ARTICLES
SPATIAL DATA COMPRESSION AND DENOISING VIA WAVELET TRANSFORMATION Biswajeet Pradhan, Institute for Advanced Technologies (ITMA), Faculty of Engineering, University Putra, Malaysia Correspondence to Biswajeet Pradhan:
[email protected] Sandeep Kumar, Department of Mechanical Engineering, Institute of Technology, Banaras Hindu University (BHU), India Shattri Mansor, Institute for Advanced Technologies (ITMA), Faculty of Engineering, University Putra, Malaysia Abdul Rahman Ramli, Institute for Advanced Technologies (ITMA), Faculty of Engineering, University Putra, Malaysia Abdul Rashid B. Mohamed Sharif, Institute for Advanced Technologies (ITMA), Faculty of Engineering, University Putra, Malaysia
A new interpolation wavelet filter for TIN data compression has been applied in two steps, namely splitting and lifting. In the splitting step, a triangle has been divided into several sub-triangles and the elevation step has been used to ‘modify’ the point values (point coordinates for geometry) after the splitting. This data set is then compressed at the desired locations by using second-generation wavelets: scalar wavelets constructed by using a lifting scheme. Application of the compressed data compares favourably with results derived using the original (and much larger) TIN data set.
1. INTRODUCTION Recently, most of the methods for image data compression have become based on wavelets and related techniques because these tend to outperform Fourier approaches in representing both spatially localized features, and smooth image regions. The superior compression capability of wavelets combined with their natural multiresolution structure makes them suitable for storing image data. While working with dyadic wavelet decomposition, digital images are represented by wavelet coefficients. Such types of representation are by linear decomposition over a fixed orthogonal basis. Non-linearity in the approximation of images by wavelets is introduced by the thresholding of the wavelet coefficients. This type of approximation can be viewed as mildly nonlinear. Recently, several highly nonlinear methods for capturing the geometry of images were developed, including that exemplified by use of wedgelets (Donoho, 1999); as well as edge-adapted nonlinear multiresolution and geometric spline approximation (Demaret et al., 2004). This paper presents a new approach for the non-linear image compression method using second generation wavelets. A random set of points has been approximated to represent a surface by Delaunay triangulation. The theory, computations, and applications of Delaunay triangulations and Voronoi diagrams have been described in detail in the literature (Lawson, 1972; Sibson, 1978; Lee and Schachter, 1980; Watson, 1981; Mirante and Weingarten, 1982; Macedonio and Pareschi, 1991; Kao et al., 1991; Puppo et al., 1992; Tsai and Vonderohe, 1991; Tsai 1993). The present work describes a fast algorithm based on Tsai’s Convex Hull Insertion algorithm for the construction of Delaunay triangulations and Voronoi diagrams of arbitrary collections of points on the Euclidean plane. The original algorithm has been improved further for a faster computation of geometric structures. The source code has been written in FORTRAN.
APPLIED GIS, VOLUME 2, NUMBER 1, 2006 MONASH UNIVERSITY EPRESS
06.1
A second generation wavelet is used for selection of significant points that are used for the triangulated irregular network. Multi-resolution of terrain representation is obtained by using different sets of significant pixels. Results are compared and shown at different levels of resolution.
2 TRIANGULATION (SEE FORTUNE, 1991, AND AURENHAMMER, 1991) Many researchers (Evans et al., 2001; Abásolo, et al., 2000) have suggested different ways to construct triangulations with the local equilateral property. A well known construction, called the Delaunay Triangulation, simultaneously optimizes several of the quality measures such as max-min angle, min-max circumcircle, and min-max min-containment circle. The Delaunay triangulation (DT) of a point set is the planar dual of the famous Voronoi diagram. The Voronoi diagram is a partition of the plane into polygonal cells one for each input point so that the cell for input point ‘a’ consists of the region of the plane closer to ‘a’ than to any other input point. So long as no four points lie on a common circle, then each vertex of the Voronoi diagram has degree three and the DT which has a bounded face for each Voronoi vertex and vice versa will indeed is a triangulation. If four or more points do lie on a common circle, then these points will be the vertices of a larger face that may itself be triangulated to give a triangulation containing the DT Voronoi diagrams and Delaunay triangulations that have been generalized in numerous directions. There is a nice relationship between Delaunay triangulation and three dimensional convex hulls. Each point of the input data set can be lifted to a paraboloid in three-dimensional space by mapping the point with coordinates (x, y) to the point (x, y, x2+y2). The convex hull of the lifted points can be divided into lower and upper parts: a face belongs to the lower convex hull if it is supported by a plane that separates the point set from (0,0,–8). It can be shown that the DT of the input points is the projection of the lower convex hull onto the xy-plane as depicted in Figure 1. Finally a direct characterization: if a and b are input points the DT contains the edge {a, b} if and only if there is a circle through a and b that intersects no other input points and contains no input points in its interior Moreover each circumscribing circle (circumcircle) of a DT triangle contains no input points in its interior. Figure 1 shows that the lifting transformation maps the DT to the lower convex hull.
Figure 1 The lifting transformation maps the DT to the lower convex hull.
06.2
SPATIAL DATA COMPRESSION AND DENOISING ARTICLES
The DT properties relevant to compression algorithm generation are: Let Y denote a finite planar point set.
•
•
A Delaunay triangulations D(Y) of Y is one, such that for any triangle in D(Y), the interior of its circumcircle does not contain any point from Y. This specific property is termed as Delaunay property. The Delaunay triangulation D(Y) of Y is unique, provided that no four points in Y are cocircular. Since neither the set X of pixels nor its subsets satisfy this condition, we initially perturb the pixel positions in order to guarantee unicity of the Delaunay triangulations of X and of its subsets. Each perturbed pixel corresponds to one unique unperturbed pixel. From now on, we denote the set of perturbed pixels by X, and the set of unperturbed pixels by
•
•
.
can be computed from D(Y) by a local update. This follows from the For any Delaunay property, which implies that only the cell C(y) of y in D(Y) needs to be retriangulated. Recall that the cell C(y) of y is the domain consisting of all triangles in D(Y) which and the Delaunay triangulation of its contain y as a vertex. Figure 1 shows a vertex cell C(y). D(Y) provides a partitioning of the convex hull [Y] of Y. Figure 2 shows the irregular set of points and Figure 3 shows the TIN data structure using Delaunay triangulation.
2.1 DESCRIPTION OF THE ALGORITHM: The Delaunay Convex Hull algorithm is performed in four steps:
•
Sorting the points: The bubble-sort algorithm (Knuth, 1997) is deployed to sort the set of planar points. The sort is by ascending values of the co-ordinate (x or y) corresponding to the largest variation. Dx = max( xj) – min( xj) or Dy = max( yi) – min( yi)
SPATIAL DATA COMPRESSION AND DENOISING ARTICLES
06.3
•
The convex hull of the whole point set is determined, based on the gift wrapping method (Tsai, 1993). The Delaunay triangulation of the convex hull is constructed (based on the Delaunay criterion). Then, other points which are not involved with the convex hull are added iteratively: the existing triangulations are refined by adding a new point to the triangulation process. Figure 4 shows the flow chart showing the algorithm for the triangulation.
• •
Figure 4 Flow chart showing the algorithm for the triangulation
3. THE LIFTING SCHEME (DEPLOYMENT OF SECOND-GENERATION WAVELETS) “Lifting” (Sweldens, 1996) offers a flexible method for constructing/designing scalar wavelets with desirable (definable) properties. The basic idea behind lifting is to start with simple multiresolution analysis and gradually build a multi-resolution analysis with specific, a priori defined properties. The lifting scheme is a tool for constructing second-generation wavelets (Sweldens, 1994, 1996), which are no longer, dilates and translates of one single function. In contrast to firstgeneration wavelets, which use the Fourier transform for wavelet construction, a construction using lifting is performed exclusively in the spatial domain and, thus, wavelets can be custom
06.4
SPATIAL DATA COMPRESSION AND DENOISING ARTICLES
designed for complex domains and irregular sampling. The lifting scheme can, thereby, be viewed as a process of taking an existing wavelet and modifying it by adding linear combinations of the scaling function at the same level of resolution. In the consequent subdivision process, the interpolation wavelet filter for TIN generation takes place in two steps (Kiema and Bähr, 2001). A splitting step; is followed by an elevation step. In the splitting step, a triangle is divided into several sub-triangles (Wu and Amaratunga, 2003; Dyn et al. 1990; Cohen, 2001; Mallat, 1989). The elevation step is to calculate the point values (point coordinates for geometry) after the splitting. Let us discuss this partition step mathematically. Let us consider that a signal, as represented in Figure 5 and Figure 6, can be defined mathematically as: Signal
Figure 5 Graphical representation of a signal
Figure 6 Mathematical representation of the signal
The signal value at xn,j will be Sn,j : signal value at xn,j (evenn–1 , oddn–1 ) = split (Sn) Sn = { Sn,0 , Sn,1 , Sn,2 , Sn,3 , Sn,4 , Sn,5 , Sn,6 , Sn,7 } evenn–1 = { Sn,0 , Sn,2 , Sn,4 , Sn,6 }
SPATIAL DATA COMPRESSION AND DENOISING ARTICLES
06.5
oddn–1 = { Sn,1 , Sn,3 , Sn,5 , Sn,7} In predicting step, even samples can be used to calculate the odd samples and vice versa: dn-1 = oddn–1 – P( evenn–1) In case of Haar wavelets, dn–1,j = Sn,2j+1 – Sn,2j Update ensures that the coarse signal has the same average value as the original signal. The update operator U can be written as: Sn–1 = evenn–1 + U(dn–1) In case of Haar wavelets,
3.1 FORWARD AND INVERSE TRANSFORM Figures 7 and 8 show the forward and inverse transformation for the signal. Mathematically we can represent both the forward and inverse transformation as shown in Figure 9 and Figure 10 respectively.
Figure 7 Forward transform of the signal
Figure 8 Inverse transform of the signal
06.6
SPATIAL DATA COMPRESSION AND DENOISING ARTICLES
Figure 9 Mathematical representation for the Forward transform of the signal
Figure 10 Mathematical representation for the inverse transform of the signal
In inverse transformation we can get the original signal by undo update, undo predict and merging as shown below: Undo update step: even n–1 = Sn–1 – U(dn–1) Undo predict step: oddn–1 = dn–1 + P(evenn–1) Undo merge step: Sn = merge (evenn–1 , oddn–1) Linear wavelet transformation can be represented as dn = oddn – P(evenn)
Sn = evenn + U(dn)
3.2 SECOND GENERATION BASED LIFTING SCHEME The (GIS) terrain to be encoded can be regarded as a discrete surface, i.e. a finite set of points in three-dimensional (3-D) space, by considering a non-negative discrete function of two variables F(x, y) and establishing the correspondence between the image and the surface A = {( x,y,c)|c = F(x,y)}, so that each point in A corresponds to a pixel in the image; the couple (x,y) gives the pixel’s position in the XY plane, while c is the point’s height. Our goal is to approximate A by a discrete surface B = {(x,y,d) | d = G(x,y)}, defined by means of a finite set of points. Let T be a generic triangle on the XYof vertices: and let Where P1, P2 and P3 are represented as
SPATIAL DATA COMPRESSION AND DENOISING ARTICLES
06.7
Let O be the centroids of the triangle which means a new point after adding into the triangulation (Figure 11). Let A1, A2 and A3 be the area of the three triangles. Then the total area A can be represented as: A = A1 + A2 + A3 Now, the detail coefficient can be represented as: where, dn is the detail coefficient at the nth level and is the predicted value at nth level and n z is the values at the odd samples. The equation above can be rewritten in a general form for n–1th level as where C(dn) is the correction factor for wavelet coefficients.
Figure 11 A surface showing the areas of the triangles to calculate the wavelet coefficients
4. DATA USED AND EXPERIMENTAL RESULTS Two sets of LIDAR (Light Detection and Radio imaging) data have been assembled for testing the efficiency of the compression program. The data were in ASCII format describing the x, y and z values for the points. The screenshot of the original LIDAR data in ASCII format has been shown in figure 12. Different areas were selected because of their diverse feature contents and particular density of local point clouds. All algorithms were coded in Visual Fortran and MATLAB and executed on a Pentium IV, 256 MB RAM machine. The wavelet based triangulation compression method gives better performance for the LIDAR data that we used. This section presents some results of deploying the algorithm on LIDAR data using the wavelet-based triangulation compression method at different levels of spatial resolution. In our test, at each recursion step, about 25% of the vertices are removed by using the second generation wavelet and lifting scheme. The number of triangles in the hierarchy is at most only three times larger than the number of triangles of the initial triangulation. We next examine the storage “overhead cost” caused by the maintenance of the hierarchy. The total amount of storage in bytes of the data structure described in the previous section, (it may include multiple copies
06.8
SPATIAL DATA COMPRESSION AND DENOISING ARTICLES
of the same triangle), is only 4 to 5 times larger than the storage of the initial triangulation. In our implementation, this gave a total memory requirement for the hierarchical representation of a terrain of n data points of roughly O(n log n) bytes.
Figure 12 Shows the screenshot for the original LIDAR data in ascii format
The data structure of Delaunay triangulation is different from the data structure in Wavelet compression in two ways. First, we no longer store multiple copies of the same triangle, but instead we store for each triangle the level at which it was created. Second, we introduce the intermediate nodes, which reduce the total number of pointers in the structure. Together these two changes reduce the storage requirement by one third, which means that the storage is about 3.5 times as much as the storage for the initial terrain data. Results for irregular triangulated network data for three-dimensional GIS terrain have been presented (Figure 14 and 15). The original spatial data was irregularly distributed..The Delaunay triangulation method for the TIN creation has been applied to the original data. A new algorithm has been developed for the creation of the TIN. Further, for second generation wavelet compression, the TIN was compressed using the second generation wavelets by means of a new algorithm. Figure 13 shows the original TIN data and Figure 14 shows the triangulation of the same set of data. Based on the initial configuration of the original TIN, different resolutions are constructed during wavelet analysis. Figures 15 and 17 show the results computed using the wavelet-based lifting scheme algorithm. Figure 15 shows the results after 12% compression and figure 16 shows the triangulation of the compressed data at the level of 12%. Similarly, Figure 17 shows the results after 23% compression and Figure 18 shows the triangulation of the compressed data at the same level. Another set of LIDAR data is also tested to verify the compression ratio. Figure 19 shows the original LIDAR terrain. Figure 20 shows the wavelet compression of LIDAR data to 14% and Figure 21 shows the wavelet compression of the same data to 27%. Different compression schemes, such as Huffman coding can be applied to these wavelet coefficients to further reduce the storage size. This work also provides current implementation of wavelet coefficients
SPATIAL DATA COMPRESSION AND DENOISING ARTICLES
06.9
during the compression operation. The proposed algorithm has multiresolution capability and offers ease of compression for web displaying.
Figure 13 Shows the original image
Figure 14 Shows the triangulation of the original data
06.10
SPATIAL DATA COMPRESSION AND DENOISING ARTICLES
Figure 15 The wavelet compression at the level of 12%
Figure 16 Shows the triangulation of the compressed data at the level of 12%
SPATIAL DATA COMPRESSION AND DENOISING ARTICLES
06.11
Figure 17 The wavelet compression at the level of 23%
Figure 18 Shows the triangulation of the compressed data at the level of 23%
06.12
SPATIAL DATA COMPRESSION AND DENOISING ARTICLES
Figure 19 Shows the original LIDAR terrain data
Figure 20 Shows the wavelet compression of LIDAR terrain data at the level of 14%
Figure 21 Shows the wavelet compression of LIDAR terrain data at the level of 27%
SPATIAL DATA COMPRESSION AND DENOISING ARTICLES
06.13
4.1 MODEL VALIDATION Results were evaluated by visual analysis as well as by analysis of the Peak-Signal-to-Noise Ratio (PSNR). The Peak Signal to Noise Ratio PSNR (dB) indicates the size of the error relative to the peak value of the signal. Quantitative evaluation of image quality is achieved through analysis of the gray value histograms to show the relation between the maximal gray value within the original image (peak) and the noise that is caused through the compression. PSNR is defined as
A comparative analysis has been made between the proposed Wavelet based results with the JPEG images. Figure 22 demonstrates the superiority of the wavelet method against JPEG with increasing compression ratios.
Figure 22 Shows the relationship between compression ratio and PSNR
5. CONCLUSION The construction of Triangulated Irregular Network using Delaunay triangulation has been shown. This approach uses a fast and efficient second generation wavelets algorithm for multiresolution analysis of Delaunay triangulation data. Data compression has been demonstrated. The mathematical and computational deployment of the new algorithm takes place with minimal time, irrespective of the data volume. Our algorithm scheme preserves high-gradient regions, and so compression can take place with minimum loss of terrain information. We have tested our method with two data sets. The computational cost of deploying our algorithm depends on the approach chosen. The initial triangulation can be done in O(n log n), the gradient approximation can be done in O(n log n). The individual refinement step has to check all the original data points lying in the involved triangles, so the complexity of each step is O(n). How often the iteration
06.14
SPATIAL DATA COMPRESSION AND DENOISING ARTICLES
step is executed depends on the error value given in the input. As a general rule, the authors have assumed that no more iteration should be done than there are original data sites.
ACKNOWLEDGMENTS Authors would like to thank various anonymous reviewers for their constructive feedback.
REFERENCES Abásolo, M; Blat, J; De Giusti, A. 2000. ‘A Hierarchial Tringulation for Multiresolution Terrain Models’. The Journal of Computer Science and Technology 1: 3. Aurenhammer, F. 1991. ‘Voronoi diagrams – A survey of a fundamental geometric data structure’. ACM Computing Surveys 23 (3): 345–405. Cohen, A. 2001. ‘Applied and computational aspects of nonlinear wavelet approximation’. Multivariate Approximation and Applications, edited by N. Dyn, D. Leviatan, D. Levin, and A. Pinkus. Cambridge:Cambridge University Press. pp. 188–212. Demaret, L.; Dyn, N.; Floater, M. S.; Iske, A. 2004. ‘Adaptive thinning for terrain modelling and image compression’. Advances in Multiresolution for Geometric Modelling, edited by N. A. Dodgson, M. S. Floater, and M. A. Sabin., Heidelberg: Springer-Verlag. pp. 321–340. Donoho, D. 1999. ‘Wedgelets: nearly-minimax estimation of edges’. Annals of Statistics 27: 859–897. Dyn, N; Levin, D; Gregory, J. A. 1990. ‘A butterfly subdivision scheme for surface interpolation with tension control’. ACM Transaction on Graphics 9: 160–169. Evans, W; Kirkpatrick, D; Townsend, G. 2001. ‘Right-Tringulated Irregular Networks’. Algorithmica Special Issue on Algorithms for Geographical Information Systems 30 (2): 264–286. Fortune, S. 1991. ‘Voronoi diagrams and Delaunay triangulations’. In Computing in Euclidean Geometry, edited by F. K. Hwang and D. Z DU. World Scientific. Kao, T.; Mount, D. M.; Saalfeld, A. 1991. ‘Dynamic maintenance of Delaunay triangulations’. Proceedings of Auto-Carto I0, Baltimore, Maryland, pp. 219–233. Kiema, J; Bähr, H. P. 2001. ‘Wavelet compression and the automatic classification of urban environments using high resolution multispectral imagery and laser scanning data’. Geoinformatics 5: 165–179. Knuth, D. 1997. The Art of Computer Programming: Sorting and Searching. 3rd edn. Addison-Wesley, Sorting by Insertion, 3. pp. 80–105. Lawson, C. L. 1972. ‘Generation of a triangular grid with application to contour plotting’. California Institute of Technology, Jet Pollution Laboratory, Technical Memorandum No. 299. Lee, D; Schachter, B. J. 1980. ‘Two algorithms for constructing a Delaunay triangulation’. International Journal of Computer and Information Sciences 9 (3): 219–242. Macedonio, G; Pareschi, M. T. 1991. ‘An algorithm for the triangulation of arbitrary distributed points: applications to volume estimate and terrain fitting’. Computers and Geosciences 17 (7): 859–874. Mallat, S. 1989. ‘A theory of multiresolution signal decomposition: The wavelet representation’. IEEE Trans. On Pattern Analysis and Machine Intelligence 11: 674–693. Mirante, A; Weingarten, N. 1982. ‘The radial sweep algorithm for constructing triangulated irregular networks’. IEEE Computer Graphics and Applications 2 (3): 11–21. Puppo, E.; Davis, L.; DeMenthon, D.; Teng, Y. A. 1992. ‘Parallel terrain triangulation’. Proceedings of Fifth International Symposium on Spatial Data Handling. Vol. 2. Charleston, South Carolina, pp. 632–641. Sibson, R. 1978. ‘Locally equiangular triangulations’. Computer Journal, 21 (3): 243–245. Sweldens, W. 1994. ‘Construction and Applications of wavelets in Numerical Analysis’. Unpublished PhD thesis. Dept. of Computer Science, Katholieke Universiteit Leuven, Belgium. Sweldens, W. 1996. ‘The lifting scheme: A custom-design construction of biorthogonal wavelets’. Appl. Comput. Harmon. Anal. 3(2): 186–200.
SPATIAL DATA COMPRESSION AND DENOISING ARTICLES
06.15
Tsai, V. J. D.; Vonderohe, A. P. 1991. ‘A generalized algorithm for the construction of Delaunay triangulations in Euclidean n-space’. Proceedings of GIS/LIS ’91. Vol. 2. Atlanta, Georgia, pp. 562–571. Tsai, V. J. D. 1993. ‘Fast topological construction of Delaunay triangulations and Voronoi diagrams’. Computer and Geosciences 19 (10): 1463–1474. Watson, D. F. 1981. ‘Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes’. Computer Journal 24 (2): 167–172. Wu, J; Amaratunga, K. 2003. ‘Wavelet triangulated irregular networks’. International Journal of Geographical Science 17 (3): 273–289.
Cite this article as: Pradhan, Biswajeet.; Kumar, Sandeep; Mansor, Shattri; Ramli, Abdul Rahman; Sharif, Abdul Rashid B. Mohamed. 2006. ‘Spatial data compression and denoising via wavelet transformation’. Applied GIS 2 (2). pp. 6.1–6.16. DOI: 10.2104/ag060006.
06.16
SPATIAL DATA COMPRESSION AND DENOISING ARTICLES
