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HERIOT-WATT UNIVERSITY
DEPARTMENT OF COMPUTING AND ELECTRICAL ENGINEERING
B35SD2 Matlab tt!"#al $I%a&' %!('ll#)& *#)& Ma"+!,
Ra)(!% F#'l(*
Ob'.t#,'*/
We will demonstrate how Markov Random Fields (MRF) and fractals
can be used effectivelyin image processing for modelling both for
synthesis and analysis.
During this session you will learn ! practice"#$ %sing MRF
models&$ 'utoinomial Models
R'**!".'* "'0#"'(/
%n order to carry out this session you will need to download
images and matlab files from thefollowing location"
1tt/444.''14a.+6.''7"WWWT'a.1#)&B38SD2B38SD21t%l
lease download the following functions and script"
generate*MRF.m
generate*MRF*+uick.m
generate*MRF*inomial.m
denoise*MRF.m
I%a&' M!('ll#)& /
's you will have seen in the lecture room image modelling plays
an important role in modernimage processing. %t is commonly used in
image analysis for te,ture segmentationclassification and synthesis
(game industry for instance). %t is also of common use inmultimedia
applications for compression purposes. Due to the diversity of
image types it isimpossible to have one universal model to cover
all the possible types even when limitingourselves to simple
te,ture cases. -arious models have been proposed and each has
itsadvantages and drawbacks. For te,tures co$occurrence matrices
are a standard choice butthey are in general difficult to compute
and are difficult to use for modelling.
We will concentrate here on MRF and fractal models which have
been widely and verysuccessfully used in the last # years. Fractals
have been used for compression for instance.
Ma"+!, Ra)(!% 9#'l(* /
Markov random fields belong to the statistical models of
images./ach pi,el of an image can be viewed as a random variable.'n
image can very generally be described as the realisation of a n by
m dimensional randomvariable of (in general unknown) probability
density function (pdf). 0iven the fact that n andm are the
dimensions of the image each component of the vector corresponds to
a pi,el in theimage. 1his is called a random field.
1o generate a sample image from a given random field we want to
sample p(2) where 2 isthe n,m random field. %f we assume that all
the pi,els are uncorrelated we can then simulate
#
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(or sample) p(2) as =
=n
i
ixpXp#
)()( where ,iare the pi,els of the image. 1his will not
however enable us to generate te,ture images as correlation
between pi,els is an essential partof te,ture modelling.1o
introduce correlation in its most general form we have to assume
that all the pi,els in theimage are correlated and we have to
consider all the pi,els at the same time. 1his is impossiblein
practice due to the si3e of the sampling space (&4(n5m) for a
binary image k4(n5m) for a k
grey level image).%f we now assume that a pi,el depend only on
its neighbours we have what is called a MarkovRandom Field. 1hey
are much easier to manipulate as changing one pi,el in the image
onlyaffects its neighbours. 1hey are a very powerful tool for image
analysis and *7)t1'*#*. Formore information on Markov Random Fields
please consult your notes or any relevant bookas the theory is out
of the scope of this tutorial. % strongly recommend the tutorial
from6harles ouman that can be found
at"http"77dynamo.ecn.purdue.edu78bouman7publications7mrf*tutorial7view.pdf%n
rief a random field is a Markov random field if (29,) has a 0ibbs
distribution. 1his
means that (2) can be e,pressed as" )()(
##)( )( xU
xV
e
Z
e
Z
xXP xVsxc
=
== where :(,) is an
energy term depending on the vicinity of ,. 1his can take any
form and two of them are seen isthis tutorial the inomial form seen
in class and the otts model. %n the case of the inomialmodel can be
calculated directly from the neighboorhood to give ;t and the we
candirectly sample ;t. 1his is done in the program
generate*MRF*inomial.m whichcorresponds to the model seen in
class.%n most case however we cannot find < in the e,pression of
(29,) and cannot therefore use directly. %n this case the following
procedure is followed"
6reate a random image of n grey levels.
%teratively change each pi,el and calculate the likelihood
(energy) associated to the
change. 1he choice of the neighbourhood function is the choice
of the user and willmake the specificity of a given random
field.=eep a change if the likelihood is smaller.>ave a
probability of keeping the change if the likelhood is higher. 1his
probabilityensures convergence to the global minimum of energy.
?top after n iterations or when no more changes are
accepted.
@ou can see this as an analogy with physics. %magine you heat a
crystal until it melts andbecomes li+uid (our initial random
state). Aet it now cool down as the li+uid cools down thecrystal
structure comes back (our MRF term) and slowly imposes a structure
onto the initiallyrandom arrangement of molecules (our pi,els).
1ry this now using the generate*MRF and generate*MRF*+uick
programs.'fter having tried with the current parameters edit them
analyse them and try to modify themask (energy function) and see
the effect on the generated image. 's an indication adirectional
mask will produce a directional te,ture.
's you can see changing the MRF terms (parameters) in the mask
changes the imagegenerated. 1he same parameters also generate
images that Blook the sameC. 1his can be used inte,ture synthesis.
>owever you should soon realise that it is not easy to find the
right terms togenerate a specific image type. What is normally done
is the following" from a set of imageswe want to analyse we e,tract
the parameters of the MRF (estimation phase). nce this isdone we
can now generate similar images. We can also Eust send the
parameters of say a
te,ture and regenerate it at the other end from the parameters
realising use compressionefficiency.
&
http://dynamo.ecn.purdue.edu/~bouman/publications/mrf_tutorial/view.pdfhttp://dynamo.ecn.purdue.edu/~bouman/publications/mrf_tutorial/view.pdf
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'nother are of application of MRF is ('-)!#*#)&.%f you know
the parameters of an MRF image (@: D ;1 ;//D 1 =;W 1>/%M'0/
%1?/AF) you can very effectively de$noise it as we will show
now.First create an image using a isotropic MRF model "
mask 9 G#7H #7H #7HI #7H #7H #7HI #7H #7H #7HJ in
generate*MRFIDo ima 9 generate*MRF(GH& H&J&)I
;ow dodenoise*MRF(ima)I1his programs first adds a random noise
to the image and then tries to recover the originalimage from the
noisy version assuming it does not know anything about the original
imageapart from its Markov parameters.@ou should recover the
original image. lease note that the program starts from the
noisyimage and does not know the original one>ow is this
doneKWell we now observe a noise data @ corrupted by noise n as @ 9
2Ln. We know that 2 is aMarkov random field. We want to recover 2.
1herefore we want to find the most probable 2given the observed
data @" (27@). :sing bayes we can rewrite this as (27@)8
(@72)(2).
We know (2) based on our Markovian model and (@72) 9 (@92Ln72) 9
(n9@$2). %f wehave a model for the noise (>ere we have assumed a
0aussian ;oise) we can evaluate (n)and therefore evaluate (27@) for
each possible value of 2 and choose the most probable'gain in
practice a noise model is assumed (say 0aussian) and the MRF
parameters andnoise parameters are iteratively estimated (/M
techni+ue for instance) from the data (our @here) alleviating the
need for the knowledge of the MRF parameters.
H
