American Institute of Mathematical Sciences

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February  2014, 8(1): 321-337. doi: 10.3934/ipi.2014.8.321

PHLST with adaptive tiling and its application to antarctic remote sensing image approximation

Zhihua Zhang 1, and  Naoki Saito 2,

1. 

College of Global Change and Earth System Science, Beijing Normal University, Beijing, 100875, China
2. 

Department of Mathematics, University of California, Davis, California, 95616, United States

Received  July 2011 Revised  March 2012 Published  March 2014

We propose an efficient nonlinear approximation scheme using the Polyharmonic Local Sine Transform (PHLST) of Saito and Remy combined with an algorithm to tile a given image automatically and adaptively according to its local smoothness and singularities. To measure such local smoothness, we introduce the so-called local Besov indices of an image, which is based on the pointwise modulus of smoothness of the image. Such an adaptive tiling of an image is important for image approximation using PHLST because PHLST stores the corner and boundary information of each tile and consequently it is wasteful to divide a smooth region of a given image into a set of smaller tiles. We demonstrate the superiority of the proposed algorithm using Antarctic remote sensing images over the PHLST using the uniform tiling. Analysis of such images including their efficient approximation and compression has gained its importance due to the global climate change.
Keywords: climate change., Antarctic remote sensing, image approximation, PHLST, adaptive tiling.
Mathematics Subject Classification: Primary: 42C40, 65D18; Secondary: 68W2.
Citation: Zhihua Zhang, Naoki Saito. PHLST with adaptive tiling and its application to antarctic remote sensing image approximation. Inverse Problems & Imaging, 2014, 8 (1) : 321-337. doi: 10.3934/ipi.2014.8.321
References:
[1]

A. Averbuch, M. Israeli and L. Vozovoi, A fast Poisson solver of arbitrary order accuracy in rectangular regions, SIAM J. Sci. Comput., 19 (1998), 933-952. doi: 10.1137/S1064827595288589.  Google Scholar

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R. R. Coifman and Y. Meyer, Remarques sur l'analyse de Fourier à fenêtre, Comptes Rendus Acad. Sci. Paris, Serie I, 312 (1991), 259-261.  Google Scholar

[6]

J. S. Lim, Two-Dimensional Signal and Image Processing, Prentice Hall, Englewood Cliffs, NJ, 1990. Google Scholar

[7]

G. G. Lorentz, Approximation of Functions, 2nd Ed., Chelsea Pub. Co., New York, 1986. doi: 10.2307/2008020.  Google Scholar

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H. S. Malvar, Lapped transforms for efficient transform/subband coding, IEEE Trans. Acoust., Speech, Signal Process., 38 (1990), 969-978. doi: 10.1109/29.56057.  Google Scholar

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H. S. Malvar and D. H. Staelin, The LOT: Transform coding without blocking effects, IEEE Trans. Acoust., Speech, Signal Process., 37 (1989), 553-559. doi: 10.1109/29.17536.  Google Scholar

[10]

J. R. Parker, Algorithms for Image Processing and Computer Vision, John Wiley & Sons, Inc., New York, 1997. Google Scholar

[11]

N. Saito and J.-F. Remy, The polyharmonic local sine transform: A new tool for local image analysis and synthesis without edge effect, Appl. Comput. Harmon. Anal., 20 (2006), 41-73. doi: 10.1016/j.acha.2005.01.005.  Google Scholar

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G. Strang, The discrete cosine transform, SIAM Review, 41 (1999), 135-147. doi: 10.1137/S0036144598336745.  Google Scholar

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A. F. Timan, Theory of Approximation of Functions of a Real Variable, Macmillan, New York, 1963.  Google Scholar

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A. Zygmund, Trigonometric Series, 3rd Edition, Cambridge University Press, 2003. doi: 10.2307/1989363.  Google Scholar

show all references

References:
[1]

A. Averbuch, M. Israeli and L. Vozovoi, A fast Poisson solver of arbitrary order accuracy in rectangular regions, SIAM J. Sci. Comput., 19 (1998), 933-952. doi: 10.1137/S1064827595288589.  Google Scholar

[2]

E. Braverman, M. Israeli, A. Averbuch and L. Vozovoi, A fast 3D Poisson solver of arbitrary order accuracy, J. Comput. Phys., 144 (1998), 109-136. doi: 10.1006/jcph.1998.6001.  Google Scholar

[3]

J. F. Canny, A computational approach to edge detection, IEEE Trans. Pattern Anal. Machine Intell., 8 (1986), 679-698. doi: 10.1109/TPAMI.1986.4767851.  Google Scholar

[4]

R. A. DeVore and V. N. Temlyakov, Nonlinear approximation by trigonometric sums, J. Fourier Anal. Appl., 2 (1995), 29-48. doi: 10.1007/s00041-001-4021-8.  Google Scholar

[5]

R. R. Coifman and Y. Meyer, Remarques sur l'analyse de Fourier à fenêtre, Comptes Rendus Acad. Sci. Paris, Serie I, 312 (1991), 259-261.  Google Scholar

[6]

J. S. Lim, Two-Dimensional Signal and Image Processing, Prentice Hall, Englewood Cliffs, NJ, 1990. Google Scholar

[7]

G. G. Lorentz, Approximation of Functions, 2nd Ed., Chelsea Pub. Co., New York, 1986. doi: 10.2307/2008020.  Google Scholar

[8]

H. S. Malvar, Lapped transforms for efficient transform/subband coding, IEEE Trans. Acoust., Speech, Signal Process., 38 (1990), 969-978. doi: 10.1109/29.56057.  Google Scholar

[9]

H. S. Malvar and D. H. Staelin, The LOT: Transform coding without blocking effects, IEEE Trans. Acoust., Speech, Signal Process., 37 (1989), 553-559. doi: 10.1109/29.17536.  Google Scholar

[10]

J. R. Parker, Algorithms for Image Processing and Computer Vision, John Wiley & Sons, Inc., New York, 1997. Google Scholar

[11]

N. Saito and J.-F. Remy, The polyharmonic local sine transform: A new tool for local image analysis and synthesis without edge effect, Appl. Comput. Harmon. Anal., 20 (2006), 41-73. doi: 10.1016/j.acha.2005.01.005.  Google Scholar

[12]

G. Strang, The discrete cosine transform, SIAM Review, 41 (1999), 135-147. doi: 10.1137/S0036144598336745.  Google Scholar

[13]

A. F. Timan, Theory of Approximation of Functions of a Real Variable, Macmillan, New York, 1963.  Google Scholar

[14]

A. Zygmund, Trigonometric Series, 3rd Edition, Cambridge University Press, 2003. doi: 10.2307/1989363.  Google Scholar

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