1. Error Analysis for Sparse Data Volume Visualization
Hao Sun
Good after noon every one, the topic of my today’s defense is Error Analysis for Sparse Data Volume Visualization

2. Presentation Outline Introduction Theory Results and Discussion
Conclusion
Today’s presentation includes four parts, Introduction, which introduces the concept of the sparse data, the challenge for visualizing sparse data, the theory part will discuss some theories about interpolation functions, constraints and error analysis methods.
And results and discussion, finally the conclusion part

3. Introduction Concept of Sparse Data
Data distributed sparsely and irregularly throughout the volume of interest.
Scattered nature.
Traditional grid-based volume visualization techniques do not directly apply to visualize sparse data.
Sparse data is defined as the data distributed sparsely and irregularly throughout the volume of interest. The sparse data is usually generated from many science and engineering fields.
The important nature of sparse data is his scattered nature
because of scattered nature and limited data points make it very challenge to visualize sparse data since traditional grid-based volume visualization techniques do not directly apply to visualize sparse data.

4. Introduction Two Step Approach
One of the most common used techniques for visualizing sparse data is the two-step approach:
Interpolating the sample data onto an intermediate grid;
Using grid-based volume visualization techniques to visualize interpolated data.
Which includes first, interpolating….
Second, using grid-based …..

5. Introduction key Issues
Four interpolation methods:
Metric method
Thin plate splines method
Volume splines method
Multiquadric method
Four constraints:
Local number
Local region
Local region-number
Global
In this thesis, I have studied the key issues related to the interpolation and visualization of sparse data,
such as four interpolation functions, four constraints

6. Introduction key Issues
Three error estimation methods:
Numerical error
Descritization error
Modeling error
A MFC based Windows application.
We also provided three error analysis methods to help us to evaluate the performance of the interpolation.
Finally, A windows … by using Visualization Tool Kit (VTK) in order to visualize volume data and facilitate error analysis

7. Theory - Interpolation Methods
Metric Method:
2.1
Multiquadric Method:
In the following, I will discuss the details of interpolation methods we used.
The principle of this method is to assign more weight to nearby points than to distant points.
Where di- is the weighting function,  is any real number greater than zero, vi is the data value at point (xi, yi, zi), and di is the Euclidean distance from sample point (xi, yi, zi) to interpolated point (x, y, z).
di2 = (x-xi)2 + (y-yi)2 + (z-zi)2.
Based on the similar nature of metric method and multiquadric method, most of time the show the similar pattern during error analysis. However, the multiquadric method can get the better interpolation results than metric method,
2.2

8. Theory - Interpolation Methods
Thin-Plate Splines Method
2.3
Volume Splines Method
How about the
2.4

9. Theory - Constraint Study
Local by Number Limit
Local by Region Limit
Local Region - Number limit
Global Constraint
We have four constraints studied in this thesis

10. Theory- Constraint Study
Figure 2.1 Selection of Local Sample Points.
If we select points by number limit, for example we set number limit equals to 7, the number we
Selected should be p1 tp p7.
If we select points by region limit within the bounding box like this one, the points we select should be p1-p5
And if we select points by region number limit, if the number limit we set is 4 and region limit is set by the bounding box, at this time, the points we selected should be decided by the number limit which is p1 to p4.
Figure 2.1

11. Error Analysis- Numerical Error
Difference between the sparse data value and the interpolated value at specific point.
2.5
Where vi is the sparse data value and f(x,y,z) is the interpolated value at point x,y,z

12. Error Analysis – Descritization Error
where vi is still the sparse data value at point (xi, yi, zi), however, f(xi,yi,zi) is the trilinearly interpolated value at point (xi, yi, zi) using the eight nearest grid nodes instead of the interpolated value directly using the interpolation function. (Thesis p15-16)
Trilinear Interpolation
2.6

13. Error Analysis – Modeling Error
Use Vi, sparse data value at point (xi, yi, zi), as check point, don’t use it to define interpolation function.
we randomly pick up some of sparse data as modeling data from the sparse data set. The modeling data are not used in the interpolation process to define the interpolation function.
When an interpolation function is defined using other sparse data points, the values at the modeling data points are calculated and the results are compared with the modeling data value.
The differences between them are called as modeling error.
2.7

14. Error Analysis - Root Mean Square
2.8
Where F represents the test function and A represents one of the sparse data interpolates being evaluated. Ni, Nj, Nk represent three dimensions of the grid .
The root mean square is defined, so we can get an overall view of errors generated by
by different interpolation methods and under different constraints,

15. Test Functions Nielson, 1993
Before we begin error analysis, we need select some test functions to generate the test sparse data, those functions are selected because, they are used very often to generate the test data set for testing the performance of interpolation method.
Nielson, 1993

16. Test Functions

17. Visualization Pipe Line
Sparse Data File
Interpolation
Parameters
Computational Module
Error Analysis Module
Visualization Engineer
Rendered Image
Error Analysis
Report
Vtk Data File
Input Data
Intermediate Data
Output Data
This figure describes the visualization pipe implemented in this project. The visualization pipe states sparse data file and interpolation parameters, and then the data go through the computational module which implements four interpolation methods.
The output of the computation module is the data values on grid node and can be either stored in a text file such as vtk data file or maintained in a data structure and consumed by error analysis module to generate error analysis report.
The text file format file can be used by the visualization engine module and generate different images

18. Application Architecture Document/View
The application architecture use traditional window programming Document/View architecture, the main document class is CInterpolationDoc class.
Document Classes

19. Application Architecture Document/View
The main view class is Cinterpolation View class
View Classes

20. Software Demo Lookup Table 0.00 0.25 0.50 0.75 1.00
The program implemented a visualization pipeline with visualization tool kit (VTK).
There is a menu bar of the program which can provide user three main functions
(1) User can select different interpolation methods with different constratins to interpolate sparse to vtk file which represents intermediate grid node data.
For example, if we want to interpolate sparse data with metric method, we select interpolation menu, there will be a interpolation dialog
(2) User can visualize interpolated data through Visualization menu
Vtk file option will use a cube to represent volume of interest and map the interpolated value with different color listed on look up table.
Lookup table build from blue to red which represents small value to large value.
In this case blue 0 red 1
It also implements two visualization algorithms to visualize interpolated data. Finally, it provides user a error analysis menu which let user
(3) User can do error analysis with different interpolation methods under different constraints
By selecting Error Analysis Menu
The error analysis was implemented with batch mode, this means, we can get a bunch of error analysis reports for one specific interpolation method under one specific constraints. This will make it very easy and quick to get the systematic error analysis reports.
Lookup table for color map

21. Results and Discussion
Based on the error analysis
Interpolation Accuracy of interpolation methods
Resolution
The error analysis for different interpolation methods under different constraints
The results and discussion parts will discuss interpolation accuracy of interpolation methods, resolution based on error analysis
And I will also discuss the error analysis for different interpolation methods under different constraints

22. Approximation Accuracy
Numerical Error: Global Constraints, 21*21*21
Descritization Error: Global Constraints, 21*21*21
The data demonstrates that the numerical errors are zero, regardless the test functions and interpolation algorithms used. This indicates that thin plate splines, volume spline and multiquadric method do enforce the condition of f(xi,yi,zi) = vi , i=1,2,3,…n and can exactly reproduce the sample data values at the original sample location.
Global Constraints
Resolution 21*21*21
Modeling Error: Global Constraints, 21*21*21

23. Resolution Descritization Error:
The resolution in this paper refers to the dimension of the grid node used in the interpolation process. We select different resolutions (from 5x5x5 to 80x80x80) and calculate three errors generated by the interpolation methods with region-number limit (region limit 20% and number limit 10) to study the effect of resolution on different interpolation methods.
For descritization error, with the increase of the resolution, the descritization error decreased. The descritization error in this paper is calculated by using trilinear method to interpolate the sample point value in the cube in which the sample data resides, and the size of the cube is decided by the resolution during interpolation. When the resolution is high, or the size of cube is small, the trilinear interpolation will be more accurate. This will result in a decrease of descritization error. However, the tradeoff is that it will need more computational resource to handle the increasing load of computation in high resolution.
Descritization Error:
Region-Number Limit with Region Limit 20% and Number Limit 10)

24. Error Analysis – Descritization Error
where vi is still the sparse data value at point (xi, yi, zi), however, f(xi,yi,zi) is the trilinearly interpolated value at point (xi, yi, zi) using the eight nearest grid nodes instead of the interpolated value directly using the interpolation function. (Thesis p15-16)
Trilinear Interpolation
2.6

25. Number Limit Descritization Error
A set of number limits range from 5 to 100 (global, the total number of sample is 100) were tested to see the effects of number limit on different interpolation methods.
The metric method get the highest descrization error.
The reason is: metric method is low order interpolation function and didn’t reproduce local shape properties.
For the other interpolation methods, With the increasing of number limit, the descritization error decreased first and reach the lowest value when number limit is 10 and then increasing with the increasing of number limit.
The reason is that when number limit is low, there is no enough points used to interpolate grid node and when number limit becomes high, it is hard to represent the local shape with the sample point far away the interpolated grid node.
The effect of number limit on descritization Error

26. Number Limit on Modeling Error for Function 1
Number Limit Modeling Error
The similar trends are seen in modeling error analysis, except mulitqudaric method shows a different patter. When we applied number limit on four interpolation methods, we can see that metric method and multiquadric method got the higher modeling error than those of volume splines and thin splines method.
Number Limit on Modeling Error for Function 1

27. Region Limit on Descritization Error for Function 1
Region Limit Descritization Error
For region limit, we select region bounding box of 5% of volume of interest to 100% (global constraint),
Similar to number limit constraint, the metric method also get the highest descritization error.
When region limit is low, all the interpolation method get almost the same high descritization error, the reason, when region limit is lower than 20%, there is not enough points to define the high order interpolation function
Region Limit on Descritization Error for Function 1

28. Region Limit on Modeling Error for Function 1
Region Limit Modeling Error
Similar results of modeling error under region limit
Region Limit on Modeling Error for Function 1

29. Region Number Limit on Descritization Error for Function 1
Region Number Limit Descritization Error
Region Number Limit on Descritization Error for Function 1

30. Region Number Limit on Modeling Error for Function 1
Region Number Limit Modeling Error
Region Number limit is more like region limit than number limit, may because most of time the constraints is defined by the number limit
Region Number Limit on Modeling Error for Function 1

31. Metric Method Comparison of three constraints applied on metric method
The comparison of different constraints are also studied, This figure show the comparison of three constraints applied on metric method, The 5-5% under the x coordinate means, at this point, number limit is set to 5, region limit is set to 5% and region- number limit is set to number limit 5 and region limit 5%.
number limit is select from 5, 10, to Global, and region limit is set to 5%, 10% and 100%, and the region-number limit is set to 5-5%, 10-10%, to Global constraint for all the interpolation function.
When constraint values are low, in this figure, number limit less that 10, region limit less the 10%, the applied number limit can get better performance than region limit and region number limit.
Comparison of three constraints applied on metric method

32. Multiquadric Method Descritization Error
Similar results can be obtained by comparing different constraints on other interpolation functions.
Descritization Error

33. Volume Splines Method Descritization Error
Similar results can be obtained from volume spline methods, thin splines method and multiquadric method.
Descritization Error

34. Thin Splines Method Descritization Error
Different with metric method, for thin splines, volume splines and multiquadric method, there is no big difference among the three constraints when constraint value are higher. For example number limit higher than 40.
Descritization Error

35. Conclusion
The two-step approach presents a convenient way for visualizing sparse data.
The local constraints with error estimation can also help us to select appreciate interpolation function under different conditions
Through this study, we can get the following conclusions.

36. Conclusion
Thin plate, volume splines and multiquadric method can reconstruct a test function quite accurately regardless of the type of the function based on numerical error analyis

37. Conclusion
Resolution will not affect numerical error and modeling error, However, it does affect descritization error. The larger the resolution is, the smaller the descritization error is.

38. Conclusion
In general, thin plate splines, volume splines can get a better performance that multiquadric and metric method.

39. Conclusion
When constraint values are low, the number limit constraint get the better performance.
When constraint values are high, there is no big difference of RMS among the three constraints except the metric method.

40. Acknowledgments Dr.Yincai Xiao Dr. Chien-Chung Chan Dr. Kathy Liszka
Dr. Wolfgang Pelz
Dr. Xuan-Hien Dang
Finally, I would like to thank Dr. xiao, my supervise, for his valueable advice and instruction on my thesis, also thanks Dr. Chan for his class which I studied during my M.S. study and help me a lot during my work right now. I also thank Dr. Liszka for your help to review my thesis. And I also thank Dr. Pelz and Dr. Dang and my friends and family who support me to finish my graduate study in the University of Arkon.