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Surface mosaics

Visual Comput 2006

報告者 :丁琨桓

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Introduction

Mosaics are an art form with a long history: many examples are known from Graeco-Roman times.

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Introduction

The idea is simple: an image is formed using small colored square tiles which almost touch, so as to cover some area.

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Surface mosaics

a mesh model M to be covered with tiles the size and shape of the tiles is constant small gaps are allowed between tiles

close-upSurface mosaics

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Algorithm

step1 : control vectors and features step2 : vector field and initialization step3 : tile position optimization

step1 step2 step3

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Vector field

Texture Synthesis on Surfaces, SIGGRAPH 2001.

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Initialization

The tiles size : lx × ly , lx = ηly

Number N of tiles: N= (1-g)A / lx ly

A is the surface area.

g is the fractional area of the surface between the tiles ( g = 0.1)

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Initialization

The key problem in mosaicing is positioning of the tiles.

centroidal voronoi diagram

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Centroidal voronoi diagram

a

b

cd

y

|| y – c ||2 < || y – a ||2

|| y – c ||2 < || y – b ||2

|| y – c ||2 < || y – d ||2

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Centroidal voronoi diagram

a

c d

b

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Tile position optimization

Before position optimization After position optimization

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Tile position optimization

The energy leads to a repulsive force between tiles.

The energy between any two tiles Ti and Tj is defined as :

The fall off of energy with increasing distance.

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Tile position optimization

σis the interaction radius that controls the range of the repulsive force.

A is the surface area and N is the desired number of tiles.

d(Ti,Tj) : the Manhattan distance between the centers of the tiles.

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Manhattan distance

The distance between two points in a grid based on a strictly horizontal and/or vertical path

B

A

Euclidean distances and Manhattan Distance Points with equal Manhattan distance( d = 2 )

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Tile position optimization

pi is the position of tile i

t controls step size set to 1 in practice Consider k-nearest neighbors of tile i

(k = 20 in implementation)

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Tile position optimization example

a(0,0) b(3,0)

c(0,3)

x(2,2)

-d(a,x)2 = -(42) = -16

-d(b,x)2 = -(32) = -9

-d(c,x)2 = -(32) = -9

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5 σ=2.25, 2σ2 = 10

Eax = exp( -16/10 ) = 0.2

Ebx = exp( -9/10 ) = 0.4

Ecx = exp( -9/10 ) = 0.4

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Tile position optimization example

a(0,0) b(3,0)

c(0,3)

x(2,2)

px-pa = (2,2) - (0,0) = ( 2, 2)px-pb = (2,2) - (3,0) = (-1, 2)px-pc = (2,2) - (0,3) = ( 2,-1)

Eax = 0.2

Ebx = 0.4

Ecx = 0.4

Fx = (2,2)*0.2 + (-1,2)*0.4 + (2,-1)*0.4 = (0.8,0.8)

a(0,0) b(3,0)

c(0,3) x’(2.8,2.8)

px’ = px + Fx = (2,2) + (0.8,0.8) = (2.8,2.8)

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Feature lines

tiles adjacent to each feature line to be aligned with it.

constrain the tiles to be parallel or perpendicular to each feature line.

Tiles aligned with feature line

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Result

Surface mosaics (knot) Surface mosaics (fish)

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Result

Surface mosaics (horse) Surface mosaics (cube)

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Conclusions This paper has given an efficient algorithm

for covering a mesh with a mosaic of rectangular tiles.

optimization approach and the underlying model, a perfectly even distribution is not usually possible.

Tile position optimization’s priorities and termination conditions