Compactly Supported Tight Frames Associated with Refinable Functions

Chui, Charles K.; He, Wei

doi:10.1006/acha.2000.0301

Cited by 201 publications

(221 citation statements)

References 9 publications

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“…We form tight frame filters by applying the unitary extension principle [20] resulting in the two high-pass filters [21] …”

Section: Multi-resolution Analysis Of Density Estimatesmentioning

confidence: 99%

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Incremental Computation of Feature Hierarchies

Felsberg

2010

Lecture Notes in Computer Science

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Abstract. Feature hierarchies are essential to many visual object recognition systems and are well motivated by observations in biological systems. The present paper proposes an algorithm to incrementally compute feature hierarchies. The features are represented as estimated densities, using a variant of local soft histograms. The kernel functions used for this estimation in conjunction with their unitary extension establish a tight frame and results from framelet theory apply. Traversing the feature hierarchy requires resampling of the spatial and the feature bins. For the resampling, we derive a multi-resolution scheme for quadratic spline kernels and we derive an optimization algorithm for the upsampling. We complement the theoretic results by some illustrative experiments, consideration of convergence rate and computational efficiency.

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“…We form tight frame filters by applying the unitary extension principle [20] resulting in the two high-pass filters [21] …”

Section: Multi-resolution Analysis Of Density Estimatesmentioning

confidence: 99%

“…The downsampling is fully covered in terms of the matrices (5). The upsampling of g is achieved by combining the reconstruction formula (4.13) in [21] with the iterative scheme in [20]. Plugging (6) into (8) results in…”

Section: Upsampling Channel Vectorsmentioning

confidence: 99%

Incremental Computation of Feature Hierarchies

Felsberg

2010

Lecture Notes in Computer Science

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“…A different viewpoint was proposed in [6,7] for understanding (10). By (8), the observed image g is formed by sampling and summing different blurring images…”

Section: Mathematical Model For High-resolution Image Reconstructionmentioning

confidence: 99%

“…Recently, the unitary extension principle was further extended independently by Daubechies, Han, Ron and Shen in [15] and Chui, He and Stöckler in [11] to the Oblique Extension Principle. These two principles lead to some systematic constructions of tight framelets from MRA generated by various refinable functions (see [10,11,15,22,43]). Here, we will use the unitary extension principle to design a tight framelet system from a given refinable function and a wavelet generator.…”

Section: Construction Of Tight Frameletsmentioning

confidence: 99%

High‐resolution image reconstruction with displacement errors: A framelet approach

Chan¹,

Riemenschneider²,

Shen³

et al. 2004

Int J Imaging Syst Tech

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High-resolution image reconstruction arises in many applications, such as remote sensing, surveillance, and medical imaging. The model of Bose and Boo [2] can be viewed as the passage of the high-resolution image through a blurring kernel built from the tensor product of a univariate low-pass filter of the form 1 2 + , 1, · · · , 1, 1 2 − , where is the displacement error. When the number L of low-resolution sensors is even, tight frame symmetric framelet filters were constructed in [8] from this low-pass filter using the unitary extension principle of [43]. The framelet filters do not depend on , and hence the resulting algorithm reduces to that of the case where = 0. Furthermore, the framelet method works for symmetric boundary conditions. This greatly simplifies the algorithm. However, both the design of the tight framelets and extension to symmetric boundary are only for even L and cannot be applied to the case when L is odd. In this paper, we design tight framelets and derive a tight framelet algorithm with symmetric boundary conditions that work for both odd and even L. An analysis of the convergence of the algorithms is also given. The details of the implementations of the algorithm are also given.

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“…Tight wavelet frames and tight framelet filter banks without symmetry have been extensively studied in a lot of papers, to mention only a few here, see [2,3,4,5,8,12,18,19,20] and many papers therein. Interesting examples of real-valued tight framelet filter banks with symmetry have been obtained in [3,5,6,13,14,15,16,17,18,21].…”

Section: Introductionmentioning

confidence: 99%

Matrix splitting with symmetry and symmetric tight framelet filter banks with two high-pass filters

Han¹

2013

Applied and Computational Harmonic Analysis

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Abstract. The oblique extension principle introduced in [3,5] is a general procedure to construct tight wavelet frames and their associated filter banks. Symmetric tight framelet filter banks with two high-pass filters have been studied in [13,16,17]. Tight framelet filter banks with or without symmetry have been constructed in [1]- [21] and references therein. This paper is largely motivated by several results in [11,13,17] to further study tight wavelet frames and their associated filter banks with symmetry and two high-pass filters. Our study not only leads to a simpler algorithm for the construction of tight framelet filter banks with symmetry and two high-pass filters, but also allows us to obtain a wider class of tight wavelet frames with symmetry which are not available in the current literature. The key ingredient in our investigation is a complete characterization of splitting positive semi-definite 2 × 2 matrices of Laurent polynomials with symmetry. Several examples are provided to illustrate the results and algorithms in this paper.

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Compactly Supported Tight Frames Associated with Refinable Functions

Cited by 201 publications

References 9 publications

Incremental Computation of Feature Hierarchies

Incremental Computation of Feature Hierarchies

High‐resolution image reconstruction with displacement errors: A framelet approach

Matrix splitting with symmetry and symmetric tight framelet filter banks with two high-pass filters

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