A novel variational model for image decomposition

  • Jianlou XuEmail author
  • Yan Hao
  • Min Li
  • Xiaobo Zhang
Original Paper


Image decomposition denotes a process by which an image is decomposed into several different scales, such as cartoon, texture (or noise) and edge. In order to better separate the noise and preserve the edges, one coupled variational model for image decomposition is proposed in this paper. In this coupled model, an introduced vector field and the gradient of image are intertwined and the orders of this model can be adjusted by the given parameters. To prevent image from being too smooth and edges from being damaged, one weighted function containing a Gaussian convolution is proposed. Meanwhile, considering the equivalence between the solution of the heat diffusion equation and the Gaussian convolution, we turn the convolution computation into a variational model for the introduced vector field. Different from the existing methods, the proposed model firstly contains first- and second-order regularization terms which can remove the noise better; secondly, the solution for the introduced vector field is just given Gaussian convolution. To solve the variational system, the alternating direction method, primal–dual method and Gauss–Seidel iteration are adopted. In addition, the proximal point method is designed for solving the primal variable and dual variable. Extensive numerical experiments verify that the new method can obtain better results than those by some recent methods.


Image decomposition Image restoration Total variation Texture Cartoon 



This work is supported by the National Science Foundation of China (Nos. U1504603, 61301229, 61401383) and Key Scientific Research Project of Colleges and Universities in Henan Province (Nos. 18A120002, 19A110014).


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Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsHenan University of Science and TechnologyLuoyangChina
  2. 2.College of Mathematics and StatisticsShenzhen UniversityShenzhenChina
  3. 3.Institute of Graphics and Image ProcessingXianyang Normal UniversityXianyangChina

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