Tight wavelet frames in Lebesgue and Sobolev spaces

Borup, Lasse; Gribonval, Rémi; Nielsen, Morten

doi:10.1155/2004/792493

Cited by 19 publications

(20 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Although there are more than one function whose quasi-interpolations are P q −1 S and P 0 S given as (2.4) and (2.5), we never get the underlying function S. One can only expect to obtain a better approximation P 0 S of S from P q −1 S. The approximation 123 power of P 0 S and P q −1 S and their difference can be established for smooth functions by applying the corresponding results in [19] which depend on the properties of the underlying refinable function; more general for piecewise smooth functions, it can be studied by applying results and ideas from [3] and [4] which depend on the properties of the framelets. We omit the detailed discussion here.…”

Section: Formulation In Mramentioning

confidence: 99%

Deconvolution: a wavelet frame approach

2007

View full text Add to dashboard Cite

This paper devotes to analyzing deconvolution algorithms based on wavelet frame approaches, which has already appeared in Chan et al. (SIAM J. Sci. ) as wavelet frame based high resolution image reconstruction methods. We first give a complete formulation of deconvolution in terms of multiresolution analysis and its approximation, which completes the formulation given in ). This formulation converts deconvolution to a problem of filling the missing coefficients of wavelet frames which satisfy certain minimization properties. These missing coefficients are recovered iteratively together with a built-in denoising scheme that removes noise in the data set such that noise in the data will not blow up while iterating. This approach has already been proven to be efficient in solving various problems in high resolution image reconstructions as shown by the simulation results given in

show abstract

Section: Formulation In Mramentioning

confidence: 99%

Deconvolution: a wavelet frame approach

2007

View full text Add to dashboard Cite

show abstract

“…Instead, similar to the approaches in [10,11,12,17,24], we only require the underlying solution f to have a sparse representation under the tight frame system we constructed. It is shown in [7,8] that piecewise smooth functions with a few spikes do have sparse representations by compactly supported tight frame systems. Hence, implicitly, we assume that f is piecewise smooth with possibly some spikes.…”

Section: S = F (X Y) + η(X Y T)mentioning

confidence: 99%

Restoration of Chopped and Nodded Images by Framelets

Cai¹,

Chan²,

Shen³

et al. 2008

SIAM J. Sci. Comput.

131

View full text Add to dashboard Cite

Abstract. In infrared astronomy, an observed image from a chop and nod process can be considered as the result of passing the original image through a highpass filter. Here we propose a restoration algorithm which builds up a tight framelet system that has the highpass filter as one of the framelet filters. Our approach reduces the solution of restoration problem to that of recovering the missing coefficients of the original image in the tight framelet decomposition. The framelet approach provides a natural setting to apply various sophisticated framelet denoising schemes to remove the noise without reducing the intensity of major stars in the image. A proof of the convergence of the algorithm based on convex analysis is also provided. Simulated and real images are tested to illustrate the efficiency of our method over the projected Landweber method.

show abstract

“…In order to state our result on dual Riesz wavelet bases in Sobolev spaces, let us introduce a notion µ 2 (â), which is similar and closely related to the quantity ν 2 (â) in (1.18) ( [34,35]). For a 2π-periodic functionâ, we define µ 2 (â) to be the supremum of all ν 2 …”

Section: \{0}mentioning

confidence: 99%

“…Note that this system cannot form a frame in L 2 (R ) is equivalent to its Sobolev norm (see e.g. [2,3,16,17,26,41]). This approach in the literature requires that wavelet systems have positive regularity and vanishing moments.…”

Section: Introductionmentioning

confidence: 99%

Dual Wavelet Frames and Riesz Bases in Sobolev Spaces

Han

Shen

2008

Constr Approx

View full text Add to dashboard Cite

Abstract. This paper generalizes the mixed extension principle in L 2 (R For s > 0, the key of this general mixed extension principle is the regularity of , k ∈ Z} is a wavelet frame in H s (R) for any 0 < s < m − 1/2, where B m is the B-spline of order m. This simple construction is also applied to multivariate box splines to obtain wavelet frames with short supports, noting that it is hard to construct nonseparable multivariate wavelet frames with small supports. Applying this general mixed extension principle, we obtain and characterize dual Riesz bases (. For example, all interpolatory wavelet systems in [25] generated by an interpolatory refinable function φ ∈ H s (R) with s > 1/2 are Riesz bases of the Sobolev space H s (R). This general mixed extension principle also naturally leads to a characterization of the Sobolev norm of a function in terms of weighted norm of its wavelet coefficient sequence (decomposition sequence) without requiring that dual wavelet frames should be in L 2 (R d ), which is quite different to other approaches in the literature.

show abstract

Tight wavelet frames in Lebesgue and Sobolev spaces

Cited by 19 publications

References 23 publications

Deconvolution: a wavelet frame approach

Deconvolution: a wavelet frame approach

Restoration of Chopped and Nodded Images by Framelets

Dual Wavelet Frames and Riesz Bases in Sobolev Spaces

Contact Info

Product

Resources

About