American Institute of Mathematical Sciences

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

Edge detection by using rotational wavelets

Liming Zhang 1, , Tao Qian 1, and  Qingye Zeng 2,

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.
Keywords: edge detection, Radon measure, Morlet wavelet, Rotational wavelets.
Mathematics Subject Classification: 37C45.
Citation: Liming Zhang, Tao Qian, Qingye Zeng. Edge detection by using rotational wavelets. Communications on Pure & 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 & 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 & Imaging, 2015, 9 (2) : 551-578. doi: 10.3934/ipi.2015.9.551
[3]

Audric Drogoul, Gilles Aubert. The topological gradient method for semi-linear problems and application to edge detection and noise removal. Inverse Problems & Imaging, 2016, 10 (1) : 51-86. doi: 10.3934/ipi.2016.10.51
[4]

Kazuhiro Ishige. On the existence of solutions of the Cauchy problem for porous medium equations with radon measure as initial data. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 521-546. doi: 10.3934/dcds.1995.1.521
[5]

Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina, Alberto Tesei. Signed Radon measure-valued solutions of flux saturated scalar conservation laws. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020041
[6]

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

[7]

Hildebrando M. Rodrigues, Tomás Caraballo, Marcio Gameiro. Dynamics of a Class of ODEs via Wavelets. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2337-2355. doi: 10.3934/cpaa.2017115
[8]

Michael Krause, Jan Marcel Hausherr, Walter Krenkel. Computing the fibre orientation from Radon data using local Radon transform. Inverse Problems & Imaging, 2011, 5 (4) : 879-891. doi: 10.3934/ipi.2011.5.879
[9]

Simon Gindikin. A remark on the weighted Radon transform on the plane. Inverse Problems & Imaging, 2010, 4 (4) : 649-653. doi: 10.3934/ipi.2010.4.649
[10]

Qiao-Fang Lian, Yun-Zhang Li. Reducing subspace frame multiresolution analysis and frame wavelets. Communications on Pure & Applied Analysis, 2007, 6 (3) : 741-756. doi: 10.3934/cpaa.2007.6.741
[11]

Michael Dellnitz, O. Junge, B Thiere. The numerical detection of connecting orbits. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 125-135. doi: 10.3934/dcdsb.2001.1.125
[12]

P. Cerejeiras, M. Ferreira, U. Kähler, F. Sommen. Continuous wavelet transform and wavelet frames on the sphere using Clifford analysis. Communications on Pure & Applied Analysis, 2007, 6 (3) : 619-641. doi: 10.3934/cpaa.2007.6.619
[13]

Xiaoqun Zhang, Tony F. Chan. Wavelet inpainting by nonlocal total variation. Inverse Problems & Imaging, 2010, 4 (1) : 191-210. doi: 10.3934/ipi.2010.4.191
[14]

Jan Boman. A local uniqueness theorem for weighted Radon transforms. Inverse Problems & Imaging, 2010, 4 (4) : 631-637. doi: 10.3934/ipi.2010.4.631
[15]

A. Calogero. Wavelets on general lattices, associated with general expanding maps of $\mathbf R^n$. Electronic Research Announcements, 1999, 5: 1-10.

[16]

Sunghwan Moon. Inversion of the spherical Radon transform on spheres through the origin using the regular Radon transform. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1029-1039. doi: 10.3934/cpaa.2016.15.1029
[17]

Giuseppe Alì, John K. Hunter. Orientation waves in a director field with rotational inertia. Kinetic & Related Models, 2009, 2 (1) : 1-37. doi: 10.3934/krm.2009.2.1
[18]

Mayee Chen, Junping Shi. Effect of rotational grazing on plant and animal production. Mathematical Biosciences & Engineering, 2018, 15 (2) : 393-406. doi: 10.3934/mbe.2018017
[19]

Elena Beretta, Markus Grasmair, Monika Muszkieta, Otmar Scherzer. A variational algorithm for the detection of line segments. Inverse Problems & Imaging, 2014, 8 (2) : 389-408. doi: 10.3934/ipi.2014.8.389
[20]

Heinz-Jürgen Flad, Gohar Harutyunyan. Ellipticity of quantum mechanical Hamiltonians in the edge algebra. Conference Publications, 2011, 2011 (Special) : 420-429. doi: 10.3934/proc.2011.2011.420

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences