Variable exponent functionals in image restoration

• Published in: Applied mathematics and computation Vol. 216; no. 3; pp. 870 - 882
• Main Authors: Li, Fang, Li, Zhibin, Pi, Ling
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.04.2010; Elsevier
• Subjects: BV space; Diffusion; Exact sciences and technology; Exponents; Gaussian; Heat equations; Heat flow; Image restoration; Mathematical analysis; Mathematical models; Mathematics; Numerical analysis; Numerical analysis in abstract spaces; Numerical analysis. Scientific computation; Numerical linear algebra; Optimization; Partial differential equations; Sciences and techniques of general use; Staircasing effect; Variable exponent functional; Staircasing effect; Heat flow; BV space; Variable exponent functional; Smoothing methods; Heat equation; Numerical linear algebra; Denoising; Regularization method; Experimental result; Numerical analysis; Smoothing; Applied mathematics; Ill posed problem; Regularization; Numerical stability
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2010.01.094

We study a functional with variable exponent, 1 < p ( x ) ⩽ 2 , which provides a model for image denoising and restoration. Here p ( x ) is defined by the gradient information in the observed image. The diffusion derived from the proposed model is between total variation based regularization and Gaussian smoothing. The diffusion speed of the corresponding heat equation is tuned by the variable exponent p ( x ) . The minimization problem and its associated flow in a weakened formulation are discussed. The existence, uniqueness, stability and long-time behavior of the proposed model are established in the variable exponent functional space W 1 , p ( x ) . Experimental results illustrate the effectiveness of the model in image restoration.