Niels v. Festenberg and Stefan Gumhold Ja Min Jeongkucg.korea.ac.kr/seminar/2012/ppt/ppt-2012-08-29.pdf · Making waves: Kinetic processes controlling surface evolution during low

Ja Min Jeong

2012/08/29

Korea University

Computer Graphics Lab.

Niels v. Festenberg and Stefan Gumhold

COMPUTER GRAPHICS forum

2011

Korea University Computer Graphics Lab.

Ja Min Jeong | 2012/08/29 | # 2 KUCG |

Abstract

• Both volumetric snow shapes and photorealistic texturing are computed.

• We formulate snow accumulation as a diffusive distribution process on a ground scene

• Snow bridges and overhangs are also included.


Ja Min Jeong | 2012/08/29 | # 3 KUCG |

1. Introduction

• We propose a mathematical model for the process of snow accumulation on visual scales. The model is based on a diffusion equation.

• The key contributions of our approach a mathematical framework for snow distribution

based on the physical properties of falling and fallen snow summarized as a diffusion process,

an efficient and realistic snow cover generation scheme for virtual environments that comprises the creation of snow bridges and snow overhangs

a scheme to texture the snow cover realistically


Ja Min Jeong | 2012/08/29 | # 4 KUCG |

1. Introduction

• Snow distribution

Diffusion Equation

Figure 2. Figure 3.


Ja Min Jeong | 2012/08/29 | # 5 KUCG |

1. Introduction

Snow bridges & Snow overhangs

Snow texture


Ja Min Jeong | 2012/08/29 | # 6 KUCG |

1. Introduction

• 3. Snow Physics Background We first motivate why diffusion equations are

appropriate to the description of fallen snow and support this with a set of real world observations

• 5.2. Numerical approximation for mathematical snow distribution We then introduce an efficient numerical scheme based

on a Gaussian kernel to approximate snow accumulation on any scene

• 5.4. Snow surface visualization To make the computed snow cover look real we propose

a method to naturally texture the snow surface


Ja Min Jeong | 2012/08/29 | # 7 KUCG |

2. Related Work

• Computer Modeling Of Fallen Snow

Paul Fearing (siggraph 2000)


Ja Min Jeong | 2012/08/29 | # 8 KUCG |

2. Related Work

• Heat Transfer Simulation for Modeling Realistic Winter Sceneries

N. Maréchal, E. Guérin (EUROGRAPHICS 2010)


Ja Min Jeong | 2012/08/29 | # 9 KUCG |

2. Related Work

• A Geometric Algorithm for Snow Distribution in Virtual Scenes

N. v. Festenberg (Eurographics Workshop on Natural Phenomena (2009))


Ja Min Jeong | 2012/08/29 | # 10 KUCG |

3. Snow Physics Background

• Top line: Snow surface growth

• Middle line: Typical edge profiles

• Bottom line: Snow bridge formation and smoothing.

Figure 1.


Ja Min Jeong | 2012/08/29 | # 11 KUCG |

3. Snow Physics Background

• Snow Crystals: Natural and Artificial Surprisingly its main ingredient is air contributing about

90% to the volume of freshly fallen snow.

The list of influencing factors • Snow age, Temperature, Humidity

• Fea00 : It is misleading to model snow as a standard granular material like sand or flour

• vFG09 : Similar to soft shadows Some of the relevant snow features were reproduced by

the ad hoc assumption of an edge profiles

However, Snow bridges, overhangs and particularly the smoothing qualities of snow covers were not included.


Ja Min Jeong | 2012/08/29 | # 12 KUCG |

4. Theoretical Framework for Snow Accumulation on Visual Scales

• Vertical visibility and Snow profiles near boundaries

• A consequence of the statistics of granular material deposition with strong adhesion Making waves: Kinetic processes controlling surface evolution

during low energy ion sputtering ([CC07])

• Edwards–Wilkinson equation

z is the surface height d is a measure of the diffusion strength

ξ (t , x) is the stochastic forcing term

Equation 1.


Ja Min Jeong | 2012/08/29 | # 13 KUCG |


We can neglect the stochasticity of ξ and assume continuous surface growth instead.

h(x, t) is the snow height

σ is the snowfall rate

heat conduction : h(temperature) σ (heating term) ‗ a powerful alternative to manual snow distribution [NIDN97], snow

edge profiles with global distance transforms [vFG09], recursive snow stability tests [Fea00], explicit particle simulations [MC00] and depth buffer based surface displacement methods [OS04]

• Strong adhesion between the particles > Macroscopic granular such as sand or flour

Equation 2.


Ja Min Jeong | 2012/08/29 | # 14 KUCG |


σ = const

boundary conditions • h(0, t ) = h(1, t ) = 0

D is a measure for snow edge steepness • Large Ds : wet and unstable snow

• Small Ds : dry and stable snow

Figure 2.

Equation 2.


Ja Min Jeong | 2012/08/29 | # 15 KUCG |


• Analytic solutions for diffusion equations are obtained by convoluting the initial conditions with a Gauss kernel with a width increasing with time t.

Integration of the differential equation (2) until time t can be translated to a single integral.


Ja Min Jeong | 2012/08/29 | # 16 KUCG |

5. The Algorithm for Snow Cover Generation

• For the implementation of the mathematical model we start by depth peeling the scene as described in [vFG09].

• Potential mesh inconsistencies are circumvented like this, and the method also becomes independent of mesh resolution

pijk : a future snow site Red balls are kernel members, grey balls are not, color mixtures indicate membership depending on the maximal snow height hmax. R : candidate cylinder radius, r : local snow segment edge distance, zijk : relative height difference within the kernel cijk : snow capacity of a site. 0 ~ ∞ Figure 3.


Ja Min Jeong | 2012/08/29 | # 17 KUCG |

5.1. Snow bridge generation

• Before evaluating the snow diffusion kernel for each site p we create the sites for snow bridges

these sections are shorter than a bmax

bmax = 3% of the scene

• {1, 0, 0, 0, 1} -> {1, 1, 1, 1, 1}

Figure 3.


Ja Min Jeong | 2012/08/29 | # 18 KUCG |

5.1. Snow bridge generation

Figure 6. bmax = 2.5%, 10% and 33%

Figure 4.


Ja Min Jeong | 2012/08/29 | # 19 KUCG |

5.2. Numerical approximation for mathematical snow distribution

• R : radius

• Np : a subset of the cylinder

• p’ : elements of Np

• z =(z(p’)+h0(p’))−(z(p)+h0(p))

Total initial height differences

h0(p)=min (c(p), h0), c(p) : capacity

• The computation consists of three steps: (a) performing the convolution itself to compute h,

(b) exactly fitting the boundary conditions with a factor g1(r) ∈ [0 . . 1]

(c) correcting the result with a visibility factor v ∈ [0 . . 1]

Figure 3.


Ja Min Jeong | 2012/08/29 | # 20 KUCG |

5.2. Numerical approximation for mathematical snow distribution

•

•

•

• R as 5% of the scene diameter

•

•

r = 1 : edge distance of R

Figure 11.


Ja Min Jeong | 2012/08/29 | # 21 KUCG |

5.3. Snow overhang generation

• Overhangs can arise in a snow cover in nature when the snow flakes are extremely adhesive or after snow compaction due to snow aging.

Figure 5.

bridges bridges and overhangs


Ja Min Jeong | 2012/08/29 | # 22 KUCG |

5.4. Snow surface visualization

• Bump mapping applied to a procedural 3D texture with Perlin noise produced convincing results except close to the boundaries of the snow cover as illustrated in the bottom close-up in Figure 7.

• We propose to blend the 3D material close to boundaries with an alpha mask generated from photographs of real snow boundaries.

Figure 7.

Figure 8.


Ja Min Jeong | 2012/08/29 | # 23 KUCG |

5.4.1. Triangle mesh generation

• First, all symmetric pairs {p, p} which are direct neighbors in i or j direction within the height span map are connected to a quad mesh

• Concave corners of the boundary of the quad mesh are smoothed by adding a triangle cutting the concave corner short


Ja Min Jeong | 2012/08/29 | # 24 KUCG |

5.4.2. Texturing

• The alpha mask is parameterized over the texture coordinates u and v

u runs along the boundaries

v corresponds to the distance to the boundary

The bottom half of the mask is used where snow settles only very sparsely, that is on very steep areas of the scene


Ja Min Jeong | 2012/08/29 | # 25 KUCG |

5.4.2. Texturing

• The following requirements lead to a parameterization suitable for our purpose (1) the mapping to the uv-domain should be as

isometric as possible close to the boundaries, (2) zigzag boundaries at steep slopes should not

introduce distortion in the texture map (compare Figure 9a),

(3) the v coordinate should be 0 at boundaries with thin snow cover and increase to 1/2 with increasing thickness of the snow cover (compare Figure 9b)

(4) to avoid visible seams orthogonal to the snow boundary, along each boundary loop the texture coordinate u should cover an interval [0, k] with k being an integer number.


Ja Min Jeong | 2012/08/29 | # 26 KUCG |

5.4.2. Texturing

• Initialization of boundary vertices

To support zigzag boundaries and snow covers of different thicknesses we first compute the v coordinate vi for each boundary vertex pi as the sum of two contributions vi,s and vi,γ

vi,s : qi with 1/4(qi−1 + 2 qi + qi+1).

Figure 9.


Ja Min Jeong | 2012/08/29 | # 27 KUCG |

5.4.2. Texturing

vi,γ

• we compute the angle γ between the snow surface and the ground as illustrated in Figure 9(b)

• For each edge of the quad mesh and each quad diagonal, we compute γ and keep track of the maximum value for each boundary vertex. Finally, we linearly map the angle γ to the interval [0, 1/2] by division with π

Figure 9.


Ja Min Jeong | 2012/08/29 | # 28 KUCG |

5.4.2. Texturing

• Texture coordinate propagation Computing Geodesic Distances on Triangular Meshes

[NK02]

• We label the mesh vertices as processed or unprocessed

• A priority queue is filled with all triangles that exactly have two incident processed vertices

• We keep extracting triangles with highest priority (lowest v) from the queue

• We stop propagation ‗ when the queue is empty or the lowest v coordinate is larger than 1


Ja Min Jeong | 2012/08/29 | # 29 KUCG |

5.4.2. Texturing

• This leads to a quadratic equation • It is worth mentioning that some care has to be taken

where the u coordinates wraps around, that is there are edges and triangles with a u coordinate a bit above zero on one end and a bit below the maximum value k of the boundary loop on the other end


Ja Min Jeong | 2012/08/29 | # 30 KUCG |

6. Results

• Our algorithm prototype was tested on a Intel Core2Duo CPU with 2.2 GHz, 2 GB RAM and a NVIDIA Quadro FX 570M GPU.

Table 1.


Ja Min Jeong | 2012/08/29 | # 31 KUCG |

6. Results

Figure 12.

Figure 13. grass and gold


Ja Min Jeong | 2012/08/29 | # 32 KUCG |

6. Results

Figure 14.


Ja Min Jeong | 2012/08/29 | # 33 KUCG |

7. Discussion and Limitations

• Our method is more versatile.

• Complexity

Diffusion kernel computation is about O(n2).

Our snow computation scheme can in principle be parallelized

• Besides computation times, our model performs well compared to others either in the degree of realism

• This is small kernels significantly simplify the choice of the kernel members within the candidate cylinder.

Figure 15.


Ja Min Jeong | 2012/08/29 | # 34 KUCG |

7. Discussion and Limitations

• Limitations Each of these processes is quite complex in itself,

and all interact with each other • Wind transport

• Melting due to solar radiation

• The mechanical response of the ground to the snow load

We are confident that the diffusion approach is a good starting point for the future creation of virtual snow scenes • σ(x, t ) would be computed in a wind field simulation.

• Influence of temperature ‗ Use to coefficient D

‗ But optical properties


Ja Min Jeong | 2012/08/29 | # 35 KUCG |

8. Conclusion and Future Work

• We presented a method for snow accumulation constructed on the base of a physical model for surface growth under adhesive particle impingement

• Fast and realistic support in 3D modeling tools

• Parallelizable

• Snow cover growth animation

• Future Work

The interaction of the snow mass with the ground

The optical properties of snow

Documents

Niels v. Festenberg and Stefan Gumhold Ja Min Jeongkucg.korea.ac.kr/seminar/2012/ppt/ppt-2012-08-29.pdf · Making waves: Kinetic processes controlling surface evolution during low