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

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

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

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

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

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

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

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

Denoising Algorithm: Methodology Edge points 1-2-3-4 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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