A Study in the BV Space of a Denoising—Deblurring Variational Problem
A Study in the BV Space of a Denoising—Deblurring Variational Problem
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DOI：
10.1007/s00245-001-0017-7
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发表时间：
2001-09
影响因子：
1.8
通讯作者：
L. Vese
中科院分区：
文献类型：
--
作者：
L. Vese
在本文中，我们在有界变分函数的框架内研究了图像恢复中出现的一般变分问题，如[3]中所述。我们使用测度凸函数的下半连续性结果证明了解的存在性和唯一性。我们还使用 Temam 的对偶技术来实现瞬态最小曲面理论，对泛函的次微分以及解的最优条件进行了新的精细表征。我们在非线性半群理论的背景下研究相关的演化方程，并使用 Γ 收敛给出连续变量的近似结果。最后，我们通过有限差分格式离散化问题，并提出了信号和图像重建的几个数值结果。
In this paper we study, in the framework of functions of bounded variation, a general variational problem arising in image recovery, introduced in [3]. We prove the existence and the uniqueness of a solution using lower semicontinuity results for convex functionals of measures. We also give a new and fine characterization of the subdifferential of the functional, together with optimality conditions on the solution, using duality techniques of Temam for the theory of time-dependent minimal surfaces. We study the associated evolution equation in the context of nonlinear semigroup theory and we give an approximation result in continuous variables, usingΓ-convergence. Finally, we discretize the problems by finite differences schemes and we present several numerical results for signal and image reconstruction.