Post date: Dec 12, 2010 5:20:59 AM
By the pioneering work of Nikolova, Esedoglu and Chan, TV-L1 based image approximation leads to the exact and global optimal image u(x) in {0,1} when the input image f(x) is binary, e.g. f(x) in {0,1}. In this work, we extend the Nikolova et al's elegant result both in theory and numerics. We show that when the input image f(x) is integer valued, i.e. f(x) in {l1, ..., ln} which just properly models the most common case of gray-scale images, TV-L1 image approximation also leads to the global and exact optimum u*(x) in {l1, ..., ln}. This means one can always find the exactly and globally optimal image u(x) which takes values in the same value set.
To achieve this nice result, we fist propose the equivalent dual model of TV-L1. With helps of the coarea theorem of the total-variation function, the result can be derived. In addition, the dual model of TV-L1 can naturally lead to a fast numerical algorithm based on the standard multiplier method. Potential applications can be found in image denoising and image segmentation.
Related works:
Jing Yuan, Juan Shi and Xue-Cheng Tai, A Fast and Exact TV-L1 Approach to Discrete Constrained Image Approximation (To appear soon)
Jing Yuan, Juan Shi and Xue-Cheng Tai, A Convex and Exact Approach to Discrete Constrained TV-L1 Image Approximation (UCLA CAM 10-51)
Jing Yuan, Juan Shi and Xue-Cheng Tai, A Convex and Exact Approach to Discrete Constrained TV-L1 Image Approximation (To appear in East Asian Journal on Applied Mathematics (EAJAM) )Â