# American Institute of Mathematical Sciences

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

 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.
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
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##### References:
 [1] Min-Fan He, Li-Ning Xing, Wen Li, Shang Xiang, Xu Tan. Double layer programming model to the scheduling of remote sensing data processing tasks. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1515-1526. doi: 10.3934/dcdss.2019104 [2] A Voutilainen, Jari P. Kaipio. Model reduction and pollution source identification from remote sensing data. Inverse Problems & Imaging, 2009, 3 (4) : 711-730. doi: 10.3934/ipi.2009.3.711 [3] Betseygail Rand, Lorenzo Sadun. An approximation theorem for maps between tiling spaces. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 323-326. doi: 10.3934/dcds.2011.29.323 [4] Linfei Wang, Dapeng Tao, Ruonan Wang, Ruxin Wang, Hao Li. Big Map R-CNN for object detection in large-scale remote sensing images. Mathematical Foundations of Computing, 2019, 2 (4) : 299-314. doi: 10.3934/mfc.2019019 [5] Graciela Canziani, Rosana Ferrati, Claudia Marinelli, Federico Dukatz. Artificial neural networks and remote sensing in the analysis of the highly variable Pampean shallow lakes. Mathematical Biosciences & Engineering, 2008, 5 (4) : 691-711. doi: 10.3934/mbe.2008.5.691 [6] Xuefeng Zhang, Hui Yan. Image enhancement algorithm using adaptive fractional differential mask technique. Mathematical Foundations of Computing, 2019, 2 (4) : 347-359. doi: 10.3934/mfc.2019022 [7] Yonggui Zhu, Yuying Shi, Bin Zhang, Xinyan Yu. Weighted-average alternating minimization method for magnetic resonance image reconstruction based on compressive sensing. Inverse Problems & Imaging, 2014, 8 (3) : 925-937. doi: 10.3934/ipi.2014.8.925 [8] Samuel Amstutz, Antonio André Novotny, Nicolas Van Goethem. Minimal partitions and image classification using a gradient-free perimeter approximation. Inverse Problems & Imaging, 2014, 8 (2) : 361-387. doi: 10.3934/ipi.2014.8.361 [9] Florian Bossmann, Jianwei Ma. Enhanced image approximation using shifted rank-1 reconstruction. Inverse Problems & Imaging, 2020, 14 (2) : 267-290. doi: 10.3934/ipi.2020012 [10] Michael Hintermüller, Monserrat Rincon-Camacho. An adaptive finite element method in $L^2$-TV-based image denoising. Inverse Problems & Imaging, 2014, 8 (3) : 685-711. doi: 10.3934/ipi.2014.8.685 [11] Victor Churchill, Rick Archibald, Anne Gelb. Edge-adaptive $\ell_2$ regularization image reconstruction from non-uniform Fourier data. Inverse Problems & Imaging, 2019, 13 (5) : 931-958. doi: 10.3934/ipi.2019042 [12] Ka Kit Tung. Simple climate modeling. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 651-660. doi: 10.3934/dcdsb.2007.7.651 [13] Inez Fung. Challenges of climate modeling. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 543-551. doi: 10.3934/dcdsb.2007.7.543 [14] Rafael De La Llave, A. Windsor. An application of topological multiple recurrence to tiling. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 315-324. doi: 10.3934/dcdss.2009.2.315 [15] S. Eigen, V. S. Prasad. Tiling Abelian groups with a single tile. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 361-365. doi: 10.3934/dcds.2006.16.361 [16] Susanna V. Haziot. Study of an elliptic partial differential equation modelling the Antarctic Circumpolar Current. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4415-4427. doi: 10.3934/dcds.2019179 [17] Yangyang Xu, Wotao Yin, Stanley Osher. Learning circulant sensing kernels. Inverse Problems & Imaging, 2014, 8 (3) : 901-923. doi: 10.3934/ipi.2014.8.901 [18] Vikram Krishnamurthy, William Hoiles. Information diffusion in social sensing. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 365-411. doi: 10.3934/naco.2016017 [19] Graeme Wake, Anthony Pleasants, Alan Beedle, Peter Gluckman. A model for phenotype change in a stochastic framework. Mathematical Biosciences & Engineering, 2010, 7 (3) : 719-728. doi: 10.3934/mbe.2010.7.719 [20] Diana M. Thomas, Ashley Ciesla, James A. Levine, John G. Stevens, Corby K. Martin. A mathematical model of weight change with adaptation. Mathematical Biosciences & Engineering, 2009, 6 (4) : 873-887. doi: 10.3934/mbe.2009.6.873

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