#### DMCA

## Wavelet frame based image restoration with missing/damaged pixels

Venue: | East Asia Journal on Applied Mathematics |

Citations: | 7 - 3 self |

### Citations

2600 |
Ten lectures on wavelets
- Daubechies
- 1992
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Citation Context ....g. Fourier or windowed Fourier transforms, local cosine transforms, wavelet or framelet transforms, or discrete gradient operators. In particular, the sparsity prior of images in tight frame system (=-=[24, 37]-=-) has been successfully used in many image restoration tasks, such as image inpainting ([11, 12, 16]); non-blind image deblurring ([15, 16]) and blind motion deblurring ([13, 14]). All these sparsity-... |

2267 | Nonlinear total variation based noise removal algorithms
- Rudin, Osher, et al.
- 1992
(Show Context)
Citation Context ...inferred from partial degraded information of u. There have been many image priors proposed in the past, e.g. Tikhonov functional based smoothness prior for images ([41]); total variation functional (=-=[20, 38]-=-) or Mumford-Shah functional ([33]) based piece-wise smoothness prior for cartoon images; and the exemplar-based local patch redundancy prior of nature images ([23]). In recent years, sparsity-based p... |

1361 |
Solutions of Ill-posed Problems
- Tikhonov, Arsenin
- 1977
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Citation Context ...images such that all pixels can be inferred from partial degraded information of u. There have been many image priors proposed in the past, e.g. Tikhonov functional based smoothness prior for images (=-=[41]-=-); total variation functional ([20, 38]) or Mumford-Shah functional ([33]) based piece-wise smoothness prior for cartoon images; and the exemplar-based local patch redundancy prior of nature images ([... |

1292 | Optimal Approximations by Piecewise Smooth Functions and Associated Variational Problems
- Mumford, Shah
- 1989
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Citation Context ...tion of u. There have been many image priors proposed in the past, e.g. Tikhonov functional based smoothness prior for images ([41]); total variation functional ([20, 38]) or Mumford-Shah functional (=-=[33]-=-) based piece-wise smoothness prior for cartoon images; and the exemplar-based local patch redundancy prior of nature images ([23]). In recent years, sparsity-based priors of images in certain domains... |

1056 | A fast iterative shrinkage-thresholding algorithm for linear inverse problems
- Beck, Teboulle
- 2009
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Citation Context ... 2Ghz intel CPU. As the comparison, the algorithm proposed in [11] is quite slow and requires much more computational times. We note that the recent accelerated proximal gradient (APG) algorithm (see =-=[3, 40]-=-) tremendously accelerates the balanced approach in [11] and take less time to obtain a satisfactory solution in the same hardware configuration. Hence, we also employ the APG algorithm for computatio... |

367 |
The split bregman method for l1-regularized problems
- Goldstein, Osher
(Show Context)
Citation Context ...ch to recover degraded images by enforcing the analysis-based sparsity prior of images in tight frame domain. The resulting minimization problem can be efficiently solved by the split Bregman method (=-=[28]-=-). Numerical experiments on various image restoration tasks: simultaneously image deblurring and inpainting, super-resolution imaging and image deblurring under impulsive noise, demonstrated the effec... |

365 | Region filling and object removal by exemplar-based image inpainting.
- Criminisi, Perez, et al.
- 2004
(Show Context)
Citation Context ...]); total variation functional ([20, 38]) or Mumford-Shah functional ([33]) based piece-wise smoothness prior for cartoon images; and the exemplar-based local patch redundancy prior of nature images (=-=[23]-=-). In recent years, sparsity-based priors of images in certain domains have been widely used in many image restoration tasks, which is based on the observation that images usually have sparse represen... |

201 | Framelets: MRA-based constructions of wavelet frames
- Daubechies, Han, et al.
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Citation Context ...t algorithms for both framelet decomposition and reconstruction by passing the images/coefficients through a series of discrete framelet filters with a small memory footprint. Readers are referred to =-=[17, 25]-=- for more implementation details. 2.1. Minimization formulation for image restoration with missing pixels. In the image degradation model (1.2), the availability of the index set Λ of remained image p... |

195 | An iterative regularization method for total variation-based image restoration,”Multiscale Modeling
- Osher, Burger, et al.
- 2005
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Citation Context ...t Bregman iteration first introduced by [28] showed its efficiency for solving a class of `1 related minimization problems. The Bregman iteration was first introduced to image processing community in =-=[36]-=- and was then applied to a variety of signal and image processing problems. The split Bregman iteration extends utility of the Bregman iteration for a more general class of `1 norm related minimizatio... |

183 | An accelerated proximal gradient algorithm for nuclear norm regular- ized least squares problems
- Toh, Yun
- 2009
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Citation Context ...let tight frame domain by `1-norm minimization tends to sharpen image edges. The minimization (2.1) is actually one extreme case of a more general sparsity-based minimization formulation discussed in =-=[40]-=-: min v 1 2 ‖GWT v − g‖22 + β 2 ‖(I −WWT )v‖22 + λ‖v‖1, (2.2) where G is some matrix and β ≥ 0 is a positive parameter, and v is the frame coefficient vector such that the solution u = WT v. If we set... |

161 |
Adaptive median filters: new algorithms and results
- Hwang, Haddad
- 1995
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Citation Context ...ods proposed in [8, 9] is dependent on the accuracy of detecting damaged pixels. We adopt the same median filter techniques used in 14 [8, 9] to locate damaged pixels: adaptive median filter (AMF) in =-=[30]-=- is used to detect salt-andpepper noise and adaptive center-weighted median filter (ACWMF) in [22] is used to detect random valued impulsive noise. Interested readers are referred to [22, 30] for more... |

137 |
A variational approach to remove outliers and impulse noise
- Nikolova
- 2004
(Show Context)
Citation Context ...some other applications such as image deblurring in the presence of impulsive noise. In such a case, the `1-norm based distance function is more suitable as it is known to be robust to outliers (e.g. =-=[35]-=-). Thus, both `2-norm and `1-norm based distance functions are discussed in this paper, and the choice of p is application dependent. See Section 3 for more details. The regularization term in (2.1) e... |

120 |
Handbook of Image and Video Processing
- Bovik
- 2000
(Show Context)
Citation Context ...ise. Gaussian noise caused by thermal noise is prevalent in imaging systems and impulsive noise is caused by dead pixels, analog-to-digital converter errors, bit errors in transimisson, etc (see e.g. =-=[7]-=-). Such an image degradation can be modeled as follows: f = Np(Hu+ ), (3.4) where f and u are the observed and the latent image respectively, H is the matrix of blurring, is the Gaussian noise and ... |

105 | Minimization of cost-functions involving nonsmooth data-fidelity terms. application to the processing of outliers.
- Nikolova
- 2002
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Citation Context ...er is very accurate and f |Λ is nearly free of error. In such a case, we choose p = 1 for that it allows the exact data fitting, i.e. (Hu)ij = fij if fij is not corrupted, while p = 2 can not do this =-=[34, 35]-=-. B. If there is the mixture of both salt-and-pepper noise and Gaussian noise in the image, the AMF filter still can accurately locate pixels damaged by salt-and-pepper noise. Thus f |Λ contains only ... |

104 | Split Bregman methods and frame based image restoration, Multiscale Modeling and Simulation: A
- Cai, Osher, et al.
(Show Context)
Citation Context ...rms, or discrete gradient operators. In particular, the sparsity prior of images in tight frame system ([24, 37]) has been successfully used in many image restoration tasks, such as image inpainting (=-=[11, 12, 16]-=-); non-blind image deblurring ([15, 16]) and blind motion deblurring ([13, 14]). All these sparsity-based research works motivate us to investigate the application of the sparsity prior of images in t... |

92 | Affine systems in L2(Rd): the analysis of the analysis operators
- Ron, Shen
- 1997
(Show Context)
Citation Context ....g. Fourier or windowed Fourier transforms, local cosine transforms, wavelet or framelet transforms, or discrete gradient operators. In particular, the sparsity prior of images in tight frame system (=-=[24, 37]-=-) has been successfully used in many image restoration tasks, such as image inpainting ([11, 12, 16]); non-blind image deblurring ([15, 16]) and blind motion deblurring ([13, 14]). All these sparsity-... |

87 | A framelet-based image inpainting algorithm - Cai, Chan, et al. |

86 |
Applications of Lagrangian-based alternating direction methods and connections to split Bregman
- Esser
- 2009
(Show Context)
Citation Context ...ation tasks as it converges fast, uses a small memory footprint and is easy to implement. The connection of split Bregman iteration between some existing algorithms in optimization are pointed out by =-=[26, 39]-=-. In fact, the split Bregman algorithm is equivalent to the alternating direction method of multipliers and the Douglas-Rachford splitting algorithm for the dual problem. We use the split Bregman iter... |

83 |
Tri-state median filter for image denoising”,
- Chen, Ma, et al.
- 1999
(Show Context)
Citation Context ...iable pixels. The detection of image pixels corrupted by impulsive noise has bee extensively studied in the application of removing impulsive noise from images. Median filtering based technique (e.g. =-=[21, 22, 32]-=-) is the dominant one among available techniques for detecting pixes with impulsive noise. As the damaged pixels contain little information of the original image content, a reasonable approach is to r... |

60 |
Image Processing and Analysis: variational, PDE, wavelet, and stochastic methods
- Chan, Shen
- 2005
(Show Context)
Citation Context ...inferred from partial degraded information of u. There have been many image priors proposed in the past, e.g. Tikhonov functional based smoothness prior for images ([41]); total variation functional (=-=[20, 38]-=-) or Mumford-Shah functional ([33]) based piece-wise smoothness prior for cartoon images; and the exemplar-based local patch redundancy prior of nature images ([23]). In recent years, sparsity-based p... |

58 | Geometric applications of the split bregman method: Segmentation and surface reconstruction
- Goldstein, Bresson, et al.
- 2009
(Show Context)
Citation Context ...man iteration (See [28]) has been developed to solve this type of `1-norm related minimization problems and demonstrated great performance and efficiency in many applications in imaging science (e.g. =-=[16, 27, 27]-=-). In this paper, we proposed a modified version of the split Bregman iteration as the numerical solver to (1.3). The rest of this paper is organized as follows. Section 2 is devoted to the formulatio... |

55 | Wavelet algorithms for high-resolution image reconstruction
- Chan, Chan, et al.
- 2003
(Show Context)
Citation Context ...degradations. The balanced sparsity prior in tight frame domain is used to solve the problem of image inpainting ([12]) with H = I being an identity matrix, and the problem of image super-resolution (=-=[18]-=-) with PΛ being a sub-sampling operator and H being a convolution operator with a particular low-pass filter. In [15], the synthesis based sparsity prior is proposed to deblur images with full pixels ... |

53 |
Tight frame: an efficient way for high-resolution image reconstruction
- Chan, Riemenschneider, et al.
- 2004
(Show Context)
Citation Context ...with M = Km,N = Kn. The relationship between each individual low-resolution frame and the underlying high-resolution image can be modeled by a blurring process followed by a uniform sub-sampling (see =-=[6, 18, 19]-=- for more details): fpq = PΛpqHu+ pq, p, q = 1, 2, . . . ,K, where the blurring matrix H is associated with a specific blurring kernel modeled by the tensor product of the following 1D filter h = [ 1... |

50 | An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise,”
- Wang, Yang, et al.
- 2008
(Show Context)
Citation Context ...e linear system can be efficiently in the Fourier domain by using periodic boundary condition, since the matrices HTH and WTW are circulant and can be diagonalized by fast Fourier transform (see e.g. =-=[42]-=-). However, such an approach does not work any more when there is missing data, as the circulant structure is lost even using periodic boundary extension. In our approach, the symmetric boundary exten... |

48 |
Split Bregman algorithm, Douglas-Rachford splitting and frame shrinkage
- Setzer
- 2009
(Show Context)
Citation Context ...ation tasks as it converges fast, uses a small memory footprint and is easy to implement. The connection of split Bregman iteration between some existing algorithms in optimization are pointed out by =-=[26, 39]-=-. In fact, the split Bregman algorithm is equivalent to the alternating direction method of multipliers and the Douglas-Rachford splitting algorithm for the dual problem. We use the split Bregman iter... |

42 |
High-resolution image reconstruction with multisensors
- Bose, Boo
(Show Context)
Citation Context ...solution images are captured by an array of image sensors with arranged image displacements among images. In this section, we adopt the model of the array of image sensors proposed by Bose and Boo in =-=[6]-=-. Consider a K×K sensor array in which each sensor (p, q) produce an image of resolution m× n, denoted by fpq for p, q = 1, 2, . . . ,K. The goal of super-resolution imaging is to reconstruct a high-r... |

42 | Deconvolution: A wavelet frame approach - Chai, Shen |

39 | Image deblurring in the presence of saltand-pepper noise
- Bar, Sochen, et al.
(Show Context)
Citation Context ... the content of pixels contaminated by impulsive noise contains little information of the original image content. These damaged pixels are viewed as outliers and there have been a few research works (=-=[1, 2]-=-) on developing image deblurring algorithms robust to outliers. However, when images are corrupted by significant impulsive noise, the amount of outliers is simply too many to be handled well by these... |

34 | Linearized bregman iterations for frame-based image deblurring
- Cai, Osher, et al.
(Show Context)
Citation Context ...ticular, the sparsity prior of images in tight frame system ([24, 37]) has been successfully used in many image restoration tasks, such as image inpainting ([11, 12, 16]); non-blind image deblurring (=-=[15, 16]-=-) and blind motion deblurring ([13, 14]). All these sparsity-based research works motivate us to investigate the application of the sparsity prior of images in tight frame domain on solving (1.2), ima... |

30 | Restoration of chopped and nodded images by framelets - Cai, Chan, et al. |

28 |
motion deblurring from a single image using sparse approximation
- Blind
- 2009
(Show Context)
Citation Context ...n tight frame system ([24, 37]) has been successfully used in many image restoration tasks, such as image inpainting ([11, 12, 16]); non-blind image deblurring ([15, 16]) and blind motion deblurring (=-=[13, 14]-=-). All these sparsity-based research works motivate us to investigate the application of the sparsity prior of images in tight frame domain on solving (1.2), image restoration with missing/damaged pix... |

27 | Dual wavelet frames and Riesz bases in Sobolev spaces,
- Han, Shen
- 2009
(Show Context)
Citation Context ...ose to zero). Moreover, it is known that the weighted norm of the canonical frame coefficient vector of a function is equivalent to its function norm in some spaces, e.g. Sobolev or Besov spaces (see =-=[5, 29]-=- for more details). In particular, it is shown in [5] that the `1-norm of the canonical tight frame coefficient vector of a function, i.e. the regularization term in (2.1), is equivalent to its Besov ... |

25 |
Bi-framelet systems with few vanishing moments characterize Besov spaces,
- Borup, Gribonval, et al.
- 2004
(Show Context)
Citation Context ...ose to zero). Moreover, it is known that the weighted norm of the canonical frame coefficient vector of a function is equivalent to its function norm in some spaces, e.g. Sobolev or Besov spaces (see =-=[5, 29]-=- for more details). In particular, it is shown in [5] that the `1-norm of the canonical tight frame coefficient vector of a function, i.e. the regularization term in (2.1), is equivalent to its Besov ... |

15 | Simultaneously inpainting in image and transformed domains, Num
- Cai, Chan, et al.
(Show Context)
Citation Context ...rms, or discrete gradient operators. In particular, the sparsity prior of images in tight frame system ([24, 37]) has been successfully used in many image restoration tasks, such as image inpainting (=-=[11, 12, 16]-=-); non-blind image deblurring ([15, 16]) and blind motion deblurring ([13, 14]). All these sparsity-based research works motivate us to investigate the application of the sparsity prior of images in t... |

13 |
Fast two-phase image deblurring under impulse noise
- Cai, Nikolova, et al.
(Show Context)
Citation Context ...t the noise level s = 70% are shown in 3.10. The PSNR values of the results from our algorithm under different noise level are given in the Table 3.2 and compared against that from the two methods in =-=[8, 9]-=-. It is seen from Table 3.2 that our results achieved higher PSNR values with about 2 − 3 dB gain over the one in [8] and about 1dB gain over the one in [9] when the noise level is lower than 50%. The... |

12 | Two-phase approach for deblurring images corrupted by impulse plus Gaussian noise
- Cai, Chan, et al.
- 2008
(Show Context)
Citation Context ...wever, when images are corrupted by significant impulsive noise, the amount of outliers is simply too many to be handled well by these robust methods. Recently, two-phase based approaches proposed in =-=[8, 9]-=- demonstrated better performance on deblurring images in the presence of impulsive noise. The 11 Fig. 3.6. Super-resolution image reconstruction on “Goldhill”. Columns represent (from left to right) t... |

12 |
weighted median filters and their applications to image enhancement. A new Decision Based Median Filter using Cloud Model for the removal of high density Salt and Pepper noise 53
- Center
(Show Context)
Citation Context ...iable pixels. The detection of image pixels corrupted by impulsive noise has bee extensively studied in the application of removing impulsive noise from images. Median filtering based technique (e.g. =-=[21, 22, 32]-=-) is the dominant one among available techniques for detecting pixes with impulsive noise. As the damaged pixels contain little information of the original image content, a reasonable approach is to r... |

7 |
motion deblurring using multiple images
- Blind
(Show Context)
Citation Context ...n tight frame system ([24, 37]) has been successfully used in many image restoration tasks, such as image inpainting ([11, 12, 16]); non-blind image deblurring ([15, 16]) and blind motion deblurring (=-=[13, 14]-=-). All these sparsity-based research works motivate us to investigate the application of the sparsity prior of images in tight frame domain on solving (1.2), image restoration with missing/damaged pix... |

4 |
Adaptive impulse dectection using center-weighted median filters
- Chen, Wu
(Show Context)
Citation Context ...iable pixels. The detection of image pixels corrupted by impulsive noise has bee extensively studied in the application of removing impulsive noise from images. Median filtering based technique (e.g. =-=[21, 22, 32]-=-) is the dominant one among available techniques for detecting pixes with impulsive noise. As the damaged pixels contain little information of the original image content, a reasonable approach is to r... |

2 |
Caselles, C.: Image inpainting
- Bertalmio, Sapiro, et al.
- 2000
(Show Context)
Citation Context ...g is then to fill in (or interpolate) the missing/damaged pixels based on available information of the image. The problem of image inpainting has been extensively studied since the pioneering work by =-=[4]-=-. Interesting readers are referred to [20] for a thorough review on this topic. In this section, we address a more complicated image degradation problem: image are blurred and pixels are missing in so... |

2 | Wavelet frame based scene reconstruction from range data
- Ji, Shen, et al.
(Show Context)
Citation Context ...ly outliers in the detected reliable pixels and thus `1-norm is used as the distance function in the fidelity term of (3.5). `2-norm is not a good choice in the presence of outliers as it is shown in =-=[31]-=- that the results can be either over-smoothed by using a large 16 Fig. 3.10. Restoration results for the corrupted images with the out-of-focus kernal of radius 3 and 70% saltand-pepper noise. The PSN... |

1 |
Nir Sochen. Image deblurring in the presence of impulse noise
- Bar, Kiryati
(Show Context)
Citation Context ... the content of pixels contaminated by impulsive noise contains little information of the original image content. These damaged pixels are viewed as outliers and there have been a few research works (=-=[1, 2]-=-) on developing image deblurring algorithms robust to outliers. However, when images are corrupted by significant impulsive noise, the amount of outliers is simply too many to be handled well by these... |