September  2007, 6(3): 899-915. doi: 10.3934/cpaa.2007.6.899

Edge detection by using rotational wavelets

1. 

Faculty of Science and Technology, University of Macau, Av. Padre Tomas Pereira, Taipa, Macau, Macau

2. 

Department of Image Engineering, Chinese Academy of Sciences, Beijing 100101, China

Received  February 2006 Revised  March 2007 Published  June 2007

Based on a mathematical model involving Radon measure explicit computations on convolution integrals defining continuous (integral) wavelet transformations are carried out. The study shows that the truncated Morlet wavelet significantly depends on a rotation parameter and thus lay a foundation of edge detection in pattern recognition and image processing using rotational (directional) wavelets. Experiments and algorithms are developed based on the theory. The theory is further generalized to the $n$-dimensional cases and to a large class of rotational wavelets.
Citation: Liming Zhang, Tao Qian, Qingye Zeng. Edge detection by using rotational wavelets. Communications on Pure and Applied Analysis, 2007, 6 (3) : 899-915. doi: 10.3934/cpaa.2007.6.899
[1]

Monika Muszkieta. A variational approach to edge detection. Inverse Problems and Imaging, 2016, 10 (2) : 499-517. doi: 10.3934/ipi.2016009

[2]

Yuying Shi, Ying Gu, Li-Lian Wang, Xue-Cheng Tai. A fast edge detection algorithm using binary labels. Inverse Problems and Imaging, 2015, 9 (2) : 551-578. doi: 10.3934/ipi.2015.9.551

[3]

Yuying Shi, Zijin Liu, Xiaoying Wang, Jinping Zhang. Edge detection with mixed noise based on maximum a posteriori approach. Inverse Problems and Imaging, 2021, 15 (5) : 1223-1245. doi: 10.3934/ipi.2021035

[4]

Maria Michaela Porzio, Flavia Smarrazzo, Alberto Tesei. Radon measure-valued solutions of unsteady filtration equations. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022040

[5]

Audric Drogoul, Gilles Aubert. The topological gradient method for semi-linear problems and application to edge detection and noise removal. Inverse Problems and Imaging, 2016, 10 (1) : 51-86. doi: 10.3934/ipi.2016.10.51

[6]

Kazuhiro Ishige. On the existence of solutions of the Cauchy problem for porous medium equations with radon measure as initial data. Discrete and Continuous Dynamical Systems, 1995, 1 (4) : 521-546. doi: 10.3934/dcds.1995.1.521

[7]

Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina, Alberto Tesei. Signed Radon measure-valued solutions of flux saturated scalar conservation laws. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3143-3169. doi: 10.3934/dcds.2020041

[8]

Yunjuan Jin, Aifang Qu, Hairong Yuan. Radon measure solutions for steady compressible hypersonic-limit Euler flows passing cylindrically symmetric conical bodies. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2665-2685. doi: 10.3934/cpaa.2021048

[9]

Kanghui Guo, Demetrio Labate, Wang-Q Lim, Guido Weiss and Edward Wilson. Wavelets with composite dilations. Electronic Research Announcements, 2004, 10: 78-87.

[10]

Hildebrando M. Rodrigues, Tomás Caraballo, Marcio Gameiro. Dynamics of a Class of ODEs via Wavelets. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2337-2355. doi: 10.3934/cpaa.2017115

[11]

Stefano Barbero, Emanuele Bellini, Rusydi H. Makarim. Rotational analysis of ChaCha permutation. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021057

[12]

Michael Krause, Jan Marcel Hausherr, Walter Krenkel. Computing the fibre orientation from Radon data using local Radon transform. Inverse Problems and Imaging, 2011, 5 (4) : 879-891. doi: 10.3934/ipi.2011.5.879

[13]

Simon Gindikin. A remark on the weighted Radon transform on the plane. Inverse Problems and Imaging, 2010, 4 (4) : 649-653. doi: 10.3934/ipi.2010.4.649

[14]

Alberto Ibort, Alberto López-Yela. Quantum tomography and the quantum Radon transform. Inverse Problems and Imaging, 2021, 15 (5) : 893-928. doi: 10.3934/ipi.2021021

[15]

Qiao-Fang Lian, Yun-Zhang Li. Reducing subspace frame multiresolution analysis and frame wavelets. Communications on Pure and Applied Analysis, 2007, 6 (3) : 741-756. doi: 10.3934/cpaa.2007.6.741

[16]

P. Cerejeiras, M. Ferreira, U. Kähler, F. Sommen. Continuous wavelet transform and wavelet frames on the sphere using Clifford analysis. Communications on Pure and Applied Analysis, 2007, 6 (3) : 619-641. doi: 10.3934/cpaa.2007.6.619

[17]

Xiaoqun Zhang, Tony F. Chan. Wavelet inpainting by nonlocal total variation. Inverse Problems and Imaging, 2010, 4 (1) : 191-210. doi: 10.3934/ipi.2010.4.191

[18]

Michael Dellnitz, O. Junge, B Thiere. The numerical detection of connecting orbits. Discrete and Continuous Dynamical Systems - B, 2001, 1 (1) : 125-135. doi: 10.3934/dcdsb.2001.1.125

[19]

Jan Boman. A local uniqueness theorem for weighted Radon transforms. Inverse Problems and Imaging, 2010, 4 (4) : 631-637. doi: 10.3934/ipi.2010.4.631

[20]

Giuseppe Alì, John K. Hunter. Orientation waves in a director field with rotational inertia. Kinetic and Related Models, 2009, 2 (1) : 1-37. doi: 10.3934/krm.2009.2.1

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (171)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]