Non-local sparse models for image restoration

Julien MairalFrancis BachJean PonceGuillermo SapiroAndrew Zisserman

article2009ICCV1,859 citations

Unifies non-local image self-similarity with learned dictionary sparse coding via simultaneous sparse approximations, achieving state-of-the-art performance in camera raw image denoising and demosaicking at practical computational costs.

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Consumer and mobile digital cameras frequently produce noisy raw image data when operating under low light or high shutter speeds. Reconstructing clean full-color images from this raw sensor data requires solving two core restoration problems: removing sensor noise and reconstructing full color information from incomplete sensor measurements. Prior techniques have addressed these issues either by exploiting image self-similarity to average out noise or by using learned sparse coding to approximate local image patches with a compact set of basis elements. However, each approach presents trade-offs: self-similarity methods struggle when unique image patches lack duplicates, whereas sparse coding can introduce visual artifacts because mathematically similar patches may be reconstructed using inconsistent sets of basis elements.

The article develops and evaluates a unified restoration framework called learned simultaneous sparse coding. This method forces clusters of similar image patches to share identical active elements from a learned dictionary, effectively merging non-local self-similarity with adaptive sparse representations.

To test this framework, the authors conducted quantitative benchmark experiments and qualitative assessments across image denoising and color reconstruction tasks. Denoising was tested against synthetic Gaussian noise across twelve standard test images at multiple noise levels, comparing results with leading methods such as block matching 3D filtering and previous sparse models. Color reconstruction was evaluated on a twenty-four-image benchmark dataset. To establish practical feasibility, the method was also applied directly to noisy raw photographs captured with a consumer digital camera under challenging high-sensitivity settings.

The experimental findings show that the unified framework consistently outperforms or matches established state-of-the-art baselines. First, in synthetic noise reduction, the method produced the highest overall image quality scores, outperforming leading algorithms particularly in heavy noise conditions. Second, in color reconstruction benchmarks, the approach achieved an average quality improvement of approximately 0.87 decibels over top-performing specialized baselines while eliminating visible color distortion artifacts that occur in standard sparse coding. Third, qualitative tests on raw high-sensitivity camera captures demonstrated that the framework produced cleaner results with fewer edge and color artifacts than several established commercial photographic processing tools, even when operating without camera-specific sensor noise models.

These results show that combining non-local patch grouping with dictionary learning significantly improves visual restoration quality without requiring application-specific hand-tuning. The framework offers an effective, unified mechanism for processing raw sensor data into high-quality images. The authors note that the method can be tuned flexibly: adjusting dictionary size and iteration depth allows processing time to drop by an order of magnitude (from around twenty seconds down to sub-second or low-second execution per image) with virtually no noticeable drop in visual fidelity, making it viable for practical software and hardware imaging pipelines.

For operational implementation, system designers and engineering teams should consider adopting simultaneous sparse coding as a general-purpose processing layer for computational imaging pipelines. Prior to deployment, developers should incorporate device-specific, non-uniform noise models to improve handling of non-homogeneous sensor noise and background artifacts. The authors also recommend extending the framework to related restoration challenges, including image deblurring, missing-region completion, and video restoration.

The main limitation noted in the article is the reliance on a uniform noise assumption during real-world camera testing, which occasionally left faint residual noise artifacts in complex backgrounds. Computational cost is also higher than fixed-basis transforms, requiring cluster-based approximations to achieve practical speeds. Despite these limitations, there is high confidence in the quantitative and visual gains demonstrated across standardized image benchmarks.

Ecole Normale Supérieure INRIA University of Minnesota University of Oxford classification classification methods demosaicing non-local sparse models image restoration image denoising

arXiv: 0904.3523 getspams/spams-devel

Paper: A non-local algorithm for image denoising, Antoni Buades et al. (2005). Read this first to understand the non-local means principle of grouping similar image neighborhoods that the source combines with sparse coding.
Paper: Online dictionary learning for sparse coding, Julien Mairal et al. (2009). Its online dictionary-learning method supplies the learned-dictionary foundation needed to follow the source’s simultaneous sparse-coding model.

Paper: Image Deblurring and Super-Resolution by Adaptive Sparse Domain Selection and Adaptive Regularization, Weisheng Dong et al. (2010). This later restoration framework carries adaptive sparse representations and non-local patch similarity into deblurring and super-resolution.

Cover for Non-local sparse models for image restoration

Abstract

We propose in this paper to unify two different approaches to image restoration: On the one hand, learning a basis set (dictionary) adapted to sparse signal descriptions has proven to be very effective in image reconstruction and classification tasks. On the other hand, explicitly exploiting the self-similarities of natural images has led to the successful non-local means approach to image restoration. We propose simultaneous sparse coding as a framework for combining these two approaches in a natural manner. This is achieved by jointly decomposing groups of similar signals on subsets of the learned dictionary. Experimental results in image denoising and demosaicking tasks with synthetic and real noise show that the proposed method outperforms the state of the art, making it possible to effectively restore raw images from digital cameras at a reasonable speed and memory cost.

Topics

classification classification methods demosaicing image restoration non-local means reconstruction sparse approximations

Keywords

non-local sparse models image restoration image denoising image demosaicking non-local means filtering orthogonal matching pursuit sparse decomposition joint sparsity total variation basis pursuit

1. Introduction
2. Related Work
2.1. Non-Local Means Filtering
2.2. Learned Sparse Coding
2.3. Block Matching 3D (BM3D)
3. Proposed Formulation
3.1. Simultaneous Sparse Coding
3.2. Principle of the Formulation
3.3. Practical Formulation and Implementation
3.4. Real Images and Demosaicking
4. Experimental Validation
4.1. Denoising - Synthetic Noise
4.2. Demosaicking
4.3. Denoising - Real Noise
5. Conclusion
Acknowledgments
References

Knowls

Knowl 1 — Learned Simultaneous Sparse Coding (LSSC) Formulation
model/method
Learned Simultaneous Sparse Coding (LSSC) integrates non-local self-similarities into learned dictionary sparse representations by forcing groups of similar patches to share the same dictionary elements in their sparse decompositions.

Let a noisy image of $n$ pixels be represented as overlapping patches $y_i \in \mathbb{R}^m$ of size $m$ centered at each pixel $i \in \{1, \dots, n\}$ . For each patch $y_i$ , let $S_i \subseteq \{1, \dots, n\}$ denote the set of indices of patches similar to $y_i$ . Let $D \in \mathbb{R}^{m \times k}$ be a dictionary whose columns (atoms) belong to the unit $\ell_2$ -sphere $\mathcal{C} = \{d \in \mathbb{R}^m \mid \|d\|_2 = 1\}$ , and let $A_i = [\alpha_{ij}]_{j \in S_i} \in \mathbb{R}^{k \times |S_i|}$ be the matrix of sparse codes for the patch group $S_i$ , where $\alpha_{ij} \in \mathbb{R}^k$ represents the decomposition vector of patch $y_j$ .

The dictionary $D$ and sparse code matrices $(A_i)_{i=1}^n$ are learned jointly by solving the optimization problem:

$\min_{(A_i)_{i=1}^n, D \in \mathcal{C}} \sum_{i=1}^n \frac{\|A_i\|_{p,q}}{|S_i|^p} \quad \text{s.t.} \quad \forall i, \sum_{j \in S_i} \|y_j - D\alpha_{ij}\|_2^2 \le \varepsilon_i$

where $\|A_i\|_{p,q} \triangleq \sum_{l=1}^k \|\alpha_i^l\|_p^q$ is the grouped-sparsity (pseudo) matrix norm with $\alpha_i^l$ denoting the $l$ -th row of $A_i$ , and $\varepsilon_i$ is a group-level residual tolerance. The factor $|S_i|^p$ normalizes the objective across groups of varying sizes.

In practice, dictionary learning is performed using the convex $\ell_{1,2}$ norm $((p,q) = (1,2))$ , which can be optimized online, while signal reconstruction is performed using the $\ell_{0,\infty}$ pseudo-norm $((p,q) = (0,\infty))$ , counting the number of nonzero rows in $A_i$ and solved approximately via simultaneous orthogonal matching pursuit.
Knowl 2 — Patch Similarity Selection and Group Residual Tolerance
equation
In Learned Simultaneous Sparse Coding (LSSC), the set $S_i$ of patches similar to the reference patch $y_i \in \mathbb{R}^m$ is constructed according to Euclidean distance:

$S_i \triangleq \{j \in \{1, \dots, n\} \mid \|y_i - y_j\|_2^2 \le \xi\}$

where $\xi$ is a distance threshold set empirically for pixel intensities in the range $[0, 255]$ by:

$\xi = \frac{(32\sigma)^2}{m}$

with $\sigma$ being the noise standard deviation and $m$ the patch dimension (number of pixels in a patch).

For each group $S_i$ , the residual fitting error threshold $\varepsilon_i$ in the joint reconstruction constraint $\sum_{j \in S_i} \|y_j - D\alpha_{ij}\|_2^2 \le \varepsilon_i$ is determined based on the statistical behavior of Gaussian noise. Because the normalized residual sum $\sum_{j \in S_i} \|y_j - D\alpha_{ij}\|_2^2 / \sigma^2$ follows a chi-squared distribution $\chi^2_{m|S_i|}$ with $m|S_i|$ degrees of freedom, the tolerance is computed as:

$\varepsilon_i = \sigma^2 F_{m|S_i|}^{-1}(\tau)$

where $F_{m|S_i|}$ is the cumulative distribution function of $\chi^2_{m|S_i|}$ , $F_{m|S_i|}^{-1}$ is its inverse, and $\tau \in (0, 1)$ is a confidence threshold set to $\tau = 0.8$ for synthetic and real denoising experiments.
Knowl 3 — Pixel Averaging and Image Reconstruction in Simultaneous Sparse Coding
equation
Once the dictionary $D \in \mathbb{R}^{m \times k}$ and the sparse code vectors $\alpha_{ij} \in \mathbb{R}^k$ for all patch groups $S_i$ have been computed, every pixel in the image receives multiple estimates coming from all patches and similarity groups containing that pixel. The final reconstructed image $x \in \mathbb{R}^n$ is obtained by taking the weighted average of all reconstructed patch estimates:

$x = \operatorname{diag}\left(\sum_{i=1}^n \sum_{j \in S_i} R_j \mathbf{1}_m\right)^{-1} \sum_{i=1}^n \sum_{j \in S_i} R_j D \alpha_{ij}$

where:
- $R_j \in \mathbb{R}^{n \times m}$ is the binary placement matrix that maps a patch of size $m$ at position $j$ into its corresponding coordinates in the full image vector of size $n$ .
- $\mathbf{1}_m \in \mathbb{R}^m$ is an $m$ -dimensional vector filled with ones.
- $\operatorname{diag}\left(\sum_{i=1}^n \sum_{j \in S_i} R_j \mathbf{1}_m\right)$ is an $n \times n$ diagonal matrix whose $l$ -th entry equals the total number of estimates generated for pixel $l$ .
- When each group is a singleton ( $S_i = \{i\}$ ), this formulation reduces to standard patch-averaging sparse coding.
Knowl 4 — Simultaneous Sparse Coding Demosaicking Algorithm
algorithm
The demosaicking algorithm reconstructs a full-color RGB image from a single-channel Bayer-mosaicked sensor image $y$ by combining an offline natural image dictionary $D_0$ with an image-adapted dictionary $D_1$ under simultaneous sparse coding constraints.
Input: Mosaicked image vector $y \in \mathbb{R}^n$ , Bayer sampling binary diagonal masks $M_j \in \mathbb{R}^{m \times m}$ for each patch $j$ , offline pre-trained dictionary $D_0 \in \mathbb{R}^{m \times k}$ , similarity threshold $\xi$ , regularization parameters $\varepsilon_i$
Output: Demosaicked full-color image $x \in \mathbb{R}^n$
1. Group similar patches: For each patch $y_i$ in $y$ , determine its similarity cluster $S_i = \{j \mid \|y_i - y_j\|_2^2 \le \xi\}$
2. Initial reconstruction with $D_0$ : For all $i \in \{1, \dots, n\}$ , solve for $A_i = [\alpha_{ij}]_{j \in S_i}$ :
    $\min_{A_i \in \mathbb{R}^{k \times |S_i|}} \|A_i\|_{0,\infty} \quad \text{s.t.} \quad \forall j \in S_i, M_j(y_j - D_0 \alpha_{ij}) = 0$
   Average the resulting patch estimates via placement matrices $R_j$ to obtain an intermediate demosaicked estimate $\bar{x}$
3. Dictionary adaptation: On the estimate $\bar{x}$ , train an adaptive dictionary $D_1 \in \mathbb{R}^{m \times k}$ using the LSSC formulation with large group error tolerances $\varepsilon_i$
4. Final reconstruction with concatenated dictionary: Construct $D_2 = [D_0, D_1] \in \mathbb{R}^{m \times 2k}$ and solve for each cluster:
    $\min_{A_i} \|A_i\|_{0,\infty} \quad \text{s.t.} \quad \forall j \in S_i, M_j(y_j - D_2 \alpha_{ij}) = 0$
5. Aggregate estimates: Compute the final demosaicked image $x$ via weighted averaging:
    $x = \operatorname{diag}\left(\sum_{i=1}^n \sum_{j \in S_i} R_j \mathbf{1}_m\right)^{-1} \sum_{i=1}^n \sum_{j \in S_i} R_j D_2 \alpha_{ij}$
return $x$
Typical hyperparameters are patch size $m = 8 \times 8$ (with 3 color channels, $m = 8 \times 8 \times 3 = 192$ ), dictionary size $k = 256$ , and $\xi = 3 \times 10^4$ for pixel values in $[0, 255]$ .
Knowl 5 — Implementation Optimizations for Scalable Simultaneous Sparse Coding
model/method
Directly solving simultaneous sparse coding across all pairs of $n$ patches scales quadratically ( $n^2$ code vectors $\alpha_{ij}$ ). The following algorithmic components reduce the computational complexity and memory footprint to scale linearly in $n$ :
1. Semi-local grouping: Patch similarity search for $y_i$ is restricted to a local window of size $w \times w$ around pixel $i$ (with $w \le 64$ ), reducing the worst-case number of code vectors to $n w^2$ .
2. Disjoint patch clustering: Pixels are partitioned into disjoint clusters $C_k$ such that all pixels $i \in C_k$ share an identical patch set $S_i$ . Each pixel belongs to exactly one cluster, requiring the computation of only $n$ code vectors $\alpha_{ij}$ in total.
3. Offline dictionary pre-training: Initial dictionaries $D_0$ are pre-computed offline on $2 \times 10^7$ natural image patches randomly sampled from the 10,000 images of PASCAL VOC'07 using online dictionary learning, providing a superior starting point compared to smaller batch methods.
4. Two-stage matching: An initial baseline denoising step (such as classical patch-wise sparse coding) is applied to the noisy image before patch grouping, significantly improving patch matching accuracy.
5. Patch mean normalization: The mean intensity (or RGB color vector) of each patch is subtracted prior to sparse coding and re-added after estimation, improving numerical stability and visual quality.
6. On-the-fly code evaluation: Online dictionary updates do not store codes globally in memory; the maximum number of code vectors stored at any time is bounded by the size of the largest cluster $|C_k|$ .

Knowl 6 — Denoising Benchmark Comparison on Additive White Gaussian Noise

data/table

Denoising performance was quantitatively benchmarked on 12 standard grayscale test images corrupted by synthetic zero-mean additive white Gaussian noise with standard deviation $\sigma \in \{5, 10, 15, 20, 25, 50, 100\}$ . Performance was evaluated using Peak Signal-to-Noise Ratio (PSNR in dB, defined as $\text{PSNR} = 10 \log_{10}(255^2 / \text{MSE})$ ), averaged over 5 independent noise realizations.

The evaluated methods include:

GSM: Gaussian Scale Mixtures in the wavelet domain (Portilla et al., 2003)
FoE: Fields of Experts (Roth & Black, 2005)
K-SVD: Adaptive sparse coding via K-SVD (Elad & Aharon, 2006)
BM3D: Block-Matching and 3D Filtering (Dabov et al., 2007)
SC: Fixed dictionary learned offline on $2 \times 10^7$ PASCAL VOC'07 patches, without grouping
LSC: Learned sparse coding adapted to the test image via $\ell_1$ dictionary learning
LSSC: Learned simultaneous sparse coding combining learned dictionaries with joint patch sparsity

$\sigma$	GSM	FoE	K-SVD	BM3D	SC	LSC	LSSC
5	37.05	37.03	37.42	37.62	37.46	37.66	37.67
10	33.34	33.11	33.62	34.00	33.76	33.98	34.06
15	31.31	30.99	31.58	32.05	31.72	31.99	32.12
20	29.91	29.62	30.18	30.73	30.29	30.60	30.78
25	28.84	28.36	29.10	29.72	29.18	29.52	29.74
50	25.66	24.36	25.61	26.38	25.83	26.18	26.57
100	22.80	21.36	22.10	23.25	22.46	22.62	23.39

LSSC matches or outperforms BM3D across all noise levels, with the performance advantage increasing at higher noise levels (e.g., +0.19 dB over BM3D and +0.96 dB over K-SVD at $\sigma = 50$ ; +0.14 dB over BM3D and +1.29 dB over K-SVD at $\sigma = 100$ ).

Knowl 7 — Per-Image Denoising Performance of LSSC on Standard Benchmarks

data/table

The quantitative denoising performance of Learned Simultaneous Sparse Coding (LSSC) was measured across 12 standard benchmark images at noise standard deviations $\sigma \in \{5, 10, 15, 20, 25, 50, 100\}$ . Hyperparameters used were $k = 512$ atoms; patch sizes $m = 9 \times 9$ for $\sigma \le 25$ , $m = 12 \times 12$ for $\sigma = 50$ , and $m = 16 \times 16$ for $\sigma = 100$ ; $\tau = 0.8$ ; and $\xi = (32\sigma)^2/m$ . Results are given in PSNR (dB), averaged over 5 noise realizations.

Image	$\sigma = 5$	$\sigma = 10$	$\sigma = 15$	$\sigma = 20$	$\sigma = 25$	$\sigma = 50$	$\sigma = 100$
house	39.93	36.96	35.35	34.16	33.15	30.04	25.83
peppers	38.18	34.80	32.82	31.37	30.21	26.62	23.00
cameraman	38.32	34.21	32.01	30.57	29.51	26.42	23.08
lena	38.69	35.83	34.15	32.90	31.87	28.87	25.82
barbara	38.48	34.97	33.00	31.57	30.47	27.06	23.59
boat	37.35	34.02	32.20	30.89	29.87	26.74	23.84
hill	37.17	33.67	31.89	30.71	29.80	27.05	24.44
couple	37.45	33.98	32.06	30.69	29.61	26.30	23.28
man	37.89	34.06	32.01	30.64	29.63	26.69	24.00
fingerprint	36.70	32.57	30.31	28.78	27.62	24.25	21.26
bridge	35.78	31.22	28.92	27.46	26.42	23.68	21.46
flintstones	36.13	32.46	30.78	29.63	28.71	25.16	21.10
Average	37.67	34.06	32.12	30.78	29.74	26.57	23.39

Knowl 8 — Color Demosaicking Benchmark Performance on Kodak PhotoCD Dataset

data/table

Color demosaicking performance was evaluated on the 24 RGB images ( $512 \times 768$ pixels) of the standard Kodak PhotoCD benchmark with a Bayer color filter mask applied. Following standard benchmarking protocols (Paliy et al., 2007), a 15-pixel border was excluded from the evaluation to prevent boundary bias. Results are reported in PSNR (dB).

The compared algorithms are:

AP: Alternating Projections (Gunturk et al., 2002)
DL: Directional Linear Minimum Mean Square-Error Estimation (Zhang & Wu, 2005)
LPA: Local Polynomial Approximation (Paliy et al., 2007)
SC: Sparse Coding with fixed offline dictionary (Mairal et al., 2009)
LSC: Learned Sparse Coding adapted to the image (Mairal et al., 2009)
LSSC: Learned Simultaneous Sparse Coding (Mairal et al., 2009)

Image	AP	DL	LPA	SC	LSC	LSSC
1	37.84	38.46	40.47	40.84	40.92	41.36
2	39.64	40.89	41.36	41.76	42.03	42.24
3	41.40	42.66	43.47	43.15	43.92	44.24
4	39.92	40.49	40.84	41.99	42.14	42.45
5	37.28	38.07	37.51	38.72	39.15	39.45
6	38.69	40.19	40.92	41.29	41.36	41.71
7	41.75	42.35	43.06	43.30	43.59	44.06
8	35.58	36.02	37.13	37.42	37.38	37.57
9	41.84	43.05	43.50	43.17	43.74	43.83
10	41.93	42.54	42.77	43.01	43.17	43.33
11	39.25	40.01	40.51	41.19	41.29	41.51
12	42.62	43.45	44.01	44.29	44.49	44.90
13	34.28	34.75	36.08	36.16	36.29	36.35
14	35.66	36.91	36.86	37.64	38.48	38.77
15	39.17	39.82	40.09	41.04	41.24	41.74
16	42.10	43.75	44.02	44.36	44.42	44.91
17	41.23	41.68	41.75	41.75	41.86	41.98
18	37.31	37.64	37.59	38.05	38.27	38.38
19	39.99	41.01	41.55	41.58	41.71	42.31
20	40.63	41.24	41.48	41.95	42.25	42.27
21	38.72	39.10	39.61	40.55	40.59	40.65
22	37.63	38.37	38.44	38.73	38.97	39.24
23	41.93	43.22	43.92	43.47	43.93	44.34
24	34.74	35.55	35.44	35.59	35.85	35.89
Average	39.21	40.05	40.52	40.88	41.13	41.39

LSSC outperforms the state-of-the-art LPA method by 0.87 dB on average. When evaluated over the full image (including borders), SC achieves 40.72 dB, LSC achieves 40.98 dB, and LSSC achieves 41.24 dB (compared to 39.56 dB for fixed-dictionary SC and 40.32 dB for adaptive LSC in Mairal et al., 2008), while eliminating typical demosaicking color fringe and zipper artefacts.

Knowl 9 — Raw Sensor Denoising and Processing Pipeline for Digital Cameras
model/method
To restore images corrupted by real sensor noise (e.g., Canon Powershot G9 at 1600 ISO), denoising is performed directly on the raw CFA (Color Filter Array) mosaicked sensor data before running internal camera rendering pipelines.

The pipeline operates in five successive steps:
1. Raw data extraction: Extract the raw un-demosaicked Bayer array directly from the camera sensor data (e.g., using dcraw).
2. Channel scaling: Manually or calibrationally scale the R, G, and B sensor values so that their noise variances visually match, producing approximately uniform noise across the mosaicked array.
3. Mosaicked denoising: Apply LSSC denoising directly to the single-channel mosaicked Bayer array using patch size $m = 8 \times 8$ , $k = 256$ atoms, user-estimated uniform noise standard deviation $\sigma$ , and threshold $\xi = (32\sigma)^2/m$ .
4. Demosaicking: Apply the simultaneous sparse coding demosaicking algorithm to reconstruct full RGB triplets at every pixel.
5. Color post-processing: Apply white balancing, color space conversion to sRGB, gamma correction, and contrast enhancement.
Denoising the raw mosaicked array prior to demosaicking avoids amplifying sensor noise through interpolation and non-linear color transformations, yielding sharper results with fewer artefacts than commercial pipelines (e.g., NoiseWare 4.2 or Adobe Camera Raw).
Knowl 10 — Spatially Homogeneous Noise Assumption in Raw Camera Restoration
limitation
The restoration model assumes that noise in the raw sensor data is independent and identically distributed additive white Gaussian noise with a single uniform standard deviation $\sigma$ across the entire sensor grid. In physical digital cameras, however, sensor noise is signal-dependent, spatially non-uniform, and channel-dependent (varying with pixel brightness, ISO amplification, and sensor temperature).

Although scaling the R, G, and B channels mitigates global variance differences, assuming strict spatial uniformity can lead to suboptimal restoration in certain image regions (for instance, partially reconstructing noise in flat background areas of high-ISO photographs).

Coverage note — None was omitted; all primary contributions (LSSC formulation, grouping and tolerance equations, pixel reconstruction, demosaicking algorithm, implementation speedups, synthetic denoising benchmarks, Kodak demosaicking benchmarks, real RAW camera restoration pipeline, and noise model limitations) are included.

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Citation

MLA

Jenatton, R., et al. “Structured Variable Selection with Sparsity-Inducing Norms”. Journal of Machine Learning Research 12 (2011) 2777-2824, 2009, http://arxiv.org/abs/0904.3523v3.

APA

Jenatton, R., Audibert, J.-Y., & Bach, F. (2009). Structured Variable Selection with Sparsity-Inducing Norms. Journal of Machine Learning Research 12 (2011) 2777-2824. http://arxiv.org/abs/0904.3523v3

Chicago

Jenatton, R., J.-Y. Audibert, and F. Bach. 2009. “Structured Variable Selection with Sparsity-Inducing Norms”. Journal of Machine Learning Research 12 (2011) 2777-2824. http://arxiv.org/abs/0904.3523v3.

Harvard

Jenatton, R., Audibert, J.-Y. and Bach, F. (2009) “Structured Variable Selection with Sparsity-Inducing Norms”, Journal of Machine Learning Research 12 (2011) 2777-2824 [Preprint]. Available at: http://arxiv.org/abs/0904.3523v3.

Vancouver

1. Jenatton R, Audibert J-Y, Bach F (2009) Structured Variable Selection with Sparsity-Inducing Norms. Journal of Machine Learning Research 12 (2011) 2777-2824

BibTeX

@article{jenatton2009structured,
  title = {Structured Variable Selection with Sparsity-Inducing Norms},
  author = {Jenatton, Rodolphe and Audibert, Jean-Yves and Bach, Francis},
  year = {2009},
  journal = {Journal of Machine Learning Research 12 (2011) 2777-2824},
  url = {http://arxiv.org/abs/0904.3523v3},
  eprint = {0904.3523}
}

Metadata:arXiv

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Non-local sparse models for image restoration

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Knowls

Knowl 1 — Learned Simultaneous Sparse Coding (LSSC) Formulation

Knowl 2 — Patch Similarity Selection and Group Residual Tolerance

Knowl 3 — Pixel Averaging and Image Reconstruction in Simultaneous Sparse Coding

Knowl 4 — Simultaneous Sparse Coding Demosaicking Algorithm

Knowl 5 — Implementation Optimizations for Scalable Simultaneous Sparse Coding

Knowl 6 — Denoising Benchmark Comparison on Additive White Gaussian Noise

Knowl 7 — Per-Image Denoising Performance of LSSC on Standard Benchmarks

Knowl 8 — Color Demosaicking Benchmark Performance on Kodak PhotoCD Dataset

Knowl 9 — Raw Sensor Denoising and Processing Pipeline for Digital Cameras

Knowl 10 — Spatially Homogeneous Noise Assumption in Raw Camera Restoration

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