Multiscale representation of surfaces by tight wavelet frames with applications to denoising

Dong, Bin; Jiang, Qingtang; Liu, Chaoqiang; Shen, Zuowei

doi:10.1016/j.acha.2015.03.005

Cited by 28 publications

(21 citation statements)

References 67 publications

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“…They have been successfully used in video processing [26], image segmentation [27,28] and classifications [29,30]. More recently, wavelet frames are constructed on non-flat domains such as surfaces [31,32] and graphes [33][34][35][36] with applications to denoising [31,32,36] and classifications [36].…”

mentioning

confidence: 99%

CT Image Reconstruction by Spatial-Radon Domain Data-Driven Tight Frame Regularization

Zhan¹,

Dong²

2016

SIAM J. Imaging Sci.

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This paper proposes a spatial-Radon domain CT image reconstruction model based on data-driven tight frames (SRD-DDTF). The proposed SRD-DDTF model combines the idea of joint image and Radon domain inpainting model of [1] and that of the data-driven tight frames for image denoising [2]. It is different from existing models in that both CT image and its corresponding high quality projection image are reconstructed simultaneously using sparsity priors by tight frames that are adaptively learned from the data to provide optimal sparse approximations. An alternative minimization algorithm is designed to solve the proposed model which is nonsmooth and nonconvex. Convergence analysis of the algorithm is provided. Numerical experiments showed that the SRD-DDTF model is superior to the model by [1] especially in recovering some subtle structures in the images.

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confidence: 99%

CT Image Reconstruction by Spatial-Radon Domain Data-Driven Tight Frame Regularization

Zhan¹,

Dong²

2016

SIAM J. Imaging Sci.

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“…Since tight wavelet frames can provide redundant representations of image data and exhibit substantial ability for feature/texture extraction, they have been successfully applied to various research areas, such as image segmentation [16], [28], image denoising [29]- [31], image restoration [25], [32], and mesh surface reconstruction [33], [34]. For simplicity, we present the main idea of a tight wavelet frame transform concisely.…”

Section: B Tight Wavelet Frame Transformmentioning

confidence: 99%

Sparse Regularization-Based Fuzzy C-Means Clustering Incorporating Morphological Grayscale Reconstruction and Wavelet Frames

Wang

Pedrycz

Zhou

et al. 2021

IEEE Trans. Fuzzy Syst.

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Instead of directly utilizing an observed image including some outliers, noise or intensity inhomogeneity, the use of its ideal value (e.g. noise-free image) has a favorable impact on clustering. Hence, the accurate estimation of the residual (e.g. unknown noise) between the observed image and its ideal value is an important task. To do so, we propose an ℓ0 regularization-based Fuzzy C-Means (FCM) algorithm incorporating a morphological reconstruction operation and a tight wavelet frame transform. To achieve a sound trade-off between detail preservation and noise suppression, morphological reconstruction is used to filter an observed image. By combining the observed and filtered images, a weighted sum image is generated. Since a tight wavelet frame system has sparse representations of an image, it is employed to decompose the weighted sum image, thus forming its corresponding feature set. Taking it as data for clustering, we present an improved FCM algorithm by imposing an ℓ0 regularization term on the residual between the feature set and its ideal value, which implies that the favorable estimation of the residual is obtained and the ideal value participates in clustering. Spatial information is also introduced into clustering since it is naturally encountered in image segmentation. Furthermore, it makes the estimation of the residual more reliable. To further enhance the segmentation effects of the improved FCM algorithm, we also employ the morphological reconstruction to smoothen the labels generated by clustering. Finally, based on the prototypes and smoothed labels, the segmented image is reconstructed by using a tight wavelet frame reconstruction operation. Experimental results reported for synthetic, medical, and color images show that the proposed algorithm is effective and efficient, and outperforms other algorithms.

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“…The K-SVD algorithm is able to learn a frame ψ ksvd ; however, it has no control over the bounds, due to the fact that it imposes no constraints on frame bounds (the largest and smallest singular values of ψ ksvd ψ ⊤ ksvd ). Parseval tight frames with frame bounds of 1 have been widely used in signal processing [29]- [31]. Thus, we re-formulated the K-SVD optimization problem in our development of a learning algorithm to learn the optimal Parseval tight frame as well as the sparse coefficients from a set of observations.…”

Section: Learning Parseval Frames For Sparse Representationmentioning

confidence: 99%

Frame-Based Sparse Analysis and Synthesis Signal Representations and Parseval K-SVD

Hwang

Huang

Kung

et al. 2019

IEEE Trans. Signal Process.

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Frames are the foundation of the linear operators used in the decomposition and reconstruction of signals, such as the discrete Fourier transform, Gabor, wavelets, and curvelet transforms. The emergence of sparse representation models has shifted of the emphasis in frame theory toward sparse l 1 -minimization problems. In this paper, we apply frame theory to the sparse representation of signals in which a synthesis dictionary is used for a frame and an analysis dictionary is used for a dual frame. We sought to formulate a novel dual frame design in which the sparse vector obtained through the decomposition of any signal is also the sparse solution representing signals based on a reconstruction frame. Our findings demonstrate that this type of dual frame cannot be constructed for over-complete frames, thereby precluding the use of any linear analysis operator in driving the sparse synthesis coefficient for signal representation. Nonetheless, the best approximation to the sparse synthesis solution can be derived from the analysis coefficient using the canonical dual frame. In this study, we developed a novel dictionary learning algorithm (called Parseval K-SVD) to learn a tight-frame dictionary. We then leveraged the analysis and synthesis perspectives of signal representation with frames to derive optimization formulations for problems pertaining to image recovery. Our preliminary, results demonstrate that the images recovered using this approach are correlated to the frame bounds of dictionaries, thereby demonstrating the importance of using different dictionaries for different applications.

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Multiscale representation of surfaces by tight wavelet frames with applications to denoising

Cited by 28 publications

References 67 publications

CT Image Reconstruction by Spatial-Radon Domain Data-Driven Tight Frame Regularization

CT Image Reconstruction by Spatial-Radon Domain Data-Driven Tight Frame Regularization

Sparse Regularization-Based Fuzzy C-Means Clustering Incorporating Morphological Grayscale Reconstruction and Wavelet Frames

Frame-Based Sparse Analysis and Synthesis Signal Representations and Parseval K-SVD

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