Digital surfaces are sets of surfels with specific properties. For
instance, the boundary of a set of voxels in 3D (defined as the cube
faces that separates object voxels from background voxels) is a
digital surface. Digital surfaces are defined in arbitrary dimension
and can be seen as n-1-cells in the cubic cellular decomposition of
Rn.
Digital surfaces can be used to extract isosurfaces in images. They
can also be used to approach implicit surfaces by tracking changes of
sign of the implicit surface. They can be efficiently
implemented in arbitrary dimension. We are currently investigating the
extraction of singular surfaces with digital surfaces and cubical
complexes.
[ Lachaud07b , Lachaud03c Lachaud03b , ]
Sphere | Whitney's Umbrella | Four lines example |
f(x,y,z)=x^2+y^2+z^2-1 | f(x,y,z)=x^2-y^2*z | f(x,y,z)=x*y*(y-x)*(y-x*z) |
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Some examples are given below. The meshing algorithm correctly detect singularities of implicit surfaces. The handle of Whitney's umbrella is detected {z<0, x=y=0}, as well as the intersection of two sheaves around {z>=0, x=y=0}.
Sphere | Whitney's Umbrella | Four lines example |
f(x,y,z)=x^2+y^2+z^2-1 | f(x,y,z)=x^2-y^2*z | f(x,y,z)=x*y*(y-x)*(y-x*z) |
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Implicit surface | Cubical complex {F=0} inter {t=0} | After collapse | After projection | Raycasting rendering | |
Whitney umbrella f(x,y,z)=x^2-y^2*z | ![]() |
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Crixxi f(x,y,z)=(y^2+z^2-1)^2 +(x^2+y^2-1)^3 = 0 | ![]() |
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Buggle f(x,y,z)=x^4y^2+y^4x^2-x^2y^2+z^6 = 0 | ![]() |
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Lachaud07b , Lachaud03c , Lachaud03b
J.-O. Lachaud and R. Malgouyres, Géométrie discrète et images numériques, chapter 3. Topologie, courbes et surfaces discrètes. Traité IC2. Hermès, 2007. In french.
[BibTeX reference] [chapter PS gzipped] [Lavoisier]
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Abstract:
Keywords:
digital topology, digital surface, surface tracking, homotopy, topological invariant
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J.-O. Lachaud and A. Vialard, Geometric measures on arbitrary dimensional digital surfaces. In Proc. Int. Conf. Discrete Geometry for Computer Imagery (DGCI'2003), Napoli, Italy, November 2003, volume 2886 of Lecture Notes in Computer Science, pages 434-443, 2003. Springer.
[BibTeX reference] [paper PS gzipped] [slides PDF] [LNCS]
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Abstract:
Keywords:
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J.-O. Lachaud. Coding cells of digital spaces: a framework to write generic digital topology algorithms. In Proc. Int. Work. Combinatorial Image Analysis (IWCIA'2003), Palermo, Italy, volume 12 of Electronic Notes in Discrete Mathematics, 2003.
[BibTeX reference] [paper PS gzipped] [slides PDF] [ENDM]
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Abstract:
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