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1 Three dimensional mosaics with variable- sized tiles Visual Comput 2008 報告者 : 丁琨桓.
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Transcript of 1 Three dimensional mosaics with variable- sized tiles Visual Comput 2008 報告者 : 丁琨桓.
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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.