Transcript Folie 1 - Computer graphics (images)

A AQuadrilateral
Quadrilateral
Rendering
Primitive
Rendering Primitive
Kai Hormann · Marco Tarini
Visual Computing Group · CNR · Pisa
Motivation
our method
diagonal split
quadrilaterals
triangles
modelling primitive
rendering primitive
A Quadrilateral Rendering Primitive
How to render a quad ?
diagonal split
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two triangles
linear in each triangle
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diagonal visible (C0)
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A Quadrilateral Rendering Primitive
How to render a quad ?
diagonal split

two triangles
linear in each triangle

diagonal visible (C0)

A Quadrilateral Rendering Primitive
How to render a quad ?
two-fold linear
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linear along edges
linear along horizontal
spans
two horizontal lines
visible (C0)
strongly view dependent
A Quadrilateral Rendering Primitive
How to render a quad ?
projective map
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unique mapping from
unit square to quad
false depth-impression
not linear along edges
convex quads only
A Quadrilateral Rendering Primitive
How to render a quad ?
bilinear interpolation
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unique mapping from
unit square to quad
costly geometry
refinement required
fold-overs for flat and
non-convex shapes
A Quadrilateral Rendering Primitive
How to render a quad ?
A Quadrilateral Rendering Primitive
General concept
3D
vertices
vertex
process
fragment final
process pixels
rasterizer
2D screen
quad
fragments
A Quadrilateral Rendering Primitive
General concept
3D
vertices
vertex
process
fragment final
process pixels
rasterizer
2D screen
quad
fragments
remains the same
project
vertices
define vertex attributes
etc
A Quadrilateral Rendering Primitive
General concept
3D
vertices
vertex
process
fragment final
process pixels
rasterizer
2D screen
quad
fragments
remains the same
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depthtest
test
depth
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shading
shading
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textureaccesses
accesses
texture
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etc.
etc.
A Quadrilateral Rendering Primitive
General concept
3D
vertices
vertex
process
fragment final
process pixels
rasterizer
2D screen
quad
fragments
a new quad rasterizer
determine
fragments to be covered
interpolate attributes per fragment
A Quadrilateral Rendering Primitive
Determine fragments to be covered
possible shapes of 2D screen quads
convex
concave
(v-shape)
self-intersecting
(cross-shape)
rasterization
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general scan-line algorithm
via edge-functions
A Quadrilateral Rendering Primitive
General concept
3D
vertices
vertex
process
fragment final
process pixels
rasterizer
2D screen
quad
fragments
a new quad rasterizer
determine
fragments to be covered
interpolate attributes per fragment
A Quadrilateral Rendering Primitive
Interpolating vertex attributes
triangles
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barycentric coordinates
A(v, vi+1,vi+2)
λi(v) =
A(v0, v1,v2)

v1
v
v2
weights for linear interpolation
v0
2
a(v) = Σ λi(v) ai
i=0
A Quadrilateral Rendering Primitive
Generalized barycentric coordinates
Quadrilaterals

Mean value coordinates
αi-1
αi
tan
+ tan
2
2
μi(v) =
ri
λi(v) =
v
μi
Σ j=0 μj
(Floater 2003)
vi+1
αi
αi-1
3
vi-1
ri
vi
A Quadrilateral Rendering Primitive
Generalized barycentric coordinates
properties
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v
αi-1
Lagrange property
linear along edges
positive
work for all shapes
smooth (C∞ inside)
support front/back-facing
reproduce triangle coordinates
not too costly to compute
vi-1
vi+1
αi
ri
vi
A Quadrilateral Rendering Primitive
Example
interpolating texture coordinates
texture
A Quadrilateral Rendering Primitive
Perspective correction
3
a(v) = Σ λi(v) ai
i=0
without
ai
a(v) = Σ λi(v)
wi
i=0
3
/
3
1
λi(v)
Σ
wi
i=0
with
A Quadrilateral Rendering Primitive
Works in all cases
possible shapes of 2D screen quads
convex
concave
(v-shape)
self-intersecting
(cross-shape)
A Quadrilateral Rendering Primitive
Points behind the eye
all vertices in front
it also works for exterior quads
some vertices behind
A Quadrilateral Rendering Primitive
Points behind the eye
1.
4.
2.
5.
3.
6.
A Quadrilateral Rendering Primitive
Interpolating depth
A Quadrilateral Rendering Primitive
Discussion
blends well with current architecture
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modifies the rasterizer only
vertex and fragment ops unchanged
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properties of triangles carry over
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depth calculation
behaviour along edges
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allows mixed triangle/quad meshes
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OpenGL and DirectX friendly
A Quadrilateral Rendering Primitive
Discussion
rasterization cost
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generalized barycentric coordinates
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for example, 19 ARB instructions
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5 instead of 3
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1 quad instead of 2 triangles
edge functions for membership test
fewer and larger primitives
A Quadrilateral Rendering Primitive
Discussion
increased quality
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quad meshes look much better
almost identical to a refined mesh
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view-dependency
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A Quadrilateral Rendering Primitive
Future work
improving the method
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extend to other primitives?
simplify formulas?
avoid view-dependency?
turn it into hardware
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discuss feasibility with
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graphics card developers
DirectX developers
OpenGL community
A Quadrilateral Rendering Primitive
Application front-end
OpenGL
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quads are part of current syntax
our quads comply with the specs
reimplement GL_QUADS in the API
DirectX
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need to add a new primitive type
A Quadrilateral Rendering Primitive
a lot of triangles
more quads
few quads
few
triangles
quality
Why don’t we just use more triangles?
complexity
A Quadrilateral Rendering Primitive