Eksik Veri Temininde Shearlet Dönüşümünün Asimptotik Analizi

Year 2018, Volume: 22 Issue: 6, 1544 - 1551, 01.12.2018

Hülya Kodal Sevindir , Cüneyt Yazıcı , Süleyman Çetinkaya

https://doi.org/10.16984/saufenbilder.377081

Abstract

Asimptotik analiz bir algoritmanın performansı hakkında bize
yol gösterir. Matematiksel ispatlarla yapılmış bir asimptotik analiz,
algoritmaların performansını anlamlandırmada oldukça güçlü bir yöntemdir.
Shearlet dönüşümü ise dalgacık dönüşümünün etkili bir geliştirmesi olarak son
yıllarda ortaya çıkmış matematiksel bir yöntemdir. Bu amaçla bu makalede,
görüntülerdeki olası kayıp verinin yeniden temini açısından shearlet dönüşümü,
dalgacık dönüşümü ile asimptotik olarak kıyaslanmıştır. Görüntü işleme
uygulamaları içerisinde, görüntüler üzerinde eksik (kayıp) olan verinin yeniden
elde edilmesi amacıyla, eksik verinin şeklen yatay konumlanmış dikdörtgen
şeklinde olması durumunda Shearlet dönüşümü için asimptotik analizi
yapılmıştır. 

Keywords

Shearlet Dönüşümü, Dalgacık Dönüşümü, Eksik Verinin Elde Edilmesi, Asimptotik Analiz

Asymptotic Analysis of Shearlet Transfom for Inpainting

Abstract

Supply of missing data, also known as inpainting, is an important application of image processing.
Wavelets are commonly used for inpainting algorithms. Shearlet transform which is an affine
transformation is the improvement of the wavelet transform. An asymptotic analysis may help to evaluate
the performance of an algorithm. In this article we compare the asymptotical analysis for wavelet and
shearlet transforms in the case of inpainting where the missing data is shaped like a rectangle.  

Keywords

Shearlet Transform, Wavelet Transform, Inpainting, Asymptotic Analysis

References

• [1] R. H., Bamberger and M. J. T., Smith, “A Filter Bank for the Directional Decomposition of Images: Theory and Design”, IEEE Trans. Signal Process., 1992, 40, 882–893.
• [2] J. P., Antoine, P., Carrette, R., Murenzi and B.,Piette, “Image Analysis with Two-dimensional Continuous Wavelet Transform”, Signal Process., 1993, 31, 241–272.
• [3] E. J., Cand`es and D. L., Donoho, “New Tight Frames of Curvelets and Optimal Representations of Objects with Piecewise C2 Singularities”, Comm. Pure Appl. Math., 2002, 57, 219–266.
• [4] M. N., Do and M., Vetterli, “The Contourlet Transform: an Efficient Directional Multiresolution Image Representation”, IEEE Trans. Image Process., 2005, 14, 2091–2106.
• [5] K., Guo, G., Kutyniok and D., Labate, “Sparse Multidimensional Representations Using Anisotropic Dilation and Shear Operators, Editors: G. Chen and M. J., Lai ” Wavelets and Splines: Athens 2005, Nashboro Press, Nashville, 189–201, 2006.
• [6] D., Labate, W. Q., Lim, G., Kutyniok, and G., Weiss, “Sparse Multidimensional Representation Using Shearlets”, in Wavelets XI, Edited by M. Papadakis, A. F. Laine, and M. A. Unser, SPIE Proc., 2005, 5914, 254–262.
• [7] K., Guo, D., Labate, W. Q., Lim, G., Weiss and E., Wilson, “Wavelets with Composite Dilations”, Electron. Res. Announc. Amer. Math. Soc., 2004, 10, 78–87.
• [8] K., Guo, D., Labate, W. Q., Lim, G., Weiss and E., Wilson, “The Theory of Wavelets with Composite Dilations, Editors: C. Heil”, Harmonic Analysis and Applications, Birkhauser, Boston, 231–250, 2006.
• [9] K., Guo, W. Q., Lim, D., Labate, G., Weiss and E., Wilson, “Wavelets with Composite Dilations and Their MRA Properties”, Appl. Comput. Harmon. Anal., 2006, 20, 220–236.
• [10] G., Kutyniok and D., Ch., Labate, “Introduction to Shearlets”, Shearlets: Multiscale Analysis for Multivariate Data, Birkhäuser, Boston, 1–38, 2012.
• [11] www.shearlab.org
• [12] G. R., Easley, D., Labate and F., Colonna, “Shearlet-Based Total Variation for Denoising”, IEEE Trans. Image Processing, 2009, 18(2), 260–268.
• [13] Q., Guo, S., Yu, X., Chen, C., Liu and W., Wei, “Shearlet-based Image Denoising Using Bivariate Shrinkage with Intra-band and Opposite Orientation Dependencies”, IEEE Conference Publications, 2009, 1, 863–866.
• [14] G. R., Easley and D., Labate, “Image Processing Using Shearlets”, Editors: G., Kutyniok and D., Labate Shearlets: Multiscale Analysis for Multivariate Data, Birkhäuser, Boston, 283–325, 2012.
• [15] S., Häuser and J., Ma, “Seismic Data Reconstruction via Shearlet-Regularized Directional Inpainting”, http://www.mathematik.uni-kl.de/uploads/tx_sibibtex/seismic.pdf
• [16] E. J., King, G., Kutyniok and X., Zhuang, “Analysis of Data Separation and Recovery Problems Using Clustered Sparsity”, SPIE Proceedings: Wavelets and Sparsity XIV, 2011, 8138, 1-11.
• [17] E. J., King, G., Kutyniok and Zhuang X., “Analysis of Inpainting via Clustered Sparsity and Microlocal Analysis”, J. Math. Imaging Vis., 2014, 48, 205–234.
• [18] E. J., King, G., Kutyniok and W.Q, Lim, “Image Inpainting: Theoretical Analysis and Comparison of Algorithms”, SPIE Proceedings, 2013, 8858, 1-11.
• [19] C., Yazıcı, “Shearlet Teorisi ve Medikal Verilere Uygulaması”, Kocaeli University, Unpublished Ph.D. Thesis for Mathematics Degree, 2015.
• [20] I., Daubechies, “Ten lectures on Wavelets”, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1992.
• [21] W. Q., Lim, “The Discrete Shearlet Transform: A New Directional Transform and Compactly Supported Shearlet Frames”, Image Proc. IEEE Transactions on, 2010, 19(5), 1166–1180.
• [22] K.,Guo and D., Labate, “Optimally Sparse Multidimensional Representation Using Shearlets”, SIAM J. Math. Anal., 2007, 39, 298–318.
• [23] G., Kutyniok and W. Q., Lim, “Image Separation Using Wavelets and Shearlets”, Curves and Surfaces, 2012, 6920, 416-430.
• [24] G., Kutyniok and D., Ch., Labate, “Introduction to Shearlets”, Shearlets: Multiscale Analysis for Multivariate Data, Birkhäuser, Boston, 1–38, 2012.