letter generation of organized connectivity (soccnm

Discrete Dynamics & Nature and Society, Vol. 6, pp. 225-228Reprints available directly from the publisherPhotocopying permitted by license only

(C) 2001 OPA (Overseas Publishers Association) N.V.Published by license under

the Gordon and Breach Science Publishers imprint,member of the Taylor & Francis Group.

Letter to the Editor

Generation of Self Organized CriticalConnectivity Network Map (SOCCNM)

of Randomly SituatedWater Bodies During Flooding Process

B. S. DAYA SAGAR*

Centre for Remote Sensing & Information Systems, Department of Geoengineering,Andhra University, Visakhapatnam-530 003, India

(Received in finalform 13 April 2000)

This letter presents a brief framework based on nonlinear morphological transforma-tions to generate a self organized critical connectivity network map (SOCCNM) in2-dimensional space. This simple and elegant framework is implemented on a sectionthat contains a few simulated water bodies to generate SOCCNM. This is based ona postulate that the randomly situated surface water bodies of various sizes and shapesself organize during flooding process.

Keywords: Water bodies; Mathematical morphology; SOCCNM; Nonlinear transformations

1. INTRODUCTION

During flooding process the surface water bodiestend to self organize. This study that relates toknow how the surface water bodies contact eachother on a 2-dimensional space during floodingprocess is dealt with. Floods of different intensitiesgenerate typical surface water body organization

network. The intuitive argument is that the smallerwater bodies are more homogeneously distributedover a landmass than that of larger water bodies.A case study using real world data is tested in ourearlier study that supports that this argument isjustified [1]. Small water bodies will tend to mergeeach other with flood of small intensity, and thehigh degree of flood intensity makes the larger

* Present Address: Centre for Remote Imaging Sensing Processing (CRISP), Faculty of Science, The National University ofSingapore, Lower Kent Ridge Road, Singapore 119260. e-mail: [email protected], [email protected]

225

226 B.S. DAYA SAGAR

water bodies contact together. This is the postu-late. For instance, how two closely situated surfacewater bodies self organize with small intensity offlooding can be visualized in spatial domain.Similarly randomly situated surface water bodiesof several sizes and shapes will be self organizeddue to flooding. The critical map in 2-dimensionalspace of such self organization process of waterbodies under the influence of flooding is simulatedby following nonlinear mathematical morphologi-cal transformations. Such a map is visualized interms of one pixel wide caricature. The random-ness in the spatial distribution of surface waterbodies decides the spatial structure of self orga-nized critical connectivity network map. In a wayit depicts the extinguishing points of self organizedwater bodies at critical state. The critical state isdefined by the degree of flooding intensity. Thismap is termed in the present study as a selforganized critical connectivity network map(SOCCNM).

In brief the second section enables the simpleframework based on mathematical morphologicaltransformations [2]. In third section, that follows,includes a sample study that consists of randomlydistributed simulated water bodies. In what fol-lows includes the mathematical framework.

2. MATHEMATICALMORPHOLOGICAL FRAMEWORKTO EXTRACT SOCCNM

supremums and infemums. Prior to understand theframework from geometrical point of view mor-

phological dilations and erosions are defined as settransformations that expand and contract a set.The morphological operators can be visualized as

working with two images. The image being pro-cessed is referred to as the set or set complement,and other image being a structuring template.Each structuring template has a designed shapethat can be thought of as a probe of the image. Theimage can be decomposed by probing it withvarious structuring templates. The four basicmorphological transformations dilation, erosion,and cascade of erosion-dilation and closing are

defined with respect to the structuring element Swith scaling factor e, image M and point A0 1R2.

Dilation:

6f (M)(A0) MAXA e A0+e.S(A0)(M(A))e 1,2,...,n

Erosion:

se(M)(Ao) MINA Ao+e.S(Ao)(M(A)) (2)

By duality, the opening of M by S comes fromerosion first and then dilation. Thus the opening isdefined as

Opening:

-y(M) 6e( (M)) (3)

Henceforth, the section containing surface waterbody data is referred to as M that is a discretebinary level image consists of water bodies and no

water body region. Dilation, erosion, opening andclosing [2] are simplest quantitative morphologicalset transformations. The discrete binary levelimage, M, is defined as a finite subset of Euclideantwo dimensional space, 1R2 that can admit values(water body or set) and 0 (no-water body region orset complement). Surface water bodies hereafterrefer to as M testify the presence of zones of

The dilation followed by erosion is called closingtransformation. These transformations can becarried out according to the multiscale approach[3]. In the multiscale approach, the size of thestructuring template will be increased from itera-tion to iteration. Due to the noninvertible propertyof the superficially simple erosion and openingtransformations and a logical operation SOCCNMcan be generated.

These transformations are systematically used toremove the supremums "A" in the set complement

LETTER TO THE EDITOR 227

(MC). These topographically significant featuresenable the structural composition of theSOCCNM. This structural composition is sensitiveto changes that occur due to both exogenic andendogenic processes. This self organized criticalconnectivity network map acts as a barrier thatobstructs the further diffusion of water bodies,which are treated as source points from whichrainfall diffuses and propagates towards theSOCCNM, during flooding. This propagation isof anisotropic in nature. The framework thusdeveloped in this section is intended to isolateSOCCNM on a section, containing simulatedwater bodies, of size 200 x 200 pixels. In what fol-lows in this section includes mathematical repre-sentation of sequential steps involved.SOCCNM subsets of nth order: In the process

of extraction of SOCCNM from the image, theopened version of each eroded version ofM needsto be subtracted from the corresponding erodedversion. This process is mathematically shown asfollows:

SOCCNMe(Mc) [(MC)]/{,,/s[(MC)]} (4)

Once, several orders of subsets of SOCCNM areisolated from the image, they need to be madeunion by following the logical AND operation.The mathematical representation of this process to

see the SOCCNM of M is as follows:

SOCCNM(M)- U[SOCCNMe(MC)]e=l

The above transformation needs to be performedto generate the SOCCNM.

3. A SAMPLE STUDY

As a sample study, a section that contains a fewsimulated water bodies (M) (Fig. a) has beenused. This study is emphasized on the methodol-ogy and further scope to develop quantitativemethods. The SOCCNM (Fig. b) is generatedfrom the M. The higher the spatial resolution thehigher is the intricacy of the SOCCNM, vice versa.

By following the mathematical framework thusdeveloped on the data SOCCNM has beengenerated and shown in the Figure lb.

Such an analysis can be acquired out on multi-date data of a region (e.g., landform, river networksetc.) to understand the complex limnologicalprocesses. From the present study it is deducedthat even minor degree of deformation that maybe reflected on the SOCCNM structural composi-tion can be extracted by following the procedurethus developed and tested. Moreover, spatio-temporal distribution of various geologically and

FIGURE (a) A few simulated water bodies (M), and (b) the SOCCNM of the M.

228 B.S. DAYA SAGAR

geomorphologically complex features can be stu-died further to develop cogent models of interestto geophysicists. Present study is an attempt totest the developed mathematical framework to ex-tract SOCCNM from the section ofsimulated waterbodies. However, these network maps across timeintervals enable the spatio-temporal distributionthat has not been covered in the present letter.

References

[1] Sagar, B. S. D., Is the spatial distribution of larger waterbodies heterogeneous? (in press), International Journal ofRemote Sensing.

[2] Serra, J. (1982). Image Analysis and Mathematical Mor-phology, Academic Press, London, p. 610.

[3] Maragos, P. A. and Schafer, R. W. (1986). Morphologicalskeleton representation and coding of binary images,LE.E.E. Transactions on Acoustics, Speech, and SignalProcessing, ASSP-34, 5, 1228-1244.

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