html5-img
1 / 17

HARMONIC ANALYSIS OF BV

HARMONIC ANALYSIS OF BV. Ronald A. DeVore. Ronald A. DeVore. HARMONIC ANALYSIS OF BV. Industrial Mathematics Institute Department of Mathematics University of South Carolina. BV ( ) SPACE OF FUNCTIONS OF BOUNDED VARIATION. WHAT IS BV?. WHY BV?. • BV USED AS A MODEL FOR REAL IMAGES

jean
Télécharger la présentation

HARMONIC ANALYSIS OF BV

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. HARMONIC ANALYSIS OF BV Ronald A. DeVore Ronald A. DeVore HARMONIC ANALYSIS OF BV Industrial Mathematics Institute Department of Mathematics University of South Carolina

  2. BV ( ) SPACE OF FUNCTIONS OF BOUNDED VARIATION WHAT IS BV? WHY BV? •BV USED AS A MODEL FOR REAL IMAGES • BV PLAYS AN IMPORTANT ROLE IN PDES

  3. •REPLACE BV BY BESOV SPACE EXTREMAL PROBLEM FOR BV •DENOISING •STATISTICAL ESTIMATION •EQUIVALENT TO MUMFORD-SHAH •EXAMPLE OF K-FUNCTIONAL •LIONS-OSHER-RUDIN PDE APPROACH

  4. WAVLET ANALYSIS -DYADIC CUBES -DYADIC CUBES OF LENGTH IS A COS

  5. = 0 1 HAAR FUNCTION +1 0 1 -1

  6. DAUBECHIES WAVELET

  7. WAVELET COEFFICIENTS DEFINE

  8. THIS MAKES IT EASY TO SOLVE This decouples and the solution is given by (soft) thresholding. Coefficients larger than in others into .

  9. CAN WE REPLACE BY BV • BV HAS NO UNCONDITIONAL BASIS THEOREM(Cohen-DeV-Petrushev-Xu) SANDWICH THEOREM-AMER J. 1999 • IS WEAK THE PROBLEM ALSO SOLVED BY THRESHOLDING AT . SIMPLE NON PDE SOLUTION TO OUR ORIGINAL PROBLEM

  10. POINCARÉ INEQUALITIES Simplest case nice domain; • Does not scale correctly for modulation • Replace THEOREM (Cohen-Meyer) • Scales correctly for both modulation and dilation SPECIAL CASE MEYER’S CONJECTURE: ABOVE HOLDS FOR ALL

  11. (1,1) - BV (1/2,0) L2 B-1  (0,-1) Smoothness (1/q, )  Lq 1/q Lq Space

  12. THEOREM (Cohen-Dahmen- Daubechies-DeVore) FOR ALL •Gagliardo-Nirenberg

  13. THESE THEOREMS REQUIRE FINER STRUCTURE OF BV LET New space THIS IS EQUIVALENT TO

  14. THEOREM (Cohen-Dahmen- Daubechies-DeVore) i. If , then implies ii. Counterexamples for • is original weak result. • solves Meyer conjecture.

  15. DYADIC CUBES BAD CUBES BAD CUBES GOOD CUBES

  16. IDEA OF PROOF GOOD CUBE: COLLECTION OF GOOD CUBES THE COLLECTION OF BAD CUBES IF BV, THEN IS IN

  17. CONCLUDING REMARKS • FINE STRUCTURE OF BV • NEW SPACES • NEW INTERPOLATION THEORY • CARLESON MEASURE • Sandwich Theorem • spaces • RELATED PAPERS: DEVORE-PETROVA: AVERAGING LEMMAS-JAMS 2001 COHEN-DEVORE-HOCHMUTH-RESTRICTED APPROXIMATION -ACHA 2001 COHEN-DEVORE-KERKYACHARIAN-PICARD: MAXIMAL SPACES FOR THRESHOLDING ALGORITHMS -ACHA 2001

More Related