Why did the authors conduct a test on reconstructing complex-valued images?

Motivated by the fact that most of the MR images are complex-valued, the authors conducted a test on reconstructing complex-valued images.

What is the weight update for (2.2)?

(c) Weight update for (2.2): gij ← 0 ∀(i, j) ∈ Ek; gij ← 1 ∀(i, j) ∈ Ek. (d) k ← k + 1.In (2.2), μ and λ are two weight parameters that balance TV and wavelet sparsity.

What is the augmented Lagrangian of (4.1)?

The augmented Lagrangian of (4.1) is(4.2) L(u,w, z) := Jg(w) + μβ 2 ‖w −Du− dw‖22 +K(z) + λβ 2 ‖z − Φu− dz‖22 +H(u, b),where dw and dz are Lagrange multipliers of proper sizes.

How many steps did EdgeCS take to reconstruct an image?

One can see that even after 2000 iterations EdgeCS with Canny still failed to reconstruct an accurate image, while EdgeCS with either Canny∩LMI or the wavelet method succeeded after only 300 steps.

What are some other methods used to reduce false positives?

Besides the above standard steps, postprocesses such as nonmaximum suppression and smoothing are sometimes conducted to further reduce false positives due to noise.

What is the simplest way to minimize the u, w, and z?

The problems of minimizing L(u,w, z) with respect to each of u, w, and z while fixing the rest are called the u-, w-, and z-subproblems:u-subproblem: min uμβ2 ‖w −Du− dw‖22 +λβ2 ‖z − Φu− dz‖22 +H(u, b),(4.3)w-subproblem: min wJg(w) + μβ2 ‖w −Du− dw‖22,(4.4)z-subproblem: min zK(z) + λβ2 ‖z − Φu− dz‖22.(4.5)The ADMM applied to (4.1) or, equivalently, (2.3) is as follows.

What is the difference between Canny and LMI?

Since Canny returns thin boundaries and is better on the image background (see Figure 4(d)), intersecting Canny and LMI, written as Canny∩LMI, leads to more reliable edges (set E in EdgeCS), as shown in (f).

What is the definition of dyadic wavelet transform?

For dyadic wavelet transforms, the scale s is chosen as s = 2j , j = 1, 2, . . ..Lipschitz regularity is difficult to verify directly and thus not practically useful, but the following theorem relates local Lipschitz regularity with the dyadic wavelet transform.

Open AccessJournal Article10.1137/110837309

Edge Guided Reconstruction for Compressive Imaging

Weihong Guo, +1 more

- 03 Jul 2012

- Siam Journal on Imaging Sciences

- Vol. 5, Iss: 3, pp 809-834

TL;DR: The edge detector of EdgeCS is designed to faithfully return partial edges from intermediate image reconstructions even though these reconstructions may still have noise and artifacts.

Abstract: We propose EdgeCS---an edge guided compressive sensing reconstruction approach---to recover images of higher quality from fewer measurements than the current methods. Edges are important image features that are used in various ways in image recovery, analysis, and understanding. In compressive sensing, the sparsity of image edges has been successfully utilized to recover images. However, edge detectors have not been used on compressive sensing measurements to improve the edge recovery and subsequently the image recovery. This motivates us to propose EdgeCS, which alternatively performs edge detection and image reconstruction in a mutually beneficial way. The edge detector of EdgeCS is designed to faithfully return partial edges from intermediate image reconstructions even though these reconstructions may still have noise and artifacts. For complex-valued images, it incorporates joint sparsity between the real and imaginary components. EdgeCS has been implemented with both isotropic and anisotropic discret...

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Most frequently asked questions

1. What contributions have the authors mentioned in the paper "Edge guided reconstruction for compressive imaging" ?

The authors propose EdgeCS—an edge guided compressive sensing reconstruction approach—to recover images of higher quality from fewer measurements than the current methods.

2. Why is gij 0 assigned to pixels on edges?

The motivation of assigning zero TV weight (gij ← 0) to pixels on edges is to free them from the TV minimization and thus encourage the edges to form in the recovery.

3. What is the main contribution of EdgeCS?

Given undersampled and/or noisy measurements, EdgeCS reconstructs images with fewer errors and artifacts than the state-of-the-art methods.

4. What is the common method of removing noise?

Removing noise by, for example, convolving with a low-pass filter, e.g., Gaussian filterGσ = e−(x 2+y2)/2σ22πσ2 or a median filter.

Figure 7. Comparison of four reconstruction methods on measurements taken on 15, 8, and 7 radial lines (sample rates 6.44%, 3.98%, and 3.03%, respectively) with no noise added, and parameter μ = 10−10. Note: SNR 38dB of anisotropic EdgeCS at 7 lines can be significantly improved if more iterations are allowed.

Citations

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Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity, by Jean-Luc Starck, Fionn Murtagh, and Jalal M. Fadili

Mrityunjay Kumar

- 01 Jan 2007

- Journal of Electronic Imaging

TL;DR: This PDF file contains the editorial “Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity," by Jean-Luc Starck, Fionn Murtagh, and Jalal M. Fadili for JEI Vol.

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Ting Wang, +3 more

- 08 Sep 2017

- IEEE Transactions on Nuclear Science

TL;DR: This paper proposes a new iteratively reweighted anisotropic TV (ATV) method, in which a reweighting technique is incorporated into the idea of ATV, and is able to approximate the most direct measure of sparsity.

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- 22 Aug 2017

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TL;DR: The difference of convex functions algorithm (DCA) is applied to efficiently solve truncated $l_{1-2}$ minimization with guaranteed convergence and is validated on sparse vector recovery, matrix completion, and magnetic resonance imaging (MRI) reconstruction, with performance comparable to the state of the art.

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Yaghoub Rahimi, +3 more

- 19 Nov 2019

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References

Journal Article•10.1109/TPAMI.1986.4767851

A Computational Approach to Edge Detection

John Canny

- 01 Jun 1986

- IEEE Transactions on Pattern Analysis an...

TL;DR: There is a natural uncertainty principle between detection and localization performance, which are the two main goals, and with this principle a single operator shape is derived which is optimal at any scale.

...read moreread less

29.9K

•Journal Article•10.1109/TIT.2005.862083

Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information

Emmanuel J. Candès, +2 more

- 01 Feb 2006

- IEEE Transactions on Information Theory

TL;DR: In this paper, the authors considered the model problem of reconstructing an object from incomplete frequency samples and showed that with probability at least 1-O(N/sup -M/), f can be reconstructed exactly as the solution to the lscr/sub 1/ minimization problem.

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Journal Article•10.1016/0167-2789(92)90242-F

Nonlinear total variation based noise removal algorithms

Leonid I. Rudin, +2 more

- 01 Nov 1992

- Physica D: Nonlinear Phenomena

TL;DR: In this article, a constrained optimization type of numerical algorithm for removing noise from images is presented, where the total variation of the image is minimized subject to constraints involving the statistics of the noise.

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•Journal Article•10.1109/83.902291

Active contours without edges

Tony F. Chan, +1 more

- 01 Feb 2001

- IEEE Transactions on Image Processing

TL;DR: A new model for active contours to detect objects in a given image, based on techniques of curve evolution, Mumford-Shah (1989) functional for segmentation and level sets is proposed, which can detect objects whose boundaries are not necessarily defined by the gradient.

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•Book

Image Analysis and Mathematical Morphology

Jean Serra

- 11 Feb 1984

TL;DR: This invaluable reference helps readers assess and simplify problems and their essential requirements and complexities, giving them all the necessary data and methodology to master current theoretical developments and applications, as well as create new ones.

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10.1K

Edge Guided Reconstruction for Compressive Imaging

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AI Agents for this Paper

Most frequently asked questions

1. What contributions have the authors mentioned in the paper "Edge guided reconstruction for compressive imaging" ?

2. Why is gij 0 assigned to pixels on edges?

3. What is the main contribution of EdgeCS?

4. What is the common method of removing noise?

5. Why did the authors conduct a test on reconstructing complex-valued images?

6. What is the weight update for (2.2)?

7. What is the augmented Lagrangian of (4.1)?

8. How many steps did EdgeCS take to reconstruct an image?

9. What are some other methods used to reduce false positives?

10. What is the simplest way to minimize the u, w, and z?

11. What is the difference between Canny and LMI?

12. What is the definition of dyadic wavelet transform?

Figures

Citations

Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity, by Jean-Luc Starck, Fionn Murtagh, and Jalal M. Fadili

Reweighted Anisotropic Total Variation Minimization for Limited-Angle CT Reconstruction

Truncated $l_{1-2}$ Models for Sparse Recovery and Rank Minimization

A Scale-Invariant Approach for Sparse Signal Recovery

Nonlocal Gradient Sparsity Regularization for Image Restoration

References

A Computational Approach to Edge Detection

Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information

Nonlinear total variation based noise removal algorithms

Active contours without edges

Image Analysis and Mathematical Morphology

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