Friday, July 26, 2013
Abstract:
The phenomenon of erosion of peaks
in an uneven terrain by natural factors like wind etc., has been used to evolve
a technique for characterization of images especially cloud IR images. A
mathematical technique has been used to analyze the phenomenon of percolation
the pith of the technique is projecting an image inside a parallelepiped in a
3D format the position of a pixel point in the image is given by two horizontal
coordinates where as third one is the Gray value. From that 3D format we start
slicing the image into one unit thick slabs from the top of the body and for
each slab keep a count of sites where mass of gray value column exists. As we
go down a critical point will come where the two opposite faces of the image
become connected with masses. We argue that the study at this stage is
important since beyond this critical point the image no longer remains
independent but gets connected with some extended image vis-à-vis some external
causes. The processes of calculating fractal and chemical dimensions are used
to draw out implications on the cloud thermodynamics.
INTRODUCTION:
One of the recent methods for
characterization of atmospheric images is to estimate the fractal and multi
fractal dimensions of the surface created by the gray values of an image. We
take up the study of IR images of cloud to show how the concept of percolation
in conjunction with fractal dimension may be used to characterize thermodynamic
properties of cloud.
We build a 3-D dimensional object
from the data set representing a monochrome image by putting a solid column at
every pixel point, the height of the column being equal to the grey value of
the image at that set pixel location. Viewed from the top of the solid object
would look like a relief map of a rugged terrain with irregular ups and downs.
Each point on the surface is represented by a triplet, the third element of the
triplet being the grey value generally called as a pixel value and the first
two being the positions of the horizontal plane. A point is situated at a point
higher than another one if the pixel value of the former is greater than the
latter.
It must be noted that in the
given surface there is no scope for tunnel or adit, meaning that P(a,b,c) and
Q(a,b,c’) can not be two different points and always c=c’.
BASIC FORMULATIONS:
The parallelepiped we constructed
possesses numbers of mini cube sites. But not all sites are filled with the
object or have mass. There after the object is cut horizontally to slices of
equal thickness. This is to replicate the phenomenon of erosion by wind and
other climatic forces.
We suppose that the thickness
of each slice is equal to the total
thickness of l number of mini cube sites .Generally l should be a factor of
number of slices forming the height of parallelepiped .We assumed that the base
is a square and a slice contains say N2
Sites of l x l
x l size. We assign value 0 to a site
with out mass and a value 1 with mass.
Generally, a few slices at the top will
have 0-sites only. As we go down we would encounter a slice where for the first
time 1-site would appear. We number the slices from N to 1 starting from the
top and let us suppose that Nth to (N-s) th slices have
no 1-site. In the sth slice the first 1-site appears. It may so
happen that in the sth slice 1-sites appear in large numbers and at
different locations. Lower the position of the slice more may be the total
number of 1-sites.
TERMINOLOGY:
·
Neighbor: Two one sites will be called
neighbors if they have common edge.
·
Cluster: A set of 1-sites having at least
one element, which has, more than one neighbor is a cluster.
·
Border site: A site that does not have more than
three neighbors is a border site.
·
P- is the probability that a site is a 1- site and is given by the
ratio between the number of 1-sites and the total number of sites.
·
Infinite cluster: A cluster that has at
least one 1-site at each of the opposite boundaries of the image is called
infinite cluster.
·
Finite cluster: A cluster that is not an
infinite cluster is finite cluster.
·
Size of cluster (x): First we find all pairs
of border sites lying in the same row or same column in a particular cluster.
The distance between the sites of each pair is estimated. The average of all
those distances is x
.
CLUSTER MATHEMATICS:
In the subsequent analysis we would frequently follow
the concept of percolation model adequately described by Bundle et al [1].
In terms of above discussion as
we go down the number of 1-sites may increase and accordingly their sizes may
grow larger and larger. At some point of time a cluster may so large that it
connects two opposite sides of image plane and we get an infinite cluster.
As per the above definition
the probability of appearance of a 1-site in a slice is given as
P
= number of 1- sites (2)
N2
As we go to the lower slice
the value of p may increase. At some stage an infinite cluster appears and
there p=pc.
We call p
c the critical concentration of 1-sites. So we may say that as long p
< pc a slice is composed
of a set of finite clusters only. At p=pc infinite cluster appears.
Appearance of infinite
cluster is important in many respects. It is linked with the critical
phenomenon of the system.
In this paper we shall study some thermo
dynamical features of cloud for its IR image. In the image plane many of its
characteristics near pc , may be described by power law and critical
exponents. We give definitions of such exponents.
P¥
~ (p-pc)β
x
~ |p-pc|-v (3)
Here P¥ is order parameter i.e.
the probability that the site belongs to an infinite cluster and x the correlation length i.e. mean
size of finite clusters near the critical point .β and v are two exponents. It may be observed that v remains same whether p approaches pc
from below or above.
The fractal dimension of the
image at the critical point is given as
df = d
- β/v (4)
d is the topological dimension
and in the present case it is equal to 3.This process is suggested by Bunde [1].
In percolation
study the average shortest path between two sites in a cluster is generally
termed as chemical distance.
Here we
calculate the chemical distance by following procedure:
Step1:
Identify al finite clusters just before reaching the critical stage .
Step2: For
each finite cluster we find the range of shortest paths .
Step3: Take the mid value of the range instead of taking
average shortest path. Thus we get a set
of mid values.
Step4: l is the arithmetic average of these
mid values. We assume that l scales with some r < x
Step5:
We write the scaling law as l ~ xdmin (5)
Where inverse relation may be
given as
x ~ l1/dmin (6)
Step6: Now we define the chemical distance as dl = df / dmin (7)
The chemical dimension dl
is important, because it distinguishes one fractal structure from other even if
they have same fractal dimension.
EXISTING ALGORITHM:
The existing algorithm goes like this:
3D-Erosion Technique()
STEP1: Read the image
STEP2:
For I=1 to m
//Where m is no of slices
Construct
slices() //Construct all the clusters in
that slice
Is
infinite() //Check whether
it is infinite cluster
If not so
continue
STEP3:At the infinite
cluster find the values of dmin , df , And so on.
STEP4:Draw
the inferences that can be drawn
STEP5:Stop
PROPOSED ALGORITHM:
The modification we proposed are:
Transformed
technique()
STEP1: Read the image
STEP2:Initialise the
process indication image
STEP3:
For Every Zero element in process
indication image
Form_cluster() //
Form a cluster in the image at
that value
Construct_info_node() //
END
Node
info: n[
] //Contains no of 1 sites
Size of cluster[ ] // Contains the size of each cluster
encountered
Cluster_ value [ ] // Conains the cluster value
STEP4:
From the
nodes info find out probabilities p[],β,v, dmin , df
STEP5:
Draw the inferences from the values obtained
STEP6:Stop
5. PERFORMANCE ANALYSIS:
Space Complexity:
·
Existing Algorithm:For constructing the 3d parellopiped N3
pixel are required for representing it in memory and the memory for the data
structures.
Best case =Worst Case =N3
Space Complexity is O(N3)
·
Proposed Algorithm:As only 2 images of N2 size are used and
in worst case the additional N2 data elements to be stored At most 3N2 memory
locations are used.
Best case=2N2 Locs ; Worst Case =3N2 Locs
Space complexity=O(N2)
Time Complexity:
·
Existing algorithm:
ü Just we have to process m
slices
ü For each slice we have to
process N2 pixels of image
ü For pixel that belongs to
the slice value that we process now then by using image-matching technique we
have to construct cluster. For that we need 8 N2 comparisons .
ü So totally m*N2 +8
N2 comparisons .
·
Proposed Algorithm:
ü Once we have to process
the process the process_indication_image once .requires N2 comparisons .
ü For each pixel in image 8
comparisons are required for constructing cluster
ü So totally N2 +8 N2 =9N2
Performance:Here
irrespective of the no of slice (fray values) the comparisons are same
Application of cluster Mathematics to IR
images of the cloud
In IR images greater grey value
means higher temperature. The way we have made slices have all high temperature
zones. As we go down a slice includes zones of temperature, which we call the
base temperature. So lower we go the lower is the base temperature. As long as
there is no infinite cluster we have all the zones restricted with in the cloud
or its image. But once we encounter an infinite cluster the cloud under
reference becomes a part of larger cloud vis-à-vis a larger thermo dynamical
system.
So if the appearance of an infinite
cluster occurs at a lower level the cloud may be connected to a weaker thermo
dynamical system. There is small possibility of its being connected to a strong
source of energy .On the other hand, if it occurs at substantially higher level
the cloud may be connected to a stronger source of thermal energy .It may be
noted in this context that the study of a small cloud may not give a reliable
picture since an infinite cluster may be able to give some idea of the nearest
neighborhood only. For wider information we may have to study a large area
comprising of a large section of cloud formation.
However we would like to conjecture that
connection of a system with a stronger or weaker thermo dynamical system means
transfer of energy from one system to the other at the critical level.
Transfer process may occur in
different ways it may be a simple gradient transfer, a linear process .It may
also be a random process of cascading of energy from one scale to the other or
a turbulent transfer. The degree of turbulent transfer of energy may be
quantified by the fractal dimension df of the cluster near the critical zone.
If df is large thermal
turbulence will be vigorous which means the transfer of thermal energy may take
place in almost all scales. In addition if the critical stage is at higher
level there is a possibility of vigorous thermal activity inside the cloud,
which may lead to rapid climatic change.
Two separate systems may
have same fractal dimension but may be different in many respects .We have
tried to distinguish between two such systems with the help of chemical
dimension. This dimension as has been stated earlier is a result of relation
between Pythagorean distance and Euclidean distance between two sides. If a
cluster had got large number pseudo podia like structures Euclidean distance
would be very much different from Pythagorean distance and as a result dmin
would be much different from unity. And so would be the chemical
dimension from the fractal dimension.
CASE STUDY:
IR Image of a cloud The image drawn at critical stage
The critical values for previous image are:
df = 2.19; dL = 1.778
Inferences:
Low df indicate a non random
structure and absence of turbulent transfer of energy.
Low dL indicate presence of
articulated branching.
Implementation:
3D Erosion Package
Conclusion:
To sum it up proposed technique can be used to find
out effectively the effect of a complete
cloud on the environment. The IR image of the cloud has been successfully
analysed and the implication are drawn .Further its advancement can be proposed
by artificial intelligence techniques to inhibit the human intervention in the
whether forecasting in near future.
REFERENECES:
[1]. A. Bunde and S.Havlin
,Fractals and disorder systems
[2]. A. Chanda ,A.K.De and
J.Das ,Fractal charecterization of
temperature in the convective boundary
layer ,Fractals ,vol5, II,1997
