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Progress in Natural Science 18 (2008) 1409-1415

Progress in Natural Science

www.elsevier.com/locate/pnsc

NURBS reconstruction of digital terrain for hydropower engineering based on TIN model

Denghua Zhong, Jie Liu, Mingchao Li*, Caiwei Hao

School of Civil Engineering, Tianjin University, 92 Weijin Road, Nankai District, Tianjin 300072, China Received 27 November 2007; received in revised form 16 May 2008; accepted 20 May 2008

Abstract

Digital terrain model (DTM) has played an important role in 3D designing, visual analysis and 3D geological modeling in large-scale hydropower engineering. As the pivotal base of 3D visualization and modeling, DTM should be characterized by high precision, less storage and well interactivity during graphic operation. Considering the diversity of data source and taking advantage of two data structures, triangulated irregular network (TIN) and non-uniform rational B-splines (NURBS), a novel methodology is presented for reconstructing engineering terrain of hydropower project. With integration of multi-source data, enhanced Delaunay algorithm is introduced to rebuild the TIN-DTM, which is a terrain surface in TIN and a faithful depiction of complex topography but in low-memory efficiency. Based on the TIN model, applying section scanning sampling and linear interpolation, the transformation from discrete, irregular and diverse data to continuous and regular sampling cross-sectional curve sequence, is realized. The appropriate compression of the sampling data is also imposed to be performed for guaranteeing the following reconstruction work. Eventually, employing the NURBS technique and skinning method, the NURBS-DTM, which represents a NURBS surface and satisfies the requirement after precision assess with weighted errors, is reconstructed with the intermediate data. Meanwhile, there is another achievement that two databases of terrain data, one from initial data and the other from sampling data, are established for repeatable reconstruction with different demands. With the successful application of the presented method, a stable foundation is laid for 3D engineering geological modeling, visual designing and analysis of the hydropower projects. © 2008 National Natural Science Foundation of China and Chinese Academy of Sciences. Published by Elsevier Limited and Science in China Press. All rights reserved.

Keywords: Terrain reconstruction; TIN; NURBS; Data compression; Weighted error; Hydropower engineering

1. Introduction

Digital terrain model (DTM), which mathematically describes spatial position information and topography of interest, is widely employed in the scientific and practical applications, such as surveying and mapping, geographic information system (GIS), civil and hydropower engineering, and topography analysis.

The terrain, where large-scale hydropower engineering is located, is normally wide and significantly complicated. As the rapid development of hydropower project construction in China and corresponding proposition of 3D visual

* Corresponding author. Tel.: +86 22 27890911; fax: +86 22 27890910. E-mail address: lmc@tju.edu.cn (M. Li).

designing, DTM has become the pivotal foundation of 3D geological modeling. Additionally, it is able to be utilized for hydrological analysis [1], amount computation of excavation and filling, stability analysis of landslide [2], etc. Therefore, DTM to be applied in 3D visual analysis should be characterized by a high precision, less storage and well interactivity during graphic operation.

Currently, contour lines from the topographical map are still the most common form of elevation data for terrain surface, and the triangulated irregular network (TIN) is all through the basic data structure for DTM. Many researches focusing on the field are carried out [3-12]. Zhang [3] introduced the tiling rules to reconstruct 3D terrain only based on contours, which guaranteed arbitrary branching terrain to be divided into correct topology. Jin

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and Science in China Press. All rights reserved.

doi:10.1016/j.pnsc.2008.05.015

[6] obtained high data of discrete point from contour map, then generated Delaunay triangulation model, which make up three-dimensional realistic terrain. Richard [7] used a variance-bias criterion to select the optimal areas for the triangular facets of the mesh. In this way, the mesh adapts itself to offer the best tradeoff between increasing the facet area to minimize the noise variance and decreasing the facet area to minimize the bias of the fitted facet parameters. Gregory [8] described a new set of data structures and algorithms for dynamic terrain modeling using a triangulated irregular network, and its basic data structure consists of three interconnected data objects: triangles, nodes, and directed edges.

However, as the supplementary for contour lines, feature-specific (F-S) data, such as geologic investigation point, hereafter referred to as geo-point, and characteristic lines, are also very important terrain data, for they can remedy the defect that topography information from contour data in deep gully or steep cliff region is insufficient. Hence, the integration of multi-source data is imposed to done for DTM modeling. Moreover, there exists intrinsic complicacy in the management of TIN model, and its memory storage is very huge for large-scale hydropower engineering region. Thus, a lot of difficulties will be raised for the following 3D geological modeling or other visual analysis, and, therefore, it is imperative to introduce a new data structure for DTM in hydropower industry.

To well solve the problems mentioned above, a novel approach of digital terrain reconstruction with integrated data is presented for the hydropower engineering. Taking advantage of the technical merit of two data structures, TIN and non-uniform rational B-splines (NURBS), the method perfectly handles the constrained problems that the initial data are diverse and multi-source. Then, after three procedures, data sampling from TIN model, vector data compression and NURBS skinning, the DTM meeting the requirement is reconstructed successfully. It is able to be characterized by a high precision and less storage, and also to be utilized for 3D geological modeling and other terrain analyses.

2. Integration of DTM data

2.1. Data source of DTM

In hydropower industry, topographic map and ground observed record are the main source of DTM data. Especially, contour line is the most important and frequent in use. However, the F-S data, such as geo-point, mountain peak, valley bottom, points along ridge line and gorge, breakline, road line, which are obtained by a selective surveying method, are absolutely the supplementary for terrain data. The experiment directed by Li [13] proved it right that DTM precision will be improved when F-S data are added into terrain data. Thereby, the integration of all initial data should be performed, and the procedure is defined mathematically as

D = C U P U L (1)

where D is the aggregate of all terrain data; C represents contour line from topographical map; P mainly represents geo-point, sometimes including other F-S point; L is the aggregate of F-S line; and U is the behavior of integration operation.

2.2. Point-storage method for DTM data

DTM data include the information of horizontal position and elevation, and their spatial attribution is line or point in geometry. Meanwhile, point sequence is another representation of vector line, and point is the basic unit depicting geometric object. Hence, all DTM data, namely D, can be translated into point data. The point-storage method is presented to store initial multi-source data, and the uniform database for engineering terrain data can be established.

With the limitation of manpower and material resource for reconnaissance and survey, reliability of terrain data from different region is varied. The data obtained in the engineering region are more reliable than the one that are out of or far away from the region. Hereby, terrain data should be stored with relevant reliability value to distinguish its source, which is convenient to fulfill with the point-storage method. Normally, the mode with the two level of reliability, in engineering region and out of engineering region, is adopted.

The point in point sequence from L is orderly, and point from P is individual one. So memory implementation with linking list can be introduced for DTM data, and its structural unit for point-storage is defined as follows:

Struct TerPoint {

Float x, y, z; // The coordinates of 3D

point in the terrain data. bool bIsPoint; // If the point is an

individual point from P, bIsPoint is TRUE; If the point is one of the points in the contour or feature line from L, bIsPoint is FALSE.

TerPoint * // If bIsPoint is TRUE,

next; next is set to NULL; If

FALSE, next is pointed to the next TerPoint in the same line. Notably, next for last TerPoint of line is specified by NULL. int Rel; // It represents the

reliability level the TerPoint is ranked in, and can be set from 1 to l. l is the total number of all reliability level.

Therefore, terrain initial data can be divided into several aggregations Xi (i = 1, 2,..., /), in which all TerPoint are flagged by reliability level i. This storage method has three advantages listed below: (1) Coupling multi-source terrain data and the uniform database for sharing initial data is easy to be built; (2) It is convenient to modify local data and to extend or shrink the area of interest, also the continuity of line is still preserved; (3) The interface of the database is simple, so it is easy for many 3D modeling software to access.

3. Data structure for digital terrain

With the uniqueness of its structure and stability of triangle, TIN is the optimum for surface approximation, which makes it to be the most basic data structure for terrain representation. Moreover, TIN is suitable for not only regular-distributed data, but also, especially, irregular-distributed data that are diverse in type or inhomogeneous in density, and it is able to make full use of spatial geometry defined by lines or points to generate digital terrain. However, despite maintenance of high precision, its storage memory is inevitably sacrificed, namely TIN model will occupy more memory, and there also exists intrinsic complicacy in the management for TIN model. Therefore, TIN-DTM is not an advisable choice for many terrain analyses, like hydrological analysis, engineering exaction and filling, landslide prediction, and following 3D geological modeling in wide and complicate hydropower engineering region.

The NURBS technique is the only expression standard in STEP (ISO, 1991) for free-form curves and surfaces and gives the uniform mathematical expression for all graphics [14]. For the 3D geometry modeling of complex surface, it has the advantages of small memory space, efficient computer processing, convenient database management, dimensional uniqueness and geometric invariability [15], so it supply application value for representation of complex and wide terrain. However, the modeling data are imposed to be regular in NURBS technique, namely they should accord with the regulation of directions u and v, which the initial DTM data does not satisfy.

The terrain data of hydropower engineering are continuous or discrete, diverse and complicate. With the integration of multi-source data and the establishment of uniform

database, we propose a new hybrid data structure with two surface representations for complex terrain of hydropower engineering, which is composed of the triangulated irregular network mode and the non-uniform rational B-spline structure. This method makes full use of the technical merits of TIN and NURBS.

4. NURBS reconstruction for digital terrain

Beginning with the integrated data of DTM, a serial procedure to reconstruct digital terrain surface is illustrated in Fig. 1. These operations are listed below: 0 Translate from integrated data into point aggregation, and build a uniform database for initial terrain data, which has been discussed in Section 2; s Call enhanced Delaunay algorithm [13] for initial terrain data; ® Read from the uniform database, then call enhanced Delaunay algorithm for the DTM data; ® Sample sectional lines from TIN model; ® Establish a regular terrain database with the sampling data; © Perform compression for vector data, and obtain the modeling data; ® Employ NURBS skinning technique; ® Achieve an indirect transformation of terrain data, from discrete and irregular to consecutive and regular. Their detailed steps will be discussed in the following context.

4.1. TIN-DTM modeling

Applying TIN technique, the surface which is constitutive of many triangles with topology can be constructed with irregular integrated data or uniform database to approximate the terrain. What is more, when a great deal of F-S data is included, the information of terrain will be more abundant, and TIN model will be more precise. Considering the continuity of line data, enhanced Delaunay algorithm is introduced to build TIN-DTM.

The model covers all terrain information, and linear interpolation is automatically implemented when the data are absent, so the detail of terrain in TIN is precisely and objectively represented. However, because of inherent characteristic, this model occupies large memory. Generally, the memory capacitance of TIN-DTM for hydropower engineering region is up to 70 MB, and its efficiency of data description is very low, which will greatly block the further studies. Thereby, based on this

Fig. 1. Flowchart for digital terrain reconstruction.

TIN-DTM in high precision, some technical disposal should be executed to model the required terrain surface.

4.2. Sampling based on TIN model

Based on the TIN model, it will be a good way to regularly sample data for further terrain reconstruction. In this paper, sectional line sampling method with a linear interpolation is put forward.

Fig. 2 demonstrates part of TIN surface we will study on. 4P1P2P3 is a triangular facet, whose vertexes P1, P2 and P3 can be directly read from TIN model. The vertical plane s for sampling is known, and it is defined by Eq. (2) in 3D Euclidean space.

Y = kX + m (2)

The plane, where 4P1P2P3 is located, can be described by Eq. (3). Coefficients ao, a1 and a2 are unknown, but will be acquired by solving Eq. (4). It should be noticed that (xi, yi, zi) is the coordinate value of the vertex Pt (/=1, 2, 3).

Z = a0 + a1X + a2Y (3)

ao 1 x1 y1 —1 Z1

a1 = 1 x2 y2 Z2

a2 1 X3 y3. .Z3.

{y = kx + m

z = a0 + a1x + a2y (5)

Z = Z1 +(Z2 - Z1 )(x - X1)/(X2 - X1)

In Eq. (5), there exist simultaneous equations for inter-sectant points, A and B, between the plane s with the two boundaries of AP1P2P3, P1P2 and P2P3. The relevant variables (x1, z1) and (x2, z2) are assigned with the (x, z) of P1 and P2 or P2 and P3, and the coordinates of points A and B will be gained, respectively. However, it is required that equation z = z1 + (z2 — z1) (x — x1)/(x2 — x1) should be replaced by x = x1 in special condition x1 = x2.

A line segment is drawn to join the intersectant points A and B, and line segment AB is the sampling data from the triangular facet AP1P2P3 with linear interpolation. The above operations of the mathematical computation are

/ -^lll | r ^^^ ■Ti PP^"

Fig. 2. Sampling data with linear interpolation from a triangular facet (in top view).

encapsulated into the bottom layer, so that it can be implemented conveniently by calling the program.

Based on the TIN model, the stepwise procedure of regular sampling for the whole terrain region is stated as follows:

Step 1: Build a rational vertical plane s, and ensure that s is wide enough, approximately perpendicular to the flow direction of river, and located on starting point of traversal;

Step 2: Compute the intersectant points between the boundaries of TIN model and sampling plane s by calling the algorithm mentioned above;

Step 3: Join ordinally two adjacent points with a line segment, and then the data between two points are linearly interpolated, so a sectional sampling line for s is gained.

Step 4: Move parallelly the plane s with the distance of d along the direction of traversal, go to step 2 until the whole TIN-DTM is entirely covered.

When the sampling procedure is completed, the regular terrain data are gained. d is the sampling interval of two adjacent sectional lines in distance, so the smaller d is, the more detailed the sampling data will be. Generally, d is set to 1 m. Meanwhile, regular terrain database is established with the sampling data, which is what we need. Based on the database, the reconstruction of hydropower engineering terrain can be reproducible and satisfy various precision demands.

4.3. Compression of sampling data

The aim of data compression is to delete the redundant data, save the storage memory, and, what is more important, quicken the subsequent operations. Data from the sampling are not optimal, and its deficiency is represented by existence of the redundant data. To improve the efficiency of storage memory, the compression is imposed to be operated for the following NURBS surface modeling.

A new algorithm for compressing terrain sampling data is presented, differing from the commonly used Douglas-Peucker algorithm [16], with the constraint that two factors, ang and dist, should satisfy the respective given criteria. Here, ang denotes the included angle between two vectors, and dist denotes the length of a vector line.

Let S = (S1, S2,..., SN) define the locations of consecutive vertices in a polyline, which represents the original terrain sectional line from sampling. By all appearances, S1 and SN are begin-point and end-point, respectively. Constants, Ca and Cd, are the given criteria for ang and dist, respectively.

c is a variable for the amount of points deleted, and its initial value is set to zero. The algorithm of compression for a single sectional line is described as follows:

Step 1. Pick up three points from a terrain sampling line, Si, Si+1 and Si+k (i =1,2,...,N, k = 2, 3,...,N-i, and the initial value of i and k are set to 1 and 2, respectively). So, the vector V1 = SS77, V2 = , the length of V1 and V2 is defined by d1 = |V1| and d2 = |V2|, and the angle between V1 and V2 is a;

Step 2. There exist three cases. case 1, when a = 0, then k = k + 1; case 2, when a > Ca, or d1 > Cd, or d2 > Cd, then delete the point sequence from Si to Si+k-1, i = i + k, and c = c + k — 1; otherwise, k = k + 1; Step 3. Go to step 1 till i + k = N, end the algorithm and a new sectional polyline is gained.

The ratio of compression can be defined by e = (c/ N) x 100%. It is clear that the two end-points of sectional line are remained. From above algorithm, the maximal deviation from the deleted points to the new polyline is not more than L = 2 x Ca x Cd.

This new compression algorithm is suitable for any type of terrain sampling data, including the condition that it has many flections along the polyline. In the engineering application, the compression criteria (Ca, Cd) are usually set to (10 m, 0.5o), so L « 0.175 m. What is more, the ratio of compression always may be up to 70%, which greatly improves the efficiency of storage. In (Fig. 3), it is showed that part of a single sectional line is compressed. 10 points are removed for improving the storage efficiency, and their maximal deviation distance is about 0.153 m.

4.4. NURBS-DTM generation with skinning method

Surface skinning, the most important method of reconstruction for free-form surface, is a process of passing a smooth surface through a set of the so-called cross-sectional curves [17]. Using the NURBS representation, the method admits great generality in that the cross-sectional curves can be of any degree, and can be defined over different knot vectors.

Skinning is a memory hungry process, and approximation is willing to be accepted instead of interpolation. The algorithm for approximate skinning [18], presented by Les Piegl, can eliminate a significant number of control points, about 90%, from the precisely skinned surface using

a ¿7 T ^12

Sio 5n

specified tolerance. Both the cross-sectional curves and the skinned surface can be approximated. Hence, the approximate skinning is introduced to reconstruct the terrain surface of hydropower engineering.

The modeling data gained from the compression are piecewise linear curve, namely polyline, so the transformation from polyline to NURBS line is going to be executed. Meanwhile, in order to skin across the curves of various types, they all have to be made compatible, i.e. they should be all rational or non-rational, and have the same degree, or be defined over the same knot vector. Then, control points located in the surface to be reconstructed have been acquired. The back calculation [17] with NURBS technique is employed, and eventually the terrain surface approximates all cross-sectional curves. The skinning is completed, and NURBS-DTM for hydropower engineering is reconstructed successfully.

5. Accuracy evaluation indexes

Evaluation indexes, root mean square error (RMSE) and mean error (ME), are conventionally introduced to assess the accuracy of reconstructed terrain surface [19]. They are both mathematical statistics depicting deviation from the elevation of control points of DTM surface to the true value of their relevant sampling points, and are defined as follows:

RMSE — J- > '(Rk - Zk)2 1 n

ME = -V | Rk - Zk |

(6) (7)

where n is the amount of sampling points, Zk represents the elevation of sampling point Pk from the terrain uniform database, and Rk is the elevation of control point that is from the reconstructed surface and importantly with the same plane coordination as the sampling point Pk.

However, because of initial multi-source data, sampling point with different reliability levels will have different weights of contribution for whole terrain surface accuracy, and the weight value in the RMSE and ME is thought to be the same, namely 1/n. Hence, based on the terrain uniform database and classification mechanism of reliability level, weighted RMSE (WRMSE) and weighted ME (WME) are presented to assess the accuracy of reconstructed terrain surface. They are both closely related with the engineering application, and are defined as follows:

V k —1

WRMSE — JV Wk (Rk - Zk)

WME = Rk - Zk |-Wk k1

(8) (9)

Fig. 3. Compression for a polyline. (a) An initial polyline; (b) The new polyline after compression.

where wk is the weight value of Pk's contribution on the whole terrain surface accuracy, and its expression is

wk = ajiii, where nt is the point number of X,, and a, is the sum of weight value specified for all points in X,. What is more, there exist two relation equations: J2'¡=in¡ = n and Y^\=\ai = 1 (0 6 at 6 1, i = 1, 2,...,/). It is found that point ranked in the same reliability level will have the same weight value, and the sum of all weight values is equal to 1.0.

In the presented assess model, the value of ai can be set from 0.0 to 1.0 according to the engineering consideration. The reliability of terrain data from different source is varied, so WRMSE and WME are suitable for accessing the accuracy of reconstructed terrain surface of engineering with multi-source data.

6. Experimental application

The planned hydropower facility is located on the Jin-shajiang Gorge in the southwest of China, and it is the third of large hydropower station whose installed capacity exceeds 10 MKW, behind the Three Gorges hydroplant and Xiluodu hydroplant. Hyperbolic arch dam and underground structures are the main works of the engineering construction. The region of interest stretches north along the Jinshajiang River, and is a rectangle with 3000 m long and 2240 m wide. The relief is significantly complicated, with cloughs and altiplanos. The physiognomy is high in the north and low in the south, inclining to the east.

The integrated terrain data include 57,213 contour lines with one meter elevation interval, from 570 m to 1944 m, 8212 geo-points, two road lines along the river, and several breaklines. The TIN-DTM is built with a memory storage of 75.6 MB. During sampling process d is set to 2 m, and

the compression with the criteria of 10 m and 0.5°, whose e is 73%, is implemented. By the NURBS technique and skinning method, the NURBS-DTM is reconstructed, as illustrated in Fig. 4. Its storage amount is only about 7.41 Mb, decreasing from 101 Mb level to 100 Mb level, and it has only 320,000 control points in NURBS-DTM, rather than about 3,500,000 ones in TIN-DTM.

As Fig. 5 displays, 6627 checkpoints are picked up from the uniform database to access the accuracy of the reconstructed terrain surface, and they are all the geo-points from geological mapping. Checkpoints, in the red region and closely around the hyperbolic arch dam, are more reliable than the ones out of the red rectangle, and its sum weighted value is defined by a1, so a2 = 1 — a1. The accuracy assess result is given in Table 1. The first-level accuracy standard of RSME for the mountain DTM from China Bureau of Surveying and Cartography (in 1998) is

Fig. 5. Distribution of checkpoints.

Fig. 4. The reconstructed terrain surface for hydropower engineering. (a) The terrain region of interest in top view. It covers the main structures of hydropower engineering, hyperbolic arch dam and underground structures; (b) and (c) show the topography of right bank and left bank in engineering region, respectively.

Table 1

Accuracy access for NURBS-DTM

WRMSE/m WME/m

RMSE/m a1 = 0.7, a1 = 1.0, a1 = 0.0, ME/m a1 = 0.7, a1 = 1.0, a1 = 0.0,

a2 = 0.3 a2 = 0.0 a2=1.0 a2 = 0.3 a2 = 0.0 a2=1.0

0.4485 0.4337 0.4227 0.4584 0.3443 0.3394 0.3359 0.3476

2.5 m. Therefore, the terrain detail has been represented in the NURBS-DTM, and the model adequately meets the accuracy required for 3D geological modeling and visual analysis.

7. Conclusion

In the large-scale hydropower engineering, DTM has been playing an important role in 3D designing, visual analysis and 3D geological modeling. In application, it should be characterized by high precision, less storage and well interactivity. Making full use of technical merit of data structures, TIN and NURBS, the paper presents a novel method of NURBS reconstruction digital terrain for hydropower engineering based on the TIN model. This method perfectly handles the problem of terrain multi-source data, and after sampling and compression the transformation for data, from discrete, irregular and diverse to continuous, regular and uniform, is achieved. Then the NURBS-DTM satisfying the requirement is reconstructed with the intermediate data by skinning method.

However, terrain condition of hydropower engineering is diverse, and practical requirement should be taken into account. Therefore, the modeling parameter, like d, Ca and Cd, should be selected reasonably, and the result should offer best tradeoff between precision and storage for hydropower engineering.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 50539120, 50525927, 50579045) and the National Basic Research Program of China (Grant No. 2007CB714101).

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