Large-scale Terrain Level of Detail Estimation based on Wavelet

Transform

Sid'Ali Kalem and Assia Kourgli

USTHB, Faculté d'Electronique et d'Informatique, LTIR, BP32 El_ Alia Bab-Ezzouar, 16111, Algiers, Algeria

Keywords: Large Terrain, Rendering, LOD, View-dependent, GPU, Wavelet, QuadTree.

Abstract: The goal of the following paper is to point out an alternative approach to the adaptive triangulation problem.

A new technique of terrain rendering which uses wavelet transform to select appropriate LOD is described.

This technique is a region-based multi-resolution approach that partitions the terrain into tiles that can be

processed independently. The algorithm organizes the heightmap into a QuadTree of nodes and computes

maximum world-space errors for each node. World-space errors are then calculated at preprocess step. As

the datasets of realistic terrains are usually huge, we suggest using the multi-resolution wavelet

decomposition to localize the position of the maximum world-space error estimated and limit the region of

research inside the node. It permits to choose the appropriate resolution of the regular grid that will

represent the node at run time. By this way, computation load on the CPU is greatly reduced.

1 INTRODUCTION

Over recent years, terrain rendering has been used in

different fields such as movies, virtual

environments, cartography, and games. In particular,

it has been intensively developed for real-time

outdoor games including flight simulators, driving

simulators, and massive multiplayer games. The

rapid development in acquisition of topographic

maps and cartography has led to the generation of

large terrain datasets as height-maps that contain

billions of samples. Such terrains datasets exceed the

rendering capability of available graphics hardware.

Consequently, it is not possible to display 3D scenes

represented by too many details in real-time. Thus,

adaptive Level-Of-Detail (LOD) rendering is used to

simplify the geometry of the heightmap-based

terrain. Usually, LOD rendering algorithms

represent terrains as triangulated meshes which

approximate the surface of the terrain. The challenge

is to efficiently combine quality rendering and real-

time navigation.

The goal of the following paper is to point out an

alternative approach to the adaptive triangulation

problem: the usage of the wavelet transform (WT) as

a mathematical framework which localizes rough

surface approximation where error of the

approximation should be controlled. In some sense,

the WT provides a local spectral estimate of the data

and describes local variations which can be

harvested to govern the coarseness of a surface

mesh.

The organization of the paper is as follows: In

the next section, we give a brief review on terrain

representation techniques. Section 3 is dedicated to

the description of our terrain rendering algorithm. In

the following section, we describe the mathematical

framework of the 2D wavelet transform for surface

and we introduce our main contribution, the

algorithm which determines the appropriate

resolution of the grid based on the wavelet

transform. Section 5 shows the implementation.

Finally, the results of our algorithm are presented in

section 6.

2 RELATED WORK

A number of view-dependent LOD techniques for

terrain rendering have been proposed, which differ

mainly in the hierarchical structure used. Previous

work can be broadly classified into dynamic re-

meshing strategies, region-based multi-resolution

approaches and regular nested grids. All of those

approaches permit continuous LOD rendering of the

terrain geometry.

View-dependent dynamic re-meshing techniques

construct a continuous LOD triangulation in every

258

Kalem S. and Kourgli A..

Large-scale Terrain Level of Detail Estimation based on Wavelet Transform.

DOI: 10.5220/0005310402580264

In Proceedings of the 10th International Conference on Computer Graphics Theory and Applications (GRAPP-2015), pages 258-264

ISBN: 978-989-758-087-1

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

Figure 1: Rendering 3D terrain using regular grid’s meshes of different resolutions.

frame with respect to a given world-space deviation

and screen-space error tolerance. Early approaches

were based on Triangulated Irregular Networks

(TINs) as introduced by Peucker (Peucker T. K.,

Fowler R. J., Little J. J., 1978) and Fowler (Fowler

R. J., Little J. J., 1979) those approaches are well-

known by their approximation quality. Irregular

triangulations minimize the amount of triangles to be

rendered at a given approximation error, but on the

other hand they require quite elaborate data

structures that necessitate an intense CPU

processing. Consequently, more regular

triangulations have been used, for instance, bin-tree

hierarchies (Lindstrom P., Koller D., Ribarsky W.,

Hodges L. F., Faust N., Turner G. A., 1996)

(Duchaineau M., Wolinsky M., Sigeti D. E., Miller

M. C., Aldrich C., Mineev-Weinstein M. B., 1997)

and restricted quad-tree meshes (Von Herzen B.,

Barr A. H., 1987) (Pajarola R., 1998).

Region-based multi-resolution approaches

partition the terrain into tiles that can be processed

independently (Koller D., Lindstrom P., Ribarsky

W., Hodges L. F., Faust N., Turner G., 1995) (Suter

M., Nüesch D., 1995) (Blow J., 2000). To avoid

visual artifacts like popping, either geomorphs are

used (Ferguson R. L., Economy R., Kelly W. A.,

Ramos P. P., 1990) or the maximum screen-space

error is restricted to one pixel. Recent region-based

multi-resolution approaches are based on techniques

that fully exploit the power of modern graphics

hardware. BDAM (Cignoni, P., Ganovelli, F.,

Gobbetti, E., Marton, F., Ponchio, F., and Scopigno,

R, 2003) and P-BDAM (Cignoni P., Ganovelli F.,

Gobbetti E., Marton F., Ponchio F., Scopigno R.,

2003) methods proposed by Cignoni et al exploit

bintree hierarchies of pre-computed triangulations or

batches instead of individual triangles. C-BDAM

method, an extension of BDAM and P-BDAM

algorithms, was presented by Gobbetti et al in

(Gobbetti, E., Marton, F., Cignoni, P., Di Benedetto,

M., and Ganovelli, F, 2006). The method exploits a

wavelet-based two stages near-lossless compression

technique to efficiently encode the height map data.

Terrain rendering method presented by Schneider

and Westermann (Schneider, J., and Westermann, R,

2006) partitions the terrain into square tiles and

builds for each tile a discrete set of LODs using a

nested mesh hierarchy. Following this approach,

Dick et al proposed a method for tile triangulations

encoding that enables efficient GPU-based decoding

(Dick, C., Schneider, J., and Westermann, R., 2009).

Refer to a nice survey by R. Pajarola and E.

Gobbetti (Pajarola, R., and Gobbetti, E., 2007).

Losasso and Hoppe (Losasso F., Hoppe H.,

2004) even show that re-meshing can completely be

avoided by using a set of nested regular grids

centered about the viewer. As the grid resolution

decreases with increasing distance to the viewer,

approximately uniform screen-space resolution is

achieved. This technique caches the terrain in a set

of nested regular grids centered about the viewer.

Asirvatham and Hoppe further improved this

technique in (Asirvatham A., Hoppe H., 2005) to

handle most of computations on the GPU.

Thus, techniques proposed in (Losasso F., Hoppe

H., 2004) (Asirvatham A., Hoppe H., 2005) depend

only on camera position and do not take into account

local surface characteristics. We still believe that

local surface characteristic is an important

component of 3D terrain rendering process, since

different datasets have different characteristics that

should be automatically taken into account to

guarantee the quality of the rendering. Settings used

for terrain rendering should be carefully chosen to

match terrain dataset characteristics while providing

the best performance. In this context, we propose

new technique of LOD estimation based on multi-

resolution wavelet decomposition that permits to

adapt 3D terrain rendering process according to local

surface characteristics in order to reduce

computation load on the CPU.

3 ALGORITHM DESCRIPTION

Our algorithm is a region-based multi-resolution

Large-scaleTerrainLevelofDetailEstimationbasedonWaveletTransform

259

approach which partitions the terrain into tiles that

can be processed independently as illustrated in

Figure 1. The concept of the rendering method is

illustrated in Figure 2.

First, the algorithm organizes the heightmap into

a QuadTree of nodes, which is used to select

appropriate nodes from different LOD layers at run

time, as illustrated in Figure 3. The algorithm is

based on QuadTree as spatial subdivision scheme.

The QuadTree hierarchy does not store any

geometry; instead it stores the position and

dimension of each node with respect to the terrain.

The QuadTree structure is generated from the input

heightmap. It is of constant depth, predetermined by

memory and granularity requirements. Once created,

the QuadTree does not change unless the source

heightmap changes. In the QuadTree structure every

node has four child nodes and it covers four times

more area than one of its children.

Figure 2: Terrain rendering algorithm.

In the second step, the algorithm computes

maximum world-space error for each node of the

QuadTree. Object-space error is independent from

the metric used and can be computed directly from

the finest resolution grid. World-space errors are

calculated at pre-process. They are used to select the

appropriate resolution of the regular grid which will

represent at run time the node. As the datasets of

realistic terrains are so big, we use the multi-

resolution wavelet decomposition to localize the

position of the maximum world-space error and limit

the region of research inside the node, which is the

main contribution. As a result, we reduce

computation load on the CPU. Since the heightmap

is organized as QuadTree of nodes, we can calculate

only appropriate resolutions of nodes of the lowest

level in the QuadTree and afterwards deduce

resolutions of their parents.

The resolution of the regular grid chosen must

guarantee the quality of the triangulation

approximation of the node. So, the resolution of the

grid to be used for each node of the terrain at run

time is stocked in the structure QuadTree.

At run time, the third step of the rendering

process is the QuadTree nodes selection. It is

performed every time the observer moves, which

usually means during every frame. The LOD of

nodes corresponds to the QuadTree depth level (area

covered) of the nodes and the regular grid’s

resolution used to render the node. The selection is

performed according to the distance between the

node and the camera. At this stage we also perform

view-frustum culling that eliminates the rendering of

non-visible nodes. In order to know which nodes to

select where, distances covered by each QuadTree

layer are pre-calculated before the node selection

process is performed. The array of distance ranges is

thus created which it is also used to create an array

of world-space error tolerable at each layer of the

QuadTree. The array representing the world-space

error tolerable in each range of distance will help us

to choose the resolution of the regular grid for nodes

of the QuadTree.

The actual rendering is performed in step four by

rendering areas covering selected nodes using

regular grid-meshes of different resolution, reading

the heightmap in the Vertex Shader, and displacing

the mesh vertices accordingly, thus forming the

representation of the particular terrain patch.

Commonly used grid-mesh dimensions are 8x8,

16x16, 32x32, 64x64 or 128x128, depending on the

required output complexity.

4 RESOLUTION ESTIMATION

The main contribution is the idea of using the

wavelet coefficients to quickly identify tile sub-

regions that contain the largest world-space error.

Thus, we present a new method for adaptive surface

meshing which selects the appropriate resolution of

the regular grid-mesh by local estimates. The latter

are determined by a wavelet representation of the



(

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)

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(1)

GRAPP2015-InternationalConferenceonComputerGraphicsTheoryandApplications

260

surface data. So, the thought is to decompose the

initial data set by wavelet transform WT and to

analyze the resulting coefficients. In surface regions,

where the partial energy of the resulting coefficients

is high, fine grain details are localized, and

consequently, the maximum of world-space error is

also localized and can be calculated. This approach

employs the WT to expand the data and the

amplitude of the detail signals is taken as a measure

of the local frequency. Applying wavelet transform

allows an elegant and fast estimation of the local

level of detail needed.

Figure 3: Height-map organized as a QuadTree of nodes,

and corresponding arrays of distance ranges and elevation

tolerable errors.

Any finite energy function f(x,y) ∈L

) can be

approximated by the bases elucidated above.

denotes the coordinate of ƒ in functional

space with respect to the wavelet ψ

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The localization of the maximum world-space

error inside the node by detail signals of wavelet

decomposition decreases the computation. Instead of

researching the maximum world-space error over the

entire node, this limit the region of research to

already localized fine grain details as illustrated in

the figure. 4. We will see in section 6 that the

accuracy of the localization depends on the type of

the wavelet used to decompose the heightmap.

It is clear that, we can now formulate a simple

criterion for the maximum of world-space error by

introducing a threshold τ. Increasing τ will result in

increasing the error bounds of the approximation and

decreasing τ will decrease the approximation error.

According to the maximum of elevation errors of

each node of the QuadTree, and by comparing this

maximum with τ we can select the appropriate

resolution for the regular grid to guarantee the

quality of surface approximation of nodes.

According also to the array of distance ranges pre-

computed (Figure 3), the threshold tolerated for each

range of distances is calculated by the formula (3).

Thus every layer of the QuadTree has its threshold.

The quality of the approximation is guaranteed

by the use of a maximum screen space error. Screen

space error is derived at run time from a patch

bounding volume and its world-space geometry. To

approximate screen space error of the node we use

the following equation:

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Where S is the screen resolution (maximum of

horizontal and vertical resolutions),

γ is the field of

view angle,

ε is the node geometric world space

error calculated at preprocess stage, and d is the

distance from the camera to the bounding box.

Since node approximations provide guaranteed

world-space error bound, the given formula provides

guaranteed screen-space error bound of the node.

During the pre-process we increase the grid

resolution for the regions with screen space error

greater than defined threshold and decrease it where

it does not introduce intolerable error. This simple

top-down algorithm generates adaptive

approximation which satisfies user-defines screen

space error threshold.

Figure 4: Localization of the regions of maximum object-space error through wavelet decomposition inside a node.

Large-scaleTerrainLevelofDetailEstimationbasedonWaveletTransform

261

5 IMPLEMENTATION

The implementation of terrain rendering is written in

C++, using DirectX9 as the graphical API and

HLSL. It should work on most GPUs that support

vertex Shader texture sampling, ones supporting

Shader Model 3.0.

To best exploit power of modern GPUs we cache

data of terrain elevations in the fast GPU video

memory and use it across successive frames. CPU

performs QuadTree traversal and selection of

appropriate LOD for different areas of the terrain

based on node geometric world space error and

distance to camera. CPU also performs view-frustum

culling. Thus slow data transfer between CPU and

GPU occurs very rarely.

6 RESULTS AND DISCUSSION

For the following, we investigate the best type of

wavelet function, used to decompose the elevation-

map, to localize regions of maximum world-space

error. We test several types of wavelet functions

such as Haar, Daubechies and biorthogonal wavelet.

We use the dataset “Puget Sound” 16385×16385

height map sampled at 10 meters spacing, which is

used as the common benchmark, with no normal

map, no dynamic lighting, no detail map, and an

overlay color map with embedded lighting. The

dataset is large enough for realistic profiling of

performance. Nodes used for measures are nodes of

low level of the QuadTree that have 1025 × 1025

size. We investigate choice of the type of wavelet

function, used to decompose the elevation-map,

which most closely matches the surface

approximation.

6.1 Maximum World-space Error

Estimation

One of the main advantages of our method is the low

algorithmic complexity for both computation of the

respective transforms and for the QuadTree

meshing. The localization of the maximum world-

space error inside the node by D2WT benefits from

dyadic scaling and sparsity and requires only O (N

)

computations. The localization allows us to avoid

computing world-space error inside the entire node,

but focus the computation on the fine grain details

region as illustrated in figure. 4.

Table 1. depicted the substantial gain in

computation loaded on the CPU to compute the

Table 1: Reduction of computation load on the CPU.

Resolution Window

size

Errors computations

per node

Computation

gain (%)

512 3

9 99.9991

256 5

25 99.9976

128 9

81 99.9923

64 17

289 99.9725

32 33

1089 99.9

16 65

4225 99.6

8 129

16641 98.42

maximum world-space errors, in order to select

appropriate resolution for each node. The table

shows the number of error computations performed

per node instead of 1025 × 1025 computations over

the entire node. Furthermore, the table illustrated the

gain in computation compared to computation over

the entire node (1025 × 1025 computations).

We define the estimation error as the ration of

the remaining vertices whose planar approximation

error exceeds the maximum world-space error

estimated by the algorithm:

 = M N



⁄

(4)

Figure 5: Estimation error per node.

Where N

is the number of elevation samples

(1025×1025) per node and M is the number of

elevation samples whose planar approximation error

exceeds the maximum world-space error estimated

GRAPP2015-InternationalConferenceonComputerGraphicsTheoryandApplications

262

by the algorithm. Let f(x,y) be the original surface

and g(x,y) be an approximation. M is the number of

elevation samples f(x

) that:

∆

(



)

(



)

−g

(



)

>∆



(5)

The Figure. 5. illustrates the estimation error per

node of our algorithm for several types of wavelet

function. According to results depicted in Figure 5,

Haar wavelet provides the best localization with

small estimation errors. When we use Haar wavelet

(Figure. 6.), we notice that the estimation error is

negligible and can be interpreted as some kind of

coding gain, depending on the purpose.

Figure 6: Estimation error per node (Haar wavelet).

6.2 Error Analysis of Planar

Approximation

Figure 7: Root of mean square errors of the planar

approximation per node.

One important aspect, when dealing with surface

approximations, is to quantify the approximation

error of the method. In our approach, error

quantification is figured out by the following mean-

square measure. The respective reference value for

the surface f

) is obtained by bilinear

interpolation.

Let f(x,y) be the original surface and g(x,y) be

an approximation. We define the mean-square error

as:

∆

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∑

∆(

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,

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where: Δ (x

) = f(x

) - g(x

)

In Figure. 7. the root of the mean-square error is

recorded in meter for each node and for several

types of wavelet functions. We notice that the Haar

wavelet provides the best mean-square error. In the

case of Haar wavelet, we note that the mean-square

error is pretty low for high quality of rendering.

7 CONCLUSIONS AND FUTURE

WORKS

In this paper, we have presented new approach of

selecting LOD based on wavelet decomposition.

This approach uses the wavelet transform as

mathematical framework to localize fine grain

details region where we focus computation which

estimates the maximum of world-space error used to

chooses the appropriate resolution of the regular grid

mesh for each tile of the terrain. This provides a less

expensive method in computation. We also

investigated the best type of wavelet function, used

to decompose the elevation-map, to localize regions

of maximum world-space error. We found that the

Haar wavelet localizes better than other wavelets

regions of maximum world-space error.

In a future work, we will investigate choice of

the type of wavelet function, used to decompose the

elevation-map, which most closely matches the

surface approximation as triangulated meshes.

Details of wavelet decomposition would be used

directly as world-space errors. Moreover, one future

issues of our research is providing an overall

mathematical framework based on wavelet

transform improving the representation of the 3D

height-field geometry in terms of memory cost, time

performance and rendering quality.

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