
Research Article


Convex Characterization of Simulated Droughts and Floods of Water Bodies 

S. Dinesh



ABSTRACT

Convexity is considered as one of the basic descriptors of shapes. In this study, the characterization of the convexity of simulated droughts and floods of water bodies is performed. First, concepts of mathematical morphology employed to generate simulated droughts and floods of water bodies. The average convexity measures of the generated simulated droughts and floods are computed. It is observed that droughting reduces the convexity of water bodies, while flooding increases the convexity of water bodies. A power law relationship is observed between the average convexity measures of the simulated droughts/floods and the level of droughting/flooding and areas of water bodies. The scaling exponent of this power law, which is named as a fractal dimension, indicates the rate of change of convexity of simulated droughts/floods of water bodies over varying levels of droughting/flooding.





INTRODUCTION
Convexity is considered as one of the basic descriptors of shapes. Convexity in image processing has been studied for quite some time (Valentine, 1964; Stern, 1989; Boxer, 1993; Held and Abe, 1994; Popov, 1996; Zunic and Rosin, 2004; Rosin and Mumford, 2004; Rahtu et al., 2004; Kolesnikov and Fränti, 2005; Rahtu et al., 2006; Varosanec, 2007) and has numerous applications, including shape decomposition (Latecki and Lakämper, 1999; Rosin, 2000), camouflage breaking (Tankus and Yeshurun, 2000), object indexing (Latecki and Lakämper, 2000), measurement of border irregularities measurement in medical image analysis (Lee et al., 2003), handwritten word recognition (Kapp et al., 2007) and estimation of derivatives of holomorphic functions (Li, 2007).
An object P is said to be convex if it has the following property: If points A and B belong to P, then all points from the line segment [AB] belong to P as well. The smallest convex set which includes P is called the convex hull of P and is denoted as CH(P). The convexity measure C(P) is defined to be:
Convexity measures have the following properties (Zunic and Rosin, 2004):
• 
Convexity measures have the range of (0,1) 
• 
The convexity measure of a given object equals 1 if and only if this object
is convex 
• 
There are objects whose convexity measure is arbitrarily close to 0 
• 
The convexity measure of an object is invariant under similarity transformations
of the object. 
In this study, the characterization of the convexity of simulated droughts and floods of water bodies is performed. It is shown that a power law relationship exists between the average convexity measures of simulated droughts/floods of water bodies and the level of droughting/flooding. MATHEMATICAL MORPHOLOGY
Mathematical morphology is a branch of image processing that deals with the extraction of image components that are useful for representational and descriptional purposes. The fundamental morphological operators are discussed in Matheron (1975), Serra (1982) and Soille (2003). Morphological operators generally require two inputs; the input image A, which can be in binary or gray scale form and the kernel B, which is used to determine the precise effect of the operator.
Dilation sets the pixel values within the kernel to the maximum value of the pixel neighborhood. Binary dilation fills the small holes inside particles and gulfs on the boundary of objects, enlarges the size of the particles and may connect neighboring particles (Duchene and Lewis, 1996). The dilation operation is expressed as:
Erosion sets the pixels values within the kernel to the minimum value of the
kernel. Binary erosion removes isolated points and small particles, shrinks
other particles, discards peaks on the boundaries of objects and disconnects
some particles (Duchene and Lewis, 1996). Erosion is the dual operator of dilation:
where A^{c} denotes the complement of A and B is symmetric with respect to reflection about the origin. Drought and flood simulation is implemented by performing erosion and dilation, respectively, on water bodies using square kernels. Erosion reduces the area of water bodies, mimicking droughting, while dilation increases the area of water bodies, mimicking flooding. The level of droughting/flooding is indicated by the kernel size. Figure 1 shows a number of water bodies situated in the flood plain region of Gothavary River, India. The water bodies were traced from IRS 1D remotely sensed data. Due to the impracticalities of dealing with incomplete water bodies, only the complete water bodies are considered (Fig. 2). Simulated droughts (Fig. 3) and floods (Fig. 4) of the water bodies for levels of 1 to 15 are computed. The areas of the generated simulated droughts and floods are shown in Table 1.
 Fig. 1: 
Water bodies traced from IRS 1D remotely sensed data 
 Fig. 2: 
The
water bodies after removal of incomplete water bodies 
 Fig. 3: 
The generated simulated droughts of the water bodies at droughting levels
of: (a) 3 (b) 7 (c) 11 (d) 15 
 Fig. 4: 
The
generated simulated floods of the water bodies at flooding levels of:
(a) 3 (b) 7 (c) 11 (d) 15 
Table 1: 
Areas
of the generated simulated droughts and floods of the water bodies 

CHARACTERIZATION OF CONVEXITY OF WATER BODIES
The convex hulls of the generated simulated droughts (Fig. 5)
and floods (Fig. 6) of the water bodies are computed using
the convex hull computing neural network (CHCNN) algorithm proposed in Leung
et al. (1997). The algorithm is based on a twolayered neural network,
topologically similar to ART, with is an adaptive training strategy called excited
learning. CHCNN provides a parallel online and realtime processing of data
which after training, yields two closely related approximations, one from within
and one from the outside, of the desired convex hulls. The accuracy of the approximated
convex hull is approximately O[K^{1/n1)}], where K is the number of
neurons in the output layer of the CHCNN. When K is taken to be sufficiently
large, CHCNN can generate any accurate approximate convex hull.
The average convexity measures of the simulated droughts and floods are computed (Table 2). It is observed that droughting reduces the convexity of water bodies, while flooding increases the convexity of water bodies.
Loglog plots of the average convexity measures of the simulate droughts/floods
C against the level of droughting/flooding r is drawn (Fig. 7
and 8). Power law relationships are observed in both plots.
These power laws take the following form:
These power law relationships arise as a consequence of the fractal properties
of the convexity of simulated droughts and floods of water bodies.

Fig. 5: 
Convex hull of the corresponding simulated droughts in fig. 3 

Fig. 6: 
Convex hull of the corresponding simulated floods in fig. 4 
 Fig. 7: 
Loglog
plot of the average convexity measures of the simulated droughts against
the droughting level 
 Fig. 8: 
Loglog
plot of the average convexity measures of the simulated floods against
the flooding level 
Table 2: 
Average
convexity measures of the generated simulated droughts and floods of the
water bodies 

In Eq. 4, c is a constant of proportionality, while D is
the fractal dimension of the convexity of simulated droughts/floods of water
bodies, which indicates the rate of change of convexity of simulated droughts/floods
of water bodies over varying levels of droughting/flooding. D has a positive
value for flooding and a negative value for droughting.
CONCLUSION
In this study, the characterization of the convexity of simulated droughts
and floods of water bodies is performed. It was observed that droughting reduces
the convexity of water bodies, while flooding increases the convexity of water
bodies. A power law relationship was observed between the average convexity
measures of the simulated droughts/floods and the level of droughting/flooding.
The scaling exponent of this power law, which was named as a fractal dimension,
indicates the rate of change of convexity of water bodies over varying levels
of droughting/flooding.

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