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NETWORK FLOW OPTIMIZATION
FOR RESTORATION OF IMAGES
BORIS A. ZALESKY
Received 5 October 2001
work flow optimization approach is offered for restoration of
gray-scale and color images corrupted by noise. The Ising models are
used as a statistical background of the proposed method. We present the
new work flow minimum cut algorithm, which is es-
pecially efficient in identification of the maximum a posteriori (MAP)
estimates of corrupted images. The algorithm is able pute the
MAP estimates of large-size images and can be used in a concurrent
mode. We also consider
the problem
of integer minimization
of two
U ( )=λ|y − x | + β|x − x | U ( )= λ(y −
functions,
1 x i i i i,j i,j i j and 2 x i i i
2 2
xi) + i,j βi,j (xi − xj ) , with parameters λ,λi,βi,j > 0 and vectors x =
n
(x1,...,xn), y=(y1,...,yn)∈{0,...,L− 1} . Those functions constitute the
energy ones for the Ising model of color and gray-scale images. In the
case L = 2, they coincide, determining the energy function of the Ising
model of binary images, and their minimization es equivalent to
work flow minimum cut problem. The efficient integer minimiza-
tion of U1(x), U2(x) by work flow algorithms is described.
1. Introduction
We present a new multiresolution algorithm for finding the minimum
ffi
network flow cut and
methods
of e cient integer minimization
of the
U ( )=λ|y − x |+ β|x − x | U ( )= λ(y − x )2 +
function 1 x i i i i,j i,j i j and 2 x i i i i
2
i,j βi,j (xi − xj ) . The problem is posed in terms of Bayesian approach to
image restoration to have unified canvas of presentation of the results,
and since the results developed were tested and turned out efficient for
the processing of corrupted images. Nevertheless, the minimization of
Copyright c 2002 Hindawi Publishing Corporation
Journal of Applied Mathematics 2:4 (2002) 199–218
2000 Mathematics Subject Classification: 62M40, 90C10, 90C35, 68U10
URL: http://dx./
work flow optimization for resto