TY - JOUR
T1 - Finite Element Approximation of the Gradient Flow for a Class of Linear Growth Energies with Applications to Color Image Denoising
AU - X. Feng & M. Yoon
JO - International Journal of Numerical Analysis and Modeling
VL - 3
SP - 389
EP - 401
PY - 2009
DA - 2009/06
SN - 6
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/774.html
KW - Linear growth energy functionals, gradient flow, $p$-harmonic maps, BV functions, color image denoising, finite element methods.
AB - <p style="text-align: justify;">This paper concerns with the finite element approximation of a
nonlinear second order parabolic system which describes the $L^2$-gradient flow
for a class of linear growth energy functionals. Besides their appeals in differential
geometry and calculus of variations, linear growth energy functionals and
their gradient flows also arise naturally from emerging applications of image
processing such as color image denoising. In this paper, we introduce a family
of variational models for color image denoising which minimize linear growth
energy functionals of maps into the unit sphere in $\rm{R}^3$. These models generalize
the popular 1-harmonic map model which has been studied intensively
in recent years. To compute the solutions of the variational models, we first
derive their $L^2$-gradient flow equations and then introduce some fully discrete
implicit finite element method for the gradient flow equations. It is proved
that the proposed finite element method is uniquely solvable and absolutely
stable, and the finite element solution converges to the PDE solution as the
mesh sizes tend to zero. Numerical experiments are presented to demonstrate
the effectiveness of the proposed variational models for color image denoising
and to show the efficiency of the proposed finite element method. A numerical
comparison of the proposed models with the channel-by-channel model is also
presented.</p>