1. OFP Filters in the Denoising of the Significance Map
J. Nair, D. S. Thompson,
G.Craciun, R. Machiraju & J. Fowler
Evita Technical Review and Symposium May 2002

2. Generating the Significance Map : A Recap
Definition: Color/value map of values of desired feature(s)
Feature of interest: Swirl
Swirl calculations at each grid point:
Eigen values of the velocity gradient tensor
Important formulae :
torb = 2π / | im(λ1,2) |
Swirl parameter τ = L / ( torb* | Vconv | )
Evita Technical Review and Symposium May 2002

3. Generating the Significance Map: Modifications
Significant swirl features identified using Vortex Core Detection Algorithm
At least one of the neighboring grid point as candidate for each of the 4 quantization regions
Candidacy of a grid point for a quantization region if its velocity vector direction in the particular range
Evita Technical Review and Symposium May 2002

4. Evita Technical Review and Symposium May 2002
Need for Denoising
Several features in the significance map are not important, owing to their virtue of
Poor resolution
Weak swirl strengths
Small areas
Evita Technical Review and Symposium May 2002

5. Denoising Algorithm: The Underlying Principle
Multiresolutional Analysis of the significance map using wavelets :
Can be used to achieve:
Feature detection
Feature preservation
Can be also exploited for denoising
Evita Technical Review and Symposium May 2002

6. Denoising Algorithm: The Underlying Principle
Property of Discrete Wavelet Transformations (DWT) exploited:
The coarse/scaling coefficients retain the low frequency band
The strong swirl features survive through more than 1 or 2 levels of transformations
Owing to their high original velocity gradients
Evita Technical Review and Symposium May 2002

7. Denoising Algorithm: Methodology
Discrete Wavelet Transformation of each velocity component
Swirl calculations at each scale
Binary mapping of the swirl maps
Evita Technical Review and Symposium May 2002

8. Denoising Algorithm: Methodology
Swirl maps at each level are used to update the higher/finer level
Updating using the 0’s in the coarser grid
Corner points A'-B'-C'-D' updated as A-B-C-D, respectively, if the latter is ‘0’
Coarse grid cell ↔ Fine grid cell
Evita Technical Review and Symposium May 2002

9. Denoising Algorithm: Methodology
Edge points updated as the logical AND of corresponding corner points A-B-C-D
Center point 0 updated as the logical AND of all corner points A-B-C-D
Coarse grid cell ↔ Fine grid cell
Evita Technical Review and Symposium May 2002

10. Evita Technical Review and Symposium May 2002
OFP Filters
Design of Low Pass Filter of the analysis stage in each level of the multiresolutional analysis
Features:
Guarantees non-increasing variation of scaling coefficients (TVD features)
Based on Evolutionary PDE’s
Evita Technical Review and Symposium May 2002

11. OFP Filters: Key Properties
Preserves:
Feature location
Feature strength
Feature shape
Evita Technical Review and Symposium May 2002

12. Denoising : OFP Filters vs Non-OFP Filters
Linear lifting scheme: currently used as example for non-OFP filters
Similarities of the filters used:
Symmetric filters
Preserves feature location
Evita Technical Review and Symposium May 2002

13. Denoising : OFP Filters vs Non-OFP Filters
Dissimilarities
Linear lifting scheme introduces new extrema, OFP filters preserves extrema in the original map
Differences in preserving feature shape and feature size
Evita Technical Review and Symposium May 2002

14. Denoising using OFP Filters: Swirl Maps
Level 0
Level 1 (82)
Level 2 (116)
Level 3 (158)
Evita Technical Review and Symposium May 2002

15. Denoising using OFP Filters : Results
Level 0-1
Level 2-3
Evita Technical Review and Symposium May 2002

16. Denoising using Non-OFP Filters: Swirl Maps
Level 0
Level 1 (67)
Level 2 (105)
Level 3 (150)
Evita Technical Review and Symposium May 2002

17. Denoising using Non-OFP Filters : Results
Level 0-1
Level 2-3
Evita Technical Review and Symposium May 2002

18. Testing on an Isolated Vortex
Study of the frequency responses of filters on the induced velocity magnitude function
Induced velocity magnitude function:
Ω: Angular velocity,
R: Radius of core,
s: Distance of (x,y) from center of core
Evita Technical Review and Symposium May 2002

19. Results from 1-dimensional Velocity Magnitude Profile: OFP Filters
Evita Technical Review and Symposium May 2002

20. 512-pt FFT of 1-dimensional Velocity Magnitude Profile: OFP Filters
Frequency in Hz →
Evita Technical Review and Symposium May 2002

21. Results from 1-dimensional Velocity Magnitude Profile: Non-OFP Filters
Evita Technical Review and Symposium May 2002

22. Evita Technical Review and Symposium May 2002
512-pt FFT of 1-dimensional Velocity Magnitude Profile: Non-OFP Filters
Frequency in Hz →
Evita Technical Review and Symposium May 2002

23. Results from 2-dimensional Velocity Magnitude Profile: OFP Filters
Surface plot of the velocity profile
Cross-section along its center plane
Evita Technical Review and Symposium May 2002

24. Results from 2-dimensional Velocity Magnitude Profile: Non-OFP filters
Surface plot of velocity profile
Cross-section along its center plane
Evita Technical Review and Symposium May 2002

25. Observations & Analysis
Strong differences in the performance of OFP and non-OFP filters not observed in this dataset
Frequency content of a rankine vortex is mostly in lowpass band
FFT plots of the velocity magnitude profiles show:
OFP preserves most of the energy
OFP does not introduce any high pass bands
Evita Technical Review and Symposium May 2002

26. Observations & Analysis
In the NLOM dataset, OFP filters perform better denoising than the non-OFP
Velocity gradients diffused faster by OFP filters owing to their TVD characteristics
OFP filters retain the significant features, apart from removing relatively more features
Evita Technical Review and Symposium May 2002

27. Evita Technical Review and Symposium May 2002
Conclusions
Both filters effectively remove features with less area and weak strength
OFP filters better than non-OFP, as they
remove relatively more features
retain the significant features
conserve most of the energy of the original data distribution
Evita Technical Review and Symposium May 2002

28. In sequel to Denoising…
Segmentation of the map
Features are defined distinctively and numbered
Scanline Segmentation Algorithm
2-pass algorithm
Evita Technical Review and Symposium May 2002

29. In sequel to Denoising…
Ranking of features
Maximum strength in each feature
Total strength
Average strength
Feature size in pixels
Weighted average of above listed
Evita Technical Review and Symposium May 2002