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Three dimensional mosaics with variable-sized tilesVisual Comput 2008

報告者 :丁琨桓

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Introduction

Three dimensional mosaics, or surface mosaics, are a beautiful art form where a sculpture is made from putting together tiles on a given shape.

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Previous work

In computer graphics 2D mosaics have been fully explored.

centroidal Voronoi diagram

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Previous work

3D mosaics are much harder since the tiles have to be positioned on the surface of a non-planar object being decorated.

If the shape is complex, adequate tile positioning is a real challenge.

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Previous work

[Surface mosaics,2006] addressed the problem of mosaics with tiles of the same size.

Using the same tile size for the whole surface is not the best choice, since this size could be too big for some locations with high curvature.

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AlgorithmStep1: Tiles are initially distributed randomly over the surface Higher curvature places with higher density of bigger tiles Smaller curvature places with fewer bigger tiles

Step2: relaxation procedure move tiles away from one another, leaving some gap for grout and avoiding collisions among tiles.

Step3: rendering specific effects achieve a more realistic result

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Evaluating curvatures using model vertex data [ Re-tiling polygonal surfaces, SIGGRAPH

1992 ] The method gives a good approximation of the

exact curvature, using only the model’s polygonal data.

For each vertex the method finds an associated curvature of this vertex with respect to all edges connected to it.

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Evaluating curvatures using model vertex data

Approximation of the curvature in 2D

The radius of curvature r : r = tan(θ)|P-A|/2

Point C bisects the Angle APB

In 3D the normal vector at P approximates the line segment PC.

The term θ is estimated with the dotproduct between a normalized vector A – P and the normal vector at P.

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Evaluating curvatures using model vertex data

Radius of curvature (Rc) in the planeRed is mapped to vertices of higher curvature whereas blue is mapped to relatively flat regions

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Mapping curvatures into tile size

A : the total area of the object’s surface 2h : the average tile size and h is half this sizeN : user-specified number of tiles

h : half the tile sizer : the radius of the circle

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Mapping curvatures into tile size

Function for mapping curvatures into tile sizes

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Distributing random points on the surface of a polyhedral model distributed randomly over the surface polygon capacity

Ai : the area of polygon i

rci : the polygon radius of curvature

f : the mapping function

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Distributing random points on the surface of a polyhedral model Polygons with higher curvature, i.e., smaller radius of

curvatures, will receive more tiles.

distributed randomly distributed with capacity function

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Relaxation of points on the surface of the model move the tiles away from each other, to avoid int

ersections using a repulsive force repulsive force is proportional to tile size, such th

at small tiles will concentrate in strongly curved places, and big tiles will push smaller ones to curved regions

f = Kf * ( 1 – d/(r1 + r2))

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Relaxation of points on the surface of the model

f = Kf * ( 1 – d/(r1 + r2)) d is the distance between the particles r1 and r2 are the radii of the ideal circles ar

ound the tile.

r : the radius of the circle

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Relaxation of points on the surface of the model

f = Kf * ( 1 – d/(r1 + r2))

d d

r1 r2r1 r2

f > 0 f < 0

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Adjusting the orientation of the tiles

[ Texture Synthesis on Surfaces, SIGGRAPH 2001 ] Vector field

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Rendering

To make the results more visually appealing to the user, the final shape of the tiles may be controlled by four parameters

Square tiles, turned into general quadrilateral tiles

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Rendering

Comparison of tiles with and without random variation in the shape. Random variables U1, U2, V1, and V2 with valuesbetween 85% and 115% of h

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Result

# of tiles :7000Tsmin : 0.4hTsmax : 3.15hRcmin : 0Rcmax : 25h

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Result

# of tiles :7000Tsmin : 0.1hTsmax : 2.3hRcmin : 0.5Rcmax : 20h

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Result

Effect of varying the size of tiles ( number of tiles : 4000)

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Comparison

Comparison with previous result from [surface mosaic]

surface mosaic mosaics with variable-sized tiles

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Conclusion

This paper presented a solution efficiently computes the distribution, placement and rendering of tiles

Author plan to extend this work by allowing tiles of variable shapes, not only squares.