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IMIIMI
HARMONIC ANALYSIS OF BVHARMONIC ANALYSIS OF BV
Ronald A. DeVoreRonald A. DeVoreRonald A. DeVoreRonald A. DeVore
Industrial Mathematics InstituteDepartment of Mathematics
University of South Carolina
HARMONIC ANALYSIS OF BVHARMONIC ANALYSIS OF BV
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WHAT IS BV?
BV ( ) SPACE OF FUNCTIONS OF BOUNDED VARIATION
WHY BV?
• BV USED AS A MODEL FOR REAL IMAGES
• BV PLAYS AN IMPORTANT ROLE IN PDES
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EXTREMAL PROBLEM FOR BV
• DENOISING
• STATISTICAL ESTIMATION
• EQUIVALENT TO MUMFORD-SHAH
• EXAMPLE OF K-FUNCTIONAL
• LIONS-OSHER-RUDIN PDE APPROACH
• REPLACE BV BY BESOV SPACE
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WAVLET ANALYSIS
-DYADIC CUBES -DYADIC CUBES
OF LENGTH
IS A COS
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HAAR FUNCTION
0 1
0 1
+1
-1
=
Ψ = H
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DAUBECHIES WAVELET
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WAVELET COEFFICIENTS
DEFINE
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THIS MAKES IT EASY TO SOLVE
This decouples and the solution is given by (soft)
thresholding. Coefficients larger than in others into .
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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
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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
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•
Lq 1/q1/q
•
SmoothnessSmoothness
Lq Space
(1/q, (1/q, ))
• L2
(1,1) (1,1) - - BVBV •
(1/2,0(1/2,0))•
• (0,-1)(0,-1) B-1
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THEOREM (Cohen-Dahmen- Daubechies-DeVore)
• Gagliardo-Nirenberg
FOR ALL
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THESE THEOREMS REQUIRE FINER STRUCTURE OF BV
LET
New space
THIS IS EQUIVALENT TO
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THEOREM (Cohen-Dahmen- Daubechies-DeVore)
i. If , then implies
ii. Counterexamples for
• is original weak result.
• solves Meyer conjecture.
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DYADIC CUBESBAD CUBESGOOD CUBES BAD CUBES
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IDEA OF PROOF
GOOD CUBE:
COLLECTION OF GOOD CUBES
THE COLLECTION OF BAD CUBES
IF BV, THEN IS IN
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CONCLUDING REMARKS
• FINE STRUCTURE OF BV
• NEW SPACES
• NEW INTERPOLATION THEORY • CARLESON MEASURE
• RELATED PAPERS: DEVORE-PETROVA: AVERAGING LEMMAS-JAMS 2001
COHEN-DEVORE-HOCHMUTH-RESTRICTEDAPPROXIMATION -ACHA 2001
COHEN-DEVORE-KERKYACHARIAN-PICARD:MAXIMAL SPACES FOR THRESHOLDING ALGORITHMS -ACHA 2001
• Sandwich Theorem
• spaces