Abstract
We introduce new methodologies for the construction of high-performance very local Riesz wavelet bases of L 2(ℝ n) in arbitrarily high spatial dimension n. The localness L of the representation is measured as the sum of the volumes of the supports of the underlying mother wavelets; small localness number is one of the sought-for properties in wavelet constructions. Our constructs are very simple and they are based on our recent framelet construction methods: the CAMP scheme and the L-CAMP scheme. Within our general methodology, the subclass of piecewiseconstant constructions is the most local one. It includes Riesz wavelet bases with any performance grade and in any spatial dimension. In this subclass, the Riesz wavelet basis with Jackson-type performance k (namely, with k vanishing moments) has localness score L < k+1, independent of the spatial dimension n. Thus, while the number of mother wavelets grows exponentially fast with the dimension, the sum of their volumes remains bounded. In comparison, the widely used n-dimensional tensor-product Daubechies system with k vanishing moments has localness score L ≈ (4k - 2) n.
| Original language | English |
|---|---|
| Pages (from-to) | 2237-2265 |
| Number of pages | 29 |
| Journal | SIAM Journal on Mathematical Analysis |
| Volume | 44 |
| Issue number | 4 |
| DOIs | |
| Publication status | Published - 2012 Sep 17 |
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All Science Journal Classification (ASJC) codes
- Analysis
- Computational Mathematics
- Applied Mathematics
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High-performance very local Riesz wavelet bases of L 2(ℝ n). / Hur, Youngmi; Ron, Amos.
In: SIAM Journal on Mathematical Analysis, Vol. 44, No. 4, 17.09.2012, p. 2237-2265.Research output: Contribution to journal › Article
TY - JOUR
T1 - High-performance very local Riesz wavelet bases of L 2(ℝ n)
AU - Hur, Youngmi
AU - Ron, Amos
PY - 2012/9/17
Y1 - 2012/9/17
N2 - We introduce new methodologies for the construction of high-performance very local Riesz wavelet bases of L 2(ℝ n) in arbitrarily high spatial dimension n. The localness L of the representation is measured as the sum of the volumes of the supports of the underlying mother wavelets; small localness number is one of the sought-for properties in wavelet constructions. Our constructs are very simple and they are based on our recent framelet construction methods: the CAMP scheme and the L-CAMP scheme. Within our general methodology, the subclass of piecewiseconstant constructions is the most local one. It includes Riesz wavelet bases with any performance grade and in any spatial dimension. In this subclass, the Riesz wavelet basis with Jackson-type performance k (namely, with k vanishing moments) has localness score L < k+1, independent of the spatial dimension n. Thus, while the number of mother wavelets grows exponentially fast with the dimension, the sum of their volumes remains bounded. In comparison, the widely used n-dimensional tensor-product Daubechies system with k vanishing moments has localness score L ≈ (4k - 2) n.
AB - We introduce new methodologies for the construction of high-performance very local Riesz wavelet bases of L 2(ℝ n) in arbitrarily high spatial dimension n. The localness L of the representation is measured as the sum of the volumes of the supports of the underlying mother wavelets; small localness number is one of the sought-for properties in wavelet constructions. Our constructs are very simple and they are based on our recent framelet construction methods: the CAMP scheme and the L-CAMP scheme. Within our general methodology, the subclass of piecewiseconstant constructions is the most local one. It includes Riesz wavelet bases with any performance grade and in any spatial dimension. In this subclass, the Riesz wavelet basis with Jackson-type performance k (namely, with k vanishing moments) has localness score L < k+1, independent of the spatial dimension n. Thus, while the number of mother wavelets grows exponentially fast with the dimension, the sum of their volumes remains bounded. In comparison, the widely used n-dimensional tensor-product Daubechies system with k vanishing moments has localness score L ≈ (4k - 2) n.
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UR - http://www.scopus.com/inward/citedby.url?scp=84866120451&partnerID=8YFLogxK
U2 - 10.1137/110830654
DO - 10.1137/110830654
M3 - Article
AN - SCOPUS:84866120451
VL - 44
SP - 2237
EP - 2265
JO - SIAM Journal on Mathematical Analysis
JF - SIAM Journal on Mathematical Analysis
SN - 0036-1410
IS - 4
ER -