Sampling and Super Resolution of Sparse Signals Beyond the Fourier Domain

Bhandari, Ayush; Eldar, Yonina C.

doi:10.1109/tsp.2018.2890064

Cited by 13 publications

(6 citation statements)

References 81 publications

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“…These approaches are fruitful, but they still ultimately revert to the finite-dimensional setting. Taking inspiration from ℓ 1 -based methods for sparse vectors, new approaches have been proposed that go beyond the Hilbert space setting, such as [39,40,41,42].…”

Section: Related Workmentioning

confidence: 99%

Sparsest Piecewise-Linear Regression of One-Dimensional Data

Debarre¹,

Denoyelle²,

Unser³

et al. 2020

Preprint

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We study the problem of interpolating one-dimensional data with total variation regularization on the second derivative, which is known to promote piecewise-linear solutions with few knots. In a first scenario, we consider the problem of exact interpolation. We thoroughly describe the form of the solutions of the underlying constrained optimization problem, including the sparsest piecewise-linear solutions, i.e., with the minimum number of knots. Next, we relax the exact interpolation requirement, and consider a penalized optimization problem with a strictly convex data-fidelity cost function. We show that the underlying penalized problem can be reformulated as a constrained problem, and thus that all our previous results still apply. We propose a simple and fast two-step algorithm to reach a sparsest solution of this constrained problem. Our theoretical and algorithmic results have implications in the field of machine learning, more precisely for the study of popular ReLU neural networks. Indeed, it is well known that such networks produce an input-output relation that is a continuous piecewise-linear function, as does our interpolation algorithm.

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Section: Related Workmentioning

confidence: 99%

Sparsest Piecewise-Linear Regression of One-Dimensional Data

Debarre¹,

Denoyelle²,

Unser³

et al. 2020

Preprint

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show abstract

“…In general, I lr is calculated from I hr following a downsampling function I lr = D(I hr , δ) where δ represents the parameters for the downsampling function D [35]. Super-resolution has been around for decades [36], and many approaches have been tried from sparse coding [37] to deep learning methods [38]. Having a wide range of applications such as medical imaging [39] and surveillance [40], super-resolution is an important processing technique in computer vision for enhancing images.…”

Section: Up-sampling Networkmentioning

confidence: 99%

Adversarial Perturbations Prevail in the Y-Channel of the YCbCr Color Space

Pestana,

Akhtar,

Liu

et al. 2020

Preprint

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Deep learning offers state of the art solutions for image recognition. However, deep models are vulnerable to adversarial perturbations in images that are subtle but significantly change the model's prediction. In a white-box attack, these perturbations are generally learned for deep models that operate on RGB images and, hence, the perturbations are equally distributed in the RGB color space. In this paper, we show that the adversarial perturbations prevail in the Y-channel of the YCbCr space. Our finding is motivated from the fact that the human vision and deep models are more responsive to shape and texture rather than color. Based on our finding, we propose a defense against adversarial images. Our defense, coined ResUpNet, removes perturbations only from the Y-channel by exploiting ResNet features in an upsampling framework without the need for a bottleneck. At the final stage, the untouched CbCrchannels are combined with the refined Y-channel to restore the clean image. Note that ResUpNet is model agnostic as it does not modify the DNN structure. ResUpNet is trained end-toend in Pytorch and the results are compared to existing defense techniques in the input transformation category. Our results show that our approach achieves the best balance between defense against adversarial attacks such as FGSM, PGD and DDN and maintaining the original accuracies of VGG-16, ResNet50 and DenseNet121 on clean images. We perform another experiment to show that learning adversarial perturbations only for the Ychannel results in higher fooling rates for the same perturbation magnitude.

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“…where a 1 and a 2 are the amplitudes of the harmonics, f s represents the sampling frequency, v 1 and v 2 stand for the ratio of harmonic frequency to the sampling frequency. Without loss of generality, we take f s = 1 and 0 <v 1 , v 2 < 0.5 according to the Shannon sampling theorem [25]. In order to quantitatively measure the decomposition capability of different methods, the relative error of the extracted high frequency harmonic is defined as the evaluation indicator: where e represents the relative error.…”

Section: A the Decomposition Capability Analysismentioning

confidence: 99%

Adaptive Bandwidth Fourier Decomposition Method for Multi-Component Signal Processing

2019

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Variational mode decomposition (VMD) is a practical signal decomposition approach, which extracts the modes through bandwidth optimization in the frequency domain. In recent years, many efforts have been made to attenuate the effect of prior parameters, which heavily trouble the traditional VMD. However, as the core step, the bandwidth optimization algorithm including the initialization of center frequencies in VMD is rarely discussed or improved in the existing work. In practical applications, the nonconvergence or unreasonable convergence of the bandwidth optimization can lead to the failure of VMD in mode separation. Thus, in this paper, a new signal decomposition method termed adaptive bandwidth Fourier decomposition (ABFD) is proposed to separate the narrowband components from a complicated signal accurately. The proposed ABFD inherits the idea of implementing Fourier spectrum decomposition through bandwidth optimization. In particular, three significant improvements are made in this work. Firstly, in order to reduce the computation complexity, a novel bandwidth optimization algorithm termed Fourier spectrum bandwidth optimization (FSBO) is proposed. Secondly, inspired by the empirical principle proposed in the empirical wavelet transform (EWT), a novel variable initialization method based on spectral energy distribution is introduced. Finally, under the guidance of narrowband characteristics, a method for automatically detecting the appropriate mode number is developed. In order to evaluate the performance of the proposed ABFD, simulation analysis and measured signal analysis are carried out in this paper. The preliminary results indicate that the proposed ABFD can extract the single components more accurately than EMD and VMD.

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Sampling and Super Resolution of Sparse Signals Beyond the Fourier Domain

Cited by 13 publications

References 81 publications

Sparsest Piecewise-Linear Regression of One-Dimensional Data

Sparsest Piecewise-Linear Regression of One-Dimensional Data

Adversarial Perturbations Prevail in the Y-Channel of the YCbCr Color Space

Adaptive Bandwidth Fourier Decomposition Method for Multi-Component Signal Processing

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