IEEE TRANSACTIONS ON IMAGE PROCESSING,VOL.10,NO.4,APRIL 2001 493
Fast Image Recovery Using Dynamic Load
Balancing in Parallel Architectures,by Means of
Incomplete Projections
Francisco J.GonzálezCastaño,Ubaldo M.GarcíaPalomares,José L.AlbaCastro,Associate Member,IEEE,
and José M.PousadaCarballo
Abstract This paper formulates an incomplete projection algo
rithmthat is applied to the image recovery problem.The algorithm
allows an easy implementation of dynamic load balancing for par
allel architectures.Furthermore,the local computation  commu
nication load ratio can be adjusted,since each processor performs
a finite number of iterations of any projectiontype technique,and
this number can be provided as a parameter of the algorithm.Nu
merical results compare favorably with those obtained by the ex
trapolated method of parallel subgradient projections.
Index Terms Load balancing,parallel algorithms,recovery,
restoration.
I.I
NTRODUCTION
T
HE image recovery problemis the estimation of an image
that has been degraded (blurred) by a process we have
partial information about [1].In convex settheoretic image re
covery [2] the solution space is a Hilbert space,but since we are
interested in strong convergence we restrict our solution space
to a finite dimensional space.The image is described solely by
a family of convex constraints
arising
from observed data.If we define the convex sets
then the recovery problem reduces to the convex inequality
problem (CIP) of finding
in the closed convex set
.We refer the reader to the bibliography
survey in [1] for related and recent work.Other interesting
references are [3],[4],[9],[10],and [17].
Sequential and parallel projection techniques are robust
methods for solving the CIP,and literature on their properties
and convergence abounds.A recent and comprehensive survey
is [5].Given an estimate
,the key idea is to simultaneously
compute
,the projections of
on closed convex sets
and generate the next estimate
as
(1)
Manuscript received February 22,1999;revised January 12,2001.The as
sociate editor coordinating the review of this manuscript and approving it for
publication was Prof.Timothy J.Schulz.
F.J.GonzálezCastaño,J.L.AlbaCastro,and J.M.PousadaCarballo are
with the Departamento de Tecnologías de las Comunicaciones,E.T.S.I.Teleco
municación,Campus Universitario,36200 Vigo,Spain.
U.M.GarcíaPalomares is with Departamento de Procesos y Sistemas,Uni
versidad Simón Bolívar,Caracas 1080A,Venezuela.
Publisher Item Identifier S 10577149(01)027294.
The notation in (1) will be detailed later.In the sequel,we
commit a minor abuse in notation and denote
as the
sequence generated by an infinite loop of a programming lan
guage.
It is well known that,under suitable conditions,the sequence
,generated by the basic algorithmgivenbelow,converges
to some
[5].
Basic algorithm
Given an initial estimate
,
REPEAT Choose the set
,the relaxation factor
and the
weights
,and let
UNTIL Convergence
END.
Convergence results for the previous algorithm exist for
and
[5].Recent research shows that
convergence is preserved,and in fact improved,for relaxation
parameters
out of the conventional (0,2) interval [1],[9],
[12],[16].Within the basic algorithms framework,Combettes
suggested the extrapolated method of parallel subgradient
projections (EMOPSP),which outperforms previous strategies
like Successive Projection [6],SIRT [7] and Simultaneous
Projection [8].These strategies require exact projections on
,whereas EMOPSP carries out (easy) projections on affine
hyperplanes.This feature makes the performance of EMOPSP
look superior.
EMOPSP can be viewed as an acceleration of the scheme
presented by Bauschke and Borwein [5].It is worth mentioning
that
(2)
and a sufficient convergence condition is
(3)
Suitable conditions to ensure (3) are well known [1],[5].
Since our implementation slightly differs from Combettes,
we describe below one iteration of EMOPSP,for the special
case of equal
weights.Further details and generalization to
a Hilbert space can be found in [1].Given
and
,
10577149/01$10.00 © 2001 IEEE
494 IEEE TRANSACTIONS ON IMAGE PROCESSING,VOL.10,NO.4,APRIL 2001
EMOPSP ITERATION (Generate
)
1.Determine an index set
of violated constraints,i.e.,
.
2.Compute subgradients
,and define
.
3.Compute
and
define
.
4.Define
.
END OF ITERATION.
In a multiprocessor environment,Combettes suggests making
,the number of processors.Note that steps 2 and
3 can be carried out in parallel,whereas the other steps need
communication among processors.Unless the subgradient eval
uation
involves a large computational load,communi
cation costs could surpass the computational costs,which ren
ders a parallel method inefficient.
Themainobjectiveof this paper is toproposeamethodthat im
proves the local computation versus communication load ratio,
and hopefully improve the speedup.The next section formulates
the incomplete projection algorithm (IPA),which replaces the
exact projections
in (1) by points that satisfy
the Fejér condition (FC) stated below.Section III explains how
to apply IPA in order to 1) produce a complete load balance
among processors and 2) increase the processors computational
load.Numerical results inSection IVshowthat IPAis better than
EMOPSP as a fast initial approximation to the solution.
II.I
NCOMPLETE
P
ROJECTION
Inequality (2),usually coined as the Fejér property,holds for
all projectiontype methods,and is endowed with nice conver
gence properties [5].Given
we say that
in (1) sat
isfies the Fejér condition (FC) if
We next prove a lemma that will enable us to develop the
incomplete projection algorithm (IPA).Its main feature is that
points satisfying (FC) substitute exact projections.This is the
key to implement algorithms with a better computationcom
munication cost ratio.
Lemma 1:If
satisfies (FC),then
Proof:If there exists
such that
we obtain
a contradiction
Theorem 1:Let
be a nonfeasible point (
),
let
be points in
satisfying (FC),let
,and let
.
Then
Proof:By the previous lemma we have
,we can generate,either sequentially or in parallel,points
by incomplete projection,which is namely,
a finite number of projection steps (or any other technique
that ensures that (FC) is fulfilled).Then we form a linear
nonnegative combination of the directions
to
generate the next estimate
,which also satisfies the Fejér
condition (FC).
Note that
(4)
If we want to obtain optimal
values,we may solve either
the following quadratic programming problem:
Maximize
(5)
or the following fractional programming problem:
Maximize
(6)
which can be solved by a sequence of quadratic programming
problems.Problem (6) was also suggested by Kiwiel [12].We
do not proceed with this matter any further,because an effi
cient implementation of a quadratic programming algorithm is
required.Here we report results with
.
Let us mention that if exact projections are used,i.e.,
,very close results are obtained [16].In particular,
GONZÁLEZCASTAÑO et al.:FAST IMAGE RECOVERY USING DYNAMIC LOAD BALANCING 495
(7)
III.D
YNAMIC
L
OAD
B
ALANCING IN
P
ARALLEL
A
RCHITECTURES BY
M
EANS OF
I
NCOMPLETE
P
ROJECTION
Parallel architectures and,in particular,clusters of work
stations [13],[14] have been widely used in recent years.In
distributed memory parallel architectures,the scalability of a
method is especially sensitive to relative communication load.
We have stated that EMOPSP presents a large communication
load as compared to local computation load in a parallel
environment.This is mainly because each processor solely
completes one easy projection at step 3 of the EMOPSP
iteration.We intend to replace that step by a finite number
of iterations of a projectiontype method,which,as we have
pointed out,generates a
that satisfies (FC).This keeps the
desirable effect of having the processors carry out homoge
neous tasks.In order to do this effectively we need to provide a
dynamic load balance to the processors.If this is possible,the
effective execution time per task will be the same.
In general,the load balancing problemis a very difficult one.
We need to redistribute the data at the beginning of a parallel
iteration.Moreover,fast dynamic load balancing requires the
data to be easily available to all processors.In practice,in a
distributed memory architecture,this supposes that all the data
shouldbeknownandreadytobeusedbyall processors.This may
not be possible inlarge problems duetomemoryrestrictions,and
a static distribution must be imposed [15].If limited memory is
not an issue,we can use the easy dynamic load balance given in
step 1 of the dynamic IPAalgorithmbelow( DIPA).We believe
that the joint effect of the dynamic load balance and efficient
local projection methods,like those proposed in [1],[11],[18],
makes our algorithm amenable for solving the general convex
feasibility problem.In short,we do the following.
We impose some control over the local computation,
thus improving the computation versus communication
ratio.We generate
in a finite number of
iterations of a projection method.
We dedicate some communication load to the search of
a good balance load among processors,providing the
dynamic load balancing we are looking for.
We have implemented the incomplete projection algorithm
with dynamic load distribution (DIPA) given below,with ad
ditional peculiarities that will be explained in the next section.
Remarks that follow the description of the algorithm attempt
to further clarify its implementation.We do not claim that this
is the best algorithm,because it is certainly problemdepen
dent. We are assuming that we have
processors and
con
straints.
Incomplete Projection Algorithm with Dynamic Load
Distribution (DIPA).
Given an estimate
of the original image and a tolerance
,
REPEAT
1) Dynamic load:
(a) Determine an index set of violated constraints:
.
(b) Distribute the set
evenly among processors:Define
such that
,and
.
2) EMOPSP iteration:
(a)EachprocessorperformsoneEMOPSPiterationwith
.
(b) Let
.
3) Combination:
(a) Define
.
UNTIL
Remark:Optimal
values were not considered.
Remark 1a:To ensure convergence (3) must hold,that is
Remark 1b:
is desirable.All proces
sors would theoretically employ the same execution time if
they compute alike projections at step 2.Since we do not im
pose
[1],[5],[12],and the usual conver
gence test is
.In some cases,it is
advisable to state a less stringent condition,namely,
Max
,where
is the
direction generated by processor
at the
of the squared distance of
to the set
,namely,
,is known,the algorithm can
stop as soon as [12]
496 IEEE TRANSACTIONS ON IMAGE PROCESSING,VOL.10,NO.4,APRIL 2001
Fig.1.Original image.
This stopping condition is indeed a warning of infeasibility and
arises from (7) with
at all iterations.
IV.N
UMERICAL
R
ESULTS
This section applies IPA with dynamic load balancing to
a standard digital image restoration problem.We performed
numerical tests and compared them with EMOPSP,as imple
mented in [1].All images are 8bit grayscale,and have
pixels,with
.Problemlayout is similar to [1].
Original image
(Fig.1) is degraded by convo
lutional blur with an uniform
kernel.Edge effects
are taken into account.
Uniform white noise
,and
,for
,where
is the
th component of the degraded image,and
is the
row of the matrix
.We recall that
,
where
.Note that
,
but the projection of a point on
is a difficult task.On the
contrary,projections on
are straightforward [1].We
highlight
,the projection on
,because we will especially
consider it in the algorithm
(8)
where
.We used
will take significantly longer processing time
than
.Hence,we decided to avoid the computation
of
at each iteration and implemented the dynamic load bal
ancing among
.The proposed algo
rithm is as follows.
GONZÁLEZCASTAÑO et al.:FAST IMAGE RECOVERY USING DYNAMIC LOAD BALANCING 497
Adapted DIPA
Given the iteration counter
of the
original image,
REPEAT Set the iteration counter
2)
3) Generate
:
(a) Set
(b) Parallel stage.Dynamic load:
Determine the index set of violated constraints:
4) Combination:
(a) Define
.
(b) Let
.
UNTIL
or execution
time limit is reached.
END of Adapted DIPA
In order to verify the effectiveness of the dynamic load dis
tribution,we also implemented a static load.At the onset,we
evenly split the sets
among all pro
cessors,that is,step 3b is omitted.We must mention that in all
cases both the static and the dynamic algorithms reached the ex
ecution time limit.
To compare results,we defined the comparison functions
and
(9)
,
and
498 IEEE TRANSACTIONS ON IMAGE PROCESSING,VOL.10,NO.4,APRIL 2001
Fig.5.Dynamic IPA after 4 min,eight processors.
V.C
ONCLUSIONS
We have formulated the (Dynamic) Incomplete Projection
Algorithm,which theoretically does not require exact projec
tions on convex supersets.It allows a dynamic load distribu
tion in parallel computation and an efficient local computation
to communication ratio.DIPA can exploit the same EMOPSP
strategies and it is extremely flexible as well.Numerical results
showthat DIPAcan obtain a fast approximation to a discretized
image.Furthermore,it admits roomfor improvements by means
of strategies like set selection,weight optimization,or appro
priate choice of projectiontype methods.DIPA can be indeed
considered as an enhancement of EMOPSP,or any other projec
tion type technique,in parallel environments.
A
CKNOWLEDGMENT
The authors thank Prof.R.R.Rubio for helping them with
the programming interface to the ORIGIN 2000.They also ac
knowledge the careful reading by anonymous referees,who sug
gested meaningful improvements to this paper.
R
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[13] T.M.Warschko,J.M.Blum,and W.F.Tichy,The parastation project:
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[14] D.Ridge,D.Becker,and P.Merkey,Beowulf:Harnessing the power
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[15] F.J.GonzálezCastaño,Contribución al encaminamiento óptimo en
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memoria distribuida, Ph.D.dissertation,Universidad de Vigo,Spain,
1998.
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Eds.Norwell,MA:Kluwer.
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Francisco J.GonzálezCastaño received the M.Sc.
and Ph.D.degrees (cum laude) in telecommunica
tions engineering from the Universidad de Santiago,
Spain,in 1990 and the Universidad de Vigo,Spain,
in 1998,respectively.
He has written ten papers in international refereed
journals in diverse fields of computer science
and telecommunications and has participated in
several R&D projects.He is currently a Profesor
Titular with the Departamento de Tecnologías de
las Comunicaciones,Universidad de Vigo,and has
been a Visiting Assistant Professor with the Computer Science Department,
University of Wisconsin,Madison.He has one patent.
Ubaldo M.GarcíaPalomares was born in Caracas,
Venezuela.He received the B.S.degree in electrical
engineering in 1966 from Universidad Central de
Venezuela,Caracas,and the Ph.D.degree from
University of Wisconsin,Madison,in 1973.
He retired as Full Professor from the Universidad
Simón Bolívar,Venezuala,in 1993,but is still
engaged in research and teaching at that university
and at Universidad de Vigo,Spain.His main
interests include the theory and applications of
convex feasibility and optimization problems.He has
written over 40 reviewed papers on several nonlinear programming subjects.
GONZÁLEZCASTAÑO et al.:FAST IMAGE RECOVERY USING DYNAMIC LOAD BALANCING 499
José L.AlbaCastro (A96) received the M.Sc.and
Ph.D.degrees (cumlaude) intelecommunications en
gineering from the Universidad de Santiago,Spain,
in 1990 and the Universidad de Vigo,Spain,in 1997,
respectively.
His research interests include neural networks
for classification applications,image segmentation,
statistical pattern recognition,automatic speech,
and speaker recognition and biometrics.He is a
Professor of discrete signal processing and image
processing at the Universidad de Vigo.
José M.PousadaCarballo received the M.Sc.and
Ph.D.degrees (cum laude) in telecommunications
engineering from the Universidad Politécnica de
Madrid,Spain,in 1991 and the Universidad de Vigo,
Spain,in 1999,respectively.
He has written eight papers in international ref
ereed journals in diverse fields of computer science
and telecommunications,and has participated in sev
eral R&D projects.He is currently a Profesor Titular
with the Departamento de Tecnologías de las Comu
nicaciones,Universidad de Vigo.He has one patent.
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