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					SCATTERED DATA VISUALIZATION
Scattered Data: sample points distributed unevenly and non-uniformly
                throughout the volume of interest.

Example Data: chemical leakage at a tank-farm.
Method of Approach : Interpolation-based Two-step
Approach (Foley & Lane, 1990)




                  Modeling                    Rendering
     Sparse Data               Intermediate Grid          Rendered
                 Interpolation                 Grid-Based Volume
Interpolation Methods (Nielson, 1993)



Global: all sample points are used to interpolated a grid value.

Local: only nearby sample points are used to interpolated a
grid value.

Exact: the interpolation function can exactly reproduce the
data values on the sample points.

Problems: Xiao etc. 1996
               Defining a Global Exact Interpolant
               (Foley & Lane, 1990; Nielson, 1993)

 N sample points: (xi,yi,zi,vi) for i = 1,2,..n
 One interpolation function, e.g., Thin-plate spline,
                          n
            f ( x, y, z ) =  bi d log( di ) + c1 + c2 x + c3 y + c4 z
                                2

                                i
                         i =1

di is the distance between sample point i and the point to be
interpolated p(x,y,z).
               di = ((x-xi)2+(y-yi )2+(z-zi )2)1/2
bi,c1,c2,c3,c4 are n+4 constants to be solved by enforcing the
following conditions:
               f (xi,yi,zi) = vi      for i = 1,2,..n
            Global Exact Interpolation Functions
             (Foley & Lane, 1990; Nielson, 1993)

                                                         n
Thin-plate spline   f ( x, y, z ) =  bi d log( di ) + c1 + c2 x + c3 y + c4 z
                                                                      2

                                                                      i
                                                        i =1

                                                                n
Volume Spline        f ( x , y , z ) =  bi d 3 + c1 + c2 x + c3 y + c4 z
                                       n

                                      b d
                                                 3
                    f ( x, y, z ) =          i       + c1 + c2 x + c3 y  c4i ,
                                                                            z
                                                 i             i =1
                                      i =1




Multiquadric



Shepard
                                   n
                    f ( x , y , z ) =  bi d log( di ) + c1 + c2 x + c3 y + c4 z
                                         2
Thin-plate Spline                 i =1
                                         i
                                   n
Volume Spline   f ( x, y, z ) =   bd
                                  i =1
                                         i
                                             3
                                             i
                                                 + c1 + c2 x + c3 y + c4 z
                                    n                n       
                                                          1 
                                     d i vi         di 
                                          1
Shepard method   f ( x, y, z ) =
                                   i =1             i = 1    
                                                             
Deficiencies of the Interpolation-based Two-step
Approach


 Misinterpretation (Negative Concentration)
 Ambiguity in Selecting Interpolation Methods
 Inconsistent Interpolations in Modeling and Rendering
 Visualizing Secondary Data Instead of the Original Data
 No Error Estimation
 Unable to Add Known Information
 Not Efficient
          Three Dilemmas and Three Constraints
                    (Xiao & Woodbury, 1999)


 Zero-value dilemma
 Negative-value dilemma
 Correctness dilemma

                             Point Constraint
                              Value Constraint
                              Local Constraint
Point Constraint

                                   v
                                       sample points
                                                       constraining points


                                                                             d


 v       extrapolated values
      sample points




                               d
                         v min, if f ( x , y , z ) < vmin,
                     
Value Constraint   v   f ( x , y , z ),
                     
                         v max, if f ( x , y , z ) > vmax.
                                        p6
                              p1


Local Constraint        p2

                   p7   p3
                             p4    p5


                                             p8
Conclusions

• Two-step approach faces three dilemmas.
• Constrained interpolations can alleviate the dilemmas.
• The problems are far from being solved.


Data modeling is import to data visualization, just as
geometry modeling is important to geometry visualization.
Conclusions

To visualize scattered data, we are challenged to find
modeling techniques that
 preserve input data values;
 produce meaningful output values;
 provide error estimations;
 accept additional constraints;
 reduce the requirement on the sampling intensity.
A FINITE ELEMENT BASED APPROACH


     XIAO & ZIEBARTH, 2000
The Finite Element Based Approach


(1) Tessellation
(2) Computation
(3) Rendering
The Finite Element Based Approach




                Tessellation              Computation            Rendering
Sparse Data Volume           Element Network         Node Values           Rendered Volume
               Triangulation                 FEM                Element-Based
Tessellation
Three-Dimensional Triangulation: Tetrahedronization
Delaunay Triangulation: Sphere Criterion




                                                      input sample points
   input sample points        discontinuity points                             discontinuity surface
                                                      discontinuity points

   refinement points          discontinuity surface   refinement points        triangulated network




                                           Triangulation
                Data Points                                         Element Network
The Double Layer Technique




 input sample points                            input sample points

 discontinuity points   discontinuity surface                          double layers
                                                discontinuity points

 refinement points      triangulated network    refinement points      triangulated network




     Physical Discontinuity                            Logical Discontinuity
The Finite Element Method
(1) Problem Definition:   Boundary Value Problem
   Governing equation:                L  f

                                    p on S
   Boundary Condition:


(2) Element Definition:
   Shape: Tetrahedron
                                            4
                             ( x, y, z)   N e ( x, y, z) ej
                             e
 Order: Basis Function                        j
                                            j 1
The Finite Element Method
(3) System Formulation
   Ritz Method                 1   1                    
                           F ( )  2 L,   2 , f  2 f , 
                                                       1


   Galerkin's method      r  L  f
                                   

          [ K ]{}  {b}   {} = {i, i=1,2,...,n}T

                            k  p( k )
(4) Sparse Sample Data
(5) System Solution
   Gaussian Elimination
   Householder's Method
 Rendering : Modifying Conventional Methods




(1) Hexahedron => Tetrahedron
(2) (ijk) Indexing => Neighbor-to-Neighbor Traversal
Advantages of the Finite Element Based Approach

(1) Meaningful Results
   Z
          Ground Surface

                           
 2000

                           


 1000                      


                           
                    1000           Y
    0
                           
   1000


                X


A Pollution Problem                    Exact   Grid-based   FEM-based
Advantages of the Finite Element Based Approach

(2) Complicate Geometry: Non-Gridable Volumes
 Advantages of the Finite Element Based Approach


(3) Discontinuity: Internal Discontinuity Surface
 Advantages of the Finite Element Based Approach


(3) Discontinuity: Discontinuous Regions
Advantages of the Finite Element Based Approach


(4) Error Estimation and Iterative Refinement
       E e  2 | '' | h 2
              1
                                     h  2 E lim /| '' |

           4




           3




          2




           1




           0
               0              500            1000            1500            2000

                                             Z



                     h              1.0               0.5            0.25
                   Error            1.0               0.25          0.0625
 Advantages of the Finite Element Based Approach


(5) Efficient
 Add One Point => Add O(1) Tetrahedrons




 O(n2) Times More Efficient Than Grid-Based Approaches.
Advantages of the Finite Element Based Approach


(6) No Whittaker-Shannon Sampling Rate

  Interpolation Problem ==> Boundary Value Problem




(7) No Ambiguity in Selecting Modeling Methods
Advantages of the Finite Element Based Approach


(8) Honoring Original Sample Data
Advantages of the Finite Element Based Approach

(9) Flexible, Fast and Interactive
     Modification of an Existing Sample Point
Advantages of the Finite Element Based Approach


(9) Flexible, Fast and Interactive
         Addition of a New Sample Point
Advantages of the Finite Element Based Approach

(10) Consistent Basis Function


                       4
  ( x , y , z )   N e ( x , y , z ) ie
   e
                       j
                      j 1



                                  1   i j
 N e ( x j , y j , z j )   ij  
   j
                                  0   i j
Future Work
(1) Other Types of Problems: Initial Value Problems
(2) Other Types of Elements: Polyhedrons
(3) Higher-Order Elements: P-Version
(4) Automated Tessellation: Densification
(5) Thinning
(6) Curved Discontinuity Surfaces
(7) Delaunay Triangulation near Discontinuity Surfaces
(8) Higher-Order Rendering Method
(9) Fast Searching Algorithms
(10) Technique Issues (e.g., Solving Sparse Matrices, ...)
Summary

The finite element based approach is a new
framework for scattered data visualization. Many
challenging problems can be solved easily within
this framework. This approach revealed a
promising direction and brought many interesting
research topics into the field of sparse data volume
visualization.

				
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posted:1/27/2012
language:English
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