Download Scientific Visualization Data Representation Data Representation

Data Representation
Scientific Visualization
Rm
Rn
Data representation
Structures and Classification
domain
X
independent
variables
scientific data
computer graphics & visualization
data
values
dependent
variables
⊆ Rn+m
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Data Representation
- Discrete representations
- The objects we want to visualize are often ‘continuous’
continuous’
- But in most cases, the visualization data is given only at
-
discrete locations in space and/or time
Discrete structures consist of samples, from which
grids/meshes consisting of cells are generated
computer graphics & visualization
Data Representation
- Classification of visualization techniques according
to
- Dimension of the domain of the problem (independent
parameters)
- Type and dimension of the data to be visualized
(dependent parameters)
Primitives in multi dimensions
dimension
0D
1D
2D
3D
cell
mesh
points
lines (edges)
triangles, quadrilaterals (rectangles)
tetrahedra, prisms, hexahedra
polyline(–
polyline(–gon)
2D mesh
3D mesh or grid
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Data Representation
G
3D
F
determined by the positions of sample points
- Domain is characterized by
1D
C
D
0D
A
B
1D
2D
- Dimension
- Influence
- Structure
– these are of special
interest in this course
E
2D
Examples:
H
3D
computer graphics & visualization
Domain
- The (geometric) shape of the domain is
dimension of
data type
mD
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
nD
dimension
of domain
A: gas station along a road
B: map of cholera in London
C: temperature along a rod
D: height field of a continent
E: 2D air flow
F: 3D air flow in the atmosphere
G: stress tensor in a mechanical
part
H: ozone concentration in the
atmosphere
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
1
Domain
-
Influence of data points
-
Values at sample points influence the data distribution in a
certain region around these samples
To reconstruct the data at arbitrary points within the domain,
the influence of all samples has to be calculated
-
Point influence
-
Local influence
-
Global influence
- Construct a region around each sample point that covers
-
- Only influence on point itself
-
VoronoiVoronoi-diagram
CellCell-wise interpolation (see later in course)
- Each sample might influence any other point within the domain
Material properties for whole object
Scattered data interpolation
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Domain
- Scattered data interpolation
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
the domain is computed
Weighting functions determine the support of each
sample point
-
The interpolation problem
-
What is an RBF
- Radial basis functions simulate decreasing influence with
increasing distance from samples
terms of numerical operations
A weighted sum of translations of a radially symmetric basis
function
i =1
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Radial Basis Functions
- The basis function φ is a real function of positive
real r , where r is the radius from the origin
Various choices for the basic function
- Linear φ (r ) = r
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Given interpolation values f ( f1 , f 2 ,... f N ), we seek the weights λi such
that the RBF satisfies
s ( xi ) = f i
-
i = 1,..., N
Interpolation condition yields
-
2
A ⋅λ = f
where
(
)
Ai , j = φ xi − x j ,
r 2 + c 2 (fitting to topographical data)
-
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Radial Basis Functions
-
- Gaussian φ (r ) = exp(− cr ) (mainly used in neural networks)
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
-
( )
Approximate a realreal-valued function f ( x ) by s x given the set of
values f ( f1 , f 2 ,... f N ) at the points X = {x1 , x2 ,...x N } ⊂ ℜ d
- where λi is a realreal-valued weight, x − x i denote the Euclidean norm
between point x and point xi and φ is a basis function
Course WS 05/06 – Scientific Visualization
- Multiquadric φ (r ) =
-
- s(x ) is the Radial
Basis Function of the form
N
s (x ) = ∑ λiφ ( x − xi )
x ∈ ℜd
- Schemes might be nonnon-interpolating and expensive in
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
Radial Basis Functions
- At each point the weighted average of all sample points in
-
all points that are closer to that sample than to every
other sample
Each point within a certain region gets assigned the value
of the sample point
- Only within a certain region
-
-
Domain
- VoronoiVoronoi-diagram
i , j = 1,..., N
Solving the linear system determines the value of c and λ , thus
obtaining the RBF
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
2
Domain
- Example
Data Structures
- Requirements:
- Convenience of access
- Space efficiency
- Lossless vs. lossy
- Portability
- Radial basis functions with increasing support
- binary – less portable, more space/time efficient
- text – human readable, portable, less space/time efficient
- Definition
- If points are arbitrarily distributed and no connectivity
exists between them, the data is called scattered
- Otherwise, the data is composed of cells bounded by grid
lines
- Topology specifies the structure (connectivity) of the data
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Data Structures
- Some definitions concerning topology and
geometry
- In topology qualitative questions about geometrical
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
Data Structures
- Topology
- Properties of geometric shapes that remain unchanged
even when under distortion
structures are the main concern.
- Does it have any holes in it
- Is it all connected together
- Can it be separated into parts
- Underground map shows how lines a connected,
but not how far one station is from another
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Same geometry (vertex positions)
Same topology of surface
Different topology of triangles
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Data Structures
- Topologically equivalent
- Things that can be transformed into each other by
stretching and squeezing, without tearing or sticking
together bits which were previously separated
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
Data Structures
- Grid types
- Grids differ substantially in the simplicial elements (or
cells) they are constructed from and in the way the
inherent topological information is given
topologically equivalent
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
3
Data Structures
- Structured and unstructured grids can be
-
distinguished by the way the elements or cells
meet
Structured grids
- Have a regular topology and regular / irregular geometry
- Unstructured grids
- Have irregular topology and geometry
Data Structures
- Cartesian or equidistant grids
- Structured grid
- Cells and points are numbered sequentially with respect
to increasing X, then Y, then Z, or vice versa
Number of points = Nx•
Nx•Ny•
Ny•Nz
Number of cells = (Nx(Nx-1)•
1)•(Ny(Ny-1)•
1)•(Nz(Nz-1)
Y
3
P[i,j,(k)]
2
1
dy
0
0
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
1
2
3
dx = dy = dz
X
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Data Structures
- Cartesian grids
- Vertex positions are given implicitly from [i,j,k]:
- P[i,j,k].x = origin + i • dx
- P[i,j,k].y = origin + j • dy
- P[i,j,k].z = origin + k • dz
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
Data Structures
- Uniform grids
- Similar to Cartesian grids
- Consist of equal cells but with different resolution in at
least one dimension ( dx ≠ dy (≠
(≠ dz))
- Global vertex index I[i,j,k] = k•
k•Ny•
Ny•Nx + j•
j•Nx + i
-
dy
k = l / (Ny•
(Ny•Nx)
j = (l % (Ny•
(Ny•Nx)) / Nx
i = (l % (Ny•
(Ny•Nx) % Nx)
- Spacing between grid points is constant in each
-
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
dimension -> same indexing scheme as for Cartesian
grids
Most likely to occur in applications where the data is
generated by a 3D imaging device providing different
sampling rates in each dimension
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Data Structures
- Rectilinear grids
- Topology is still regular but irregular spacing between
grid points
- NonNon-linear scaling of positions along either axis
- Spacing, x_coord[L], y_coord[M], z_coord[N], must be
stored explicitly
- Topology is still implicit
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
Data Structures
- Curvilinear grids
- Topology is still regular but irregular spacing between
grid points
-
Positions are nonnon-linearly transformed
x_coord[L,M,N]
y_coord[L,M,N]
z_coord[L,M,N]
- Topology is still
implicit
- Geometric structure
M
might result in
concave grids
j
i
N
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
4
Data Structures
- Multigrids
Data Structures
- Characteristics of structured grids
- Focus in arbitrary areas to avoid wasted detail
- “blow up”
up” regions of interest, i.e. finer grid
- Easier to compute with
- Can be stored in a 2D / 3D array
- Arbitrary samples can be directly accessed by indexing
- Often composed of sets of connected parallelograms
(hexahedra)
- Cells might be distorted with respect to transformations
- May require more elements or badly shaped elements in
order to precisely cover the underlying domain
- Topology is represented implicitly by n-vector of
dimensions
- Geometry is represented explicitly by an array of points
- Every interior point has the same number of neighbors
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Data Structures
- Characteristics of structured grids
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
Data Structures
- If no implicit topological (connectivity) information
- Topological information is implicitly coded
is given the grids are termed unstructured grids
- Direct access to adjacent elements at random
- Cartesian, uniform, and rectilinear grids are necessarily
convex
- Visibility ordering of elements is given implicitly
- Curvilinear grids reveal a much more flexible alternative
to model arbitrarily shaped objects
- But makes the sorting of elements a complex procedure
- Unstructured grids are often computed using quadtrees
-
(recursive domain partitioning for data clustering), or by
triangulation of points sets
The task is often to create a grid from scattered points
- Characteristics of unstructured grids
- Grid points and connectivity must be stored
- Dedicated data structures needed to allow for efficient
traversal and thus data retrieval
- Often composed of triangles or tetrahedra
- Less elements are needed to cover the domain
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Data Structures
- Typical implementation of structured grids
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
Data Structures
- Unstructured grids
- Composed of arbitrarily positioned and connected
elements
DataType *data = new DataType[Nx•
DataType[Nx•Ny•
Ny•Nz];
val = data[i•
data[i•(Ny•
(Ny•Nz) + j•
j•Nz + k];
- Can be composed of one unique element type or they can
… code for geometry …
- Triangle meshes in 2D and tetrahedral grids in 3D are
be hybrid
most common
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
5
Data Structures
- Typical implementations of unstructured grids
Data Structures
-
- Direct form
Coords for
vertex 1
x1,y1,(z1)
x2,y2,(z2)
x3,y3,(z3)
x2,y2,(z2)
x3,y3,(z3)
x4,y4,(z4)
face 1
face 2
struct face
float verts[3][2]
DataType val;
2D
struct face
float verts[3][3]
DataType val;
3D
vertex list
face list
x1,y1,(z1)
x2,y2,(z2)
x3,y3,(z3)
x4,y4,(z4)
1,2,3
1,2,4
3,2,4
...
-
Course WS 05/06 – Scientific Visualization
Indexed face set
More efficient than direct approach in terms of memory
requirements
But still have to do global search to find local information
(i.e. what faces share an edge)
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Data Structures
- Typical implementations of unstructured grids
- WingedWinged-edge data structure [Baumgart 1975]
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
Data Structures
- WingedWinged-edge data structure
- EdgeEdge-based data structure, allows to answer queries
- Faces sharing an edge
- Faces sharing a vertex
- Walk around edges of face
previous right edge
vertex end
Indirect form
...
-
- Additionally store the data values
- Problems: storage space, redundancy
next left edge
-
Coords for
vertex 1
...
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Typical implementations of unstructured grids
- Stores for every vertex a pointer to an
edge
arbitrary edge that is incident to it
face
- Stores for every face a pointer to an
partner
edge on its boundary
vertex start
previous left edge
counterclockwise
orientation
- Implicit assumption:
- Every edge has at most two faces which
meet at edge Ù twotwo-manifold topology
next right edge
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Data Structures
- Hybrid grids
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
Data Structures
- Combination of different grid types
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
6
Data Structures
Data Structures
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Data Structures
- Example
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Data Structures
- Example
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
Data Structures
- Example
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
Data Structures
- Example
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
7
Data Structures
- Example
Data Structures
- Example
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Data Values
- Characteristics of data values
- Qualitative
- NonNon-metric
- Ordinal (order along a scale)
- Nominal (no order)
discretization)
Dimension (number of components)
Error (variance)
Structure of the data
- Quantitative
-
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
Data Values
- Range of values
- Range of values
- Data type (scalar, vector, tensor data; kind of
-
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Metric scale
Discrete
Continuous
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
Data Values
Data Classification
-
Classification of Bergeron & Grienstein 1989:
Data types
-
-
Scalar data
is given by a function f(x1,...,xn):R
):Rn→R with n independent
variables xi
Vector data
represent direction and magnitude and
is given by a n-tupel (f1,...,fn) with fk=fk(x1,...,xn), with 1≤
1≤ k ≤ n
Tensor data
for a tensor of level k is given by ti1,i2,…
(x
,
…
,x
)
1
n
i1,i2,…,ik
Structure of the data
-
Sequential (in the form of a list)
Relational (as table)
Hierarchical (tree structure)
Network structure
- On arbitrary positions (L0m)
- On a line (L1m)
- On a surface (L2m)
- On a (uniform) 3D grid (L3m)
- On a (uniform) n-dimensional grid (Lnm)
- Important aspects of data and grid types are
missing
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
- Lnm m-dimensional data on an n-dimensional grid
- Examples for m-dimensional data
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
8
Data Classification
Data Classification
Classification according to Brodlie 1992:
Classification via fiber bundles of Butler 1989:
- Underlying Field: domain of the data
- Visualizing entity (E)
- E is a function defined by domain and range of
- Fiber bundle:
data
- Independent variables: dimension and influence
- Dependent variables: dimension and data type
- base space: independent variables
- fiber space: dependent variables
- Definition of sections in fiber space
- Connection to differential geometry
Dependent variables
- Examples
E5Sn
or
EV3[3]
fiber space
Dimension of independent variables
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
base space
fiber bundle
computer graphics & visualization
Data Classification
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Specification according to Wong 1997:
Data Classification
- Example:
Dimension of the data values: dependent variables v
Dimension of domain: independent variables d
- Bergeron &
- Data with n independent variables and m
- Brodlie
ES{0}
- Butler
base = set, fiber = float:[-∞, ∞]
- Wong
0d1v
ndmv
Course WS 05/06 – Scientific Visualization
Grienstein
L01
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Data Classification
- Example:
Data Classification
- Example:
- Bergeron &
- Bergeron &
Ordered set of points with scalar values
Grienstein
computer graphics & visualization
Unordered set of points with scalar values
dependent variables:
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
section
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Scalar volume data set on a uniform grid
Grienstein
L01
L31
- Brodlie
ES[0]
- Brodlie
ES3
- Butler
base = ordered set, fiber = float:[-∞, ∞]
- Butler
base = 3D-reg-grid, fiber = char:[0,255]
- Wong
0d1v
- Wong
3d1v
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
9
Data Classification
- Example:
Data Classification
- Example:
- Bergeron &
- Bergeron &
Flow data on a curvilinear grid
Grienstein
3D volume with 3 scalar and 2 vector data values
Grienstein L39
L33
- Brodlie
EV33
- Brodlie
- Butler
base = 3D-curvilin-grid, fiber = float3:[-∞, ∞]3
- Butler
- Wong
3d3v
- Wong
Course WS 05/06 – Scientific Visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
E3S2V33
base = 3D-reg-grid, fiber = float x float x float x
float3 x float3
3d9v
Course WS 05/06 – Scientific Visualization
computer graphics & visualization
Prof. Dr. R. Westermann – Computer Graphics and Visualization Group
computer graphics & visualization
10