Theodore Kim Ming C - Korea Universitykucg.korea.ac.kr/seminar/2012/ppt/ppt-2012-03-28.pdf · Level...

Jae ho Lim

Korea University

Computer Graphics Lab.

Theodore Kim Ming C.Lim

Eurographics 2003

Korea University Computer Graphics Lab.

Jae ho Lim | 2012/3/28 | # 2 KUCG |

Abstract

• The beautiful, branching structure of ice is one of the most striking visual phenomena of the winter landscape

Little study about modeling this effect in computer graphics

• Present a novel algorithm for visual simulation of ice growth

Use the Phase field method

Acceleration technique

Novel geometric sharpening algorithm

Movie

Visual Simulation of Ice Crystal Growth.mpg


Jae ho Lim | 2012/3/28 | # 3 KUCG |

Introduction

• SWOT analysis for this paper

S (Strength : Contribution) • Physically based ice growth based on rigorous

mathematical formulations and sound physical observations

• User control

• Novel geometric processing step ‗ Introduces internal structure to the ice

• Accelerate and simplified computations for interactive simulation of modest-scale ice crystal growth


Jae ho Lim | 2012/3/28 | # 4 KUCG |

Introduction

• SWOT analysis for this paper

W & O (Weak point and Opportunities) • Relatively little research in computer graphics

• None of the previous work presents a mechanism ‗ Artist to automatically adjust the simulation parameters to achieve a

specific visual effect

T ( Technical Issue) • Ice growth simulation using Phase field

• Temperature mapping for user control

• Internal structure using Morphological operator

• Rendering for Photon mapping


Jae ho Lim | 2012/3/28 | # 5 KUCG |

Previous work

• Visual Simulation Methods for Water in Different States

Dynamics of water and steam • Animation and rendering of complex water surfaces

‗ D.Enright et al / SIGGRAPH 2002

• Computer modeling of fallen snow ‗ P.Fearing et al / SIGGRAPH 2000

Structure of Dendritic ice • Une mëthode géométrique élémentaire pour l'étude de

certaines questions de la théorie des courbes planes. ‗ Helge von Koch / Acta Mathematica 1906


Jae ho Lim | 2012/3/28 | # 6 KUCG |

Previous work

The Fractal Geometry of Nature • B.Mandelbrot / W H Freeman 1982

• Diffusion Limited Aggregation (DLA)

Fractal Concepts in Surface Growth • A. Barabási et al / Cambridge University Press 1995

Diffusion-limited aggregation, a kinetic critical phenomenon. • T. Witten et al / Physicsal Review Letters 1981


Jae ho Lim | 2012/3/28 | # 7 KUCG |

Previous work

• Simulation Techniques in Computational Physics

Introduce to the morphology of possible ice crystal shapes • Pattern formation in growth of snow crystals occurring in

the surface kinetic process and the diffusion process ‗ Kuroda et al / Physical Review A 1990

Phase field • Modeling and numerical simulations of dendritic crystal

growth. ‗ R.Kobayashi / Physica D 1993

• Adaptive mesh refinement computation of solidification microstructures using dynamic data structures.

‗ N.Provatas et al / Journal of Computational Physics 1999


Jae ho Lim | 2012/3/28 | # 8 KUCG |

Previous work

Level set method • Level Set Methods and Dynamic Implicit Surfaces

‗ S.J.Osher et al / Spinger-Verlag 2002

• Level Set Methods and Fast Marching Methods ‗ J.A.Sethian / Cambridge University 1999

• Level set method

High cost / Some artifact

• Phase field method

Low cost / Some artifact


Jae ho Lim | 2012/3/28 | # 9 KUCG |

Overview

• Simulate growth in 2D and add 3D detail later

Reduce the Time complexity • O(𝑁3) to O(𝑁2).

• Performing banded computation around the “front” of the ice and water interface


Jae ho Lim | 2012/3/28 | # 10 KUCG |

Overview

• User Control

Seed crystal and freezing temperature input into the phase field simulation

Seed Crystal • Set the initial condition

• Visually salient features of our target object

• Using edge detection

• Influence the simulation throughout by manipulating the freezing temperature


Jae ho Lim | 2012/3/28 | # 11 KUCG |

Overview

• Due to smoothing artifacts of the phase field

Performed by first computing the border and medial axis of the ice with morphological operators

Generate a constrained conforming Delaunay triangulation.

• Rendering using photon mapping


Jae ho Lim | 2012/3/28 | # 12 KUCG |

The Phase Field Method

• [4.1] – [4.3]

Overview of the method

Kobayashi formulation

• [4.4] – [4.5]

Our own analysis and optimizations


Jae ho Lim | 2012/3/28 | # 13 KUCG |

4.1. Undercooled Solidication

• An undercooled liquid is a liquid that has been cooled below its freezing temperature

cooled sufficiently slowly for it to remain in its liquid state

• Small amount of solid material “Seed Crystal”

Liquid -> Solid

Initial seed in a rapid and unstable reaction • Growth of the crystal can be influenced by small

perturbations

• Can lead to the complex branching


Jae ho Lim | 2012/3/28 | # 14 KUCG |

4.2. The Phase Field

• In the phase field method,

Undercooled liquid is represented implicitly as a two or three dimensional grid

`Eulerian' representation

• For simplicity and tractability

limit our simulations to two dimensions


Jae ho Lim | 2012/3/28 | # 15 KUCG |


• Two separate fields

Temperature T • Records the amount of heat in a given cell

Phase field p • Records the current phase of a given cell.

Grid coordinate (x,y)

• 𝑇𝑥𝑦and 𝑝𝑥𝑦 as the corresponding values in the

temperature and phase fields


Jae ho Lim | 2012/3/28 | # 16 KUCG |


• Figure 3: (a) A phase field in which white is p = 1 (ice), and black is p = 0 (water). The gray band in the middle is the section shown in profile in (b). (b) Cross section from (a) in profile. Note that while the transition from water to ice is abrupt, it is not instantaneous.


Jae ho Lim | 2012/3/28 | # 17 KUCG |

4.3. The Kobayashi Formulation

• The first paper to report successful simulation of a wide variety of ice growth patterns using phase fields

Reaction-diffusion equations • for texture synthesis in computer graphics

Diffusion Reaction


Jae ho Lim | 2012/3/28 | # 18 KUCG |


• K is latent heat constant

• R term models the process where, as water transitions to ice, it produces heat.


Jae ho Lim | 2012/3/28 | # 19 KUCG |


• Kobayashi's formulation

Diffusion term

Reaction term


Jae ho Lim | 2012/3/28 | # 20 KUCG |


• Diffusion term The standard Laplacian diffusion term is the

sum of the diagonal elements of the Hessian matrix

The Kobayashi formulation • Diffusion term is the sum of the all elements of the

Hessian matrix

• Diagonal terms are abbreviated as a gradient and divergence operator instead of a pure Laplacian

• Different anisotropy placement accounts for both the Laplacian of the phase term and the gradient of the anisotropy term.


Jae ho Lim | 2012/3/28 | # 21 KUCG |


• Kobayashi also presents a complex and general model of anisotropy

Define 𝜃 • orientation of the front at a given grid cell

‗ 3D :

‗ 2D :

The anisotropy term j : degree of anisotropy .: strength of anisotropy .: fixed reference direction .: Scaling factor


Jae ho Lim | 2012/3/28 | # 22 KUCG |


• Constant Table

• The term is also necessary in Eqn. 2, but this can be obtained by taking the analytical derivative of Eqn. 4.


Jae ho Lim | 2012/3/28 | # 23 KUCG |


• Reaction term

• energy potentials in the system

m term is defined as:

• This equation are probably of limited use to a graphics

audience


Jae ho Lim | 2012/3/28 | # 24 KUCG |


• Eqn. 5

m = 0, 0.5 < p < 1 : Positive

m = 0, 0 < p < 0.5 : Negative • Energy is in a ‘metastable‘ state where values of p are

encouraged to stay the same

m = 0.5, 0 < p < 1 : Positive • If a grid cell has m = 0.5, no matter what its p value, it is

encouraged to transition towards ice

• Temperature of a grid cell increases, its m increases towards 0:5, and it becomes more likely to transition to ice


Jae ho Lim | 2012/3/28 | # 25 KUCG |

4.4. Improved Anisotropy

• Eqn. 4 affords both simpler and richer controls for general texture synthesis than the anisotropy term described by Witkin and Kass

Limits the number of preferred growth directions

All the directions must be either parallel or orthogonal

Strength of anisotropy in parallel directions must be the same


Jae ho Lim | 2012/3/28 | # 26 KUCG |

4.4. Improved Anisotropy

• Slight modification

Accomplished by defining a separate 𝛿𝑖 for each 𝑖𝑡ℎ cosine lobe

Limiting the influence of to the range 𝛿𝑖


Jae ho Lim | 2012/3/28 | # 27 KUCG |

4.5. Possible Ice Crystal Shapes

Our simulation

Real photo

Dendritic growth Sectored Plate Isotropic growth


Jae ho Lim | 2012/3/28 | # 28 KUCG |

4.6. Banded Optimization

• Some of the optimization techniques

Phase field methods include adaptive meshes for representing the phase and temperature fields

Diffusion Monte Carlo (DMC) methods

• Accurate simulation of solidification at scales much smaller than the mesh resolution

Smaller scales are not of interest to us

• Solution

localization of computation to the grid cells along the interface (e.g “narrow band” )


Jae ho Lim | 2012/3/28 | # 29 KUCG |

4.6. Banded Optimization

• Compares banded and unbanded performance


Jae ho Lim | 2012/3/28 | # 30 KUCG |

4.7. Hardware Implementation

• The timings are all for an unbanded implementation.


Jae ho Lim | 2012/3/28 | # 31 KUCG |

5. User Control

• One of our goals is to introduce a user parametrization into the simulation

The seed crystal allows the user to guarantee that primary shape features are present in the final ice

Freezing temperature allows the user to provide further simulation hints by rating the importance of secondary features


Jae ho Lim | 2012/3/28 | # 32 KUCG |

5. User Control

a) Source Image b) Seed Crystal texture c) Simulation results after seed crystal d) Freezing temperature mapping e) Ice grown with a mapped seed crystal

and freezing temperature f) Bump mapped ice


Jae ho Lim | 2012/3/28 | # 33 KUCG |

5.1. Seed Crystal Mapping

• First, the user selects the most visually important features and maps them to the seed crystal

Choose important feature

Arbitrary feature as the most important. • Figure.5 (B)

Extracted using Canny edge detection

• Fig. 5(c), Desired shape is quickly lost.


Jae ho Lim | 2012/3/28 | # 34 KUCG |

5.2. Freezing Temperature Mapping

• By varying the freezing temperature over the temperature field

Model the presence of impurities in the undercooled liquid.

Freezing rate can controllable temperature field • Set the High temperature

‗ Promote ice growth

• Set the low temperature ‗ Suppress ice growth

• To rate the importance of regions with respect to one other, we set their freezing temperatures between 0 and 𝑇𝑒


Jae ho Lim | 2012/3/28 | # 35 KUCG |

5.2. Freezing Temperature Mapping

• Figure.5 (d)

white regions represent 𝑇𝑒 • Greyer regions represent progressively lower freezing

temperatures.

• Freeze over first before the simulation starts branching out into the greyer regions.

• Simulation can use any arbitrary texture, the user can impose any desired rating method.


Jae ho Lim | 2012/3/28 | # 36 KUCG |

6. Introducing Internal Structure

• Internal structure to the results of the physically-based simulation.

Produce triangles from the results of the simulation

Photon map renderer

• The phase field method provides the position of a growing ice border

Increase the scale of the simulation • Details are quickly lost

• Creating unnaturally at ice.


Jae ho Lim | 2012/3/28 | # 37 KUCG |

6.1. Naïve bump mapping

• In order to capture this internal detail

Bump mapped the ice according to the 𝜕𝑝

𝜕𝑡of the ice

As water transitions to ice • Expands slightly

• Each time step by increasing the height of the ice by the amount of phase transition

• Modeling the expansion coefficient is still an open problem in chemistry

Add a phenomenological step


Jae ho Lim | 2012/3/28 | # 38 KUCG |

6.2. Adding Subdivision Creases

• Once the simulation has run to completion

Internal structures by inserting creases into the ice at visually expected locations • Subdivision surfaces

• Repeatedly subdividing ‗ Border and along the medial axis.

Border generated by the simulation

Creases at the medial axis • sharply faceted

• visually interesting features in other natural growth phenomena


Jae ho Lim | 2012/3/28 | # 39 KUCG |

6.3. Morphological Operators

• Isolate both the border and medial axis through the use of morphological operators

morphological operators • Guarantee connectivity properties

• Greatly simplify the construction of a subdivision control mesh

Final ice is stored as a nearly binary raster image

Using threshold • P >= 0.5 : 1

• P < 0.5 : 0


Jae ho Lim | 2012/3/28 | # 40 KUCG |


• Basic convolution kernel

“structuring elements” • Jahne’s article

• Erosion

Single iteration of erosion on an image • single layer of white pixels around all white regions is

deleted


Jae ho Lim | 2012/3/28 | # 41 KUCG |


• Use morphological operators to extract the medial axis of the ice

Advantage : Very simple and guarantees the same connectivity properties

Disadvantage : Slower than other medial axis algorithms

Original Our


Jae ho Lim | 2012/3/28 | # 42 KUCG |


• In morphological terms

Isolation of the medial axis is known as the “skeletonization” • Convolving by all eight structuring elements

• include “don't care” pixels

• “don't care” pixels are denoted with empty pixels.


Jae ho Lim | 2012/3/28 | # 43 KUCG |


• The structuring elements in Fig. 9 are each run repeatedly until no further changes take place

Pixels that remain are those along the medial axis


Jae ho Lim | 2012/3/28 | # 44 KUCG |

6.4. Control Mesh Segment Generation

• To construct the control mesh for the subdivision surface

Extract a set of line segments from the border and medial axis images • line segments will be the crease edges in the subdivision

surface

Extraction is accomplished by performing a depth-first search of the white pixels in the images


Jae ho Lim | 2012/3/28 | # 45 KUCG |

• Three different vertex types at the endpoints of subdivision creases

According to Hoppe

Dart Crease Corner


Jae ho Lim | 2012/3/28 | # 46 KUCG |

• We run a similar algorithm on the border image

Not guaranteed to have any dart pixels • start the traversal from any white pixel

• Since there are not many darts or corners

Insert new vertices • according to a “stride” length


Jae ho Lim | 2012/3/28 | # 47 KUCG |

6.5. Triangulation Generation

• Input a set of line segments

Generate a triangulation that contains both the points and lines. • Generated many “needle” triangles


Jae ho Lim | 2012/3/28 | # 48 KUCG |

6.6. Height Field Generation

• Two dimensional triangulation of the ice

Assign height values to the vertices in the triangulation

• Original bump map is very smooth

values can also be very smooth

• In order to guarantee the appearance of creases in the limit surface

According to a linear interpolation that approximates a faceted surface


Jae ho Lim | 2012/3/28 | # 49 KUCG |

6.6. Height Field Generation

• Calculating the distance

Nearest border and medial axis pixels for all pixels.

linearly weighting the heights of the nearest border and medial pixels • The heights of the border and medial pixels are taken from

the original bump map

linear interpolation between the medial axis and border contours


Jae ho Lim | 2012/3/28 | # 50 KUCG |

6.7. Crease Generation

• Linear interpolation is used to set the height values of the triangulation

Creases are present in the ice from the very beginning

Refined through subdivision


Jae ho Lim | 2012/3/28 | # 51 KUCG |

6.8. Rendering

• Much of the interesting visual detail of ice is contained in the caustics generated by the refracting medium.

Used photon mapping for rendering


Jae ho Lim | 2012/3/28 | # 52 KUCG |

7. Implementation and Results

• All the pipeline stages were implemented in less than 5000 lines of C++ code

Use library • Constrained Delaunay Triangulations

‗ Jonathan Shewchuk's Triangle package

• Rendering ‗ POV-Ray 3.5

‗ supports a large shading language in addition to a nice photon map implementation


Jae ho Lim | 2012/3/28 | # 53 KUCG |

7.2. Simulation Parameters

• The simulation ran successfully at the resolutions up to and including 2048 x 2048

• The time step was fixed to 0.0002 at all times

At larger steps, the numerical noise in the simulation quickly compounded


Jae ho Lim | 2012/3/28 | # 54 KUCG |

7.3. Results

• All simulations took place on a 512 x 512 grid

Exceptions of Fig. 14, which was 512 x 800

• Practically interactive rates


Jae ho Lim | 2012/3/28 | # 55 KUCG |

7.4. Discussions and Limitations

• Validating the results of our simulation is very challenging

simulation is very sensitive to noise

Very specialized equipment is necessary

• Physical validity of the phase field methods

Proven repeatedly by researchers in the computational physics and crystal growth communities


Jae ho Lim | 2012/3/28 | # 56 KUCG |

7.4. Discussions and Limitations

• Our technique and Diffusion Limited Aggregation

Deal with the same basic problem of solidification

our method can produce the same structures as DLA

DLA also does not provide any clear way to introduce a user parameterization

• Lost internal detail

Physically plausible, not physically

Necessary to validate and refine this process


Jae ho Lim | 2012/3/28 | # 57 KUCG |

8. Summary and Future Work

• Presented a simulation technique from computational physics for the growth ice crystals

• Introduced optimizations to make the technique practical and interactive for computer graphics

• Introduced a novel geometric sharpening operation

• User control

• Ice growth has not been studied much in computer graphics


Jae ho Lim | 2012/3/28 | # 58 KUCG |

8. Summary and Future Work

• Phase field method can be applied to fully 3D ice growth

phenomena as icicles

• Investigate other optimization techniques

parallel computation on a cluster of PCs

Theodore Kim Ming C - Korea Universitykucg.korea.ac.kr/seminar/2012/ppt/ppt-2012-03-28.pdf · Level...

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