Abstract
The total variation model proposed by Rudin, Osher and Fatemi for image denoising is considered to be one of the best denoising models. In this paper, we propose, analyze and test a dual based method to solve the total variation model. This method minimizes a quadratic function with separable constraints, and we make projections onto a feasible set so that it is easy to compute. Under some mild conditions, global convergence of the proposed method is established. The proposed approach could be easily implemented. Preliminary results are reported to demonstrate its performance.
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Notes
- 1.
Suppose that \(0<\tau <1/8\), then \(\frac{1}{\mu }\nabla \cdot w_k\) converges to \(\pi _{\frac{1}{\mu }\mathcal {K}}(-f)\) as \(k\rightarrow \infty \), where \(\pi _{\frac{1}{\mu }\mathcal {K}}(\cdot )\) is the orthogonal projection onto a convex set \(\frac{1}{\mu }\mathcal {K}\) with \(\mathcal {K}:=\{\nabla \cdot w: |w_{i,j} \le 1|, \forall i, j=1,2,\cdots ,n\}\).
- 2.
Some experiments show that a better convergence can be obtained when \(\tau =0.248\).
- 3.
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Liao, Y., Hua, J., Xue, W. (2016). A Dual-Based Adaptive Gradient Method for TV Image Denoising. In: Tan, T., Li, X., Chen, X., Zhou, J., Yang, J., Cheng, H. (eds) Pattern Recognition. CCPR 2016. Communications in Computer and Information Science, vol 663. Springer, Singapore. https://doi.org/10.1007/978-981-10-3005-5_20
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