文档介绍：Introduction to Partial Differential
Equations and Variational Formulations
in Image Processing
Guillermo Sapiro
Electrical puter Engineering, University of Minnesota,
Minneapolis, MN 55455, USA
e-mail: ******@
1. Introduction
The use of partial differential equations (PDEs) and curvature driven flows in image
analysis has e an interest raising research topic in the past few years. The basic
idea is to deform a given curve, surface, or image with a PDE, and obtain the desired
result as the solution of this PDE. Sometimes, as in the case of color images, a system
of coupled PDEs is used. The art behind this technique is in the design and analysis of
these PDEs.
Partial differential equations can be obtained from variational problems. Assume a
variational approach to an image processing problem formulated as
arg MinI U(u) ,
where U is a given puted over the image (or surface) (Φ) denote
the Euler derivative (first variation) of U. Since under general assumptions, a necessary
condition for I to be a minimizer of U is that F(I) = 0, the (local) minima may be
computed via the steady state solution of the equation
∂I
= F(I),
∂t
Foundations putational Mathematics
Special Volume (F. Cucker, Guest Editor) of
HANDBOOK OF NUMERICAL ANALYSIS, VOL. XI
. Ciarlet (Editor)
© 2003 Elsevier Science . All rights reserved
383
384 G. Sapiro
where t is an ‘artificial’ time marching parameter. PDEs obtained in this way have been
used already for quite some time puter vision and image processing, and the
literature is large. The most classical example is the Dirichlet integral,
U(I) = |∇I|2(x) dx,
which is associated with the linear heat equation
∂I
(t, x) = I (x).
∂t
More recently, extensive research is being done on the direct derivation of evolution
equations which are not necessarily obtained from the energy approaches. Both types
of PDEs are studied in this chapter.
Ideas on the use of PDEs in im