JINGSONG WU and KEVIN AMARATUNGA

Transcription

1 INT. J. GEOGRAPHICAL INFORMATION SCIENCE, 2003 VOL. 17, NO. 3, Research Article Wavelet triangulated irregular networks JINGSONG WU and KEVIN AMARATUNGA Department of Civil and Environmental Engineering, MIT, Cambridge, MA 02139, USA; (Received 30 November 2001; accepted 5 June 2002) Abstract. GIS applications have recently begun to emerge on the Internet. The management of three-dimensional geographic datasets in this distributed environment poses a particularly challenging problem, which highlights the need for a good data representation. This paper presents a new multiresolution data representation: the Wavelet Triangulated Irregular Network (WTIN). Compared to the traditional cell-based Digital Elevation Model (DEM) format and the Triangulated Irregular Network (TIN) format, it is more compact and suitable for scalable distributed GIS services. This format is based on the secondgeneration wavelet theory and is specially designed for geographical height field data. The modified Butterfly scheme is used for constructing the wavelet transform. For every point in the geographic surface, only a single wavelet coefficient is used, which makes the final data representation very efficient and easy to compress. Because the transform used in the data representation is a linear filter operation, the computational efficiency is better than other multiresolution data formats for terrain surfaces. Results from numerical experiments on real data are given to demonstrate that the proposed data representation can be efficiently implemented. The results show that the proposed WTIN data format can provide multiresolution data sets, which achieve significant compression while preserving geographical features. The quality is found to be quite acceptable for geographical terrain representation. 1. Introduction The popularity of the Internet and PCs has brought GIS into people s daily lives. In a network environment, interoperability is a preferred feature for GIS products. It is the most important concept in the Open GIS model proposed by the OpenGIS Consortium (OGC; Based on this model, scalable distributed GIS services can be hosted in a network environment for better use of the data. Here scalable means the same data can be scaled to serve different users, who may have different bandwidth connections, display devices, or demands. However, in order to make this practical, good management is needed for large amounts of GIS data. The most complex GIS data are three-dimensional terrain data. The United States Geological Survey (USGS; provides publicly available GIS data covering most parts of the United States in the cell-based Digital Elevation Model (DEM) format. It is impractical to use this data directly in scalable distributed GIS services due to the large size of the data sets and the inflexibility of DEM. Another International Journal of Geographical Information Science ISSN print/issn online 2003 Taylor & Francis Ltd DOI: /

2 274 J. Wu and K. Amaratunga popular data format for three-dimensional terrain data, the Triangulated Irregular Network (TIN), is also not suitable for direct use in scalable distributed GIS services because it does not have a natural multiresolution structure in its most general form a key requirement for scalable distributed GIS services. Research on multiresolution representations of GIS data can be dated back to the 1970s. Douglas and Peucker (1973) designed a method to simplify a curve for cartographic work, which is the basis of Buttenfield s (1999) research on progressive transmission of vector data over the Internet. For raster data, image processing methods can be used. Wu et al. (2002) have discussed real-time zooming and panning of GIS images based on a fast integer wavelet transform. For irregular setting surface data, such as TINs, several researchers have proposed multiresolution data formats, (De Floriani 1989, De Floriani and Puppo 1995, De Floriani et al. 2000). Their format is based on the Delaunay pyramid and the constrained Delaunay pyramid. They also discussed compression schemes for the connectivity data of TINs. Scarlatos and Pavlidis (1992) extended Douglas and Peucker s method for surface data. They constructed a hierarchical surface based on recursively inserting new vertices in the initial triangles. Jünger and Snoeyink s (1998) work is based on the Delaunay triangulation for progressive visualization. Hoppe s (1998) format is based on edge collapse, vertex split, and view-dependent level-of-detail control. Some of these models require expensive computation; some are not for general data representation but only for rendering. However, one common feature of these new formats is that they all try to use the multiresolution structure to represent complex threedimensional GIS data. When data are in multiresolution format, unnecessary data are not transmitted, so the transmission time is reduced and the bandwidth is saved. This paper presents a new multiresolution data representation, based on the wavelet theory, for irregular setting surface data. The proposed format is compact and easy to implement. Because it has the multiresolution feature, it provides a promising format for scalable distributed GIS services. This work can be used for data generalization (data reduction) in the cartography domain as well. 2. Wavelets for irregular setting data Modern wavelet analysis started in the 1980s. The basic idea is to construct a nested basis family for a function space. The nested bases have an internal hierarchical structure, which makes them more efficient for representing functions in the space. Both continuous functions and discrete signals can be decomposed into multiresolution formats in a wavelet framework. Historically, Fourier representations have been widely used for compressing digital images, such as orthophotos. The JPEG format is an example of an image compression algorithm derived from the Fourier transform. Fourier approaches work quite well because natural images often contain only a limited range of spatial frequencies. For example, a smooth region (such as an area of blue sky) contains only a single spatial frequency i.e. a frequency of zero. On the other hand, Fourier representations do not work well for images that contain a large number of spatially localized features (such as sharp peaks and edges), since spatially localized features contain many spatial frequencies. Wavelet approaches for image compression tend to outperform Fourier approaches because they can efficiently represent both spatially localized features and smooth regions in an image. The superior compression capability of wavelets combined with their natural multiresolution structure makes

3 Wavelet triangulated irregular networks 275 them a good representation for storing images. In fact, wavelets are the basis of the new JPEG-2000 image format. Although there is earlier research on multiresolution data formats in the GIS domain, wavelet analysis did not appear in this domain until recently. Morehart et al. (1999), Wu et al. (2002), and Kiema and Bähr (2001) have applied the traditional wavelet theory in different GIS areas. The traditional theory deals with regularly sampled data (which is the case for digital images) and is called the 1st generation wavelet theory. For the discrete case, the signal can be decimated and iteratively passed through low pass filters and high pass filters to get coarse resolution data (from low pass filters) and wavelet coefficients (from high pass filters). Because wavelet bases can be chosen such that the wavelet coefficients have very small magnitudes, compression schemes can be applied to compress the data in different resolutions. First-generation wavelets are generated by shifting and scaling a single function over a uniform grid. However, a uniform grid is not suitable for many GIS applications. A new theory needs to be developed for irregular setting data, such as the triangulated networks of GIS data. This leads to the development of secondgeneration wavelets, which are designed for the irregular setting and are suitable for arbitrary curve and surface data. Such wavelets are not constrained by the strict shifting and scaling laws that apply to first-generation wavelets. The concept of second-generation wavelets is described in Sweldens s research on the lifting scheme (1997). Donoho (1993) and Lounsbery (1994) have also undertaken related work. The basic idea is to use estimation techniques and optimization criteria to construct wavelets and approximations of the original data. Subdivision is usually used as an estimation technique and constructs the hierarchical structure A basic framework Consider a data set to be partitioned into two groups, called k and m. Using P to express the point coordinates, one can construct an estimation of P based on k P m. P k =E(P m ) (1) The estimation function (or filter in signal processing terminology) E can be a local estimation or a global estimation. A global estimation is generally computationally expensive; therefore, a local estimation using only neighboring points is preferred. After the estimation step, a wavelet term, W, and an approximation term, k A, for the original data can be constructed as: m W =P P k k k (2) A =P +C(W ) m m k The correction function C is a customizable function based on different optimization requirements. No matter what E and C are, an inverse transform can always be constructed as: P =A C(W ) m m k (3) P =W +E(P ) k k m If the original point set can be partitioned into a nested group, then the above process can be iteratively applied to different sets in this group. A nested group has

4 276 J. Wu and K. Amaratunga the following structure: m05m15m25 5mN minki=mi+1, i=0,..., N 1 (4) mn denotes the finest representation of the geometry. mn can be partitioned into mn 1 and kn 1; then mn 1 can be partitioned into mn 2 and kn 2, and so on, until m1 is partitioned into m0 and k0. Note that the superscripts are used to represent different resolutions ( larger numbers represent finer resolution). Based on this nested (or hierarchical) structure of the partition, one can construct wavelets and approximations of the data as: A m N =P mn Gmi=mi 1nki 1, B mi 1 =A mi (mi 1), B ki 1 =A mi (ki 1) W k i 1 =B ki 1 E(B ), i=n, N 1,..., 1 mi 1 A m i 1 =B mi 1 +C(W ki 1 ) (5) Here, B is an intermediate symbol to represent the partitioning result. A is m partitioned into two components: A i m i (mi 1) and A (ki 1), which belong to mi 1 and ki 1 respectively. Based on equation (5), the original mi data P is decomposed m into A N m 0, W k0, W k1,..., W. Equation (5) is the analysis transform, which decomposes the finer representation into a coarser representation plus details. The synthesis kn 1 transform is the inverse transform and is shown in equation (6). The reconstructed A m 0, A m1,..., A mn (=P ) yield a multiresolution representation of the original data. mn GB mi =A mi C(W ki ) B k i =W ki +E(B mi ), i=0, 1,..., N 1 mi+1=minki, A m i+1 (mi)=b mi,a mi+1 (ki)=b ki (6) P m N =A mn In the above derivation of a wavelet representation, the process does not depend on a regular setting for the data; therefore, it can be used in both the regular and irregular setting cases. This is an important advantage of the lifting scheme (Sweldens 1997). If the filters E and C are the same for every point at a given level, the scheme is a uniform scheme. If they also do not change with the resolution, i, the scheme is a stationary scheme as well. However, equations (5) and (6) are general formulas. Non-stationary and non-uniform schemes can be written in this form with indices on E and C. Nevertheless, those schemes could cost more computing resources and may be less effective for data compression in GIS applications An example Figure 1 shows an example that illustrates the basic idea of the above construction process. Here E and C depend on m0 and k0; therefore, it is a non-stationary and non-uniform transform. Given irregular data on points m1={t 0,t 1,t 2,t 3,t 4 }, the one

5 Wavelet triangulated irregular networks 277 Figure 1. Illustration of constructing wavelets based on a linear estimator and corrector. step partition and wavelet transform are: A m 1 =P m1,m0={t 0,t 2,t 4 }, k0={t 1,t 3 } m1=m0nk0,b m 0 =A m1 (m0), B k0 =A m1 (k0), l 1 = t 1 t 0 t 2 t 0, l 2 = t 3 t 2 t 4 t 2 W k 0 =B k0 E(B m0 )= C P t 1 P t3 D C 1 l 1 )=CP 0 t0 1 A m 0 =B m0 +C(W P l 1 l k0 t2 1 2 P 0 l t4d+c1 l D 2 l l l 2 2 D CP t0 P (7) t2 P t4d CW t 1 W t3 D In this example, a linear estimator and corrector are used for E and C. Ifl and 1 l are 0.5, the correction step makes the norm defined in equation (8) take a smaller 2 value than the uncorrected value. (Note that in equation (8), A m 0 (t j ) for t j 1m0 refers to the estimation value E(A m 0 ) at t.) This is an example of the optimization j in the correction step. Equation (7) is the analysis transform for the first step. The synthesis transform can be derived by replacing the estimator and the corrector in equation (6). norm= (A m 1 (t j ) A (t (8) m0 j ))2 t j µm1 The above scheme is one type of filter based on the lifting scheme. This is called an approximation filter, in which every point value will change after each iteration. The other type of filter based on the lifting scheme is an interpolation filter, in which a point value reaches its final position once it is calculated. If the correction term is omitted in equation (5), the filter becomes an interpolation filter. An approximation filter may be optimal in a defined global norm. However, interpolation filters have the advantage of interpolating point values, which may be useful in some GIS applications. Therefore, choosing the type of filter depends on the application. For example, approximation filters are generally used in processing images because the whole scene of an image is more important to the viewer than any individual pixel. For GIS terrain data, however, interpolation is generally preferred because point

6 278 J. Wu and K. Amaratunga values are often more useful than a general shape. Therefore, interpolation wavelet filters will be used in this paper for processing three-dimensional terrain data. 3. Interpolation wavelet filters for height fields Three-dimensional terrain data are often treated as surface functions of two variables x and y. This type of surface is also called a height field. A basic characteristic of this type of data is that each point (x, y) has only one associated function value f (x, y). This is the basic assumption in processing GIS surface data, whether it is in the DEM or TIN format. If the data under the surface need to be explored, either layering techniques can be used or volume models should be considered. This characteristic is a key factor that makes the proposed data representation more efficient than the general three-dimensional geometric modelling representations. In order to represent a terrain surface, a connected network typically needs to be built for nonuniformly spaced data. Triangulated networks are popular for expressing surface geometry because they typically contain fewer polygons than cellbased models, which results in faster rendering. From the discussion in the last section, it is known that the correction term in equation (5) will not appear if an interpolation scheme is required. Therefore, the most important step to construct the wavelet filter is to find a good partition method and a good estimator E. Several subdivision algorithms based on triangular networks, provide hierarchical structures and good estimators. Therefore, they are used in the wavelet construction process. Strictly speaking, a subdivision process has two steps. One is a splitting step; the other one is an evaluation step. The splitting step is more like a partition step, in which a triangle is divided into several sub-triangles. The evaluation step is to calculate the point values (point coordinates for a geometry) after the splitting. Therefore, a subdivision scheme itself can be categorized into the interpolation type and the approximation type. In an interpolation subdivision scheme, once a point value is calculated, it will not change in the future subdivision steps. In an approximation subdivision, previous point values will change in the future subdivision steps. Because the wavelet coefficients in the lifting scheme are based entirely on the estimator (see equation 5), an interpolation subdivision is more attractive than an approximation one since the wavelet coefficients do not need to be updated for further subdivision steps. This makes the computation faster T he modified Butterfly scheme A good splitting method is the quaternary triangulation, which divides a triangle into four similar triangles. A two-step splitting is shown in figure 2(a). Loop (1987) designed a smooth subdivision scheme that is based on this splitting configuration. Note that Loop s scheme is an approximation scheme although it produces a smooth surface. Dyn et al. (1990) designed the Butterfly scheme on the quaternary splitting, which is an interpolation scheme and can achieve a C1 continuous surface with a few exceptions. Zorin et al. (1996) dealt with the extraordinary points in the Butterfly scheme to produce a modified Butterfly scheme. Figures 2(b) and 2(c) respectively show the subdivision configurations for regular internal points and extraordinary internal points. The categorization of internal points is based on their valences, which is defined as the number of points connecting to the current point. A regular internal point has a valence of 6. Otherwise, it is an extraordinary internal point. From figure 2(a), it is obvious that the new internal points always have valence

7 Wavelet triangulated irregular networks 279 Figure 2. The quaternary subdivision and the modified Butterfly scheme. 6. This is the regular case of the quaternary subdivision. Figure 2(b) illustrates the filter coefficients corresponding to each point involved in obtaining point A in the Butterfly subdivision scheme. By contrast, simple mid-point (i.e. linear) subdivision would produce the reference point A as the result. The two direct parent nodes, B and C, should be regular internal vertices. Figure 2(c) gives the filter coefficients for the extraordinary case, where one direct parent of the new point is an extraordinary point. In the graph, {s j } and q are the estimator s coefficients for corresponding points. These coefficients are defined in equation (9). P is the result of applying the modified Butterfly scheme with respect to the reference point P. In equation (9), k is the valence of the extraordinary point Q, which has a coefficient q. Notice that the points where {s i } apply need to be regular internal vertices. GS 0 =5/12, S = 1/12, k=3 1,2 S =3/8, S = 1/8, S =0, k= ,3 S =(1/4+cos(2pj/k)+1/2 cos(4pj/k))/k, j=0,..., k 1, k5 j q=1 k 1 S j j=0 Each time a new internal point is added, an appropriate filter from figures 2(b) and 2(c) will be applied based on the neighboring configuration. For the case where the two direct parents of a new node are two extraordinary internal vertices, a simple average will be used on the results that are obtained from plugging each parent in figure 2(c) A discussion on boundaries of height fields In GIS applications, a rectangular boundary is often used. In the modified Butterfly scheme, Zorin et al. (2000) presented several cases for processing boundary points. However, the formula for the extraordinary boundary points ( boundary points with valences other than 4) is not an interpolation scheme. Furthermore, the filters may result in moving the new points outside the boundary of the GIS data. The reason for this effect is that the modified Butterfly scheme tries to smooth a local region near an extraordinary point, such as the point Q in figure 2(c); while the nearby points have some tangential movement around this extraordinary point, which may cause these points move out of the boundary. This is shown in figure 3(a). In this paper, the subdivision points will be used as reference points to obtain the most detailed data ( 4). Therefore, the boundary cases need to be modified. One simple approach is to use a smaller support filter, i.e. fewer coefficients, to process (9)

8 280 J. Wu and K. Amaratunga the data, such as in figure 3(d). This can be viewed as hardening the boundary layer. In numerical experiments, it is found that the initial configuration will influence the final boundary networks. Even the filters used in plots 3(b) and 3(c) may result in a case where the new subdivision points go out of the boundary. In this case, an average filter in figure 3(d) can be used because it guarantees that the boundary condition is satisfied although the surface smoothness may be reduced. From a practical view, this reduction of smoothness and simplification do not harm the quality of the final surface. A corner type vertex is added, since for height fields a corner vertex cannot be treated as a general boundary point. The neighbouring vertices of a corner vertex should be processed separately because they do not lie on the same boundary curve; however, the neighboring vertices of a general boundary point can be processed using a curve approximation scheme, such as in figures 3( f ) and 3(g). A complete set of rules for dealing with internal and boundary points is given in table 1. Figure 3 illustrates several boundary cases. There are total of 15 Figure 3. Illustration of boundary cases. Table 1. Complete rules for processing GIS terrain data by the modified Butterfly scheme. Case Rule Notes Add an internal point IR-IR 1 IR: internal regular point (valence is 6) IR-IE 2 IE: internal extraordinary point (valence is not 6) IR-BR 3 BR: boundary regular point (valence is 4) IR-BE 6 BE: boundary extraordinary point (valence is not 4) IR-C 6 C: corner point IE-IE 4 BD: intermediate point of a boundary (BR and BE) IE-BR 2 IE-BE 6 IE-C 6 rule 1: regular case in figure 2(b) BR-BR 5 or 6 rule 2: figure 2(c) BE-BE 6 rule 3: figure 3(b) BR-C 6 rule 4: apply figure 2(c) on both parent nodes and average BE-BE 6 rule 5: figure 3(c) BE-C 6 rule 6: simple average, as in figures 3(d), 3(e) and 3(h) C-C 6 rule 7: figure 3( f ) Add a boundary point BD-BD 7 rule 8: figure 3( g) BD-C 8 C-C 6

9 Wavelet triangulated irregular networks 281 cases for adding a new internal point and three cases for adding a new boundary point. There are eight rules for processing these cases. 4. Wavelet triangulated irregular networks In accordance with the above discussion, a new multiresolution data representation can be constructed using interpolation wavelet filters, such as the one based on the modified Butterfly subdivision scheme T he computation process For a general three-dimensional object, every point has three coordinates x, y, and z. Therefore in a general three-dimensional wavelet transform, every point has three wavelet coefficients. However, a terrain surface, S, is a function of two variables, x and y. S(x, y) represents the true height value measured in the z direction. A subdivision surface is different from this true surface; every subdivision point (x 0,z 0 ) will correspond to a point (x 0,S(x 0 )) on the true surface. Note that although S(x 0 ) is not equal to z 0, these two points have the same x 0 and y 0 coordinates. The point (x 0 ) is obtained by applying the modified Butterfly scheme to a coarser triangle network. z 0 is also generated using the modified Butterfly scheme, while S(x 0 ) represents the true height value at (x 0 ). Because we choose the height value at exactly (x 0 ) to represent the terrain surface, the wavelet coefficients for the x and y directions will be zero. The only wavelet coefficient that needs to be stored is the wavelet coefficient in the z direction, S(x 0 ) z 0. This leads to a more efficient representation than those used in general three-dimensional object modeling techniques. Before the wavelet transform can be applied, there is another important step, which is to determine the coarsest resolution configuration (also called the initial configuration). The initial configuration generally can be very simple, such as the two triangles defined by the four corners of a rectangular area of interest, or some other simple triangular network defined by important points. After determining the initial configuration, the height values at corresponding subdivision points need to be obtained from the raw geometry data. GIS terrain data come from various sources, such as satellite images, aerial photos, geographical survey, and interpolation by technicians or computers. A height value at a specific point can be always obtained with reasonable accuracy by some interpolation method. The inverse distance weighted interpolator and spline interpolator are some common tools. The following step, then, is to apply a wavelet transform on the data, as described in equation (5). Since many of the wavelet coefficients output from the analysis step are very small Figure 4. The procedure for processing 3-D terrain data by 2nd generation wavelets.

10 282 J. Wu and K. Amaratunga compared to the original height data, compression schemes find their place in processing the wavelet coefficients. The wavelet coefficients can be quantized to a desired error tolerance and then compressed by traditional compression schemes such as those used in image compression. Finally, on the user side, the required data can be synthesized from those wavelet coefficients and the coarsest level data. The whole process is shown in figure 4. The wavelet transforms used in this process are given below: Analysis =C x m A m N =P N y mn m N )D, S(x m N,y mn mi=mi 1nki 1, B m i 1 =A mi (mi 1), =C B ki 1 =A mi (ki 1) B =E(B )=Cx ki 1 x ki 1 =x ki 1 ỹ B y ki 1 mi 1 k k i 1 k i 1 z ki 1D, =ỹ ki 1 S(x k i 1,y )D, i=n, N 1,..., 1 (10) ki 1 =C W k W k i 1 =B ki 1 B i 1,x =0 ) z ki 1D W ki 1 k i 1,y =0 W k i 1,z =S(x ki 1,y ki 1 A m i 1 =B mi 1 Synthesis B m i =A mi )=C 0 ki 1 B k i =W ki +E(B 0 ỹ mi k W z ki 1D=Cx k i,zd+cx =x ki 1 y k i 1 =ỹ +z ki 1D, i=0, 1,..., N 1 W k i,z mi+1=minki, =C A m i+1 (mi)=b mi,a mi+1 (ki)=b ki x m P m N =A N )D y mn m N S(x m N,y mn (11) Equation (10) is the analysis transform. It differs from equation (5) in that the wavelet coefficients in the x and y directions are always zero, which reduces the storage space of the wavelet coefficients by 2/3. The correction term disappears because an interpolation filter is the goal. Equation (11) gives the synthesis transform, which simply reverses the process in equation (10) a key advantage of the lifting scheme.

11 Wavelet triangulated irregular networks T he data format for storage and transmission Based on above discussion, a new data format Wavelet Triangulated Irregular Networks (WTIN) is designed. The storage and the transmission of GIS terrain data in this format require only four types of information as shown in table 2. The meaning of all variables in table 2 is listed below: $ V denotes the vertices in the initial configuration, which includes the three 0 coordinates of all the initial vertices. $ F0 denotes the faces in the initial configuration, which hold the labels of the three vertices. $ W denotes the wavelet coefficients for all subdivision points. For scalable distributed GIS services, W can be divided into several array objects, which correspond to different resolutions. Then the requested levels of wavelet coefficients can be transmitted separately as different objects. $ N( ) is the number of items in ( ). $ T is the type of the wavelet transform used in processing the data. This is for future extension to use other wavelet filters. In this paper, the interpolation wavelet filter described in 3 is used. $ L is the total number of resolution levels included in the data. Notice that in this format, only one wavelet coefficient, in the z ( height) direction, is stored for each new subdivision point. The reason can be clearly seen in equation (10). Because many coefficients in W are very small compared to the original data, compression schemes can be used to make the total data size smaller. A detailed analysis is given in T he data structures for computation Although the storage and transmission format of WTINs is very compact, more complex data structures need to be used during the application of the analysis and synthesis transforms in order to retrieve topological information of the subdivision. The authors propose suitable data structures to dynamically host the intermediate data, which are shown in figure 5 and figure 6. The vertex information is stored in a multi-linked list structure, and face information is stored in a quadtree-array. For a vertex, two vertex labels (pointers) for the direct parents are needed besides the coordinates. For the vertices in the initial configuration, the direct parents are null. The vertex type is also stored in order to use corresponding rules in table 2. For every face, three defining vertices, three neighboring faces (null if the neighbor is empty), and the resolution level are required. The neighboring faces permit the modified butterfly scheme to be implemented efficiently. Table 3 gives the class definitions for the vertices and faces. All the required information in the computation process can be derived from these data. Essentially, the information in table 2 will determine all the data through the wavelet transform. The intermediate data structures make the computation more efficient. Table 2. Data format for storage and transmission of WTINs. Control: N(V 0 ), N(F 0 ),N(W),T,L; integer [5] V 0 :(x 0,z 0 ); real [N(V 0 ), 3] F 0 :(v 1,v 2,v 3 ); integer [N(F 0 ), 3] W : real [N(W )]

12 284 J. Wu and K. Amaratunga Figure 5. Multi-linked list for storing vertex information of WTINs. Figure 6. Quadtree-array for storing face information in WTINs. Table 3. Class definitions for vertices and faces in WTINs. public class Vertex { public class Face { private double x, y, z; private Vertex[ ] vertex=new Vertex[3]; private int type; private int level; private Vertex[ ] parents=new Vertex[2]; private Face[ ] nb=new Face[3]; //methods... //methods Implementation and analysis In order to verify the effectiveness of the proposed WTIN format, a software prototype for three-dimensional multiresolution GIS data processing has been developed. The software was developed using the Java programming language in order to plug into online GIS applications. The software is designed using an objectoriented model; the core classes that are provided are CompactWtin, Wtin, MbutterflyFilter, MultiLinkedlist, QuadTreeArray, Vertex and Face. The original data source is in USGS DEM format. A bilinear interpolation algorithm is used to obtain values at the multiresolution points. This gives acceptable accuracy as long as the original DEM grid is sufficiently dense. Figure 7(a) shows the data at all multiresolution points. The interpolation wavelet transform based on the modified Butterfly scheme is used to process these data. The chosen initial configuration is rather simple: 9 vertices and the 12 corresponding Delaunay triangles. Figure 7(b) shows the top view of the initial configuration. Based on this initial

13 Wavelet triangulated irregular networks 285 Figure 7. Results from numerical experiments (a) Terrain defined by all multiresolution points ( b) Top view of the initial configuration (c) Histogram of the wavelet coefficients (d) Threshold compression at 3% (e) Representations using the first 2, 3, 4, and 5 levels of wavelet coefficients (the percentage number under each plot is the percentage of included wavelet coefficients). configuration, 7 resolutions are constructed during wavelet analysis. Figure 7(c) is the histogram of wavelet coefficients. From this plot, one can see that most coefficients fall within 5% of the height range (for this example, the height range is m and 94% of wavelet coefficients fall in the 5% range). Therefore, compression schemes, such as Huffman coding or Arithmetic coding, can be applied to these wavelet coefficients to reduce the storage size. In this paper, a simple threshold operation is applied to the wavelet coefficients since the focus is on the transform. The compressed

14 286 J. Wu and K. Amaratunga result using 3% of the height range as the compression threshold is shown in figure 7(d). This has a corresponding compression ratio of 11:1. A more carefully designed quantization technique can achieve better compression. In order to see the information content in each level of a WTIN, one can build a surface by using only a subset of the wavelet coefficients consisting of the first several levels. To do this, one first uses the subset of wavelet coefficients to perform the synthesis transform, and then applies pure subdivision to the result from the synthesis transform. Figure 7(e) gives four representations using the first 2, 3, 4, and 5 levels of wavelet coefficients respectively to build the terrain. The percentage of wavelet coefficients retained is given under each plot. A quantitative analysis of the quality of reconstructed surfaces is given based on Peak Signal to Noise Ratio (PSNR), which is defined in equation (12). NH2 H2 PSNR=10 log 10A N log (z z )2B=10 10A db (12) mean squared errorb 1 Here, N is the number of points, H is the height range (max height min height), z is the value of the variable of interest and is the value reconstructed after compression. The unit db, i.e. decibel, is a non-dimensional unit. It is used to express PSNR in a logarithmic scale. For example, a PSNR of 30 db corresponds to a mean squared error of H2/1000. The PSNR versus threshold plot is shown in figure 8(a). The plot of PSNR vs. compression ratios obtained using different thresholds is shown in figure 8(b). Here the compression ratio is the total number of wavelet coefficients Figure 8. Quantitative compression results.

15 Wavelet triangulated irregular networks 287 divided by the number of wavelet coefficients left after the threshold operation. Figures 8(a) and 8(b) show that a 5% threshold applied to all seven levels results in a PSNR of 30 db and a compression ratio of 25:1. Figure 8(c) gives the plot of PSNR versus the number of levels of wavelet coefficients included. A number 5 on the horizontal axis means that the first five levels of wavelet coefficients are included in the synthesis transform. Figure 8(d) presents the result of applying the 5% threshold operation on the wavelet coefficients used in figure 8(c). 6. Conclusions and future work This paper has proposed a new multiresolution data format, WTIN, for threedimensional GIS terrain surface data. The numerical results from our study demonstrate that the WTIN data format is suitable for multiresolution representation and compression of GIS data. This data format is simpler than the hierarchical data formats proposed by heuristic methods, such as Hoppe s method (1998). The final compressed data sets fit the requirements of three-dimensional scalable distributed GIS services. Compared to pure subdivision schemes, the wavelet analysis based on the lifting scheme provides complete information; while pure subdivision schemes provide only low frequency information without any detail information. By analyzing the particular features of height field data, a WTIN holds a single wavelet coefficient for every subdivision point. This is much simpler than the general wavelet-based 3-D geometric modeling techniques, which require three wavelet coefficients for each point. In summary, the proposed WTIN format has the following advantages: $ A solid mathematical foundation from wavelet and subdivision theory, $ Multiresolution capability, suitable for scalable distributed GIS services, $ Fast transforms, since the filter is a recursively linear operation (only constant matrix multiplications are involved, excluding the thresholding operation) and has local support, $ Easy to compress due to the large number of wavelet coefficients with small magnitudes, and $ An interpolation scheme, suitable for many GIS applications. The approach for building the WTIN format opens a new direction for processing GIS terrain data. WTIN technology is still in its infancy and further research opportunities exist to make it more mature. Below are some thoughts on future research. $ A first direction is to combine WTIN with rendering level techniques to achieve adaptive level-of-detail control. Directly implementing the adaptive feature in WTIN may not be worthwhile because it will destroy the internal structure and increase the storage and transmission of information, which deviates from the requirements of saving bandwidth and space. Therefore, the authors propose to separate the rendering layer and the storage and transmission layer, so that each step yields the best result for its particular focus. There is a considerable amount of research on the topic of rendering level-of-detail control. Hoppe (1998) provides a good source for this. Note that even in the current implementation, smooth regions will result in small wavelet coefficients, which will be discarded during the compression operation. Therefore, while

16 288 J. Wu and K. Amaratunga adaptive subdivision can improve rendering performance, it is not expected to significantly improve the compression performance. $ Develop more efficient quantization schemes than the simple threshold operation to compress data. Threshold compression is currently used to verify the idea and the result is clearly promising. Further research can be focused on post processing the wavelet coefficients. This will lead good coding schemes to encode the compressed data. $ A third research opportunity is to design better schemes to construct the initial configuration. The quality of the initial configuration affects the quality of the final surface. The triangular shape in the initial configuration also affects the boundary networks. Therefore, a good initial configuration is important. One approach is to use edge collapse and vertex merge to gradually reduce the raw triangulated data. But this approach is computationally expensive. Further research in this direction is therefore desirable. $ For some GIS applications, a better approximation scheme instead of an interpolation scheme is preferred. An example of the approximation wavelet transform is given in 2 of this paper. Special requirements in a problem can result in different wavelet filters due to the constraints on the corrector C in equation (2). Acknowledgments The authors work is supported by a grant from the Suksapattana Foundation, Bangkok, Thailand to the Intelligent Engineering Systems Laboratory, Massachusetts Institute of Technology. References BUTTENFIELD, B. P., 1999, Sharing Vector Geospatial Data on the Internet. In Proceedings of 18th International Cartographic Conference, Ottawa, Canada, August 1999 (International Cartographic Association), pp DE FLORIANI, L, 1989, A pyramidal data structure for triangle-based-surface description. IEEE Computer Graphics and Applications, 9, DE FLORIANI, L., and PUPPO, E., 1995, Hierarchical triangulation for multiresolution surface description. ACM T ransaction on Graphics, 14, DE FLORIANI, L., MAGILLO, P., and PUPPO, E., 2000, Compressing triangulated irregular networks. Geoinformation, 4, DONOHO, D. L., 1993, Unconditional bases are optimal bases for data compression and for statistical estimation. Applied and Computational Harmonic Analysis, 1, DOUGLAS, D. H., and PEUCKER, T. K., 1973, Algorithms for the reduction of the number of points required to represent a digitized line or its caricature. T he Canadian Cartographer, 10, DYN, N., LEVIN, D., and GREGORY, J. A., 1990, A butterfly subdivision scheme for surface interpolation with tension control. ACM T ransaction on Graphics, 9, HOPPE, H., 1998, Smooth view-dependent level-of-detail control and its application to terrain rendering. In Proceedings of IEEE V isualization 1998, Research Triangle Park, North Carolina, October 1998, IEEE, pp JÜNGER, B., and SNOEYINK, J., 1998, Selecting independent sets for terrain simplification. Proceedings of WSCG 98, Plzen, Czech Republic, February 1998, University of West Bohemia, pp KIEMA, J. B. K., and BÄHR, H.-P., 2001, Wavelet compression and the automatic classification of urban environments using high resolution multispectral imagery and laser scanning data. GeoInformatica, 5, LOOP, C., 1987, Smooth Subdivision Surfaces based on Triangles, Master s thesis, University of Utah, Department of Mathematics.

17 Wavelet triangulated irregular networks 289 LOUNSBERY, M., 1994, Multiresolution Analysisfor surface of Arbitrary Topological Type, PhD thesis, Dept. of Computer Science and Engineering, University of Washington. MOREHART, M., MURTAGH, F., and STARCK, J.-L., 1999, Spatial representation of economic and financial measures used in agriculture via wavelet analysis. International Journal of Geographical Information Science, 13, SCARLATOS, L., and PAVLIDIS, T., 1992, Hierarchical triangulation using cartographic coherence. CVGIP: Graphical Models and Image Processing, 54, SWELDENS, W., 1997, The lifting scheme: a construction of second generation wavelets. SIAM Journal on Mathematical Analysis, 29, WU, J., AMARATUNGA, K., and CHITRADON, R., 2002, Design of a distributed, interactive online GIS viewer using wavelets. Forthcoming ASCE Journal of Computing in Civil Engineering, April. ZORIN, D., SCHRÖDER, P., and SWELDENS, W., 1996, Interpolating subdivision for meshes with arbitrary topology. ACM SIGGRAPH 96 Proceedings, pp ZORIN, D., SCHRÖDER, P., DEROSE, A., KOBBELT, L., LEVIN, A., and SWELDENS, W., 2000, ACM SIGGRAPH 2000 Course Notes, pp