Bi-framelet systems with few vanishing moments characterize Besov spaces

Borup, Lasse; Gribonval, Rémi; Nielsen, Morten

doi:10.1016/j.acha.2004.01.004

Cited by 48 publications

(76 citation statements)

References 24 publications

(30 reference statements)

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“…with f being the original image, f (∞) the reconstructed image by (5), and N the number of pixels in f (∞) . The initial seed f (0) is chosen as zero.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Since canonical coefficients of a transform is often linked to the smoothness of the underlying function, e.g. a weighted norm of the canonical framelet coefficients is equivalent to some Besov norm of the underlying function, see [5], the second term together with the third term guarantee the regularity of f . Altogether, we see that the cost functional (35) balances the data fidelity in both the image and the transformed domains, and the regularity and sparsity of the limit y * in the transformed domain, which in turn guarantee the regularity of solution in the image domain.…”

Section: Convergence In the Transformed Domainmentioning

confidence: 99%

“…Hence the second term is related to the smoothness of f * , since canonical coefficients of a transform is often linked to the smoothness of the underlying function. For example, some weighted norm of the canonical framelet coefficients is equivalent to some Besov norm of the underlying function, see [5]. Therefore, the second term together with the third term guarantee the regularity of f .…”

Section: Introductionmentioning

confidence: 99%

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Simultaneously inpainting in image and transformed domains

et al. 2009

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In this paper, we focus on the restoration of images that have incomplete data in either the image domain or the transformed domain or in both. The transform used can be any orthonormal or tight frame transforms such as orthonormal wavelets, tight framelets, the discrete Fourier transform, the Gabor transform, the discrete cosine transform, and the discrete local cosine transform. We propose an iterative algorithm that can restore the incomplete data in both domains simultaneously. We prove the convergence of the algorithm and derive the optimal properties of its limit. The algorithm generalizes, unifies, and simplifies the inpainting algorithm in image domains given in [8] and the inpainting algorithms in the transformed domains given in [7,16,19]. Finally, applications of the new algorithm to super-resolution image reconstruction with different zooms are presented.

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“…with f being the original image, f (∞) the reconstructed image by (5), and N the number of pixels in f (∞) . The initial seed f (0) is chosen as zero.…”

Section: Numerical Resultsmentioning

confidence: 99%

Section: Convergence In the Transformed Domainmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Simultaneously inpainting in image and transformed domains

et al. 2009

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“…The second term measures the distance from α to the range of A. By the framelet theory in [2,24], if the coefficient α is in the range of A, then the (weighted) 1 norm α is equivalent to some Besov norm of the image A T α. Therefore, the second term links the (weighted) 1 norm to the regularity of the restored image in some sense.…”

Section: Algorithms With Tight Frame Denoising Schemementioning

confidence: 99%

Tight Frame Based Method for High-Resolution Image Reconstruction

Cai

Chan

Shen

et al. 2010

Series in Contemporary Applied Mathematics

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We give a comprehensive discussion on high-resolution image reconstruction based on tight frame. We first present the tight frame filters arising from the problem of high-resolution image reconstruction and the associated matrix representation of the filters for various boundary extensions. We then propose three algorithms for high-resolution image reconstruction using the designed tight frame filters and show analytically the properties of these algorithms. Finally, we numerically illustrate the efficiency of the proposed algorithms for natural images. High-Resolution Image Reconstruction ModelThe problem of high-resolution image reconstruction is to reconstruct a high-resolution (HR) image from multiple, under-sampled, shifted, degraded and noisy frames where each frame differs from the others by some sub-pixel shifts. The problem arises in a variety of scientific, medical, and engineering applications. The earliest study of HR image reconstruction was motivated by the need to improve the resolution of images from Landsat image data. In [28], Huang and Tsay used the frequency domain approach to demonstrate the improved reconstruction image from several down-sampled noise-free images. Later on, Kim el al. [30] generalized this idea to noisy and blurred images. Both methods in [28,30] are computational efficiency, but, they are prone to model errors, and that limits their use [1]. Statistical methods have appeared recently for super-resolution image reconstruction problems. In this direction, tools such as a maximum a posteriori (MAP) estimator with the Huber-Markov random field prior and a Gibbs image prior are proposed in [25,43]. In particular, the task of simultaneous image registration and super-resolution image reconstruction are studied in [25,45]. Iterative spatial domain methods are one popular class of methods for solving the problems of resolution enhancement [3,21,22,23,27,31,32,36,38,39,41]. The problems are formulated as Tikhonov regularization. A great deal of work has been devoted to the efficient calculation of the reconstruction and the estimation of the associated hyperparameters by taking advantage of the inherent structures in the HR system matrix. Bose and Boo [3] used a block semi-circulant matrix decomposition in order to calculate the MAP reconstruction. Ng et al. [36] and Ng and Yip [37] proposed a fast discrete cosine transform based approach for HR image reconstruction with Neumann boundary condition. Nguyen et al. [40,41] also addressed the problem of efficient calculation. The proper choice of the regularization

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“…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

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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.

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Bi-framelet systems with few vanishing moments characterize Besov spaces

Cited by 48 publications

References 24 publications

Simultaneously inpainting in image and transformed domains

Simultaneously inpainting in image and transformed domains

Tight Frame Based Method for High-Resolution Image Reconstruction

Dual Wavelet Frames and Riesz Bases in Sobolev Spaces

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