## Sparse representation for color image restoration (2007)

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Venue: | the IEEE Trans. on Image Processing |

Citations: | 106 - 27 self |

### BibTeX

@INPROCEEDINGS{Mairal07sparserepresentation,

author = {Julien Mairal and Julien Mairal and Michael Elad and Michael Elad and Guillermo Sapiro and Guillermo Sapiro},

title = {Sparse representation for color image restoration},

booktitle = {the IEEE Trans. on Image Processing},

year = {2007},

pages = {53--69},

publisher = {ITIP}

}

### OpenURL

### Abstract

Sparse representations of signals have drawn considerable interest in recent years. The assumption that natural signals, such as images, admit a sparse decomposition over a redundant dictionary leads to efficient algorithms for handling such sources of data. In particular, the design of well adapted dictionaries for images has been a major challenge. The K-SVD has been recently proposed for this task [1], and shown to perform very well for various gray-scale image processing tasks. In this paper we address the problem of learning dictionaries for color images and extend the K-SVD-based gray-scale image denoising algorithm that appears in [2]. This work puts forward ways for handling non-homogeneous noise and missing information, paving the way to state-of-the-art results in applications such as color image denoising, demosaicing, and inpainting, as demonstrated in this paper. EDICS Category: COL-COLR (Color processing) I.

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Citation Context ...d ones as described above, that leads to sparse representations on training signals drawn from Γ. This algorithm uses either Orthogonal Matching Pursuit (OMP) [20], [21], [22], or Basis Pursuit (BP), =-=[23]-=-, as part of its iterative procedure for learning the dictionary. The follow-up work reported in [3], [2] proposes a novel and highly effective image denoising algorithm for the removal of additive wh... |

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Citation Context ...the case where Γ is the set of natural images, dictionaries such as wavelets of various sorts [4], curvelets [5], [6], contourlets [7], [8], wedgelets [9], bandlets [10], [11], and steerable wavelets =-=[12]-=-, [13], are all attempts to design dictionaries that fulfill the above model assumption. Indeed, these various transforms have led to highly effective algorithms in many applications in image processi... |

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Citation Context ...noising as described in [3], [2]. The alternative path, a non-parametric learning, use image examples directly within the reconstruction process, as practiced by Efros and Leung for texture synthesis =-=[30]-=-; by Freeman et. al. for super-resolution [31], [32]; and by several follow-up works [33], [34], [35], [36] for super-resolution and inpainting. Interestingly, most of the above direct methods avoid t... |

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Citation Context ...ained with the global approach and the adaptive one permits to see the improvements in the learning process. We chose to evaluate our algorithm on some images from the Berkeley Segmentation database, =-=[51]-=-, presented in Figure 6. This data selection allows us to compare our results with the relevant work on color image denoising reported in [25], which as mentioned before, is an extension of [26]. As t... |

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Citation Context ... previously selected atoms. This orthogonalization is important since it gives more stability and a faster convergence for this greedy algorithm. For details, the reader should refer to [21]–[23] and =-=[48]-=-. An additional, more formal way to explain the lack of colors and the color bias in the reconstruction is to note that the OMP does not guarantee that the reconstructed patch will maintain the averag... |

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Citation Context ...se where Γ is the set of natural images, dictionaries such as wavelets of various sorts [4], curvelets [5], [6], contourlets [7], [8], wedgelets [9], bandlets [10], [11], and steerable wavelets [12], =-=[13]-=-, are all attempts to design dictionaries that fulfill the above model assumption. Indeed, these various transforms have led to highly effective algorithms in many applications in image processing, su... |

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Citation Context ... 3, 6 × 6 × 3, 7 × 7 × 3 and 8 × 8 × 3, on 200000 patches taken from a database of 15000 images with the patch-sparsity parameter L = 6 (6 atoms in the representations). We used the database LabelMe, =-=[50]-=-, to build our image database. Then we 17strained each dictionary with 600 iterations. This provided us a set of generic dictionaries that we used as initial dictionaries in our denoising algorithm. C... |

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Citation Context ...fficient algorithms for handling such sources of data. In particular, the design of well adapted dictionaries for images has been a major challenge. The K-SVD has been recently proposed for this task =-=[1]-=-, and shown to perform very well for various gray-scale image processing tasks. In this paper we address the problem of learning dictionaries for color images and extend the K-SVD-based grayscale imag... |

373 | Image inpainting
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Citation Context ... as a long concatenated RGB vector, fail respectively in one of these two challenges. Moreover, for classical color image processing applications such as demosaicing, denoising, and image inpainting, =-=[43]-=-, spatial and/or spectral non-uniform noise has to be handled. These challenges are addressed next. IV. SPARSE COLOR IMAGE REPRESENTATION We now turn to detail the proposed fundamental extensions to t... |

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Citation Context ... fulfill the above model assumption. Indeed, these various transforms have led to highly effective algorithms in many applications in image processing, such as compression [14], denoising [15], [16], =-=[17]-=-, [18], inpainting [19], and more. Common to all these pre-defined dictionaries is their analytical nature, and their reliance on the geometrical nature of natural images, especially piece-wise smooth... |

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Citation Context ...tead of exploiting pre-defined ones as described above, that leads to sparse representations on training signals drawn from Γ. This algorithm uses either Orthogonal Matching Pursuit (OMP) [20], [21], =-=[22]-=-, or Basis Pursuit (BP), [23], as part of its iterative procedure for learning the dictionary. The follow-up work reported in [3], [2] proposes a novel and highly effective image denoising algorithm f... |

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Citation Context ...4] and later in [25], [26]; the wavelet based image prior as appears in [27]; the Tikhonov regularization proposed by Haber and Tenorio [28]; the super-resolution approach adopted by Baker and Kanade =-=[29]-=-; and the recent K-SVD denoising as described in [3], [2]. The alternative path, a non-parametric learning, use image examples directly within the reconstruction process, as practiced by Efros and Leu... |

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Citation Context ...ll the above model assumption. Indeed, these various transforms have led to highly effective algorithms in many applications in image processing, such as compression [14], denoising [15], [16], [17], =-=[18]-=-, inpainting [19], and more. Common to all these pre-defined dictionaries is their analytical nature, and their reliance on the geometrical nature of natural images, especially piece-wise smooth ones.... |

267 |
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Citation Context ...appears in [30]; the Tikhonov regularization proposed by Haber and Tenorio [31]; the super-resolution approach adopted by Baker and Kanade [32]; and the recent K-SVD denoising as described in [2] and =-=[3]-=-. The alternative path, a nonparametric learning, uses image examples directly within the reconstruction process, as practiced by Efros and Leung for texture synthesis [33], by Freeman et al. for supe... |

263 | A review of image denoising algorithms, with a new one, Multiscale Modeling and Simulation 4 (2
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Citation Context ...versal denoiser of images, which learns the posterior from the given image in a way inspired by the Lempel-Ziv universal compression algorithm. Another path of such works is the Non-Local-Means [38], =-=[39]-=- and related works [40], [41]. Interestingly, the work in [2] belongs to this family as well, as the dictionary can be based on the noisy image itself. Most of the above methods deploy processing of s... |

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Citation Context ...ive path, a non-parametric learning, use image examples directly within the reconstruction process, as practiced by Efros and Leung for texture synthesis [30]; by Freeman et. al. for super-resolution =-=[31]-=-, [32]; and by several follow-up works [33], [34], [35], [36] for super-resolution and inpainting. Interestingly, most of the above direct methods avoid the prior and target instead the posterior dens... |

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Citation Context ...hows the behavior of our algorithm when removing data from the castle image. It is presented on Figure 10. The second example, Figure 11, is a classical example of text removal, [43], and was used in =-=[26]-=- in order to evaluate their model compared to the pioneer work from [43]. In [26], the Field of Experts model achieves 32.23 dB using their algorithm on the YCbCr space and 32.39 dB on the RGB space. ... |

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Citation Context ... a universal denoiser of images, which learns the posterior from the given image in a way inspired by the Lempel-Ziv universal compression algorithm. Another path of such works is the Non-Local-Means =-=[38]-=-, [39] and related works [40], [41]. Interestingly, the work in [2] belongs to this family as well, as the dictionary can be based on the noisy image itself. Most of the above methods deploy processin... |

194 | Image compression via joint statistical characterization in the wavelet domain
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Citation Context ...directs the learning process to tune the prior parameters 3sbased on examples. Such is the case with the MRF prior learned in [24] and later in [25], [26]; the wavelet based image prior as appears in =-=[27]-=-; the Tikhonov regularization proposed by Haber and Tenorio [28]; the super-resolution approach adopted by Baker and Kanade [29]; and the recent K-SVD denoising as described in [3], [2]. The alternati... |

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Citation Context ...age examples directly within the reconstruction process, as practiced by Efros and Leung for texture synthesis [30]; by Freeman et. al. for super-resolution [31], [32]; and by several follow-up works =-=[33]-=-, [34], [35], [36] for super-resolution and inpainting. Interestingly, most of the above direct methods avoid the prior and target instead the posterior density, from which reconstruction is easily ob... |

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Citation Context ... The parametric path suggests an analytical expression for the prior, and directs the learning process to tune the prior parameters 3sbased on examples. Such is the case with the MRF prior learned in =-=[24]-=- and later in [25], [26]; the wavelet based image prior as appears in [27]; the Tikhonov regularization proposed by Haber and Tenorio [28]; the super-resolution approach adopted by Baker and Kanade [2... |

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Citation Context ... this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIP.2007.911828 1057-7149/$25.00 © 2007 IEEE [6], contourlets [7], [8], wedgelets [9], bandlets [10], =-=[11]-=-, and steerable wavelets [12], [13], are all attempts to design dictionaries that fulfill the above model assumption. Indeed, these various transforms have led to highly effective algorithms in many a... |

141 |
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Citation Context ...y, instead of exploiting pre-defined ones as described above, that leads to sparse representations on training signals drawn from Γ. This algorithm uses either Orthogonal Matching Pursuit (OMP) [20], =-=[21]-=-, [22], or Basis Pursuit (BP), [23], as part of its iterative procedure for learning the dictionary. The follow-up work reported in [3], [2] proposes a novel and highly effective image denoising algor... |

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Citation Context ...etails, one can still use it for filling small holes, as long as their sizes are smaller than the size of the atoms. For larger holes, iterative and/or multiscale or texture synthesis methods like in =-=[45]-=- are needed. The idea for extending the previously described work for inpainting is quite simple. If one considers holes as areas with infinite power noise, this leads to some βij coefficients equal t... |

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Citation Context ...NG ALGORITHMS ON THE KODAK DATA SET.SOME RESULTS ARE TAKEN FROM THE PAPER [51]. BI REFERS TO A SIMPLE BILINEAR INTERPOLATION, THEN, K, AP, OR, SA, CC REFER TO THE ALGORITHMS, RESPECTIVELY, FROM [51], =-=[52]-=-, [53], [54], AND [59]. D REFERS TO THE RESULTS OBTAINED WITH A GLOBALLY TRAINED DICTIONARY (600 ITERATIONS WITH v aIHON 200 000 DIFFERENT 6 2 62 3 PATCHES). THEN, DLREFER TO THE RESULT OBTAINED WITH ... |

126 |
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Citation Context ...y D is redundant in describing x. If we consider the case where Γ is the set of natural images, dictionaries such as wavelets of various sorts [4], curvelets [5], [6], contourlets [7], [8], wedgelets =-=[9]-=-, bandlets [10], [11], and steerable wavelets [12], [13], are all attempts to design dictionaries that fulfill the above model assumption. Indeed, these various transforms have led to highly effective... |

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Citation Context ...e is the same as El and where each column corresponding to an index [i, j] is Rijβ. This problem is known as a weighted one-rank approximation matrix, is not simple and has not an unique solution. In =-=[44]-=-, Srebro and Jaakkola put forward a simple iterative algorithm which gives an approximated solution of a local minimum. Nevertheless, this algorithm requires a SVD for each of its iterations, being re... |

123 |
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Citation Context ...l assumption. Indeed, these various transforms have led to highly effective algorithms in many applications in image processing, such as compression [14], denoising [15], [16], [17], [18], inpainting =-=[19]-=-, and more. Common to all these pre-defined dictionaries is their analytical nature, and their reliance on the geometrical nature of natural images, especially piece-wise smooth ones. In [1], the auth... |

91 | Demosaicing: image reconstruction from color ccd samples
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Citation Context ...tern, GRGRGR. . . on odd lines and BGBGBG. . . on even ones. Several algorithms have been developed in recent years to produce high quality full color images from the mosaic sensor, e.g., [46], [47], =-=[48]-=-, [49]. Although color demosaicing is becoming less relevant with the on-going development of sensor and camera technology, addressing it remains a challenging task that helps to test the effectivenes... |

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Citation Context ...e a separate corpus of training images for learning the prior (or its parameters). The alternative option is to use examples from the corrupted image itself. This surprising idea has been proposed in =-=[37]-=- as a universal denoiser of images, which learns the posterior from the given image in a way inspired by the Lempel-Ziv universal compression algorithm. Another path of such works is the Non-Local-Mea... |

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Citation Context ...tionary, instead of exploiting pre-defined ones as described above, that leads to sparse representations on training signals drawn from Γ. This algorithm uses either Orthogonal Matching Pursuit (OMP) =-=[20]-=-, [21], [22], or Basis Pursuit (BP), [23], as part of its iterative procedure for learning the dictionary. The follow-up work reported in [3], [2] proposes a novel and highly effective image denoising... |

68 | Optimal Spatial Adaptation for Patch-Based Image Denoising
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Citation Context ...es, which learns the posterior from the given image in a way inspired by the Lempel-Ziv universal compression algorithm. Another path of such works is the Non-Local-Means [38], [39] and related works =-=[40]-=-, [41]. Interestingly, the work in [2] belongs to this family as well, as the dictionary can be based on the noisy image itself. Most of the above methods deploy processing of small image patches, a t... |

62 | Fast image and video denoising via nonlocal means of similar neighborhoods
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Citation Context ...ich learns the posterior from the given image in a way inspired by the Lempel-Ziv universal compression algorithm. Another path of such works is the Non-Local-Means [38], [39] and related works [40], =-=[41]-=-. Interestingly, the work in [2] belongs to this family as well, as the dictionary can be based on the noisy image itself. Most of the above methods deploy processing of small image patches, a theme t... |

61 | An overview of JPEG-2000
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Citation Context ... to design dictionaries that fulfill the above model assumption. Indeed, these various transforms have led to highly effective algorithms in many applications in image processing, such as compression =-=[14]-=-, denoising [15]–[18], inpainting [19], and more. Common to all these predefined dictionaries is their analytical nature, and their reliance on the geometrical nature of natural images, especially pie... |

50 | Framing pyramids
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Citation Context ...g that the dictionary D is redundant in describing x. If we consider the case where Γ is the set of natural images, dictionaries such as wavelets of various sorts [4], curvelets [5], [6], contourlets =-=[7]-=-, [8], wedgelets [9], bandlets [10], [11], and steerable wavelets [12], [13], are all attempts to design dictionaries that fulfill the above model assumption. Indeed, these various transforms have led... |

49 | Recovering edges in ill-posed inverse problems optimality of curvelet frames
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Citation Context ...me that k > n, implying that the dictionary D is redundant in describing x. If we consider the case where Γ is the set of natural images, dictionaries such as wavelets of various sorts [4], curvelets =-=[5]-=-, [6], contourlets [7], [8], wedgelets [9], bandlets [10], [11], and steerable wavelets [12], [13], are all attempts to design dictionaries that fulfill the above model assumption. Indeed, these vario... |

47 | Image denoising via learned dictionaries and sparse representation
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(Show Context)
Citation Context ...arious gray-scale image processing tasks. In this paper we address the problem of learning dictionaries for color images and extend the K-SVD-based grayscale image denoising algorithm that appears in =-=[2]-=-. This work puts forward ways for handling nonhomogeneous noise and missing information, paving the way to state-of-the-art results in applications such as color image denoising, demosaicing, and inpa... |

43 |
Multidimensional Gaussian Distributions
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Citation Context ...sHere, the choice of the parameter C is very important and depends on the dimension of the patches: If w ′ is a n-dimensional Gaussian vector, ||w ′ ||2 is distributed by the generalized Rayleigh law =-=[42]-=- which leads to the following result: P(||w ′ ||2 ≤ √ nCσ) = 1 Γ( n 2 ) � nC 2 2 z=0 z n 2 −1 e −z dz. In [2], C was tuned empirically to C = 1.15 for n = 8 × 8 = 64. Here we choose the rule which pro... |

43 | Noise estimation from a single image
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Citation Context ...tive path, a nonparametric learning, uses image examples directly within the reconstruction process, as practiced by Efros and Leung for texture synthesis [33], by Freeman et al. for super-resolution =-=[34]-=-, [35], and by several follow-up works [36]–[39] for super-resolution and inpainting. Interestingly, most of the above direct methods avoid the prior and target instead the posterior density, from whi... |

34 |
Demosaicing by successive approximation
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Citation Context ...he average result on this standard dataset when compared to the best demosaicing algorithm so far reported. The fact that we did not introduce any special procedure 22sIm BI K [48] AP [47] OR [49] SA =-=[52]-=- CC [46] D1 D1L D2 D2L 1 26.21 24.64 37.70 34.66 38.32 38.53 37.17 38.19 37.77 38.97 2 33.08 32.29 39.57 39.22 39.95 40.43 38.06 39.68 39.59 40.81 3 34.45 33.84 41.45 41.18 41.18 42.54 40.89 42.94 41.... |

30 |
Sparse decomposition of stereo signals with matching pursuit and application to blind separation of more than two sources from a stereo mixture
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Citation Context ...channel signals, with fixed dictionaries and standard Euclidean metric, for various other modalities and applications, e.g., for compression of color images [24] and for audio source separation [25], =-=[26]-=-. In this paper, our main aim is to extend the algorithm reported in [2] to color images (and to vector-valued images, in general), and then show the applicability of this extension to other inverse p... |

26 | Weighted overcomplete denoising
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(Show Context)
Citation Context ...s proposed in [2]. that finding some other way of performing this average can further improve the results and it is topic of current research. Some ideas along this direction can be found in [43] and =-=[45]-=-. Here, the choice of the parameter is very important and depends on the dimension of the patches: If is a -dimensional Gaussian vector, is distributed by the generalized Rayleigh law [46] which leads... |

21 | Color demosaicing using variance of color differences
- Chung, Chan
- 2006
(Show Context)
Citation Context ...he Bayer pattern, GRGRGR. . . on odd lines and BGBGBG. . . on even ones. Several algorithms have been developed in recent years to produce high quality full color images from the mosaic sensor, e.g., =-=[46]-=-, [47], [48], [49]. Although color demosaicing is becoming less relevant with the on-going development of sensor and camera technology, addressing it remains a challenging task that helps to test the ... |

20 | Demosaicing using optimal recovery
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(Show Context)
Citation Context ...GRGRGR. . . on odd lines and BGBGBG. . . on even ones. Several algorithms have been developed in recent years to produce high quality full color images from the mosaic sensor, e.g., [46], [47], [48], =-=[49]-=-. Although color demosaicing is becoming less relevant with the on-going development of sensor and camera technology, addressing it remains a challenging task that helps to test the effectiveness of d... |

17 |
Example-based regularization deployed to super-resolution reconstruction of a single image
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(Show Context)
Citation Context ... directly within the reconstruction process, as practiced by Efros and Leung for texture synthesis [30]; by Freeman et. al. for super-resolution [31], [32]; and by several follow-up works [33], [34], =-=[35]-=-, [36] for super-resolution and inpainting. Interestingly, most of the above direct methods avoid the prior and target instead the posterior density, from which reconstruction is easily obtained. The ... |

15 | Bandelet image approximation and compression
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- 2005
(Show Context)
Citation Context ...nt in describing x. If we consider the case where Γ is the set of natural images, dictionaries such as wavelets of various sorts [4], curvelets [5], [6], contourlets [7], [8], wedgelets [9], bandlets =-=[10]-=-, [11], and steerable wavelets [12], [13], are all attempts to design dictionaries that fulfill the above model assumption. Indeed, these various transforms have led to highly effective algorithms in ... |