19] D. P. Huijsmans and A. W. M. Smeulders, Eds., Visual Information and Information Systems, ser. Lecture Notes in Computer Science-Proceedings of the Third International Conference, VISUALâ˘A ´ Z99. Amsterdam, The Netherlands: Springer, 1999, vol. 1614.

[20] J. H. Pujar and L. M. Kadlaskar, “A new lossless method of image compression and decompression using Huffman coding techniques,” J. Theor. Appl. Inf. Technol., vol. 15, no. 01, pp. 18–23, 2010.

[21] D. Salomon, A Concise Introduction to Data Compression. Berlin: Springer, 2008.

[22] V. Wickerhauser, “Wavelet analysis and its applications,” in Proc. Third International Conference on Wavelet Analysis and Its Applications WAA, Chongqing, PR China, 2003.

[23] A. Meyer-Bäse, Pattern Recognition for Medical Imaging. Elsevier Academic Press, 2004.

[24] G. K. Smyth, Polynomial Approximation in Encyclopedia of Biostatistics. John Wiley & Sons, Ltd, 2005.

[25] L. Wojakowski, “Moments of measure orthogonalizing the 2-dimensional Chebyshev polynomials,” in Quantum probabilit. Warsaw: Polish Academy of Sciences, Institute of Mathematics, 2006, pp. 429–433.

[26] M. Cheong and K.-S. Loke, “Textile recognition using Tchebichef moments of co-occurrence matrices,” in Advanced Intelligent Computing Theories and Applications. With Aspects of Theoretical and Methodological Issues, ser. Lecture Notes in Computer Science, D.-S. Huang, D. Wunsch, D. Levine, and K.-H. Jo, Eds. Springer Berlin Heidelberg, 2008, vol. 5226, pp. 1017–1024.

[27] H. Zhu, H. Shu, T. Xia, L. Luo, and J. L. Coatrieux, “Translation and scale invariants of Tchebichef moments,” Pattern Recogn., vol. 40, no. 9, pp. 2530 – 2542, 2007.

[28] H. Rahmalan, N. Abu, and S. Wong, “Using Tchebichef moment for fast and efficient image compression,” Pattern Recogn. Image Anal., vol. 20, no. 4, pp. 505–512, 2010.

[29] P.-T. Yap, P. Raveendran, and S. Ong, “Chebyshev moments as a new set of moments for image reconstruction,” in Proc. of Int. Joint Conf. on Neural Networks (IJCNN), vol. 4, 2001, pp. 2856–2860.

[30] R. Mukundan, S. Ong, and P. Lee, “Image analysis by Tchebichef moments,” IEEE Trans. Image Process., vol. 10, no. 9, pp. 1357–1364, 2001.

[31] S. T. Welstead, Fractal and Wavelet Image Compression Techniques, ser. Tutorial Text Series. SPIE Optical Engineering Press, 1999.

[32] G. G. L. Jr. and J. Rissanen, “Compression of black-white images with arithmetic coding,” IEEE Trans. Commun., vol. 29, no. 6, pp. 858–867, 1981.

[33] P. G. Howard and J. S. Vitter, “New methods for lossless image compression using arithmetic coding,” Inf. Process. Manag., vol. 28, no. 6, pp. 765 – 779, 1992, special Issue: Data compression for images and texts.

[34] A. Said and W. A. Pearlman, “A new, fast, and efficient image codec based on set partitioning in hierarchical trees,” IEEE Trans. Circuits Syst. Video Technol., vol. 6, no. 3, pp. 243–250, 1996.

[35] M. Nelson and J. L. Gailly, The Data Compression Book. Wiley, 1995.

[36] E. Chassande-Mottin and P. Flandrin, “On the timeâA˘ S¸ frequency detection of chirps1,” Appl. Comput. Harmon. Anal., vol. 6, no. 2, pp. 252 – 281, 1999.

[37] M. Barni, “Fractal image compression,” in Document and Image Compression, ser. Signal Process. Commun., D. Saupe and R. Hamzaoui, Eds. CRC Press, 2006, pp. 145 –175.

[38] T. A. C. M. Claasen and W. F. G. Mecklenbrauker, “The Wigner distribution-a tool for time frequency signal analysis, part ii: Discrete time signals,” Philips J. Res., vol. 35, no. 4/5, pp. 276–300, 1980.