Domain-Modeling Techniques
Domain-Modeling
Discrete dataset
f) C,
(D,
D
}) ({ p
i}, {C
i}, {f
i},{Φ
iks
D
Continuous dataset Visualization is to operate on the domain
representation, e.g., the sampling points and grids, but not on the sampled attribute values.
outline
• Cutting & Selection
• Grid Construction from Scattered Points
• Grid-Processing Technique
outline
• Cutting & Selection
• Grid Construction from Scattered Points
• Grid-Processing Technique
Cutting
D D'
a nd
}) ({
: Da ta s et
Ta rge
}) ({
: Da ta s et
Source
' '
' '
'
i i
i i
i i
i i
},{Φ }, {f
}, {C p
},{Φ }, {f
}, {C p
s s
D D
Target domain is a subset of the source domain
• Target domain has the same dimension of the source domain
• Extracting a volume of interest (VOI)
} {p '}
{p i i
Brick Extracting
Slicing
• Target domain has a smaller dimension: d-1
• Fix the coordinate in the slicing dimension
Along X-axis Sagittal slice
Along Y-axis axial slice
Along Z-axis coronal slice
Implicit Function Cutting
• Generalize the axis-aligned slicing
0
plane arbitrary
an along
cutting E.g.,
By Cz D
Ax
Selection
• Cutting explicitly specifies the topology of the target domain
• Selection explicitly specify the attribute value of the target dataset.
} )
(
|
'
{
true p
s D
p
D
• The output of selection is an unstructured grid
• E.g., contouring operation, iso-surface
• E.g., thresholding or segmentation
outline
• Cutting & Selection
• Grid Construction from Scattered Points
• Grid-Processing Technique
Scattered Points
12,772 3-D points represent a human face
Gridless point cloud
Grid Construction
• Build an unstructured grid from scattered points
• Most visualization software package requires data to be in a grid-based representation.
• Triangulation methods are the most-used class of
methods for constructing grids from scattered points
{p C } }
{p i i , i
Delaunay Triangulation
• Constructed
triangles covers the convex hull of the point set
• No point lies inside the circumscribed circle of any
constructed triangles
Delaunay Triangulation
Voronoi diagram:
It is associated with Delaunay
triangulation.
The vertices of the
Voronoi cells are the centers of the
circumscribed circles of the triangles.
Delaunay Triangulation
600
random 2-D
points
Delaunay Triangulation
Angle-
constrained Delaunay
Triangulation.
20° < α <
140°
Added 361 extra points
Delaunay Triangulation
Area-
constrained Delaunay
Triangulation.
a < amax Added 1272 extra points
Surface Reconstruction
• Render a 3-D surface from a point cloud
Radial Basis Function method
1. Computing a 3D volumetric dataset, using the 3D RBF (radial basic function) for construction.
2. Find the isosurface (f=1) using marching cube algorithm 3. This method is problematic, e.g., partition of unity
R r
R r
e
x x
T x
f
kr
i
,
0
,
)), (
( )
~ (
2
1 3
R
Φ: reference RBF R: support radius
K: decay speed coefficient
T-1: word-to-reference coordinate transform
Signed Distance method
1. Computing a tangent plane Ti that approximates the local surface in the neighborhood of pi
2. Calculate the distance function f between a given point (x) and the tangent plane at the sample point closest to (x) 3. The surface is simply the isosurface as f=0
Find Local Tangent Plane
i i
i N p i
i i
i
R N
N p c
n c
T
i
ra di us s upport
wi thi n
poi nts nei bouri ng
of s et
the
|
|
norma l a nd
center )
(geometri c i ts
by defi ned i s
pl a ne Ta ngent
Find Local Tangent Plane
i i
k i j k
i j
N p
n c
x x
f
c p
c p
i
) ( ).
(
functi on Di s ta nce
The
ma tri x the
of ei genva l ue s ma l l es t
the to
i ng corres pond r
ei genvecto the
i s n norma l The
) )(
(
a a a
a a a
a a a )
(a A
Ma tri x Cova ri a ne
: norma l the
fi nd To
i
33 32
31
23 22
21
13 12
11 jk
Principal Component Analysis (PCA)
A matrix H: 2 X 2, or 3 X 3 Hs= λs
s: eigenvector λ: eigenvalue
E.g., Hessian matrix
Eigenvector is of extreme curvature
PCA
) orthogona l a nd
d (norma l i ze s
fi nd
then ,
Fi nd
) det(
t Determi na n
0 )
det
ma tri x.
i denti fy i s
I where
0 )
(
21 12
22 11
22 21
12 11
h h
h h
H
h h
h H h
I (H-
s I H
PCA
If we order the eigenvalue in decreasing order:
r ei genvecto mi nor
: e
r ei genvecto medi um
: e
r ei genvecto ma jor
: e
3 2 1
3 2
1
In case of curvature:
• e1 the direction of maximal curvature
• e2 the direction of minimum curvature
• e3: the direction of surface normal
Local Triangulation Method
• Constructing the unstructured triangle mesh from local 2D Delaunay triangulation
1. Computing a tangent plane Ti that approximates the local surface in the neighborhood of pi
2. Project its neighbor set Ni on Ti, and compute 2D Delaunay triangulation
3. Add to the mesh those triangles that have pi as a vertex
Local Triangulation Method
Mesh Construction
All previous methods are to create a triangular (or polygon) mesh to represent the 3-D surface: radial basis; distance;
local triangulation)
Mesh detail
Surface Splatting Method
• Simple, fast, “dirty” method without using mesh
1. Draw discs of radius Ri, centered at every sample point pi and oriented in the local tangent plane.
2. The disc is as a 2D transparency texture Φ: opaque (= 1), total transparent (= 0)
3. Transparency is calculated from the radial basis function 4. The texture that encodes a 2D RBF is called “splat”
5. Splat is a rendering element for a 3D surface that is analogous to the pixel in a 2D image.
Surface Splatting Method
Point-based splatting Mesh reconstruction
outline
• Cutting & Selection
• Grid Construction from Scattered Points
• Grid-Processing Technique
Grid-Processing Techniques
• Grid-processing techniques change
1. The grid geometry: location of grid sample points 2. The grid topology: the grid cells
Geometric Transformation
• change the position of the sample points; not modify the cells, attributes, and basis functions
• Affline operation: carry straight lines into straight lines and parallel lines into parallel lines.
• Translation; rotation; scaling
• Nonaffine operation:
• Bending, twisting, and tapering
• Based on attributes
• Warping, height plot, and displacement plot
• Based on grid
• Grid-smoothing technique
Grid Simplification
• Reducing the number of sampling points (but need to maintain the reconstruction quality)
• Uniform sampling yields too many grid points
• Adaptive sampling
• fewer points in area of low surface curvature
• More points in area of high surface curvature
Triangle Mesh Decimation
• Recursively reduce the vertex and its triangle fan if the resulting surface lies within a user-specified distance from the un-simplified surface
36000 grid points 3600 grid points
Vertex Clustering
• By clustering or collapsing vertices
• Assign each vertex an important value, based on the curvature (high curvature more important)
• Overlaid a grid onto the mesh
• All vertices within the grid cell are collapsed to the most important vertex within the cell
Grid Refinement
• An opposite operation of grid simplification
• Generate more sample points
• A refined grid gives better results when applying grid manipulation operation
• Inserting more points when the surface varies more rapidly
Grid Smoothing
• Question: how to reduce geometric noise?
• Modify the positions of the grid points such that the reconstructed surface becomes smoother
• Smoothing removes the high frequency, small scale variation, e.g, small spikes
Laplacian Smoothing
Before smoothing After smoothing
Laplacian Smoothing
N
j
n
qj
N1 1 ( ) b
Laplacian process shifts grid points toward the barycenter of its neighboring point set
Laplacian Smoothing
N
j
n i n
j n
i n
i p k q p
p
z u y
u x
u u u
u t k
u
1 1
2 2 2
2 2
2
