Mathematical Morphology in Geosciences and GISci: An Illustrative Review

Georges Matheron and Jean Serra of the Centre of Mathematical Morphology, Fontainebleau founded Mathematical Morphology (MM). Since the birth of MM in the mid 1960s, its applications in a wide ranging disciplines have illustrated that intuitive researchers can find varied application-domains to extend the applications of MM. This chapter provides a concise review of application of Mathematical Morphology in Geosciences and Geographical Information Science (GISci). The motivation for this chapter stems from the fact that Mathematical Morphology is one of the better choices to deal with highly intertwined topics such as retrieval, analysis, reasoning, and simulation and modeling of terrestrial phenomena and processes. This chapter provides an illustrative review of various studies carried out by the author over a period of 25 years—related to applications of Mathematical Morphology and Fractal Geometry—in the contexts of Geosciences and Geographical Information Science (GISci). However, the reader is encouraged to refer to the cited publications to gather more details on the review provided in an abstract manner.


Introduction
A basic understanding of many geoscientific and geoengineering challenges across multiple spatial and/or temporal scales of terrestrial phenomena and processes is among the greatest of challenges facing contemporary sciences and engineering. Many space-time models explaining phenomena and processes of terrestrial relevance were descriptive in nature. Earlier, several toy models were developed via classical mathematics to explain possible phases in dynamical behaviors of complex systems. With the advent of computers with powerful graphics facilities, about three decades ago the interplay between numerical methods (generated via classical equations explaining the behaviors of dynamical systems) and graphics was shown to exist. That progress provided the initial impetus to visualize the systems' spatial and/or temporal behaviors that exhibit simple to complex patterns on graphical screens. One of the efficient ways of understanding the dynamical behavior of many complex systems of nature, society and science is possible through data acquired at multiple spatial and temporal scales. Data related to terrestrial (geophysical) phenomena at spatial and temporal intervals are available in numerous formats. The utility and application of such data could be substantially enhanced through related technologies documented in edited volumes and monographs of the recent past (Sagar 2001a(Sagar , b, c, d, 2005a(Sagar , b, 2009(Sagar , 2013Sagar and Rao 2003;Sagar and Bruce 2005;Sagar and Serra 2010;Najman et al. 2012).
To understand the dynamical behavior of a phenomenon or a process, development of a good spatiotemporal model is essential. To develop a good spatiotemporal model, well-analyzed and well-reasoned information that could be extracted/retrieved from spatial and/or temporal data are important ingredients. Figure 35.1 shows a schematic illustrating the key links between the various phases where the involvement of Mathematical Morphology becomes obvious from the studies to be shown later in the chapter.
Mathematical Morphology-founded by Georges Matheron (1975) and Jean Serra (1982) has shown great impact in various fields including Geosciences and GISci-is one of the better choices to deal with all these key aspects mentioned. Mathematical morphology was founded by Georges Matheron (Agterberg 2001(Agterberg , 2004Serra 1982Serra , 1988. There are numerous representative publications related to mathematical morphology, to name a few: Serra (1982Serra ( , 1988, Sternberg (1986), Beucher (1990Beucher ( , 1999, Soille (2003), Najman and Talbot (2010), Sagar (2013). Most notably, the comment on the issue of "What do Mathematical Geoscientists  ?" made by Harbaugh (2014) includes the importance of mathematical morphology of geological features in making predictions. In this chapter we outline the successful applications of the most important concepts of mathematical morphology (Table 35.1) in the context of geosciences and Geographical Information Science (GISci).

Terrestrial Pattern Retrieval
Retrieving relevant information from precisely acquired spatial-temporal data of varied types about a specific complex system is a basic prerequisite to understand the spatial-temporal behavior of a system. Retrieval of information from a available spatiotemporal data acquired from a wide range of sources and a variety of formats, opens new horizons to the spatial statistical and geoscience communities. We have developed original spatial algorithms based on non-linear morphological transformations for retrieval of unique geophysical networks, mountain objects, segmentation of various geophysical objects, and pairing the geophysical spatial fields based on certain similarities (Sagar et al. , 2003aSagar and Chockalingam 2004;Sathymoorthy et al. 2007;Chockalingam and Sagar 2003;Lim and Sagar 2008a, b;Lim et al. 2009Lim et al. , 2011Sagar and Lim 2015a, b;Danda et al. 2016).

Mathematical Morphology in Extraction of Unique Topological Networks
In contrast to other recent works, which have focused on extraction of channel networks via algorithms that fail to precisely extract networks from non-hilly regions (e.g. tidal regions), the algorithms we proposed can be generalized for application to both hilly (e.g. fluvial) and non-hilly (e.g. tidal) terrains, and also pore connectivity networks. These algorithms concerning the framework to extract multiscale geomorphologic networks via systematically decomposing elevation surfaces and/or decomposed threshold elevation regions into their abstract structures lead to valley and ridge connectivity networks. We proposed a framework to first decompose a binary fractal basin into fractal DEM from which two unique topological connectivity networks are extracted. These networks facilitate to segment Fractal-DEM ( Fig. 35.2a) into sub-basins ranging from first to highest order ( Fig. 35.2c). Results derived from a synthetic DEM ( Fig. 35.2a) by applying one of these algorithms include unique topological connectivity networks similar to valley and ridge connectivity networks ( Fig. 35.2b) and the hierarchically partitioned watersheds ( Fig. 35.2c). We demonstrated the superiority of these stable algorithms which can be generalized to terrestrial surfaces of both fluvial and tidal types. This work helps to solve basic problems that algorithms meant for extraction of unique terrestrial connectivity networks have faced for over three decades.

Retrieval of Morphologically Significant Regions
Algorithms meant for morphological segmentation were demonstrated on a DEM, and mapped the physiographic features such as mountains, basins, and piedmont slopes from DEM ( Fig. 35.3a); and the results are compared with that of other popular approaches ( Fig. 35.3b). Further, multiscale morphological opening was employed to segment binary fractal basins ( Fig. 35.4a-c) that mimic geophysical basins, and cloud fields  Mountain pixels are the pixels in white, the piedmont pixels are the pixels in gray, and the basin pixels are the pixels in black. a The results obtained using the newly developed algorithm. b The results obtained in Miliaresis and Argialas (1999). (From Sathymoorthy et al. 2007) isolated from MODIS data into topologically prominent regions ( Fig. 35.4d-f). We proposed granulometry-based segmentation of geophysical fields (e.g. DEMs, clouds, etc.) with demonstration on binary fractals of deterministic and random types , and on cloud fields ( Fig. 35.4d-f) that have different compaction properties with varied cloud properties.
The approach based on computation of complexity measures of morphologically significant zones decomposed from binary fractal sets via multiscale convexity analysis-which can be implemented on several geophysical and geomorphologic fields (e.g. DEMs, clouds, binary fractals etc.) to segment them into regions of varied topological significance-has been demonstrated on cloud fields derived from MODIS data to better segment the regions within the cloud fields that have different compaction properties with varied cloud properties. This approach of fundamental importance can be extended to several geophysical and geomorphologic fields to segment them into regions of varied topological significance.

Ranking of Best Pairs of Spatial Fields
A new metric to quantify the degree of similarity between any two given spatial fields is proposed (Sagar and Lim 2015a, b). This metric based on morphological operations can be used for image classification, in particular hyperspectral image classification, to derive best pair(s) of spatial fields from among a large number of spatial fields available in a database. In this proposed approach to compute the ranks for every possible pair of spatial fields (grayscale images) in a database, the two major computations involved include (i) estimation of grayscale morphological distance between the source and target spatial fields, and (ii) the ratios between the areas of infima and suprema of source and target spatial fields. Using this approach, four spatial elevation fields ( Fig. 35.5b-e), in other words four quadrants partitioned from Fig. 35.5a could be paired into best pair ( Fig. 35.6a), medium best pair ( Fig. 35.6b), and the least best pair (Fig. 35.6c).

Terrestrial Pattern Analysis
Quantitative analyses of terrestrial phenomena and processes is one of the innovative new directions of geoscientific research. Analysis of terrestrial patterns-that include water bodies, valley and ridge connectivity networks, watersheds, hillslopes, mountain objects, elevation fields-at various spatial and temporal scales is an important aspect to better understand the dynamical behaviors of various terrestrial processes and surfaces. Over the decades, various quantitative approaches have been developed and successfully demonstrated. Some of these approaches include morphometric analysis of river networks, hypsometry, allometry, and granulometric analyses, and geodesic spectrum based analysis. In this section, we show some results through illustrations arrived at via applications of mathematical morphology in (i) morphometric and allometric analyses of river networks and water bodies and their corresponding zones of influence, (ii) deriving scale-invariant but shape-dependant power laws, (iii) deriving basin-specific geodesic spectrum, and (iv) DEM analysis.

Morphometry and Allometry of Networks
Towards analyzing terrestrial surfaces we have shown unique ways to quantitatively characterize the spatiotemporal terrestrial complexity via scale-invariant measures that explain the commonly sharing physical mechanisms involved in terrestrial phenomena and processes. These contributions (Sagar and Rao 1995a, b, c, d;Sagar 1996Sagar , 1999aSagar 2000aSagar , b, 2001aSagar , b, c, d 2007Sagar et al. 1998aSagar et al. , b, 1999Sagar and Tien 2004;Chockalingam and Sagar 2005;Tay et al. 2005a, b, c) highlighted the evidence of self-organization via scaling laws-in networks, hierarchically decomposed subwatersheds, and water bodies and their zones of influence, which evidently belong to different universality classes-which possess excellent agreement with geomorphologic laws such as Horton's Laws, Hurst exponents, Hack's exponent, and other power-laws given in non-geoscientific contexts. A host of allometric power-law relationships were derived that were in good accord with other established network models and real networks (Figs. 35.7, 35.8 and 35.9).

Allometry of Water Bodies and Their Zones of Influence
Topologically, water bodies ( Fig. 35.10a) are the first level topographic regions that get flooded, and as the flood level gets higher, adjacent water bodies merge. The looplike network that forms along all these merging points represents zones of influence ( Fig. 35.10b) of each water body. The geometric organizations of these two phenomena are respectively sensitive and insensitive to perturbation due to exogenic processes. To demonstrate the allometric relationships of water bodies and their zones of influence, a large number of surface water bodies (irrigation tanks), situated in the floodplain region of certain rivers of India, which are retrieved from multi-date remotely sensed data were analyzed in 2-D space (Sagar et al. 1995a, b). Basic measures of these water bodies obtained by morphological analysis were employed to show fractal-length-area-perimeter relationships. We found that these phenomena follow the universal scaling laws (Sagar et al. 2002;Sagar 2005a, b) found in other geophysical and biological contexts. In this work, universal scaling relationships among basic measures such as area, length, diameter, volume, and information about networks are exhibited by several natural phenomena to further retrieve and understand the common principles underlying organization of these phenomena. Some of the recent findings on universal scaling relations include relationships between brain and body, length and area (or volume),

Morphometry of Non-network Space: Scale Invariant but Shape-Dependent Dimension
In sequel works on terrestrial analysis, we argued that the universal scaling laws shown as examples in earlier section possess limited utility in exploring possibilities to relate them with geomorphologic processes. These arguments formed the basis for alternative methods (Radhakrishna et al. 2004;Teo et al. 2004;Sagar and Chockalingam 2004;Chockalingam and Sagar 2005;Tay et al. 2005aTay et al. , b, 2007. Shape and scale based indexes provided to analyze and classify non-network space (hillslopes) (Sagar and Chockalingam 2004;Chockalingam and Sagar 2005), and terrestrial surfaces (Tay et al. 2005a(Tay et al. , b, 2007 received wide attention. These methods that preserve the spatial and morphological variability yield quantitative results that are scale invariant but shape dependent, and are sensitive to terrestrial surface variations. "Fractal dimension of non-network space of a catchment basin", provides an approach to show basic distinction between the topologically invariant geomorphologic basins. It introduced morphological technique for hillslope decomposition that yields a scale invariant, but shape dependent, power-laws ( Fig. 35.11a, b). Varied degrees of topographically convex regions within a catchment basin represent varied degrees of hill-slopes. The non-network space, the characterization of which we focused on in our investigations, is akin to the space that is achieved by subtracting channelized portions contributed due to concave regions from the watershed space. This non-network space is akin to non-channelized convex region within a catchment basin. We proposed an alternative shape-dependent quantity akin to fractal dimension to characterize this non-network space (e.g.: Fig. 35.12a). Towards this goal, non-network space is decomposed, in two-dimensional discrete space, into simple non-overlapping disks (NODs) of various sizes by employing mathematical morphological transformations and certain logical operations ( Fig. 35.12b). Furthermore, number of NODs of lesser than threshold radius is plotted against the radius, and computed the shape-dependent fractal dimension of non-network space. This study was extended to derive shape dependent scaling laws as the laws derived from network measurements are shape independent for realistic basins (Fig. 35.12). The relationship between number of NODs and the radius of the disk provides an alternative fractal-like dimension that is shape dependent. This was done with the aim to relate shape dependent power laws with geomorphic processes such as hill-slope processes and erosion.
Applications of mathematical morphology transformations are shown to decompose fractal basins (e.g.: Fig. 35.11a) into non-overlapping disks of various sizes ( Fig. 35.11b) further to derive fractal power-laws based on number-radius relationships.

Geodesic Spectrum
We have provided a novel geomorphologic indicator by simulating geodesic flow fields ( Fig. 35.13d-f) within basins ( Fig. 35.13a-c) consisting of spatially distributed elevation regions (Lim and Sagar 2008a, b), further to compute a geodesic spectrum that provides a unique one-dimensional geometric support.
This one-dimensional geometric support, in other words geodesic spectrum, outperforms the conventional width-function based approach which is usually derived from planar forms of basin and its networks-construction involves basin as a random elevation field (e.g. Digital Elevation Model, DEM) and all threshold elevation regions decomposed from DEM for understanding the shape-function relationship much better than that of width function.

Granulometric and Anti-granulometric Analysis of Basin-DEMs
Granulometric indexes derived for spatial elevation fields also yield scale invariant but shape-dependent measures (Tay et al. 2005a(Tay et al. , b, c, 2007. DEMs are analyzed by following granulometries via multiscale opening ( Fig. 35.14 upper panel), and antigranulometries ( Fig. 35.14 lower panel) to derive shape-size complexity measures of foreground and background respectively that provide new indices to understand the terrestrial surfaces further to relate with several geomorphic processes.

Geomorphologic Modeling and Simulation
Simulations allow us to gain a significantly good understanding of complex geomorphologic systems in a way that is not possible with lab experiments. Effectively attaining these goals presents many computational challenges, which include the development of frameworks. The robustness of mathematical morphological operators combined with concepts of fractal geometry (Mandelbrot 1982) in The phenomena and processes given emphasis in this section include geomorphologic features, basins and channel networks, landscapes, water bodies, symmetrical folds and ideal sand dunes. Besides providing approaches to simulate fractal-skeletal based channel network model and fractal landscapes, we have shown via the discrete simulations the varied dynamical behavioral phases of certain geoscientific processes (e.g. water bodies, ductile symmetric folds, sand dunes, landscapes) under nonlinear perturbations due to endogenic and exogenic nature of forces. For these simulations we employed nonlinear first order difference equations, bifurcation theory, fractal geometry, and nonlinear morphological transformations as the bases. The three complex systems that we focus on include the channelization process, surface water bodies, and elevation structures.

Geomorphologic Modeling: Concept of Discrete Force
Concept of discrete force was proposed from theoretical standpoint to model certain geomorphic phenomena, where geomorphologically realistic expansion and contractions, and cascades of these two transformations were proposed, and five laws of geomorphologic structures are proposed (Sagar et al. 1998a, b). A possibility to derive a discrete rule from a geomorphic feature (e.g. lake) undergoing morphological changes that can be retrieved from temporal satellite data was also proposed in this work, and explained ( Fig. 35.15). Laws of geomorphic structures under the perturbations are provided and shown, through interplay between numerical simulations and graphic analysis as to how systems traverse through various behavioral phases.

Fractal-Skeletal Based Channel Network Model
Our work on channel network modelling Gastner and Newman (2004) and Sagar (2001c) represents unique contributions to the literature, which until recently were dominated by the classic random model. Fractal-skeletal based channel network model (F-SCN) was proposed by following certain postulates. We developed the Fractal-Skeletal Channel Network (F-SCN) model by employing morphological skeletonization to construct other classes of network models, which can exhibit various empirical features that the random model cannot. In the F-SCN model that gives rise to Horton laws, the generating mechanism plays an important role. Homogeneous and heterogeneous channel networks can be constructed by symmetric generator with non-random rules, and symmetric or asymmetric generators with random rules. Subsequently, F-SCNs (Fig. 35.16d-f) in different shapes of fractal basins (Fig. 35.16a-c) are generated and their generalized Hortonian laws ( Fig. 35.16g, h) are computed which are found to be in good accord with other established network models such as Optimal Channel Networks (OCNs), and realistic rivers. F-SCN model is extended to generate more realistic dendritic branched networks.

Fractal Landscape via Morphological Decomposition
By applying morphological transformations on fractals of varied types are decomposed into topologically prominent regions (TPRs) (Fig. 35.17a) and each TPR is coded and a fractal landscape organization that is geomorphologically realistic is simulated (Fig. 35.17b) (Sagar and Murthy 2000).

Discrete Simulations and Modeling the Dynamics of Small Water Bodies, Symmetrical Folds, and Sand Dunes
In this subsection we show the fusion of computer simulations and modeling techniques in order to better understand certain terrestrial phenomena and processes with the ultimate goal of developing cogent models in discrete space further to gain a significantly good understanding of complex terrestrial systems in a way that is not possible with lab experiments. The three synthetic phenomena that are explained by generating attractors considered include water bodies (Sagar and Rao 1995a, b, c), symmetrical folds (Sagar 1998), and sand dunes (Sagar 1999b(Sagar , 2000a(Sagar , b, 2001a(Sagar , 2005aSagar and Venu 2001;Sagar et al. 2003a, b).

Discrete Simulations and Modeling the Dynamics of Small Water Bodies
Spatio-temporal patterns of small water bodies (SWBs) under the influence of temporally varied streamflow discharge behaviors are simulated in discrete space by employing geomorphologically realistic expansion and contraction transformations ( Fig. 35.18). Expansions and contractions of SWBs to various degrees (e.g. Fig. 35.18B g-l), which are obvious due to fluctuations in streamflow discharge pattern ( Fig. 35.18A, a-f), simulate the effects respectively owing to streamflow discharge that is greater or less than mean streamflow discharge. The cascades of expansion-contraction are systematically performed by synchronizing the streamflow discharge (Fig. 35.18A, a-f), which is represented as a template with definite characteristic information, as the basis to model the spatio-temporal organization of randomly situated surface water bodies of various sizes and shapes. We have shown the varied dynamical behavioral phases of certain geoscientific processes (e.g. water bodies) under nonlinear perturbations via the discrete simulations.

Ductile Symmetrical Fold Dynamics
Under various possible time-dependent and time-independent strength of control parameter, in other words nonlinear perturbations, the three-limb symmetrical folds are transformed in a time sequential mode to simulate various possible fold dynamical behaviors ( Fig. 35.19a, b) synchronizing trajectory behavior simulated via logistic equation with strength nonlinearity parameters 3.9 and 2.8 ( Fig. 35.20a, b). We employed normalized fractal dimension values, and interlimb angles (IAs) as parameters along with strength of nonlinear parameters in this study. Bifurcation Fig. 35.18 A Streamflow discharge behavioral pattern at different environmental parameters. a-f λ = 1, 2, 3, 3.46, 3.57 and 3.99, and B Spatio-temporal organization of the surface water bodies under the influence of various streamflow discharge behavioral patterns at the environmental parameters at a-f λ = 1, 2, 3, 3.46, 3.57, and 3.99 are shown up to 20 time steps. In all the cases, the considered initial MSD, A0 = 0.5 (in normalized scale) is considered under the assumption that the water bodies attain their full capacity. It is illustrated only for the overlaid outlines of water bodies at respective time-steps with various λs diagrams are constructed for both time-dependent and time-independent fold dynamical behaviors, and the equations to compute metric universality by considering the interlimb angles computed at threshold strengths of nonlinearity parameters are proposed (Sagar 1998).

Symmetrical Sand Dune Dynamics
Certain possible morphological behaviors with respective critical states represented by inter-slip face angles of a sand dune under the influence of non systematic processes are qualitatively illustrated by considering the first order difference equation that has the physical relevance to model the morphological dynamics of the sand dune evolution as the basis. It is deduced that the critical state of a sand dune under dynamics depends on the regulatory parameter that encompasses exodyanmic processes of random nature and the morphological configuration of sand dune. With the aid of the regulatory parameter, and the specifications of initial state of sand dune, morphological history of the sand dune evolution can be investigated. As an attempt to furnish the interplay between numerical experiments and theory of morphological evolution, the process of dynamical changes ( Fig. 35.21) in the sand dune with a change in threshold regulatory parameter (e.g. Fig. 35.22) is modeled qualitatively for a better understanding. An equation to compute metric universality by considering attracting interslipface angles is also proposed. Avalanche size distribution in such a numerically simulated sand dune dynamics have also been studied.

Geospatial Computing and Visualization
Mathematical morphology not only provides robust solutions in terrestrial pattern retrieval, analysis, and modeling and simulations but also provides numerous insights worth exploring to find solutions for the challenges encountered in GISci. In recent works-that include (i) binary and grayscale morphological interpolations, SKIZ, WSKIZ and applications in spatiaotemporal visualizations, conversion of point-specific variable data into contiguous zonal maps (Rajashekara et al. 2012), morphing (Sagar and Lim 2015a, b) and variable-specific cartogram generation (Sagar 2014a, b), (ii) volumetric visualization of topologically significant components such as pore-bodies, pore-throats, and pore-channels (Teo and Sagar 2005Sagar , 2006, and (iii) spatial reasoning, planning, and interactions Vardhan et al. 2013;Sagar 2018)-one can realize on how robust approaches could be developed by considering mathematical morphological transformations. Fig. 35.22 a A 1-D map plotted between θ t+1 versus θ t for sand dune case λ = 4 and b return map plotted between θ t+1 − θ t versus θ t+2 − θ t+1 for sand dune case with λ = 4. (From Sagar et al. 2003a, b;Sagar and Venu 2001)

Morphological Interpolations
This subsection provides the applications of binary and grayscale morphological interpolations in hierarchical computation of morphological medians and in morphing, and the applications of SKIZ and WSKIZ in conversion of point-specific variable data into contiguous zonal map, and generation of variable-specific contiguous cartograms.

Computation of Hierarchical Morphological Medians
Hausdorff-distance based (i) spatial relationships between the maps possessing bijection for categorization and (ii) nonlinear spatial interpolation in visualization of spatiotemporal behavior are proposed and demonstrated. This work (Sagar 2010(Sagar , 2014aChalla et al. 2016) concerns the development of frameworks with a goal to understand spatial and/or temporal behaviors of certain evolving and dynamic geomorphic phenomena. In (Sagar 2010), we have shown (i) how Hausdorff-Dilation and Hausdorff-Erosion metrics could be employed to categorize the time-varying spatial phenomena, and (ii) how thematic maps in time-sequential mode ( Fig. 35.23a) can be used to visualize the spatiotemporal behaviour of a phenomenon, by recursive generation of median elements (Fig. 35.23b). Spatial interpolation, that was earlier seen as a global transform, is extended in Lim and Sagar (2008) by introducing bijection to deal with even connected components. This aspect solves problems of global nature in spatial-temporal GIS. Spatial Interpolation technique is found useful for spatial-temporal GIS and is demonstrated with validation on epidemic spread maps collected for eleven years between 1896 and1906 (Fig. 35.23a-k, upper left panel). Morphological medians are computed between the epidemic spread maps staggered at two-year interval upper right panel). Further morphological medians are computed in a hierarchical manner between every two epidemic spread maps of successive years ( Fig. 35.23a, b in the lower panel).

Grayscale Morphological Interpolation and Morphing
The computation of morphological medians between the thematic maps (binary images) demonstrated in the earlier subsection could be extended to the spatial fields (functions, e.g.: DEMs). This extended version is termed as grayscale morphological interpolation. We have demonstrated the application of grayscale morphological interpolations, computed hierarchically between the spatial fields ( Fig. 35.24), to metamorphose a source-spatial field into a target-spatial field. Grayscale morphological interpolations are computed in a hierarchical manner ( Fig. 35.25) with respect to non-flat structuring element, and found that the morphing, shown for transform source-spatial field into target-spatial field, created with respect to non-flat structuring element is more appropriate as the transition of source-spatial field into the target-spatial field across discrete time steps is smoother than that of the morphing shown with respect to flat structuring element (Sagar and Lim 2015a, b). This morphing shown via nonlinear grayscale morphological interpolations is of immense value in geographical information science, and in particular spatiotemporal geo-visualization.

Point-to-Polygon Conversion via WSKIZ
Data about many variables are available as numerical values at specific geographical locations in a noncontiguous form. We develop a methodology based on mathematical morphology to convert point-specific data into polygonal data. This methodology relies on weighted skeletonization by zone of influence (WSKIZ). This WSKIZ determines the points of contact of multiple frontlines propagating, from various points (e.g.: gauge stations) spread over the space, at the travelling rates depending upon the variable's strength. We demonstrate this approach for converting rainfall data available at specific rain gauge locations (points) (Fig. 35.26a) into a polygonal map (Fig. 35.26b) that shows spatially distributed zones of equal rainfall in a contiguous form (Rajashekara et al. 2012).

Cartograms via WSKIZ
Visualization of geographic variables as spatial objects of size proportional to variable strength is possible via generating variable-specific cartograms. We developed a methodology based on mathematical morphology to generate contiguous cartograms. This approach determines the points of contact of multiple frontlines propagating, from centroids of various planar sets (states), at the travelling rates depending upon the variable's strength (Fig. 35.27a-d).
The contiguous cartogram generated via this algorithm preserves the global shape, and local shapes, and yields minimal area-errors. It is inferred from the comparative error analysis that this approach could be further extended by  1 -A 34 ), b Rainfall zonal map generated by having various possible propagation speeds, and the variable strengths in terms of propagation speeds exploring the applicability of additional characteristics of structuring element, which controls the dilation propagation speed and direction of dilation while generating variable-specific cartograms, to minimize the local shape errors, and area-errors. This algorithm addresses a decade-long problem of preservation of global and local shapes of cartograms. This approach was extended to generate a cartogram for a variable population to demonstrate the proposed approach. Further, the population cartograms for the USA generated via four other approaches (Kocmoud 1997;Keim et al. 2004;Gastner and Newman 2004;Gusein-Zade and Tikunov 1993) are compared with the morphology-based cartogram (Fig. 35.28a-f) in terms of errors with respect to area, local shape, and global shape. This approach for generating cartograms preserves the global shape at the expense of compromising with area-errors. It is inferred from the comparative error analysis that the proposed morphology-based approach could be further extended by exploring the applicability of additional characteristics of probing rule, which controls the dilation propagation speed and direction of dilation while performing WSKIZ, to minimize the local shape errors, and area-errors.

Visualization of Topological Components in a Volumetric Space
Heterogeneous material is one that is composed of domains of different materials (phases). The aim of this module is to show how geometric descriptors derived via mathematical morphology and fractal analysis vary between the porous phases isolated from varied types of rocks at various spatial and spectral scales. It is evident from the recent works on Fontainebleau sandstone that the characteristics derived through computer assisted mapping and computer tomographic analysis were well correlated with the physical properties such as porosity, permeability, and conductance. Whatever the physical processes involved in altering the porous phase of material, we propose to emphasise quantifying the complexity of porous phase in both 2-D and 3-D domains. From a petrologic study perspective, such a quantitative characterization in both two-and three-dimensional spaces is of current interest.
Just like how CT scanning mechanism is employed to scan the brain to study several neurophysiologic processes, one can also employ such a CT-scanning mechanism, besides already existing scanning methods, to scan the rock bodies and store the scanned information in layered forms. Each layer depicts rock's cross sectional information at specific depth. Retrieval of three significant geometric and/ or topologic components, describing organisation of porous medium, that include (a) pore channel, (b) pore throat, and (c) pore body in both 2-D and 3-D spaces is an important task. A 3-D fractal pore ( Fig. 35.29a, b) simulated in such a way that it mimics the stacked layers of pore sections is converted into 3-D pore channel  (Kocmoud 1997), c cartodraw (Keim et al. 2004), d Gastner-Newman cartogram (Gastner and Newman 2004), e Area cartogram of the United States, with each county rescaled in proportion to its population (Gusein-Zade and Tikunov 1993), and f morphology-based cartogram (Sagar 2014a, b). U.S. population cartogram by Gusein-Zade and Tikunov (e: Reproduced with permission from Gusein-Zade andTikunov 1993, page 172, Fig. 35.1, © 1993 American Congress on Surveying and Mapping). The color coding given in Fig. 35.28a is similar to that of Fig. 35.28f network ( Fig. 35.29c, d), 3-D pore throats ( Fig. 35.29e, f) and 3-D pore bodies ( Fig. 35.29g, h). These decomposed pore features that are of topological importance would shed the light to derive geometric relations which further can be related with that of physical properties of porous structure.

Spatial Reasoning and Planning
Mathematical morphology based algorithms developed and demonstrated shown in this subsection include to determine (i) strategically significant set(s) for spatial reasoning and planning, (ii) directional spatial relationship between areal objects (e.g.: lakes, states, sets) via origin-specific dilations, and (iii) spatial interactions via modified gravity model.

Strategically Significant State (S)
Identification of a strategically significant set from a cluster of adjacent and/or non-adjacent sets depends upon the parameters that include size, shape, degrees of adjacency and contextuality, and distance between the sets. An example of cluster of sets includes continents, countries, states, cities, etc. The spatial relationships, deciphered via the parameters cited above, between such sets possess varied spatial complexities. Hausdorff dilation distance between such sets is considered to derive automatically the strategic set among the cluster of sets. The (i) dilation distances, (ii) length of boundary being shared, and (iii) degrees of contextuality and adjacency between origin-set and destination sets, which together provide solutions to derive strategically significant sets with respect to distance, degree of contextuality, degree of adjacency and length of boundary being shared. Simple mathematical morphologic operators and certain logical operations are employed in this study. Results drawn (Fig. 35.30)-by applying the proposed framework on a case study that involves spatial sets (states) decomposed from a spatial map depicting the country of India-are shown in Fig. 35.30. This approach has been applied on data depicting randomly spread surface water bodies ( Fig. 35.31a, b) and their corresponding zones of influence ( Fig. 35.31c, d) within a subbasin to detect the strategically significant water body and zone of influence ( Fig. 35.32a, b).

Directional Spatial Relationship
We provide an approach to compute origin-specific morphological dilation distances between planar sets (e.g.: areal objects, spatially represented countries, states, cities, lakes) to further determine the directional spatial relationship between sets. Origin chosen for a structuring element that yields shorter dilation distance than that of the other possible origins of structuring element determines the directional spatial relationship between A i (origin-set) and A j (destination set). We demonstrate this approach on a cluster of spatial sets (states) decomposed from a spatial map depicting country India (Fig. 35.33a). This approach has potential to extend to any number (type) of sets on Euclidean space.

Spatial Interactions
Hierarchical structures include spatial system (e.g. river basin), clusters of a spatial system (e.g. watersheds of a river basin), zones of a cluster (e.g. subwatersheds of a watershed), and so on. Variable-specific classification of the zones of a cluster of zones within a spatial system is the main focus of this work on spatial interactions. Variable-specific (e.g. resources) classification of zones is done by computing the levels of interaction between the ith and jth zones. Based on a heuristic argument,  Figure, first 10 strategically significant states (please refer to the legend on each panel) are shown in different colors. These strategically significant sets with respect to a boundary being shared, b shortest distance from origin to destination states, c shortest total distance from destination states to origin state, d contextuality, e Hausdorff dilation distance, f spatial complexity involved in length of the boundary being shared, g spatial complexity in terms of contextuality, h spatial complexity in terms of distance from origin to destination states, i spatial complexity in terms of distance from destination states to origin state, j spatial complexity in terms of Hausdorff dilation distance from origin state to destination states. States with color-codes denote first ten strategically significant states, and the region with white space represents the states that are strategically non-significant with ranks starting from eleven to twenty eight  we proposed a modified gravity model for the computations of levels of interaction between the zones. This argument is based on the following two facts: (i) the level of interaction between the ith and jth zones, with masses m i and m j is direction-dependent, and (ii) the level of interactions between the ith and jth zones with corresponding masses, situated at strategically insignificant locations would be much different (lesser) from that of the ith and jth zones with similar masses but situated at strategically highly significant locations. With the support of this argument, we provide a modified gravity model by incorporating the asymmetrical distances, and the product of location significance indexes of the corresponding zones. This modified gravity model yields level of interaction between the two zones that satisfies the realistic characteristic that is level of interaction between the zones is direction-dependent.
Each state of India is designated with ranks in terms of its (i) location significance index, (ii) strengths of interaction of all states with a specific state, (iii) strengths of interaction with other states, and (iv) strength out of (ii) and (iii) (Fig. 35.34a-d). Further by employing a modified gravity model, 28 states (X 1 to X 28 ) of India (Fig. 35.30A) are paired into best interacting to least interacting pairs with respect to areal extents of states as a variable (Fig. 35.35a-j).

Conclusions
From our attempts since early 1990s, we could clearly see a great potential for mathematical morphological transformations in the three aspects (retrieval, analysis and reasoning, and modeling) of relevance to geosciences and GISci. This chapter provided a brief illustrative review on how mathematical morphology could be applied to deal with varied topics of relevance to mathematical geosciences and geographical information science communities. Reader is encouraged to dig cited references for more details. Our studies show that there exist several open problems of relevance to the mathematical geosciences community. These open problems could be well-handled by mathematical morphology. Some of the recent advances of mathematical morphology and their applications in spatial data segmentation and morphological clustering were discussed. Applications of both classical and modern mathematical morphological transformations in geosciences and GISci are yet to be seen in full-length. It is our hope that most visible and highly distinguished scientists who are active in the IAMG activities would spread a word wide across and would spur the interest of young researchers to take the strides forward.
Teo LL, Sagar BSD (2005) Reconstruction of pore space from pore connectivity network via morphological transformations. J Microsc (Oxford) 219(Pt 2):76-85 Teo LL, Sagar BSD (2006) Modeling, characterization of pore-channel, throat and body. Discret Dyn Nat Soc (an International Journal) (Article ID 89280) 1-24 Teo LL, Radhakrishnan P, Sagar BSD (2004) Morphological decomposition of sandstone pore-space: fractal power-laws. Chaos Solitons & Fractals Vardhan SA, Sagar BSD, Rajesh N, Rajashekara HM (2013) Automatic detection of orientation of mapped units via directional granulometric analysis. IEEE Geosci Remote Sens Lett 10(6):1449-1453 Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made. The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.