A SKETCHING INTERFACE FOR TERRAIN MODELING. Nayuko Watanabe. A Senior Thesis

Transcription

1 A SKETCHING INTERFACE FOR TERRAIN MODELING by Nayuko Watanabe A Senior Thesis Submitted to the Department of Information Science the Faculty of Science, the University of Tokyo on February 9, 2004 in Partial Fulfillment of the Requirements for the Degree of Bachelor of Science Thesis Supervisor: Takeo Igarashi Assistant Professor of Information Science

2

3 ABSTRACT Three-dimensional terrain models are used not only for geographical analysis but also for entertainment application such as movies and video games. These applications do not require exact numerical values for models. It is only necessary to make terrain model look natural. However most existing methods for terrain modeling require precise numerical control. Some systems allow the user to paint 2D height fields for terrain models, but it is not easy to edit 2D height field appropriately to obtain a scene that has a specific appearance from a given viewpoint. We developed a system where the user can design the terrain model more intuitively. The user edits 3D geographical model by drawing 2D strokes on the model directly. Whenever a stroke is drawn, this system generates a terrain model whose silhouette conforms to the stroke on the screen. The system also supports the construction of more natural look by adding a noise D 2 1 2

4 Acknowledgements We would like to thank to our thesis Supervisor Takeo Igarashi for his support in doing this work. He gave us many hints, useful advice about this thesis. We also would thank members of Igarashi laboratory. It was not possible to complete this work without their encouragement.

5 Contents 1 Introduction 1 2 Related Work D modeling from 2D input Terrain modeling Noise User Interface 9 4 Implementation Detail Data structure Terrain modeling from 2D input Stroke recognition of mountain Stroke recognition of river Noise addition Result 25 6 Limitations and Discussion 28 7 Future Work 29 i

6 List of Figures 1.1 Overview of the system Terragen : input (height-field) and output Composition of noise function in Perlin noise [8] First view of the system Stroke(upper) and top view of mountain(lower) Stroke which has neither local maximum nor minimum point Stroke which has one local maximum point Stroke which has two or more local maximum and local minimum point Round and sharp stroke and mountains generated by these strokes Creating a river Stroke of a river over the mountain and bridging gaps Strokes and mountain generated by the strokes Effect of noise addition on existing terrain Terrain modeling from 2D input altitude of mountain Filling a gap between start and end point Stroke and foot of mountain generated by the stroke Stroke and altitude of mountain generated by the stroke Terrain drawn by author Terrain drawn by test users ii

7 Chapter 1 Introduction Three-dimensional terrain modeling requires considerable labor for data entry. One reason is that designer is required to enter numerical values, and another is that he or she is unable to do modeling interactively. The designer must input numerical values, render the scene, adjust values, and repeat these operations. We therefore develop a pen-based system that reconstruct a terrain from novel viewpoints, given the terrain drawn in two-dimension from a single point of view. There are many researches about 3D modeling from 2D input [7] [5] [15] [10]. Among them, Harold excels at ability of modeling terrain [6]. We extend Harold s algorithm to recognize strokes used for terrain modeling, producing more natural terrain model. In our system, strokes are projected to the vertical plane interpolating the first point on the ground. and then the system generates a terrain model whenever a stroke is drawn. In the terrain model, whole slope of mountain reflects the shape of corresponding stroke. The user draws a stroke like in Figure 1.1(a) and gets terrain model such as Figure 1.1(b). We add noise to the terrain model so that the terrain looks natural. Simple random noise however at times does not look natural. Fractals are more suitable for natural object whereas they take time to generate. So we use noise function developed by Perlin [8] that is able to generate fractal-like noise. Perlin Noise is a noise function that adds noises with various frequencies and amplitudes. This noise function generates very natural noise. 1

8 (a) The user s stroke (b) A mountain generated by the stroke Figure 1.1: Overview of the system We can draw terrain intuitively and interactively, and make a terrain model look natural. The rest of this thesis is organized as follows. Related work section mentions 3D modeling from 2D input, terrain modeling, and noise function. Interface section explains how to use our system. In implementation detail section, we describe data structure for terrain model, stroke recognition algorithm, and noise addition algorithm. Limitations and discussion section describes problems of our method, and in future work section we provides solution to limitations and directions for future work. 2

9 Chapter 2 Related Work This chapter describes related research in 3D modeling from 2D input, terrain modeling, and noise functions D modeling from 2D input A typical procedure for geometric modeling is to start with a simple primitive such as a cube or a sphere, and gradually construct a more complex model through successive transformations or a combination of multiple primitives. Today, interactively constructing 3D objects and scenes from 2D input is an active research area. Cohen et al. described this approach as the inverse of traditional computer graphics [6]. The system tries to construct a geometric model of the object from given rendering. And new renderings can then be obtained from arbitrary viewpoints. Quick-sketch is a 2D and 3D modeling tool for pen-based computers [7]. The user is able to sketch 3D solid objects and B-spline surfaces in Quick-sketch, by sweeping a 2D profile along a straight line (solid objects) or interpolating between two curves (B-spline surfaces). For modeling 3D curves, there is a method using shadow drawing [5]. The SKETCH allows the user to create a 3D object by drawing as 2D representation in similar way to draw with a paper and a pencil [15]. Teddy is the system 3

10 (a) Input to terragen (b) Output of terrain Figure 2.1: Terragen : input (height-field) and output that constructs 3D geometric object from 2D free form strokes/gestures interactively and intuitively [10]. Cohen developed Harold by which the user can draw 3D world and explore in the world [6]. Harold can do terrain modeling interactively. We can explore and expand 3D world in Harold by drawing 2D strokes. A drawback of Harold is that the generated mountains tend to be thin. Modeling natural objects is difficult because of their complexity. Okabe et al. modeled 3D tree from 2D strokes [13]. Tree modeling requires the adjustment of parameters and randomization, in addition to stroke recognition. This is similar to terrain modeling, however tree modeling algorithm cannot be applied to terrain modeling. Elevation grids are frequently used for data structure of terrain model. We use it for reasons of its simplicity and versatility. Terragen TM is a software for terrain modeling using an elevation grid [14]. The user gives 2D height field, and it become height of grid. Figure 2.1(b) is output of Figure 2.1(a) with view point placed in the south. A limitation of this is unable to specify the height interactively. 2.2 Terrain modeling Terrain generation mainly produces a height map (a 2D array of height values), then uses the map to draw the actual terrain. Methods to create a height map 4

11 are generating algorithmically or using real data. There are many algorithms that generate terrain model algorithmically such as Fractional Brownian motion [4], area subdivision (midpoint displacement) [3], function composition and real world based approaches. The key concept of fractional Brownian motion, area subdivision, and function composition is self-similarity (fractal). An object is said to be self-similar when magnified subsets of the object look a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a smaller copy of the whole. Fractals are mathematical functions proposed by Mandelbrot to describe natural phenomena such as coast lines, mountains, branching patterns of trees and rivers, clouds, and earthquakes. Fractional Brownian motion (called fbm) [4], area subdivision [3] and function composition [8] are algorithms for creating self-similar pattern automatically. The fbm expresses mountains vertical section form by using fact that behavior of Brownian motion is similar to the form of mountains. Area subdivision (or midpoint displacement) is a recursive approach. It is not fractal in a precise sense but is a fractal-like method. In one dimension, it takes a horizontal line segment, finds midpoint, then displaces it by a random amount, and repeat these operations. Area subdivision has been used by N.Wiener in 1920 s in order to create Brownian motion. And its efficiency of the calculation is higher than fbm s efficiency. There are many algorithms that process more or less the same way as area subdivision. For instance, Diamond Square algorithm, the Fault algorithm, Circle algorithm, and Hill algorithm. A drawback of these methods is that the computation is dynamic, and it is impossible to directly compute the height of a point. The function composition algorithms use functions to generate curve and compose them to form the surface. And it is similar to Fourier. The algorithms evaluate functions at each point of surface, repeating with lower amplitude, higher frequency for several octaves. There are two classic algorithms: Brown noise and Perlin noise [8]. Details of these noises are described in the next noise section 2.3. In this section, we mention terrain generated by using these noises. 5

12 Brown noise which follows fbm generates results that are highly correlated and good for mountains. Perlin noise can generate terrain more easily as well as textures. Its advantage is that the height at a given point is calculated without need of storing it in memory like subdivision methods. Many computer-generated landscapes use Perlin noise. Other direction of generating terrain model is using real data [11] [9] [2]. An approach is generating 3D model from 2D map with contours, by using a data structure TIN (Triangulated Irregular Network). TIN is a vector-based structure, which treats topographic surface as a surface consisting of number of non-overlapping triangles. TIN can render various geographical features that is impossible to render with DEM (Digital Elevation Model). 2.3 Noise Many people have used random number generators in their programs to create unpredictability, make the motion and behavior of objects appear more natural, or generate textures. At times, random number generators output can be too harsh to appear natural. On the other hand, there are many noise functions besides white noise whose power density is constant over a finite frequency range. Color noises [1] are used in data transmission: white noise, pink noise, red noise, orange noise, green noise, blue noise, purple noise, grey noise, and black noise. Brown noise is also used for terrain modeling. Brown does not means the color brown and the name comes from Brownian motion which produces random signals with the spectrum that has a falloff of 6dB/octave. It is called random walk or drunkard s walk noise. Perlin noise [8] composes noise functions generating curve at a range of different scales. See Figure 2.2. Noise functions (Figure 2.2(a)-2.2(e)) that have various amplitude and frequency are composed and Figure 2.2(f) is output. The output shows large, medium and small variations and looks a little like a mountain range. 1D output is shown but Perlin noise can make 2D noise. 6

13 (a) (b) (c) (d) (e) (f) Figure 2.2: Composition of noise function in Perlin noise [8] 7

14 We use Perlin Noise in order to add noise to a given terrain. 8

15 Chapter 3 User Interface We now describe the system from the user s point of view. Initially the system shows a ground which is a planar region of the xz-plane (Figure 3.1). Figure 3.1: First view of the system The interaction operation in this system is drawing with a stylus pen. A mouse can also be used. The user occasionally clicks buttons or moves slide bars. There are some buttons: undo, redo, clear, noise initialization, save BMP. And there is a checkbox to switch whether adding noise or not. Two slide bars are for adjusting noise parameters. We explain how to use our system in detail. The user first draws a stroke by dragging the cursor with the pen or mouse. When the button is released (or pen 9

16 Figure 3.2: Stroke(upper) and top view of mountain(lower) is released), a mountain is generated. The stroke is interpreted as the ridge of a mountain. The horizontal extent of a generated mountain corresponds to the length of the stroke s shadow (Figure 3.2). In this figure, upper area shows a 2D stroke drawn by the user, and lower area shows top view of mountains generated the stroke. When user s stroke has neigher local maximum point nor local minimum point except for the start and end point of the stroke, a bump between the start and end point of the stroke is generated (Figure 3.3). When user s stroke has only one local maximum point, a mountain is generated (Figure 3.4). 10

17 (a) Stroke which has neither local maximum nor minimum point (b) Terrain generated by the stroke (c) Terrain with noise Figure 3.3: Stroke which has neither local maximum nor minimum point, and a bump generated by the stroke (a) Stroke which has one local maximum point (b) A mountain generated by the stroke (c) A mountain with noise Figure 3.4: Stroke which has one local maximum point, and a mountain generated by the stroke 11

18 Otherwise, two or more mountains are generated (Figure 3.5). (a) Stroke which has two or more local maximum and local minimum point (b) Mountains generated by the stroke (c) Mountains with noise Figure 3.5: Stroke which has two or more local maximum and local minimum point, and mountains generated by the stroke Form of the mountain generated corresponds to form of stroke (Figure 3.6). When user s stroke is round / sharp, whole mountain is round / sharp. If there is a height gap between start and end point of user s stroke, our system fills the gap by adjusting height of terrain. View point can be changed by dragging with the shift key pressed. A river can be created by pressing a control key while drawing a stroke. The stroke should be drawn on the ground (see Figure 3.7). If the user s stroke runs a silhouette of the terrain, the system bridge any gaps with curves that look from above like straight line segments (see Figure 3.8). If the computer without a keyboard is used, shift and control keys are replaced by the function of View, Terrain, River buttons. After buttons are pressed, a stroke is drawn. Then mode returns to the terrain mode. The user can repeat these operations to obtain scene with various mountains. Figure 3.9 shows an example. First, a stroke is drawn (Figure 3.9(a)) and mountains are generated (Figure 3.9(b)). The next stroke is drawn on the mountains, that starts on the ground beyond the mountains (Figure 3.9(b)). The user then obtains a mountain and draws the next stroke in front of the mountains (Figure 12

19 (a) Round stroke (b) Sharp stroke (c) A mountain generated (d) A mountain generated (e) A mountain with noise (f) A mountain with noise Figure 3.6: Round and sharp stroke and mountains generated by these strokes 13

20 (a) A stroke drawn on the ground (b) A river generated by the stroke (c) Top view of the river Figure 3.7: Creating a river (a) A stroke drawn over the mountain (b) Top view of the river Figure 3.8: Stroke of a river over the mountain and bridging gaps 14

21 (a) First stroke (b) Mountains generated by a stroke in Figure 3.9(a) and second stroke (c) Mountains generated by a stroke in Figure 3.9(b) and third stroke (d) Mountains generated by a stroke in Figure 3.9(c) Figure 3.9: In Figure 3.9(a)-3.9(c), strokes are drawn, and by these strokes mountains are generated (Figure 3.9(d)) 15

22 3.9(c)). These three strokes generate mountains in Figure 3.9(d). Optionally, the user can add noise to make the terrain look natural. It is simply done by clicking a check box. The noise is controlled by two slide bars. The strength of existing terrain is for setting how strongly existing terrain effects. The smaller it is, the flatter the terrain become. The strength of noise is for setting how strongly noises affects. When its value is large, noise becomes harsh. See Figure Noise of various strength is added to original terrain (Figure 3.10(a)). In this way, one can obtain mountains with various smoothness of surface starting from the same original terrain. Our system can also add colors to terrains. 16

23 (a) Original terrain (b) Weak original terrain and (c) Strong original terrain weak noise and strong noise (d) Week original terrain and (e) Strong original terrain strong noise and week noise Figure 3.10: Effect of noise addition on existing terrain 17

24 Chapter 4 Implementation Detail 4.1 Data structure We first describe our data structures. The data consists of a geometry of a ground surface, strokes of mountain, and noise map. A ground surface is built by using a grid-based DEM (Digital Elevation Model). Grid-based DEM consists of a raster grid of regularly spaced elevation values. There are some reasons to use grid-based DEM; easiness and simplicity of implementation. Grid-based DEM is appropriate for real time application because complex computation is not required. Additionally, each gridpoint stores the corresponding height of strokes and a flag for a river. Each stroke of mountain is a sequence of 3D points, and has two height maps of mountain generated by this stroke. One map is for height of inside of the stroke, the other is for height of outside of the stroke. These data are created every time one stroke is drawn. Noise map is a grid-based DEM whose range of height is from -0.5 to 0.5. This noise is generated by Perlin Noise function [8] at the beginning. 4.2 Terrain modeling from 2D input In this section, we explain method for recognizing strokes. 18

25 4.2.1 Stroke recognition of mountain There are 5 steps to complete recognizing a stroke for mountain (Figure 4.1). Step 1, we project the start point of 2D stroke to the ground surface. Step 2, heights of other points of 2D stroke are determined. As a result of Step 2, there is a gap between heights of the start point and the end point of the stroke. We fill this gap in Step 3. In Step 4, a foot of mountain is determined. Finally, we complete mountains in Step 5. (a) Step 1 (b) Step 2 (c) Step 3 (d) Step 4 (e) Step 5 Figure 4.1: Terrain modeling from 2D input Step 1 : 2D points to 3D points A stroke for generating mountain is sequence of 3D points. The user inputs sequence of 2D points, so we have to convert the 2D points to 3D points. We will call the 2D point P, and assume that P is converted to 3D point P. A 2D point is projected to 3D ground surface as follows. Four points of each grid are projected to 2D screen, and make quadrangle. We check whether each projected 2D quadrangle contains P in descending order of distance of grid from view point. And coordinate of P is coordinate of 3D grid whose projected quadrangle contains P at the end. Step 2 : Altitude of mountain A sequence of 2D points is projected to a plane. The plane is perpendicular to the horizontal plane and interpolates the first 3D point of sequence S 0. The first point of 2D sequence S is projected to ground 19

26 surface by using above method (Step 1). And we will call this projected 3D point S 0. Other points of sequence are projected to the plane as follows. Figure 4.2: Altitude of mountain : surface N is plane of projection moved so that it corresponds to xz-plane, surface N is plane to which the stroke is projected. V is view point moved, S is start point of the stroke, and P is the point on the stroke. We ll denote a 2D point that we want to project to the plane as P. At the first, we move & rotate S 0 in a such way that the view screen corresponds to the xz-plane, and line of vision to y-axis. As a result, S 0 is moved to S and view point moved to V (see Figure 4.2). Next, we solve V P S P = 0 m P = P and get P which is a 3D vector. The surface N is 2D screen. Then, P is rotated and moved in a such way that the view screen and line of vision return to where they were. Stroke is separated by local maximum points and local minimum points. We denote this separated strokes as s 0, s 1, s 2,..., s n, and distance on xz-plane between 20

27 Figure 4.3: Filling a gap between start and end point P 1 and P 2 as d(p 1, P 2 ). Start point and end point of the stroke are projected on the ground, and denoted as S, T. Step 3 : Height gap between start and end point If there is a height gap between start and end point of user s stroke, our system fills the gap. In order to fill the gap, we define a diameter and determine height of a point P. Let the point P ST be on S T (S T is ST projected to ground) so that S T and P P ST are perpendicular. Height of P is h(p ) = v 0 t(1 t) 2 + v 1 (2t + 1)(1 t) 2 + v 2 t 2 (3 2t) v 3 t 2 (1 t); When P ST lays between S and T, t = d(p, S T )/diameter. v 0 and v 3 are gradient of S and T. If P ST does not lay outside segment S T and T is nearer to P than S, t = d(p, T )/(diameter + h(t )). v 1 is height of point whose distance from the segment S T or S or T is the diameter, v 2 is height of ST or S or T (Figure 4.3). This expression enables height to interpolate v 1 and v 2, and gradient at S and T to correspond with v 0 and v 3. Step 4 : The foot of a mountain We determine the foot of a mountain created by the stroke. 21

28 If the stroke has neither local maximum point nor local minimum point, the length of foot of mountain from stroke (projected to plane) is the height of stroke (Figure 4.4(a)). Next we explain the case that the stroke has local maximum point or local minimum point. Let E a point which is foot of a perpendicular to a point F laying in segment s i. Segment s i has own start point p s i and end point pt i, we assume that p s i is higher than pt i. If d(f, p s) is less than 0.5 (the distance of between F and p s is shorter than that between F and p t ), d(f, E) = v 0 t 2 (1 t) + v 1 t 2 (3 2t) + v 2 (2t + 1)(1 t) 2 where t is d(f, p t i )/d(ps i, pt i ), v 1 is height of p t i, and v 2 is (d(p s i, pt i ) + d(ps i, pt i 1 ))/2 (if s i 1 exists). v 0 is v 2 2, if s i 1 exists (see Figure 4.4(b)). This expression is similar to Hermite curve, height is continuous at t = 0 and t = 1. (a) (b) Figure 4.4: The stroke and foot of mountain generated by the stroke. Distance between F and E is determined by t and length of s i 1, s i, and s i+1. In the case that the distance of between F and p t is shorter than that between F and p s, we compute d(f, E) using s i+1 in similar way. 22

29 Step 5 : Completion of mountain We compute height of a point P laying between F and E as follows. Let u = d(f, P )/d(f, E). We explain only the case that the distance of between F and p s is shorter than that between F and p t. In other case, we are able to compute in similar way by displace p s and s i 1 with p t and s i+1. Let h(u i ) height of a point U i, the distance d(p s i, U i) = u d(p s i, pt i ). And let h(u i 1) height of U i 1, the distance d(p s i, U i 1) = (1 u) d(p s i 1, pt i 1 ) (note that ps i and pt i 1 are the same points) (see Figure 4.5). Altitude of P is average height of h(u i ) and h(u i 1 ), that are normalized so that altitude may be height of the stroke in u = 0 and 0 in u = 1. Figure 4.5: The stroke and altitude of generated by the stroke. Height of P is determined by u Stroke recognition of river For a stroke specifying a river, all 2D points are projected to the ground surface by using above (2D points to 3D points) method. As a result, there is a sequence of 3D points. The corresponding point in the boolean map is set to be true. When the user s stroke goes from the front terrain to the back terrain, the 23

30 projection of the stroke onto the ground terrain is discontinuous. We describe the last point on the front terrain as A, and the first point on the back terrain as B. Then A and B is bridged on the xz-plane. y coordinate of the point between A and B is a unique value, because our terrain is a height field. 4.3 Noise addition Original terrain is generated by drawing strokes. The user is able to add noise on this existing terrain model. A noise map is created by using Perlin Noise, and consists of elements whose values are in the -0.5 to 0.5 range. We assume that we add noise to point (i, j), and denote h(x, y) as height of point (x, y), noise(x, y) as value of point (x, y) on the noise map. Height of point(i, j) is computed as follows, h(i, j) = noise(i, j) noise strength + a(i, j) a(i, j) = 1 6 (h i 1,j + h i+1,j + h i,j 1 + h i,j+1 ) (h i 1,j 1 + h i+1 j h i 1,j+1 + h i+1,j 1 ) h x,y is height of point whose distance from (i, j) is noise(x, y) strength of existing terrain. 24

31 Chapter 5 Result Our prototype system is developed with array of height. It is implemented on Java TM version The author and test users generated terrain by using our system. Figure 5.1 shows sample terrain models designed on our system by the author, and Figure 5.2 shows terrain models designed on our system by the test users. The caption indicates times taken to model. We asked users to use our system and provide some comments. The following is the comments by the test users. When dragging on the area where there is none, view point should be slided. There should be smoothing tool. It is desired to be able to design concave terrain. The stroke of mountain should affect only to local area. Rivers should be physically possible. It is easier to model terrain using this system than existing modeling tool, for example, LiteWave [12]. It is joyful that the user can model fantastical terrain. 25

32 (a) in 7 minutes (b) in 8 minutes (c) in 10 minutes (d) in 5 minutes Figure 5.1: Terrain drawn by author 26

33 (a) in 10 minutes (b) in 7 minutes (c) in 10 minutes (d) in 14 minutes (e) in 15 minutes Figure 5.2: Terrain drawn by test users 27

34 Chapter 6 Limitations and Discussion Main drawback of our system is that at times forms of generated mountains are not intuitive. Firstly, generated mountains are symmetric on the strokes which are drawn in order to create the mountains. It is because that their height are computed by using the same algorithm at the symmetric points on the strokes. Secondly, mountains are, furthermore, not eroded and their roots are too smooth. As a result, this system is hard to create eroded mountains that are typical be in Japan. Finally, when a stroke is drawn in order to continuous mountains, if height of local minimum points are too high, foot of mountains become unnatural. For rendering, flat shading is applied. This shading is not realistic because there are rapid changes on border lines of polygons. If the designer wants realistic rendering, he or she have to use application specialized for rendering. 28

35 Chapter 7 Future Work Our current algorithm can fail or generate appropriate results when the user draws unexpected strokes. For instance, user s stroke may imply rings, or return to a point which has same x, y-coordinate as previous point of the stroke. And we must devise more robust and flexible algorithms to handle a variety of user inputs. An obvious direction for future work is to develop additional modeling operations to support other geographical features. Our current system supports only mountain and river modeling. We will support other geographical features, for example, valley, cliff, and cave. Eroding needs to be implemented for more natural mountains. In order to create caves, however, other data structure is required. There are many possible extensions to the rendering system. One example is to add automatic level-of-detail for terrain in order to speed up the rendering system. Another is to support non-photorealistic rendering style such as Harold [6] and Teddy [10] so that our system are used for variable goal, for example, modeling terrain in computer graphics of games. Currently, ground surface is only colored, but in the future texture mapping will be done. 29

36 References [1] Kevin Bolding. Noise types. Types.htm. [2] Jan Burdziej. Gis and 3dimensional digital terrain modeling. [3] B..., [4] Philippe Carmona and Laure Coutin. Fractional brownian motion and the markov property. In Electronic Communications in Probability, pages Vol.3 no.12 p , [5] J.M. Cohen, L. Markosian, R.C. Zeleznik, J.F. Hughes, and R. Barzel. An interface for sketching 3d curves. In 1999 Symposium on Interactive 3D Graphics, pages 17 21, [6] Jonathan M. Cohen, John F. Hughes, and Robert C. Zeleznik. Harold: A world made of drawings. In To appear in Proceedings of NPAR ACM NPAR, [7] Lynn Eggli, Ching yao Hsu, Beat D Brüderlin, and Gershon Elber. Inferring 3d models from freehand sketches and constraints. Computer-Aided Design, pages 29(2): , [8] Hugo Elias. Perlin noise. perlin.htm. [9] Chistopher Gold and Maciej Dakowicz. Terrain modelling from contours. 30

37 [10] Takeo Igarashi, Satoshi Matsuoka, and Hidehiko Tanaka. Teddy: A sketching interface for 3d freeform design. In SIGGRAPH 99 Conference Proceedings, pages ACM SIGGRAPH, [11] Kazufumi Kaneda, Fujiwa Kato, Ehachiro Nakamae, Tomoyuki Nishita, Hideo Tanaka, and Takao Noguchi. Three dimensional terrain modeling and display for environmental assessment. In Computer Graphics, pages 23(3): ACM SIGGRAPH, [12] NewTek. Litewave. [13] Makoto Okabe and Takeo Igarashi. 3d modeling of trees from freehand skeches. In SHIGGRAPH 2003 conference on Sketches & applications. ACM SIG- GRAPH, [14] Planetside Software. Terragen TM. [15] Robert C. Zeleznik, Kenneth P. Herndon, and John F. Hughes. Sketch: An interface for sketching 3d scenes. In SIGGRAPH 96 Conference Proceedings, pages ACM SIGGRAPH,