Fast triangular mesh approximation of surface data using wavelet coefficients

Transcription

1 1 Introduction Fast triangular mesh approximation of surface data using wavelet coefficients Han Ju Yu, Jong Beom Ra Dept. of Electrical Engineering, Korea Advanced Institute of Science and Technology, Kusongdong Yusonggu, Taejon, Republic of Korea This paper proposes a new, fast, triangular mesh approximation method for the 3D visualization of surface data. Using spatiofrequency localization characteristics and directional information of wavelet coefficients, we determine local complexities of surface data and approximate the data to a proper triangular mesh. The proposed algorithm is quite simple, and the computational cost is low due to the direct use of wavelet coefficients for vertex removal. The computer simulation results for terrain data show that the proposed algorithm is excellent for fast 3D visualization. Key words: 3D visualization ± Triangular mesh approximation ± Vertex removal ± Wavelet transform ± Terrain data Correspondence to: J.B. Ra Recently, computer graphics applications have been extended to computer simulation, computer games, movies, broadcasting dramas, etc. The portion of realistic images generated by computer graphics is increasing. Because of the simplicity of mesh rendering in 3D visualization, commercial graphics hardware usually supports polygonal mesh rendering (especially triangular mesh). Various data sources for 3D visualization are based on triangular meshes, e.g., the 3D digitizer, terrain data from flights or satellites, and scientific data of a mathematical description. Original triangular meshes from these data sources usually have too many triangles to process in real time. Hence, there have been many studies to reduce the number of triangles in a mesh. Among them, mesh decimation (Schroeder and Zarge 199), mesh optimization (Hoppe et al. 1993), and geometric optimization (Hinker and Hansen 1993) have been proposed for triangle reduction in a complex and regularly distributed triangular mesh that is generated by initial data. These methods reduce the number of triangles by using local area characteristics of the initial mesh, or by optimizing given cost functions in changing the mesh's topology and geometry. Even though these methods provide good visual quality and work for arbitrarily shaped polygonal models, they are complex and the computational costs are enormous. In computer graphics, as well as in signal and image processing, the wavelet transform (WT) has been used as a powerful tool for multiresolution analysis (Berman et al. 1994; Finkelstein and Salesin 1994; Gross et al. 1996; Lounsbery et al. 1997; Stollnitz et al. 1995). Among them, fast mesh approximation using the WT for D data seems promising for surface data visualization (Gross et al. 1996). In this scheme, by removing insignificant wavelet coefficients compared to a given threshold, the data are smoothed, and unimportant mesh vertices in regular mesh are determined and rejected in each resolution level. The remaining vertices are then represented by a quadtree, and square cells corresponding to leaf nodes are converted to triangles. This algorithm is faster than previous algorithms and provides good visual quality. However, cumbersome iterative inverse WTs are still required for vertex removal. We propose a new algorithm for triangular mesh approximation of regular grid structured and sin- The Visual Computer (1999) 15:9±0 Springer-Verlag

2 D data WT Thresholding Quadtree construction Triangulation Triangular mesh Fig. 1. Flow chart of the proposed algorithm gle valued D data. The algorithm uses wavelet coefficients. Figure 1 shows the flow chart of the proposed algorithm. In contrast to the previous algorithm (Gross et al. 1996), our algorithm determines the local area complexity by examining the amplitude of the corresponding wavelet coefficients directly, and it removes unimportant vertices from the initial mesh. This action is based on careful observation of the characteristics of wavelet coefficients such as spatio-frequency localization and directional highpass filtering. Since the algorithm uses a simple top-down approach and does not require time-consuming inverse WTs, it provides very fast mesh decimation for 3D visualization. In Sect., we describe the characteristics of the WT and determine the relation between the wavelet coefficient and the image region complexity. Then we propose a new algorithm for triangular mesh approximation using wavelet coefficients in Sect. 3. Computer simulation results are given in Sect. 4, and finally, this paper is concluded in Sect. 5. Wavelet transform and its spatio-frequency characteristics.1 Wavelet transform (WT) The WT decomposes a finite energy function f x; y into a set of basic functions. Let us assume that a vector space V 0 covers a half-open interval 0; 1 ; and is spanned by a scaling function j : t Then the dilated and translated version of the scaling function for vector spaces V m can be represented by j m p ˆ m= t j m t p for p ˆ 0; 1;...; m 1: For a fixed m, j m p are orthonormal. Because every vector in V m is also contained in V m 1 ; the vector spaces V m have the relation of V V 1 V 0 V 1 : This nested set of spaces V m with resolution m is called a multiresolution analysis. If j is a constant function, it is called the Haar basis function. As for the orthogonal complement of V m in V m 1 ; we can define a new vector space W m spanned by y m p ˆ m= t y m t p for p ˆ 0; 1;...; m 1: In other words, W m is the space of all functions in V m 1 that are orthogonal to all functions in V m for the chosen inner product. Here, y is called the mother wavelet. These basis functions have two important properties. First, the basis functions y m p of W m ; together with the basis functions j m p of V m ; form the bases of V m 1 : Second, every basic function y m p of Wm is orthogonal to every basic function j m p of Vm for the chosen inner product. Because space V m is nested and space W m is the orthogonal complement of V m in V m 1 ; we can describe the coefficients hj m p ; f i and hym p ; f i with the coefficients hjp m 1 ; f i by the following filter operations (Mallat 1989): c m p f ˆhy m p ; f iˆx g p k a m 1 k f ; a m p f ˆhj m p ; f iˆx h p k a m 1 k f ; k k 1 where h p ˆ R 1= j t p j t dt; g p ˆ 1 h p 1 ; and hf ; giˆr f gt t dt: Due to their relation to orthonormal wavelet bases, these filters provide an exact reconstruction of lower scale coefficients, i.e.; al m 1 f ˆX hp l a m p f g p l c m p f : p p 10

3 Further f can t be reconstructed as follows. f ˆX t a M p jm p XM c m p ym p : p mˆ1 3 It has been shown that there is no orthonormal finite impulse response (FIR) filter that is inversely transformable with the linear phase property except the Haar basis function (Smith and Barnwell 1986). However, the Haar basis function is a trivial function in the viewpoint of signal processing. Biorthogonal bases having linear phase property were suggested, and the corresponding WT was implemented by employing the quadrature mirror filter (QMF) pyramids (Daubechies et al. 1988). In the biorthogonal WT, Eq. 1 is still used for decomposition, but reconstruction (Antonini et al. 199; Vetterli and Herley 1990) becomes: al m 1 f ˆX ~h p l a m p f ~g p l c m p f Š: 4 where p ~g p ˆ 1 p h p 1 ; g p ˆ 1 p ~h p 1 ; and X h p ~h p k ˆ d k;0 d : delta function 5 p In D wavelet analysis, the basic function and wavelets are obtained with the 1D separable basic function and wavelets. j mpq x; y :ˆ m j m x p j m y q mpq x; y :ˆ m y m x p j m y q y ;1 y ; y ;3 mpq x; y :ˆ m j m x p y m y q mpq x; y :ˆ m y m x p y m y q : 6 Therefore, a finite energy function f x; y can be represented as f x; y X ˆX p XM mˆ1 q h a M pq j Mpq i c m;1 pq y;1 mpq cm; pq y; mpq cm;3 pq y;3 mpq ; (7) D E where a M pq ˆ f ; j Mpq and c m pq ˆ f ; y mpq : As in the 1D case, the decomposition coefficients a m pq and c m pq can be described with the coefficients apq m 1 by the following filter operations: f ˆhj mpq ; f iˆx X h q l h p k a m 1 f ; a m pq c m;1 pq c m; pq c m;3 pq f ˆhy ;1 mpq ; f iˆx X l k f ˆhy ; mpq ; f iˆx X l k f ˆhy ;3 mpq ; f iˆx X The reconstruction becomes X f ˆX a m 1 kl q p l l k k kl h q l g p k a m 1 kl g q l h p k a m 1 kl g q l g p k a m 1 kl f ; f ; f : 8 ~h q l ~h p k a m pq f ~h q l ~g p k c m;1 pq ~g q l ~h p k c m; pq ~g q l ~g p k c m;3 pq 9 Figure shows a 1-scale implementation of D biorthogonal wavelet decomposition and reconstruction using QMF pyramids (Antonini et al. 199). It is known that high-order spline wavelets, such as cubic spline wavelet bases, have good spatio-frequency localization characteristics (Chui 199). Therefore, for QMF implementation, we use the spline lowpass filter h and highpass filter g given in Fig. 3. They are obtained from a cubic spline scaling base and its wavelet. Multiscale implementation of D wavelet decomposition is achieved by applying the unit in Fig. a repeatedly. One- and two-scale decompositions result of D images M ˆ 1; are shown in Fig. 4a, b. In Fig. 4, as M increases, the upper-top low-resolution subimage rll m is further decomposed at each stage m.. Spatio-frequency characteristics of the WT Since the WT provides a local spectral estimate of the data and describes local variations, the surface coarseness of a local area may be determined from the amplitude of the corresponding wavelet coefficient. Then, the vertex removal in a surface mesh can easily be based on the observed surface coarseness. However, since the D WT of an image pro- 11

4 Decomposition Reconstruction LL m-1 H(ω) H(ω) G(ω) LL m LH m LL m LH m ~ H ( ω ) ~ G ( ω ) ~ H ( ω ) LL m-1 G(ω) H(ω) G(ω) HL m HH m HL m HH m ~ H ( ω ) ~ G ( ω ) ~ G ( ω ) ---- :1 down sampling ---- :1 up sampling a b h 1 g a 3b h(n) g(n) c m m= m=1 LL HL LL 1 HL 1 HL 1 m= LH HH 4a LH 1 HH 1 m=1 LH 1 HH 1 4b Fig. a, b. One stage in a multiscale D wavelet decomposition (a) and reconstruction (b). This is based on a 1D filtering of rows and columns of a subimage with 1D quadrature mirror filters (QMFs); G, ~G; H, and ~H Fig. 3a±c. Graphical representation of quadrature mirror filters (QMFs) for wavelet decomposition: a lowpass filter h; b highpass filter g; c their nine-tab digital version Fig. 4a, b. Image decomposition using a D wavelet transform (WT): a one-scale decomposition (M=1); b two-scale decomposition (M=). LL denotes the low-resolution subimage. HL, LH, and HH denote horizontally, vertically, and diagonally oriented subimages, respectively 1

5 f(x) or L 0 5 H(ω) G(ω) L 1 H 1 H(ω) G(ω) L H L 0 L 1 H 1 L H WD EP 6 Fig. 5a, b. Spatio-frequency localization characteristics of the 1D wavelet transform (WT): a 1D wavelet implementation; b 1D wavelet decomposition result (WD) of eight-point data and effective positions (EP) of wavelet coefficients. n, u, and n represent coefficients of subdata H 1,L, and H, respectively Fig. 6a, b. Spatio-frequency localization characteristics of the D wavelet transform (WT): a two-scale wavelet decomposition; b effective positions of wavelet coefficients in the spatial domain. Here l, n, and s represent coefficients of subimage HL 1,LH 1,andHH 1 for scale 1, respectively, and u, l, n, and s represent coefficients of subimage LL,HL, LH, and HH, respectively, for scale vides three subimages of each scale ± and even subimages in the same scale have different directional information ± a one-to-one mapping of a wavelet coefficient and a specific vertex position is not desirable. Therefore, in order to use wavelet coefficients more efficiently in a vertex removal problem, their spatio-frequency localization characteristics should be examined precisely. As already mentioned, a wavelet coefficient has both spatial and frequency domain information. Its frequency-domain position corresponds to the scale (or resolution level), and its spatial-domain position can be estimated from the position in the corresponding subimage (or subband) of the transform domain. To understand the information that wavelet coefficients have more precisely, let us consider the two-scale 1D wavelet decomposition of eight point data f obtained x from Fig. 5a. Since most of the energy in the spline filters given in Fig. 3c is concentrated on the three filter coefficients near the center, each wavelet coefficient in L m and H m can be interpreted to represent lowpass and highpass results. Three neighboring wavelet coefficients in the previous subdata L m 1 are used to do this. In Fig. 5b, since the center position of filter g is shifted by one (see Fig. 3b), the effective coefficient location in H m is shifted by one point from the one in L m : Therefore, the result of wavelet decomposition, depending on subband, can be rearranged by considering the effective position 13

6 of wavelet coefficients shown in Fig. 5b. In the same manner, the result of the wavelet decomposition in the D domain and the rearrangement of wavelet coefficients depending on their effective positions are given in Fig. 6a, b. The physical meaning of wavelet coefficients can be examined in Fig. 6b. As was mentioned, if we consider that the energy of cubic spline wavelet filters is concentrated on three neighboring filter coefficients, a wavelet coefficient in scale m can be interpreted to represent the high-frequency component (or the degree of local variation) of a 3 3 wavelet coefficient region in subimage LL m 1 : Since coefficient values in LL m 1 are determined by (m 1)times lowpass filtering and a two-toone subsampling with the same lowpass filter given in Fig. 3a, a coefficient in scale m can be considered to represent overall local variation over the area of m 1 m 1 pixels in the original image (or LL 0 ). It also contains the directional information of local variation according to the kind of subimage (i.e., HL, LH, or HH) to which it belongs. 3 Triangular mesh approximation using wavelet coefficients Based on the wavelet coefficient characteristics discussed, we now introduce a square cell division and triangulation scheme for surface data meshing. Since wavelet coefficients are divided into several frequency subbands (or subimages), a certain rule is needed to use them effectively for surface meshing. One of the most common aspects in an image model is the ªdecaying spectrum,º which means that an image power spectrum is concentrated on low frequency and becomes smaller as the frequency gets higher. The ªdecaying spectrumº model is also effective for surface data. Thus, we assume that the amplitude of wavelet coefficients is proportional to scale m. In other words, the higher the scale m, the larger the coefficient values are. Because of this observation, we adopt a top-down approach to construct square cells efficiently. In this section, we first introduce a square cell division scheme that is common for every scale, and then provide the overall triangular surface approximation scheme by combining square cell division and triangulation with a top-down approach. 3.1 Square cell division Using the observation in Sect., we devise a modality for constructing a square cell and a rule for dividing it into smaller square cells for cell division in the next scale. In scale m, a square cell unit (which covers m 1 m 1 pixels in the original image) is made with five coefficients centered at the location of a diagonal coefficient (see Fig. 7). Here, wavelet coefficient l in subimage HL represents local variation of the 3 3 pixel region in LL m 1 along the horizontal direction. If the value of this coefficient is larger than a threshold value, we can consider that a vertical edge centered at the position of the coefficient exists. (In this case, only one-half the edge is included in the square cell as shown in Fig. 7.) Similarly, wavelet coefficient n in subimage LH represents a local variation along the vertical direction. We can determine the existence of horizontal edges by examining these values. Finally, wavelet coefficient s in subimage HH represents a local variation along the diagonal direction. We can determine the existence of diagonal edges by examining this coefficient. Figure 8 shows the square cell division rule depending on the position of significant wavelet coefficients that are larger than the given threshold value t. If the number of significant coefficients is more than one, corresponding diagrams are combined for proper cell division. Figure 9 shows all the possible cases of cell division and triangulation in scale m from the result of Fig. 8. For shaded blocks, cell division and triangulation are to be repeated in the next steps. Since the white blocks will not be considered in the following steps, they are to be triangulated. If the values of the diagonal wavelet coefficient are less than a given threshold, 0±1 or 0± is selected. Otherwise, 0±3 is selected. As shown in Fig. 9, the look-up table has only 1 components, and the cell division and triangulation step is simple and efficient. 3. Triangular mesh approximation By assuming a decaying spectrum model, square cell division starts at the highest scale M and pro- 14

7 HL LH HH LH 7 HL Fig. 7. A square cell unit consisting of five coefficients at scale m. The values of coefficients l, n, and s indicate the existence of edges (depicted by the dashed lines) = = = = = = = = = = = = 1111 Fig. 8a±e. Basic schemes for dividing a square cell to be chosen according to the position of the significant wavelet coefficients. Scheme e is implemented only in the triangulation step. The same dividing scheme is implemented in each level Fig. 9. A look-up table for cell division and triangulaiton. A shaded square block is to be divided in the same way through the next hierarchical steps ceeds to lower scales. First, we make square cells, which are surrounded by four wavelet coefficients at scale M as shown in Fig. 7. Each square cell covers M 1 M 1 pixels centered at diagonal wavelet coefficients in the original image. According to the significance of its four surrounding coefficients, we perform cell division and triangulation using Figs. 7 and 8. Then, we repeat this operation for the shaded blocks with coefficients of scale M 1: This repetition continues as m decreases down to 1. In scale 1, all the shaded blocks are triangulated into either 0±1, 0±, or 0±3 in Fig. 9, depending on the values of their diagonal coefficients. In Fig. 10, we demonstrate an example of triangulation for M ˆ : In this example, since only the top coefficient is significant among surrounding coefficients at scale, we first divide the square cell as in Fig. 10b. Then, since two coefficients are significant in scale 1, two upper child square cells are divided and triangulated as in Fig. 10c. Finally, the resulting small square cells are divided 15

8 (a) (b) (c) (d) 10 Cell division and triangulation at scale Cell division and triangulation at scale 1 Final triangulation 11a 11b Fig. 10a±d. An example of cell division and triangulation using a recursive procedure. The magnitude of the wavelet coefficients and their directional information are used for proper division: a selection of significant wavelet coefficients (surrounded by ) that are larger than a given threshold; b, c hierarchical square cell division and triangulation; d resulting triangular meshes Fig. 11. a A crack occurring at the boundary of square cells with different resolutions and its removal; b crack removal procedures according to crack patterns for all possible triangle shapes into triangles as shown in Fig. 10d. In the final triangulation step, however, simple application of Fig. 9 is not enough for taking care of cracks. Cracks can occur at the boundary of two neighboring square cells with different resolutions (see Fig. 11a as an example). Since we have two kinds of triangles according to the look-up table given in Fig. 9, crack removal procedures for these two triangle shapes are considered in Fig. 11b. The number of edges having cracks is the determining factor. In both shapes, if only one side of a triangle has cracks, we divide the triangle by connecting the cracks with the triangle vertex point on the opposite side. If two sides of a triangle have cracks, we connect opposite side cracks in a zigzag way. Crack searching based on the dyadic dependency of cracks, and crack removal have been performed for the final triangulation result in Fig. 10d. 4 Simulation results The proposed algorithm is implemented in the C language and OpenGL (Neider et al. 1993) on an SGI Indigo High Impact with a R4400 processor. Regular-grid digital terrain data with a size of and a bit resolution of 8 (Fig. 1a) are used for simulation. It produces triangles in the initial mesh. We use nine-tab Daubechies spline wavelet filters (Fig. 3c) for WT. The terrain 16

9 1a 1b 13a 13b 13c 13d Fig. 1. a Digital terrain data used for a computer simulation. The size is and the resolution is 5000:1; b its Gouraud shaded image with 5488 triangles Fig. 13a±d. Simulation results of the proposed algorithm with the terrain data of Fig. 1: a the ratio of remaining triangles are plotted as a funciton of a threshold value; b peak signal-to-noise ratio (PSNR), c mean square error (MSE) and d computational time are plotted as a function of the ratio of the remaining triangles 17

10 a b c d e f Fig. 14a±f. Wireframes and Gouraud shaded images: a, b the number of triangles T=468 and remaining ratio R=47.3%; c, d T=86508, R=16.5%; e, f T=07, R=4.% 18

11 Table 1. Computer simulation results Threshold value No. of triangles No. of vertices PSNR (db) MSE Remaining ratio (%) Cell division time in seconds Triangulation time in seconds PSNR, Peak signal-to-noise ratio; MSE, mean square error data are wavelet transformed and stored for thresholding. Graphs in Fig. 13 and Table 1 show some computer simulation results. As is expected, the number of triangles in a mesh monotonically decreases as the threshold value for the wavelet coefficient increases. Here, the mean square error (MSE) and peak signal-to-noise ratio (PSNR) are computed with the errors on all regular grid points. The PSNR is given by: PSNR = 10 log ((dynamic range of data) /MSE). (10) The remaining ratio R in Table 1 is obtained by R = (No. of triangles)/ (Initial no. of triangles) 100 [%]. (11) As shown in the graphs in Fig. 13b, c, a single threshold for wavelet coefficients controls the image quality well. The proposed scheme provides a high PSNR even for small numbers of triangles. Table 1 shows that the proposed algorithm is fast enough for real-time rendering. Since the cell division process is very fast, a large portion of the computational time can be assigned to triangulation. It is also interesting to note that the inverse WT of a image [which is not required in the proposed method, but is essential to the method proposed by Gross et al. (1996)] takes about 0.3 s. This computational time itself is comparable to the sum of cell division and triangulation time required for a triangle mesh generation in the proposed method. Figure 14 shows wireframe meshes and corresponding Gouraud shaded images constructed by the proposed algorithm with various threshold values. By examining the wireframe meshes, we can see that triangles are properly allocated by local area complexity. The resulting Gouraud shaded images show good visual quality and reserve detail edges in the terrain data. 5 Conclusions This paper proposes a new, fast and efficient algorithm for triangular mesh decimation using WT coefficients. Based on spatio-frequency localization characteristics of wavelet coefficients, it directly determines local area complexity in an image domain and divides square cells depending on complexity. For cell division, it adopts a top-down approach using the ªdecaying spectrumº model, which is proper for general images or surface data. In the cell division and triangulation step, it needs a very simple look-up table with only 1 components. The algorithm is much faster than previous triangular mesh decimation methods. It can also provide decimated meshes preserving details because it uses the inherent directional information of wavelet coefficients. Computer simulation results show that the proposed algorithm is very suitable for 3D visualization of terrain data. Resulting images provide good visual quality and a high PSNR. This mesh approximation scheme based on WT may be combined with a WT-based image-compression scheme to manage large numbers of data for visualization effectively. 19

12 References Antonini M, Barlaud M, Mathieu P, Daubechies I (199) Image coding using WT. IEEE Trans Image Processing :05±0 Berman D, Bartell J, Salesin D (1994) Multiresolution painting and composition. Proceedings of SIGGRAPH'94. Orlando, Florida 85±90 Chui C (199) An introduction to wavelets. Academic Press, Sandiego, California Daubechies I, Grossman A, Mayer Y (1988) Orthonormal bases of compactly supported wavelets. Commun Pure Appl Math 41:909±996 Finkelstein A, Salesin D (1994) Multiresolution curves. Proceedings of SIGGRAPH'94. Orlando, Florida 61±68 Gross M, Staadt O, Gatti R (1996) Efficient triangular surface approximations using wavelets and quadtree data structure. IEEE Trans Visualization Comput Graph :130±143 Hinker P, Hansen C (1993) Mesh optimization. Proceedings of Visualization'93. Anaheim, California 189±195 Hoppe H (1996) Progressive mesh. Proceedings of SIG- GRAPH'96. New Orleans, Louisiana 99±108 Hoppe H, DeRose T, Duchamp T, McDonald J, Stuetzle W (1993) Mesh optimization. Proceedings of SIGGRAPH'93. Anaheim, California 19±6 Lounsbery M, DeRose T, Warren J (1997) Multiresolution analysis of surfaces of arbitrary topology type. ACM Trans Graph 16:34±73 Mallat S (1989) A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans Patt Anal Machine Intell 11:674±693 Neider J, Davis T, Woo M (1993) OpenGL Programming Guide. Addison Wesley, Reading, Mass Stollnitz E, DeRose T, Salesin D (1995) Wavelet for computer graphics: A primer, part. IEEE Comput Graph Appl 15:75±85 Schroeder W, Zarge J (199) Decimation of triangle meshes. Proceedings of SIGGRAPH'9. Chicago, Illinois 65±70 Smith M, Barnwell D (1986) Exact reconstruction of tree structured subband coders. IEEE Trans Acoust Signal Processing 34:434±441 Vetterli M, Herley C (1990) Wavelet filter banks: relationships and new result. Proceedings of the IEEE ICASSP, Albuqueque. New Mexico 173±176 JONG BEOM RA received the B.S. degree in electronic engineering from the Seoul National University, Seoul, Korea, in 1975 and the M.S. and Ph.D. degrees in electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Taejon, Korea, in 1977 and 1983, respectively. From 1983 to 1987, he was a member of the faculty at Columbia University, New York, and engaged in the development of medical imaging systems such as high field MRI and spherical PET systems. He joined the Department of Electrical Engineering at KAIST in July 1987, where he is now a Professor. His research interests are digital image processing, video signal processing and its hardware implementation, and 3-D visualization. HAN JU YU received the B.S. degree in electronic engineering from the HanYang University, Seoul, Korea, in 1991, and the M.S. degree in electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Taejon, in He is currently working toward the Ph.D. degree in the area of image processing and 3D visualization. His research interests include the image processing, image compression, 3D visualization and geometric modeling. His work is mainly concerned with geometric modeling of D terrain data using image processing techniques and real time rendering system for very large terrain data. 0