- Email: [email protected]
Digital Elevation Model Data Analysis Using the Contact Surface Area
GRAPHICAL MODELS AND IMAGE PROCESSING
Vol. 60, No. 2, March, pp. 166–172, 1998 ARTICLE NO. IP970463
NOTE Digital Elevation Model Data Analysis Using the Contact Surface Area Ernesto Bribiesca Department of Computer Science, Institute of Research in Applied Mathematics and Systems, Universidad Nacional Aut´onoma de M´exico, Apdo. 20-726, M´exico, D.F., 01000 E-mail: [email protected] Received May 29, 1997; accepted December 5, 1997
We present an approach for analyzing digital elevation model (DEM) data using the concept contact surface area and mathematical morphology. DEMs are digital representations of the Earth’s surface. Generally speaking a DEM is generated as a uniform rectangular grid organized in profiles. In order to analyze DEM data by means of binary morphology, the models are represented as binary solids composed of regular polyhedrons (voxels). In the content of this work, we use morphological operators to erode DEMs, simplify binary solid data, preserve essential shape characteristics, understand shape in terms of a decomposition, and identify object features. This is shown by means of some simple examples. We define the contact surface area for DEMs composed of voxels. The contact surface area corresponds to the sum of the contact surface areas of the neighboring voxels of DEMs. A relation between the area of the surface enclosing the volume and the contact surface area is presented. The definition of contact surface area permits us to obtain a fast and efficient method for plotting models composed of a large number of voxels. °c 1998 Academic Press
1. INTRODUCTION Mathematical morphology plays an important role in computer vision [1–5]. This work deals with binary morphology and digital elevation model (DEM) data. DEMs have a large number of applications [6]. Some of these applications include production of slope, aspect, hill-shaded maps, engineering calculations, line-of-sight calculations, urban and regional planning, navigation, geomorphology, components in complex models, and geographic information systems. When we use morphological operations on DEM data the number of applications may be increased; this is due to morphological operators permit to erode DEMs, simplify DEM data, preserve essential shape characteristics, understand shape in terms of a decomposition, and identify model features. Thus, the importance of mathematical morphology on DEM data is evident. In order to use morphological operators on DEMs it is necessary to select an appropriate representation. In this work the 166 1077-3169/98 $25.00 c 1998 by Academic Press Copyright ° All rights of reproduction in any form reserved.
DEMs are represented as three-dimensional (3D) arrays of cells (voxels) which are marked as filled with matter [7]. Recently, a method for measuring 3D shape similarity using this representation was presented in Ref. [8]. Several authors have been using different kinds of representations for solids: rigid solids represented by their boundaries or enclosing surfaces are shown in Refs. [9] and [10]; constructive solid geometry schemes are presented in Refs. [11] and [12]; generalized cylinders as 3D volumetric primitives are shown in Refs. [13–15]; and superquadrics are shown in Ref. [16]. In order to plot our results efficiently, we present the definition of contact surface area for DEMs as composed of a larger number of voxels. Furthermore, we present the relation between the area of the surface enclosing the volume and the contact surface area. This definition of area is the extension in 3D domain of the concept termed contact perimeter [17]. The organization of the paper is as follows: Section 2 contains a set of definitions which describe the contact surface area and the method used for plotting, Section 3 presents DEM data characteristics, Section 4 describes binary morphology applied to DEMs and some simple examples, and finally, Section 5 presents conclusions. 2. CONCEPTS AND DEFINITIONS First, we present the definition of contact surface area and subsequently the morphological operations used on DEMs. We use volumetric representation for DEMs by means of spatial occupancy arrays. Thus, the DEMs are represented as 3D arrays of voxels which are marked as filled with matter. Furthermore, shape is referred to as shape-of-object and object is considered as a geometric solid composed of voxels. In the content of this work, area is a numerical value expressing 2D extent in a plane, or sometimes it is used to mean the interior region itself [18]. Another consideration is the assumption that an entity has been isolated from the real world. This is called the DEM and is defined as a result of previous processing. Also, the length of all edges of voxels is considered equal to 1.
167
DIGITAL ELEVATION MODEL DATA ANALYSIS
or faces of the regular polyhedron (voxel) times the area of the used face. Geometrically, it means that the sum of two times the contact surface area plus the enclosing surface area is equal to the total sum of the polygon areas of all the voxels of the DEM. By Eq. (1), the contact surface area is defined as follows: Ac =
FIG. 1. An example of a binary solid: (a) the binary solid composed of 9572 voxels; (b) the contact surfaces of the solid; (c) the binary solid without inner faces.
2.1. Surfaces In this section, we define the contact surfaces for DEMs composed of voxels. Also, we define the relation between the contact surface area and the area of the surface enclosing the volume. This relation between the areas of the surfaces can be used in different polyhedrons, which cover up space. In this case, we present the above mentioned using regular hexahedrons (voxels). Figure 1a displays a binary solid composed of 9572 voxels. 2.1.1. The area of the enclosing surface. The area A of the enclosing surface of a DEM composed of a finite number n of voxels corresponds to the sum of the areas of the external plane polygons of the voxels which form the visible faces of the DEM. The binary solid shown in Fig. 1a has an enclosing surface area equal to 5820. 2.1.2. The contact surface area. The contact surface area Ac of a DEM composed of a finite number n of voxels corresponds to the sum of the areas of the contact surfaces which are common to two voxels. Figure 1b shows the contact surfaces of the binary solid shown in Fig. 1a; its contact surface area is equal to 25,806. 2.1.3. The relation between the areas of the enclosing surface and the contact surface. For any DEM composed of n voxels, the equation 2Ac + A = Fn
(1)
is satisfied, where Ac is the contact surface area, A is the area of the enclosing surface, and F is the number of the plane polygons
(Fn − A) . 2
(2)
In the binary solid shown in Fig. 1a, n = 9572, F = 6 (this is due to the voxel having six faces), and A = 5820. Substituting these values in Eq. (2) we obtain Ac = 25,806, which is shown in Fig. 1b. The enclosing surface area of a solid corresponds to the sum of the areas of visible faces of the solid. On the other hand, the contact surfaces correspond to the hidden faces of the solid. Therefore, when a solid is plotted, the contact surfaces must be eliminated from the plotting, this decreases greatly the computation. Thus, the number of hidden faces which must be eliminated from any plotting is equal to 2Ac . Subsequently, the visible faces of the solid are represented in a standard vector file format. This produces a wireframe plotting, where all lines are represented, including those that would be hidden by faces. To eliminate these hidden lines from the plotting, the HIDE command is used (which is programmed in most 3D CAD software). The above mentioned produces objects which are plotted as solids. Therefore, for the binary solid shown in Fig. 1a, the number of hidden faces which is eliminated from the plotting is equal to 51,612. Notice that the binary solid presented in Fig. 1c was cut across to show its interior without inner faces. The method mentioned above is only for plotting; the morphological operators used consider all information contained in the binary solids. 3. DIGITAL ELEVATION MODEL DATA In recent years, the Instituto Nacional de Estad´ıstica, Geograf´ıa e Inform´atica, M´exico (INEGI) has built up digital files corresponding to DEM data of the whole country, which were stored in just one compact disk. The digitalization of these models is based on 1 : 250,000 scale contours. In the content of this work, we use DEM data provided by INEGI. Figure 2 illustrates the DEM of El Valle de M´exico using a 3D mesh of 150 × 150 elements. The left-hand side of the figure corresponds to the north of El Valle de M´exico, the right-hand side to the south, the upper side to the east, and the lower side to the west. On the left-hand side there is a group of hills called La Sierra de Guadalupe, and on the right-hand side there are more hills which correspond to El Ajusco. The elevation data values of the models presented in this study were increased to enhance their characteristics. Figure 3a shows the DEM of La Sierra de Guadalupe represented by a 3D mesh of 150 × 150 elements. Figure 3b illustrates La Sierra de Guadalupe by means of voxels, where n = 150048, A = 59944, Ac = 420172, and the number of hidden faces is 840,344. The method for transforming DEM data
168
ERNESTO BRIBIESCA
FIG. 2. The DEM of El Valle de M´exico represented by a 3D mesh of 150 × 150 elements.
into voxels is as follows: (1) calculate the minimum elevation of the given DEM; (2) subtract the minimum elevation from all elevations of the model and increase them by one; and (3) generate a 3D array of voxels considering the same resolution of the model, where each elevation value is equivalent to the number of voxels in that position, which are located at spatial coordinates (row, column, slide). Thus, each profile of a given DEM corresponds to a slide of its 3D array of voxels.
4. BINARY MORPHOLOGY APPLIED TO DEMs In this section binary morphological transformations are applied to sets of three dimensions. Thus, sets in Euclidean 3-space denote binary solids as DEMs. Jain [19] stated that “morphological processing refers to certain operations where an object is hit with a structuring element and thereby reduced to a more revealing shape.” The two basic morphological operations are
FIG. 3. The DEM of La Sierra de Guadalupe (a) represented by a 3D mesh of 150 × 150 elements and (b) represented by voxels, composed of 150,048 voxels.
169
DIGITAL ELEVATION MODEL DATA ANALYSIS
dilation and erosion [1]; these operations are related to shape properties. The reader can find a systematic theoretical treatment of mathematical morphology in Refs. [1, 2]. In the content of this work, the morphological operation notation used is based on Ref. [3].
A⊕B =
[
Ba .
(4)
a∈A
In our case, the set A corresponds to the DEM and set B corresponds to the structuring element. 4.2. Binary Erosion
4.1. Binary Dilation If A and B are sets in E 3 with elements a and b, respectively, a = (a1 , a2 , a3 ) and b = (b1 , b2 , b3 ), then the dilation is the set of all possible vector sums of pairs of elements. Thus, the dilation of A by B (A ⊕ B) is defined by A ⊕ B = {c ∈ E | c = a + b for some a ∈ A and b ∈ B}. (3) 3
Dilation is also expressed as a union of translations of the structuring element, i.e.,
If A and B are sets in E 3 with elements a and b, respectively, a = (a1 , a2 , a3 ) and b = (b1 , b2 , b3 ), then the erosion of A by B (A ª B) is the set of all elements x for which x + b ∈ A for every b ∈ B and is defined by A ª B = {x ∈ E 3 | x + b ∈ A for every b ∈ B}.
(5)
Also, the erosion operation can be expressed as the intersection of translations of A by the elements −b with b ∈ B. Symbolically,
FIG. 4. Erosions: (a) the structuring element; (b) the erosion of the DEM presented in Fig. 3b; (c) the erosion of (b); (d) the erosion of (c).
170
ERNESTO BRIBIESCA
FIG. 5. Boundary voxels: (a) the structuring element; (b) the boundary voxels of the DEM obtained in Fig. 4c.
AªB =
\
A−b ,
(6)
erosion of the DEM shown in Fig. 3b. Figures 4c and 4d present the next consecutive steps of erosions.
where the set A corresponds to the DEM and set B corresponds to the structuring element. Erosion is the morphological dual of dilation. Figure 3b displays the DEM with corresponds to the set A and Fig. 4a shows the structuring element which corresponds to the set B. The origin of the structuring element shown in Fig. 4a is the central voxel; i.e., this central voxel will be eliminated when any neighboring voxel is zero. Figure 4b shows the
4.2.1. Boundary voxels of DEMs. Using binary erosion it is possible to find the boundary voxels of DEMs. Figure 5a shows the structuring element; in this case the central voxel will be eliminated when all its neighboring voxels have a value of 1. Figure 5b shows the boundary voxels of the DEM obtained in Fig. 4c. Notice that the DEM shown in Fig. 5b was cut across to present its boundary voxels and its interior. Figure 6 shows a
b∈B
FIG. 6. A zoom view of the DEM shown in Fig. 5b; notice that this view is presented in perspective.
171
DIGITAL ELEVATION MODEL DATA ANALYSIS
FIG. 7. An inner view of the DEM shown in Fig. 5b; notice that this view is presented in perspective.
zoom view of the DEM shown in Fig. 5b; notice that this view is presented in perspective. Finally, Fig. 7 presents an inner view of this model.
ACKNOWLEDGMENTS I thank Mar´ıa Garza for her suggestions for improving the manuscript. DEM data used in this study was provided by INEGI.
5. CONCLUSIONS
REFERENCES
In the content of this work, we presented a concept termed contact surface area, which is the extension in 3D domain of the contact perimeter defined in Ref. [17]. The contact surface area allowed us to obtain a fast and efficient method for plotting DEMs composed of a large number of voxels. Using the definition of contact surface area and the classical concepts of mathematical morphology, we obtained an interesting and powerful tool for processing DEM data. Some simple examples of basic morphological operations were presented, which showed that it is possible to increase and expand DEM applications into more complex models. All the examples presented were processed by a PC with a 80486 processor. Suggestions for further work include simulation of natural events and the use of hit-and-miss transform on DEM data for performing template matching, thinning, thickening, and centering.
1. J. Serra, Image Analysis and Mathematical Morphology, Vol. 1, Academic Press Ltd., London, 1982. 2. J. Serra, Image analysis and mathematical morphology, in Theoretical Advances (J. Serra, Ed.), Vol. 2, Chaps. 1 and 2, Academic Press Ltd., London, 1988. 3. R. M. Haralick and L. G. Shapiro, Computer and Robot Vision, Vol. I, Addison–Wesley, Reading, MA, 1992. 4. R. M. Haralick, S. R. Sternberg, and X. Zhuang, Grayscale morphology, in Proc. IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Miami Beach, 1986, pp. 543–550. 5. T. R. Crimmins and W. R. Brown, Image algebra and automatic shape recognition, IEEE Trans. Aerospace Electron. Systems AES-21, 1985, 60–69. 6. L. Kikuchi, J. A. Guevara, D. Mark, and D. F. Marble, Rapid display of digital elevation models in a mini-computer environment, in Proc. ISPRS Commission IV, Environmental Assessment and Resource Management, Crystal City, Virginia, 1982, pp. 297–307. 7. D. H. Ballard and C. M. Brown, Computer Vision, Prentice-Hall, Englewood Cliffs, NJ, 1982.
172
ERNESTO BRIBIESCA
8. E. Bribiesca, Measuring 3-D shape similarity using progressive transformations, Pattern Recognition 29, 1996, 1117–1129.
14. B. I. Soroka and R. K. Bajcsy, Generalized cylinders from serial sections, in Proc. 3rd IJCPR, 1976, pp. 734–735.
9. A. A. G. Requicha, Representations of rigid solid objects, Comput. Surv. 12, 1980, 4.
15. R. Brooks, Model-based 3-D interpretations of 2-D images, IEEE Trans. Pattern Anal. Mach. Intelligence 5(2), 1983, 140–150.
10. P. Besl and R. Jain, Three-dimensional object recognition, ACM Comput. Surv. 17(1), 1985, 75–145.
16. A. Pentland, Perceptual organization and the representation of natural form, Artificial Intelligence 28, 1986, 293–331.
11. H. B. Voelcker and A. A. G. Requicha, Geometric modeling of mechanical parts and processes, Computer 10, 1977, 48–57.
17. E. Bribiesca, Measuring 2D shape compactness using the contact perimeter, Comput. Math. Appl. 33, 1997, 1–9.
12. J. W. Boyse, Data structure for a solid modeller, NSF Workshop on the Representation of Three-Dimensional Objects, Univ. Pennsylvania, 1979. 13. B. I. Soroka, Generalised cylinders from parallel slices, in Proc. PRIP, 1979, pp. 421–426.
18. W. Karush, Webster’s New World Dictionary of Mathematics, Simon & Schuster, New York, 1989. 19. A. K. Jain, Fundamentals of Digital Image Processing, Prentice–Hall, Englewood Cliffs, NJ, 1989.
Vol. 60, No. 2, March, pp. 166–172, 1998 ARTICLE NO. IP970463
NOTE Digital Elevation Model Data Analysis Using the Contact Surface Area Ernesto Bribiesca Department of Computer Science, Institute of Research in Applied Mathematics and Systems, Universidad Nacional Aut´onoma de M´exico, Apdo. 20-726, M´exico, D.F., 01000 E-mail: [email protected] Received May 29, 1997; accepted December 5, 1997
We present an approach for analyzing digital elevation model (DEM) data using the concept contact surface area and mathematical morphology. DEMs are digital representations of the Earth’s surface. Generally speaking a DEM is generated as a uniform rectangular grid organized in profiles. In order to analyze DEM data by means of binary morphology, the models are represented as binary solids composed of regular polyhedrons (voxels). In the content of this work, we use morphological operators to erode DEMs, simplify binary solid data, preserve essential shape characteristics, understand shape in terms of a decomposition, and identify object features. This is shown by means of some simple examples. We define the contact surface area for DEMs composed of voxels. The contact surface area corresponds to the sum of the contact surface areas of the neighboring voxels of DEMs. A relation between the area of the surface enclosing the volume and the contact surface area is presented. The definition of contact surface area permits us to obtain a fast and efficient method for plotting models composed of a large number of voxels. °c 1998 Academic Press
1. INTRODUCTION Mathematical morphology plays an important role in computer vision [1–5]. This work deals with binary morphology and digital elevation model (DEM) data. DEMs have a large number of applications [6]. Some of these applications include production of slope, aspect, hill-shaded maps, engineering calculations, line-of-sight calculations, urban and regional planning, navigation, geomorphology, components in complex models, and geographic information systems. When we use morphological operations on DEM data the number of applications may be increased; this is due to morphological operators permit to erode DEMs, simplify DEM data, preserve essential shape characteristics, understand shape in terms of a decomposition, and identify model features. Thus, the importance of mathematical morphology on DEM data is evident. In order to use morphological operators on DEMs it is necessary to select an appropriate representation. In this work the 166 1077-3169/98 $25.00 c 1998 by Academic Press Copyright ° All rights of reproduction in any form reserved.
DEMs are represented as three-dimensional (3D) arrays of cells (voxels) which are marked as filled with matter [7]. Recently, a method for measuring 3D shape similarity using this representation was presented in Ref. [8]. Several authors have been using different kinds of representations for solids: rigid solids represented by their boundaries or enclosing surfaces are shown in Refs. [9] and [10]; constructive solid geometry schemes are presented in Refs. [11] and [12]; generalized cylinders as 3D volumetric primitives are shown in Refs. [13–15]; and superquadrics are shown in Ref. [16]. In order to plot our results efficiently, we present the definition of contact surface area for DEMs as composed of a larger number of voxels. Furthermore, we present the relation between the area of the surface enclosing the volume and the contact surface area. This definition of area is the extension in 3D domain of the concept termed contact perimeter [17]. The organization of the paper is as follows: Section 2 contains a set of definitions which describe the contact surface area and the method used for plotting, Section 3 presents DEM data characteristics, Section 4 describes binary morphology applied to DEMs and some simple examples, and finally, Section 5 presents conclusions. 2. CONCEPTS AND DEFINITIONS First, we present the definition of contact surface area and subsequently the morphological operations used on DEMs. We use volumetric representation for DEMs by means of spatial occupancy arrays. Thus, the DEMs are represented as 3D arrays of voxels which are marked as filled with matter. Furthermore, shape is referred to as shape-of-object and object is considered as a geometric solid composed of voxels. In the content of this work, area is a numerical value expressing 2D extent in a plane, or sometimes it is used to mean the interior region itself [18]. Another consideration is the assumption that an entity has been isolated from the real world. This is called the DEM and is defined as a result of previous processing. Also, the length of all edges of voxels is considered equal to 1.
167
DIGITAL ELEVATION MODEL DATA ANALYSIS
or faces of the regular polyhedron (voxel) times the area of the used face. Geometrically, it means that the sum of two times the contact surface area plus the enclosing surface area is equal to the total sum of the polygon areas of all the voxels of the DEM. By Eq. (1), the contact surface area is defined as follows: Ac =
FIG. 1. An example of a binary solid: (a) the binary solid composed of 9572 voxels; (b) the contact surfaces of the solid; (c) the binary solid without inner faces.
2.1. Surfaces In this section, we define the contact surfaces for DEMs composed of voxels. Also, we define the relation between the contact surface area and the area of the surface enclosing the volume. This relation between the areas of the surfaces can be used in different polyhedrons, which cover up space. In this case, we present the above mentioned using regular hexahedrons (voxels). Figure 1a displays a binary solid composed of 9572 voxels. 2.1.1. The area of the enclosing surface. The area A of the enclosing surface of a DEM composed of a finite number n of voxels corresponds to the sum of the areas of the external plane polygons of the voxels which form the visible faces of the DEM. The binary solid shown in Fig. 1a has an enclosing surface area equal to 5820. 2.1.2. The contact surface area. The contact surface area Ac of a DEM composed of a finite number n of voxels corresponds to the sum of the areas of the contact surfaces which are common to two voxels. Figure 1b shows the contact surfaces of the binary solid shown in Fig. 1a; its contact surface area is equal to 25,806. 2.1.3. The relation between the areas of the enclosing surface and the contact surface. For any DEM composed of n voxels, the equation 2Ac + A = Fn
(1)
is satisfied, where Ac is the contact surface area, A is the area of the enclosing surface, and F is the number of the plane polygons
(Fn − A) . 2
(2)
In the binary solid shown in Fig. 1a, n = 9572, F = 6 (this is due to the voxel having six faces), and A = 5820. Substituting these values in Eq. (2) we obtain Ac = 25,806, which is shown in Fig. 1b. The enclosing surface area of a solid corresponds to the sum of the areas of visible faces of the solid. On the other hand, the contact surfaces correspond to the hidden faces of the solid. Therefore, when a solid is plotted, the contact surfaces must be eliminated from the plotting, this decreases greatly the computation. Thus, the number of hidden faces which must be eliminated from any plotting is equal to 2Ac . Subsequently, the visible faces of the solid are represented in a standard vector file format. This produces a wireframe plotting, where all lines are represented, including those that would be hidden by faces. To eliminate these hidden lines from the plotting, the HIDE command is used (which is programmed in most 3D CAD software). The above mentioned produces objects which are plotted as solids. Therefore, for the binary solid shown in Fig. 1a, the number of hidden faces which is eliminated from the plotting is equal to 51,612. Notice that the binary solid presented in Fig. 1c was cut across to show its interior without inner faces. The method mentioned above is only for plotting; the morphological operators used consider all information contained in the binary solids. 3. DIGITAL ELEVATION MODEL DATA In recent years, the Instituto Nacional de Estad´ıstica, Geograf´ıa e Inform´atica, M´exico (INEGI) has built up digital files corresponding to DEM data of the whole country, which were stored in just one compact disk. The digitalization of these models is based on 1 : 250,000 scale contours. In the content of this work, we use DEM data provided by INEGI. Figure 2 illustrates the DEM of El Valle de M´exico using a 3D mesh of 150 × 150 elements. The left-hand side of the figure corresponds to the north of El Valle de M´exico, the right-hand side to the south, the upper side to the east, and the lower side to the west. On the left-hand side there is a group of hills called La Sierra de Guadalupe, and on the right-hand side there are more hills which correspond to El Ajusco. The elevation data values of the models presented in this study were increased to enhance their characteristics. Figure 3a shows the DEM of La Sierra de Guadalupe represented by a 3D mesh of 150 × 150 elements. Figure 3b illustrates La Sierra de Guadalupe by means of voxels, where n = 150048, A = 59944, Ac = 420172, and the number of hidden faces is 840,344. The method for transforming DEM data
168
ERNESTO BRIBIESCA
FIG. 2. The DEM of El Valle de M´exico represented by a 3D mesh of 150 × 150 elements.
into voxels is as follows: (1) calculate the minimum elevation of the given DEM; (2) subtract the minimum elevation from all elevations of the model and increase them by one; and (3) generate a 3D array of voxels considering the same resolution of the model, where each elevation value is equivalent to the number of voxels in that position, which are located at spatial coordinates (row, column, slide). Thus, each profile of a given DEM corresponds to a slide of its 3D array of voxels.
4. BINARY MORPHOLOGY APPLIED TO DEMs In this section binary morphological transformations are applied to sets of three dimensions. Thus, sets in Euclidean 3-space denote binary solids as DEMs. Jain [19] stated that “morphological processing refers to certain operations where an object is hit with a structuring element and thereby reduced to a more revealing shape.” The two basic morphological operations are
FIG. 3. The DEM of La Sierra de Guadalupe (a) represented by a 3D mesh of 150 × 150 elements and (b) represented by voxels, composed of 150,048 voxels.
169
DIGITAL ELEVATION MODEL DATA ANALYSIS
dilation and erosion [1]; these operations are related to shape properties. The reader can find a systematic theoretical treatment of mathematical morphology in Refs. [1, 2]. In the content of this work, the morphological operation notation used is based on Ref. [3].
A⊕B =
[
Ba .
(4)
a∈A
In our case, the set A corresponds to the DEM and set B corresponds to the structuring element. 4.2. Binary Erosion
4.1. Binary Dilation If A and B are sets in E 3 with elements a and b, respectively, a = (a1 , a2 , a3 ) and b = (b1 , b2 , b3 ), then the dilation is the set of all possible vector sums of pairs of elements. Thus, the dilation of A by B (A ⊕ B) is defined by A ⊕ B = {c ∈ E | c = a + b for some a ∈ A and b ∈ B}. (3) 3
Dilation is also expressed as a union of translations of the structuring element, i.e.,
If A and B are sets in E 3 with elements a and b, respectively, a = (a1 , a2 , a3 ) and b = (b1 , b2 , b3 ), then the erosion of A by B (A ª B) is the set of all elements x for which x + b ∈ A for every b ∈ B and is defined by A ª B = {x ∈ E 3 | x + b ∈ A for every b ∈ B}.
(5)
Also, the erosion operation can be expressed as the intersection of translations of A by the elements −b with b ∈ B. Symbolically,
FIG. 4. Erosions: (a) the structuring element; (b) the erosion of the DEM presented in Fig. 3b; (c) the erosion of (b); (d) the erosion of (c).
170
ERNESTO BRIBIESCA
FIG. 5. Boundary voxels: (a) the structuring element; (b) the boundary voxels of the DEM obtained in Fig. 4c.
AªB =
\
A−b ,
(6)
erosion of the DEM shown in Fig. 3b. Figures 4c and 4d present the next consecutive steps of erosions.
where the set A corresponds to the DEM and set B corresponds to the structuring element. Erosion is the morphological dual of dilation. Figure 3b displays the DEM with corresponds to the set A and Fig. 4a shows the structuring element which corresponds to the set B. The origin of the structuring element shown in Fig. 4a is the central voxel; i.e., this central voxel will be eliminated when any neighboring voxel is zero. Figure 4b shows the
4.2.1. Boundary voxels of DEMs. Using binary erosion it is possible to find the boundary voxels of DEMs. Figure 5a shows the structuring element; in this case the central voxel will be eliminated when all its neighboring voxels have a value of 1. Figure 5b shows the boundary voxels of the DEM obtained in Fig. 4c. Notice that the DEM shown in Fig. 5b was cut across to present its boundary voxels and its interior. Figure 6 shows a
b∈B
FIG. 6. A zoom view of the DEM shown in Fig. 5b; notice that this view is presented in perspective.
171
DIGITAL ELEVATION MODEL DATA ANALYSIS
FIG. 7. An inner view of the DEM shown in Fig. 5b; notice that this view is presented in perspective.
zoom view of the DEM shown in Fig. 5b; notice that this view is presented in perspective. Finally, Fig. 7 presents an inner view of this model.
ACKNOWLEDGMENTS I thank Mar´ıa Garza for her suggestions for improving the manuscript. DEM data used in this study was provided by INEGI.
5. CONCLUSIONS
REFERENCES
In the content of this work, we presented a concept termed contact surface area, which is the extension in 3D domain of the contact perimeter defined in Ref. [17]. The contact surface area allowed us to obtain a fast and efficient method for plotting DEMs composed of a large number of voxels. Using the definition of contact surface area and the classical concepts of mathematical morphology, we obtained an interesting and powerful tool for processing DEM data. Some simple examples of basic morphological operations were presented, which showed that it is possible to increase and expand DEM applications into more complex models. All the examples presented were processed by a PC with a 80486 processor. Suggestions for further work include simulation of natural events and the use of hit-and-miss transform on DEM data for performing template matching, thinning, thickening, and centering.
1. J. Serra, Image Analysis and Mathematical Morphology, Vol. 1, Academic Press Ltd., London, 1982. 2. J. Serra, Image analysis and mathematical morphology, in Theoretical Advances (J. Serra, Ed.), Vol. 2, Chaps. 1 and 2, Academic Press Ltd., London, 1988. 3. R. M. Haralick and L. G. Shapiro, Computer and Robot Vision, Vol. I, Addison–Wesley, Reading, MA, 1992. 4. R. M. Haralick, S. R. Sternberg, and X. Zhuang, Grayscale morphology, in Proc. IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Miami Beach, 1986, pp. 543–550. 5. T. R. Crimmins and W. R. Brown, Image algebra and automatic shape recognition, IEEE Trans. Aerospace Electron. Systems AES-21, 1985, 60–69. 6. L. Kikuchi, J. A. Guevara, D. Mark, and D. F. Marble, Rapid display of digital elevation models in a mini-computer environment, in Proc. ISPRS Commission IV, Environmental Assessment and Resource Management, Crystal City, Virginia, 1982, pp. 297–307. 7. D. H. Ballard and C. M. Brown, Computer Vision, Prentice-Hall, Englewood Cliffs, NJ, 1982.
172
ERNESTO BRIBIESCA
8. E. Bribiesca, Measuring 3-D shape similarity using progressive transformations, Pattern Recognition 29, 1996, 1117–1129.
14. B. I. Soroka and R. K. Bajcsy, Generalized cylinders from serial sections, in Proc. 3rd IJCPR, 1976, pp. 734–735.
9. A. A. G. Requicha, Representations of rigid solid objects, Comput. Surv. 12, 1980, 4.
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