Adaptive Directional Haar Tight Framelets on Bounded Domains for Digraph Signal Representations

Xiao, Yuchen; Zhuang, Xiaosheng

doi:10.1007/s00041-021-09816-3

Cited by 9 publications

(6 citation statements)

References 44 publications

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“…We first introduce the concept of hierarchical partitions given in [30]. For a compact subset Ω ⊆ R d , let {B j } j∈N0 with N 0 := N∪{0} be a family of subsets of 2 Ω .…”

Section: A Construction Of Haar Tight Frameletsmentioning

confidence: 99%

“…Obviously, Theorem 1 recovers results in [30, Theorem 1] since its matrix A is a special case of (4). Hence, our result can be applied for digraph signal representation, which is not the focus of this paper and we refer to [30] for more details.…”

Section: A Construction Of Haar Tight Frameletsmentioning

confidence: 99%

“…In this paper, we aim at the construction and applications of simple but efficient Haar-type systems for data defined on a non-Euclidean domain and, specially, on the 2-sphere S 2 . Our work is inspired by [30], in which the authors extended the most simple yet elegant Haar orthonormal wavelet system [10] to the construction of Haar tight framelets on any compact set Ω ⊆ R d , from the point of view of the underlying hierarchical partition. Such a construction can be built easily with flexibility and it brings many advantages especially for directional and (di)graph representations [12], [16], [30].…”

Section: Introductionmentioning

confidence: 99%

“…Our work is inspired by [30], in which the authors extended the most simple yet elegant Haar orthonormal wavelet system [10] to the construction of Haar tight framelets on any compact set Ω ⊆ R d , from the point of view of the underlying hierarchical partition. Such a construction can be built easily with flexibility and it brings many advantages especially for directional and (di)graph representations [12], [16], [30]. By further exploiting the hierarchical partition, in this paper, we present a general framework for the construction of Haar-type tight framelet systems on Ω.…”

Section: Introductionmentioning

confidence: 99%

“…[1,1,1] . One can easily verify that A defines a tight frame with frame bound 1, which is different from the classical one and that in[30]. Figure3visualizes this example.…”

mentioning

confidence: 95%

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Convolutional Neural Networks for Spherical Signal Processing via Spherical Haar Tight Framelets

Li¹,

Hong²,

Zhuang³

2022

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In this paper, we develop a general theoretical framework for constructing Haar-type tight framelets on any compact set with a hierarchical partition. In particular, we construct a novel area-regular hierarchical partition on the 2-sphere and establish its corresponding spherical Haar tight framelets with directionality. We conclude by evaluating and illustrate the effectiveness of our area-regular spherical Haar tight framelets in several denoising experiments. Furthermore, we propose a convolutional neural network (CNN) model for spherical signal denoising which employs the fast framelet decomposition and reconstruction algorithms. Experiment results show that our proposed CNN model outperforms threshold methods, and processes strong generalization and robustness properties.

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“…We first introduce the concept of hierarchical partitions given in [30]. For a compact subset Ω ⊆ R d , let {B j } j∈N0 with N 0 := N∪{0} be a family of subsets of 2 Ω .…”

Section: A Construction Of Haar Tight Frameletsmentioning

confidence: 99%

Section: A Construction Of Haar Tight Frameletsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…[1,1,1] . One can easily verify that A defines a tight frame with frame bound 1, which is different from the classical one and that in[30]. Figure3visualizes this example.…”

mentioning

confidence: 95%

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