Fast Image Deconvolution using Hyper-Laplacian Priors

Dilip KrishnanR. Fergus

article2009NeurIPS1,482 citations

Proposes an alternating minimization algorithm for non-blind image deconvolution under non-convex hyper-Laplacian priors that uses analytical polynomial roots and lookup tables to accelerate restoration speeds by several orders of magnitude without sacrificing quality.

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Modern digital cameras frequently capture high-resolution images containing tens of megapixels, making fast and effective image deblurring critical for computational photography, medical imaging, and computer vision. Recovering sharp details from blurry images typically relies on statistical models of natural scenes known as hyper-Laplacian priors, which accurately represent image edges. However, applying these priors creates non-convex mathematical problems that are computationally prohibitive, often requiring tens of minutes per image on standard hardware.

The article demonstrates a highly efficient non-blind image deconvolution framework that significantly accelerates processing speeds while preserving the superior restoration quality provided by hyper-Laplacian priors.

To achieve this, the authors restructured the optimization task using an alternating minimization framework that splits the overall problem into two simpler alternating phases. The first phase isolates the non-convex mathematical challenge into independent, per-pixel sub-problems, which are resolved either through precomputed lookup tables or through exact closed-form algebraic solutions for specific prior exponents. The second phase is a standard quadratic problem that is rapidly solved in the frequency domain using Fast Fourier Transforms. The approach was validated on real-world benchmark images degraded by camera shake blur and noise, and tested across image dimensions up to 3072×3072 pixels.

The findings show that the proposed algorithm runs roughly 70 to 350 times faster than the conventional Iteratively Reweighted Least Squares benchmark without sacrificing restoration quality. A one-megapixel image was restored in less than three seconds compared to approximately twenty minutes for the standard baseline. The lookup-table method was about five times faster than the analytic solver while delivering essentially identical signal-to-noise ratio performance. Furthermore, the hyper-Laplacian model consistently outperformed standard Gaussian and total variation approaches by delivering sharper restorations with fewer visual artifacts across various blur kernel sizes.

These results establish that high-quality, non-convex image restoration can now operate in near real-time, removing a major processing bottleneck for large-sensor devices. Organizations can achieve state-of-the-art deblurring at a fraction of the computational and energy cost, making advanced restoration feasible within interactive consumer tools, embedded camera systems, and large-scale automated imaging workflows.

Decision-makers and engineering teams should consider adopting the lookup-table implementation for production pipelines due to its speed and flexibility across different prior exponents. Edge-tapering routines should be incorporated into deployments to mitigate potential boundary artifacts caused by frequency-domain processing. Further development should explore adapting this splitting scheme to related ill-posed tasks, including super-resolution, image denoising, and fully blind deconvolution where blur kernels are unknown.

The reported benchmarks are based on linear grayscale images with known blur kernels and synthetic camera shake perturbations. While confident in the mathematical formulation and robust speed gains across varied image sizes, practitioners should validate performance on color images and complex real-world blur scenarios with estimated kernels before full deployment.

New York University look-up table markov random fields superresolution fast image deconvolution hyper-Laplacian priors natural image statistics

Krishnan et al (2009).pdf

No sufficiently relevant recommendations were found.

Paper: Image smoothing via L0 gradient minimization, Li Xu et al. (2011). Extends the half-quadratic splitting optimization paradigm of this paper to discrete L0 gradient minimization for structural edge-preserving image smoothing.
Paper: Learning Deep CNN Denoiser Prior for Image Restoration, Kai Zhang et al. (2017). Generalizes classical variable-splitting optimization frameworks for image restoration by replacing analytical gradient priors with learned deep CNN denoisers.

Cover for Fast Image Deconvolution using Hyper-Laplacian Priors

Abstract

The heavy-tailed distribution of gradients in natural scenes have proven effective priors for a range of problems such as denoising, deblurring and super-resolution. These distributions are well modeled by a hyper-Laplacian (p(x) ∝ e^{-k|x|^α}), typically with 0.5 ≤ α ≤ 0.8. However, the use of sparse distributions makes the problem non-convex and impractically slow to solve for multi-megapixel images. In this paper we describe a deconvolution approach that is several orders of magnitude faster than existing techniques that use hyper-Laplacian priors. We adopt an alternating minimization scheme where one of the two phases is a non-convex problem that is separable over pixels. This per-pixel sub-problem may be solved with a lookup table (LUT). Alternatively, for two specific values of α, 1/2 and 2/3 an analytic solution can be found, by finding the roots of a cubic and quartic polynomial, respectively. Our approach (using either LUTs or analytic formulae) is able to deconvolve a 1 megapixel image in less than ∼3 seconds, achieving comparable quality to existing methods such as iteratively reweighted least squares (IRLS) that take ∼20 minutes. Furthermore, our method is quite general and can easily be extended to related image processing problems, beyond the deconvolution application demonstrated.

Topics

look-up table markov random fields superresolution blind deconvolution image restoration

Keywords

fast image deconvolution hyper-Laplacian priors natural image statistics total variation compressive sensing noise removal

1 Introduction
1.1 Related Work
2 Algorithm
2.1 x sub-problem
2.2 w sub-problem
2.2.1 Lookup table
2.2.2 Analytic solution
2.3 Summary of algorithm
3 Experiments
4 Discussion
References

Knowls

Knowl 1 — Alternating Minimization Algorithm for Fast Hyper-Laplacian Deconvolution
algorithm
The overall deconvolution framework solves the half-quadratic penalty formulation by alternating between updating the latent image $x$ in the Fourier domain and updating the auxiliary gradient variables $w$ per pixel, while gradually increasing the continuation weight $\beta$ .
Input: Blurred image $y$ , blur kernel $k$ , regularization weight $\lambda$ , prior exponent $\alpha \in (0, 1]$ , initial penalty $\beta_0 = 1$ , scaling factor $\beta_{\text{Inc}} = 2\sqrt{2}$ , maximum penalty $\beta_{\text{Max}} = 256$ , inner iterations $T = 1$ .
Output: Deconvolved image estimate $x$ .
$\beta \leftarrow \beta_0$
$x \leftarrow y$
Precompute Fourier terms $\mathcal{F}(f_1)$ , $\mathcal{F}(f_2)$ , $\mathcal{F}(k)$ , $\mathcal{F}(y)$ , $|\mathcal{F}(f_1)|^2$ , $|\mathcal{F}(f_2)|^2$ , and $|\mathcal{F}(k)|^2$
while $\beta < \beta_{\text{Max}}$ do
    for $\text{iter} \leftarrow 1$ to $T$ do
        For each pixel $i$ and filter $j \in \{1, 2\}$ , compute $v = (x \oplus f_j)_i$ and solve $w_i^j = \arg\min_w |w|^\alpha + \frac{\beta}{2}(w - v)^2$ via lookup table (LUT) or analytic root-finding
        Given $w^1$ and $w^2$ , solve for $x$ in closed form via 2D FFT
    end for
     $\beta \leftarrow \beta_{\text{Inc}} \cdot \beta$
end while
return $x$
Although the overall objective is non-convex, each conditional subproblem is solved to its exact global minimum (or high-accuracy LUT approximation). Setting $T = 1$ is sufficient in practice across the entire continuation path.
Knowl 2 — Half-Quadratic Splitting for Hyper-Laplacian Image Deconvolution
model/method
Non-blind image deconvolution with a hyper-Laplacian gradient prior $p(x) \propto e^{-c|x|^\alpha}$ ( $0 < \alpha \le 1$ ) is formulated as finding the Maximum A Posteriori (MAP) image estimate $x \in \mathbb{R}^N$ given blurred observation $y \in \mathbb{R}^N$ and blur kernel $k$ :

$\min_x \sum_{i=1}^N \left( \frac{\lambda}{2} (x \oplus k - y)_i^2 + \sum_{j=1}^J |(x \oplus f_j)_i|^\alpha \right)$

where $\oplus$ is 2D convolution, $i$ indexes pixels, $f_j$ are discrete derivative filters (typically horizontal $f_1 = [1, -1]$ and vertical $f_2 = [1, -1]^T$ ), $\lambda > 0$ controls regularization strength, and $\alpha < 1$ makes the regularizer non-convex.

To decouple the non-convex penalty from the spatial convolution operations, half-quadratic splitting introduces auxiliary variables $w_i^j$ approximating $(x \oplus f_j)_i$ :

$\min_{x, w} \sum_{i=1}^N \left( \frac{\lambda}{2} (x \oplus k - y)_i^2 + \frac{\beta}{2} \sum_{j=1}^J ((x \oplus f_j)_i - w_i^j)^2 + \sum_{j=1}^J |w_i^j|^\alpha \right)$

where $\beta > 0$ is a penalty parameter. As $\beta \to \infty$ , the solution to this split objective converges to that of the original MAP cost.
Knowl 3 — Closed-Form Image Subproblem Solution via Fast Fourier Transform
equation
For fixed auxiliary variables $w = \{w^1, \dots, w^J\}$ , the subproblem for updating the image $x$ is quadratic:

$\left( \sum_{j=1}^J (F^j)^T F^j + \frac{\lambda}{\beta} K^T K \right) x = \sum_{j=1}^J (F^j)^T w^j + \frac{\lambda}{\beta} K^T y$

where $K$ and $F^j$ are matrix representations of 2D convolution with blur kernel $k$ and derivative filter $f_j$ , respectively. Under circular boundary conditions, 2D Fast Fourier Transforms (FFTs) diagonalize these convolution matrices, yielding the exact solution:

$x = \mathcal{F}^{-1} \left( \frac{\sum_{j=1}^J \mathcal{F}(F^j)^* \circ \mathcal{F}(w^j) + \frac{\lambda}{\beta} \mathcal{F}(K)^* \circ \mathcal{F}(y)}{\sum_{j=1}^J \mathcal{F}(F^j)^* \circ \mathcal{F}(F^j) + \frac{\lambda}{\beta} \mathcal{F}(K)^* \circ \mathcal{F}(K)} \right)$

where $\mathcal{F}$ and $\mathcal{F}^{-1}$ denote the 2D forward and inverse discrete Fourier transforms, $*$ denotes complex conjugation, $\circ$ represents element-wise multiplication, and division is performed component-wise. Computing $x$ requires only 3 FFT operations per iteration (forward FFTs of $w^1$ and $w^2$ , and one inverse FFT), as the other Fourier transforms and power spectra are precomputed once.
Knowl 4 — Separable 1D Auxiliary Variable Subproblem and Look-Up Table Solution
model/method
Given a fixed image estimate $x$ , the auxiliary variable optimization decouples across all $N$ pixels and $J$ filter directions into $J \times N$ independent 1D scalar minimization problems of the form:

$w^* = \arg\min_{w \in \mathbb{R}} \left( |w|^\alpha + \frac{\beta}{2}(w - v)^2 \right)$

where $v = (x \oplus f_j)_i \in \mathbb{R}$ is the target gradient value at pixel $i$ for filter $f_j$ , $\alpha \in (0, 1]$ is the prior exponent, and $\beta > 0$ is the current continuation weight.

For a fixed $\alpha$ , $w^*$ depends solely on the two scalar variables $v$ and $\beta$ . It can therefore be precomputed off-line into a 2D look-up table (LUT). The table is generated by numerically optimizing the 1D cost over $10{,}000$ uniformly sampled values of $v \in [-0.6, 0.6]$ across discrete $\beta$ values equal to integer powers of $\sqrt{2}$ between $1$ and $256$ . During optimization, solving for $w^*$ at each pixel is performed by constant-time table lookup and linear interpolation for any $\alpha > 0$ .
Knowl 5 — Optimal Root Selection Criterion for the 1D Hyper-Laplacian Subproblem
theoretical result
For the 1D scalar minimization problem $w^* = \arg\min_{w \in \mathbb{R}} \left( |w|^\alpha + \frac{\beta}{2}(w - v)^2 \right)$ with $\alpha \in (0, 1)$ , $\beta > 0$ , and target value $v \in \mathbb{R}$ , any non-zero critical point $r \neq 0$ satisfies the first-order condition:

$\alpha |r|^{\alpha - 1} \operatorname{sign}(r) + \beta(r - v) = 0$

A non-zero real root $r$ yields a lower objective value than the boundary solution $w = 0$ (which has cost $\frac{\beta}{2}v^2$ ) if and only if:

$|r| > 2 |v| \frac{1 - \alpha}{2 - \alpha} \quad \text{and} \quad \operatorname{sign}(r) = \operatorname{sign}(v)$

Specialized to specific exponents:
- For $\alpha = 1/2$ , a non-zero real root $r$ is optimal over $w = 0$ if and only if $\frac{2}{3}|v| < |r| < |v|$ .
- For $\alpha = 2/3$ , a non-zero real root $r$ is optimal over $w = 0$ if and only if $\frac{1}{2}|v| < |r| < |v|$ .
This criterion enables exact global minimum selection among candidate polynomial roots without evaluating fractional powers $|r|^\alpha$ .
Knowl 6 — Analytic Solution of the 1D Subproblem for Exponent $\alpha = 1/2$
algorithm
For $\alpha = 1/2$ , the 1D subproblem $\arg\min_w |w|^{1/2} + \frac{\beta}{2}(w - v)^2$ reduces to finding the roots of the cubic polynomial $w^3 - 2vw^2 + v^2 w - \frac{\operatorname{sign}(v)}{4\beta^2} = 0$ and selecting the global minimum using Cardano's formula.
Input: Target gradient $v \in \mathbb{R}$ , penalty parameter $\beta > 0$ , tolerance $\epsilon = 10^{-6}$ .
Output: Optimal value $w^*$ .
$m \leftarrow -\operatorname{sign}(v) / (4\beta^2)$
$t_1 \leftarrow 2v / 3$
$t_2 \leftarrow \left(-27m - 2v^3 + 3\sqrt{3}\sqrt{27m^2 + 4mv^3}\right)^{1/3}$
$t_3 \leftarrow v^2 / t_2$
$r_1 \leftarrow t_1 + \frac{1}{3 \cdot 2^{1/3}} t_2 + \frac{2^{1/3}}{3} t_3$
$r_2 \leftarrow t_1 - \frac{1 - \sqrt{3}i}{6 \cdot 2^{1/3}} t_2 - \frac{1 + \sqrt{3}i}{3 \cdot 2^{2/3}} t_3$
$r_3 \leftarrow t_1 - \frac{1 + \sqrt{3}i}{6 \cdot 2^{1/3}} t_2 - \frac{1 - \sqrt{3}i}{3 \cdot 2^{2/3}} t_3$
for each root $r_k \in \{r_1, r_2, r_3\}$ do
     $c_1 \leftarrow |\operatorname{Im}(r_k)| < \epsilon$
     $c_2 \leftarrow \operatorname{Re}(r_k)\operatorname{sign}(v) > \frac{2}{3}|v|$
     $c_3 \leftarrow \operatorname{Re}(r_k)\operatorname{sign}(v) < |v|$
     $\text{valid}_k \leftarrow c_1 \land c_2 \land c_3$
end for
if any root is valid then
     $w^* \leftarrow \operatorname{sign}(v) \max_{k: \text{valid}_k} (\operatorname{Re}(r_k)\operatorname{sign}(v))$
else
     $w^* \leftarrow 0$
end if
return $w^*$
Knowl 7 — Analytic Solution of the 1D Subproblem for Exponent $\alpha = 2/3$
algorithm
For $\alpha = 2/3$ , setting the derivative of the 1D subproblem to zero and cubing leads to the quartic polynomial $w^4 - 3vw^3 + 3v^2 w^2 - v^3 w + \frac{8}{27\beta^3} = 0$ , whose roots are found in closed form via Ferrari's method.
Input: Target gradient $v \in \mathbb{R}$ , penalty parameter $\beta > 0$ , tolerance $\epsilon = 10^{-6}$ .
Output: Optimal value $w^*$ .
$m \leftarrow 8 / (27\beta^3)$
$t_1 \leftarrow -\frac{9}{8} v^2$
$t_2 \leftarrow v^3 / 4$
$t_3 \leftarrow -\frac{1}{8} m v^2$
$t_4 \leftarrow -t_3 / 2 + \sqrt{-m^3 / 27 + m^2 v^4 / 256}$
$t_5 \leftarrow (t_4)^{1/3}$
$t_6 \leftarrow 2\left(-\frac{5}{18} t_1 + t_5 + \frac{m}{3 t_5}\right)$
$t_7 \leftarrow \sqrt{t_1 / 3 + t_6}$
$r_1 \leftarrow \frac{3}{4} v + \frac{1}{2}\left(t_7 + \sqrt{-(t_1 + t_6 + t_2 / t_7)}\right)$
$r_2 \leftarrow \frac{3}{4} v + \frac{1}{2}\left(t_7 - \sqrt{-(t_1 + t_6 + t_2 / t_7)}\right)$
$r_3 \leftarrow \frac{3}{4} v + \frac{1}{2}\left(-t_7 + \sqrt{-(t_1 + t_6 - t_2 / t_7)}\right)$
$r_4 \leftarrow \frac{3}{4} v + \frac{1}{2}\left(-t_7 - \sqrt{-(t_1 + t_6 - t_2 / t_7)}\right)$
for each root $r_k \in \{r_1, r_2, r_3, r_4\}$ do
     $c_1 \leftarrow |\operatorname{Im}(r_k)| < \epsilon$
     $c_2 \leftarrow \operatorname{Re}(r_k)\operatorname{sign}(v) > \frac{1}{2}|v|$
     $c_3 \leftarrow \operatorname{Re}(r_k)\operatorname{sign}(v) < |v|$
     $\text{valid}_k \leftarrow c_1 \land c_2 \land c_3$
end for
if any root is valid then
     $w^* \leftarrow \operatorname{sign}(v) \max_{k: \text{valid}_k} (\operatorname{Re}(r_k)\operatorname{sign}(v))$
else
     $w^* \leftarrow 0$
end if
return $w^*$

Knowl 8 — Deconvolution Reconstruction Quality and Runtime Benchmark

data/table

Deconvolution performance was evaluated on 7 grayscale real-world images of size $576 \times 864$ degraded with a $19 \times 19$ camera-shake blur kernel, $1\%$ Gaussian noise, and 8-bit quantization. Algorithms were provided with slightly perturbed versions of the true blur kernel to reflect real-world kernel estimation errors. Reconstruction quality is measured by Signal-to-Noise Ratio (SNR) in dB: $10 \log_{10} (\|\hat{x} - \mu(\hat{x})\|^2 / \|\hat{x} - x\|^2)$ , where $\hat{x}$ is the uncorrupted ground truth and $x$ is the reconstructed image.

Image	Blurry	$\ell_2$	Lucy	TV	$\ell_1$	IRLS $\alpha=1/2$	IRLS $\alpha=2/3$	IRLS $\alpha=4/5$	Ours $\alpha=1/2$	Ours $\alpha=2/3$
1	6.42	14.13	12.54	15.87	16.18	14.61	15.45	16.04	16.05	16.44
2	10.73	17.56	15.15	19.37	19.86	18.43	19.37	20.00	19.78	20.26
3	12.45	19.30	16.68	21.83	22.77	21.53	22.62	22.95	23.26	23.27
4	8.51	16.02	14.27	17.66	18.02	16.34	17.31	17.98	17.70	18.17
5	12.74	16.59	13.28	19.34	20.25	19.12	19.99	20.20	21.28	21.00
6	10.85	15.46	12.00	17.13	17.59	15.59	16.58	17.04	17.79	17.89
7	11.76	17.40	15.22	18.58	18.85	17.08	17.99	18.61	18.58	18.96
Av. Gain	-	6.14	3.67	8.05	8.58	7.03	7.98	8.48	8.71	8.93
Av. Time (s)	-	79.85	1.55	0.66	0.75	354	354	354	L:1.01, A:5.27	L:1.00, A:4.08

The alternating minimization approach with $\alpha = 2/3$ achieves the highest average SNR gain ( $8.93\text{ dB}$ ), outperforming Gaussian $\ell_2$ ( $6.14\text{ dB}$ ), TV ( $8.05\text{ dB}$ ), $\ell_1$ ( $8.58\text{ dB}$ ), and IRLS $\alpha = 4/5$ ( $8.48\text{ dB}$ ). The LUT method executes in $\sim 1.0\text{ s}$ per image, providing a $\sim 350\times$ speedup over IRLS ( $354\text{ s}$ ) while matching or exceeding its restoration quality.

Knowl 9 — Runtime Scaling of Deconvolution Across Image Resolutions

data/table

Running times in seconds were measured across image sizes from $256 \times 256$ (0.06 MP) to $3072 \times 3072$ (9.4 MP) using a $13 \times 13$ blur kernel on a 4-core CPU.

Image size	$\ell_1$	IRLS ( $\alpha=4/5$ )	Ours (LUT, $\alpha=2/3$ )	Ours (Analytic, $\alpha=2/3$ )
$256 \times 256$	0.24	78.14	0.42	0.70
$512 \times 512$	0.47	256.87	0.55	2.28
$1024 \times 1024$	2.34	1281.3	2.78	10.87
$2048 \times 2048$	9.34	4935	10.72	44.64
$3072 \times 3072$	22.40	-	24.07	100.42

The LUT implementation achieves running times comparable to $\ell_1$ total variation methods (which use standard shrinkage) and is over $100\times$ to $400\times$ faster than IRLS across all resolutions. For a 1 megapixel image ( $1024 \times 1024$ ), the LUT method takes $2.78\text{ s}$ compared to over $21\text{ minutes}$ ( $1281.3\text{ s}$ ) for IRLS.

Knowl 10 — Circular Boundary Condition Artifacts in Fourier Inversion
limitation
The closed-form frequency domain solution for the image subproblem assumes circular (periodic) boundary conditions across the image borders. Real-world photographs do not possess periodic boundaries, which can introduce boundary ringing artifacts into the deconvolved image. Although these artifacts can be partly mitigated by edge tapering and occupy a decreasing proportion of total image area on multi-megapixel images, the algorithm does not inherently handle non-periodic boundary conditions.

Coverage note — Table 3 (SNR comparison across 8 individual blur kernel sizes for a single 512x512 image) was omitted because Table 1 already comprehensively covers SNR and runtime across methods, while Table 2 covers runtime scaling across image dimensions.

References

1.R. Chartrand. Fast algorithms for nonconvex compressive sensing: Mri reconstruction from very few data. In IEEE International Symposium on Biomedical Imaging (ISBI), 2009.
2.R. Chartrand and V. Staneva. Restricted isometry properties and nonconvex compressive sensing. Inverse Problems, 24:1–14, 2008.
3.R. Fergus, B. Singh, A. Hertzmann, S. T. Roweis, and W. Freeman. Removing camera shake from a single photograph. ACM TOG (Proc. SIGGRAPH), 25:787–794, 2006.
4.D. Field. What is the goal of sensory coding? Neural Computation, 6:559–601, 1994.
5.D. Geman and G. Reynolds. Constrained restoration and recovery of discontinuities. PAMI, 14(3):367–383, 1992.
6.D. Geman and C. Yang. Nonlinear image recovery with half-quadratic regularization. PAMI, 4:932–946, 1995.
7.N. Joshi, L. Zitnick, R. Szeliski, and D. Kriegman. Image deblurring and denoising using color priors. In CVPR, 2009.
8.D. Krishnan and R. Fergus. Fast image deconvolution using hyper-laplacian priors, supplementary material. NYU Tech. Rep. 2009, 2009.
9.A. Levin. Blind motion deblurring using image statistics. In NIPS, 2006.
10.A. Levin, R. Fergus, F. Durand, and W. Freeman. Image and depth from a conventional camera with a coded aperture. ACM TOG (Proc. SIGGRAPH), 26(3):70, 2007.
11.A. Levin and Y. Weiss. User assisted separation of reflections from a single image using a sparsity prior. PAMI, 29(9):1647–1654, Sept 2007.
12.A. Levin, Y. Weiss, F. Durand, and W. T. Freeman. Understanding and evaluating blind deconvolution algorithms. In CVPR, 2009.
13.S. Osindero, M. Welling, and G. Hinton. Topographic product models applied to natural scene statistics. Neural Computation, 1995.
14.J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli. Image denoising using a scale mixture of Gaussians in the wavelet domain. IEEE TIP, 12(11):1338–1351, November 2003.
15.W. Richardson. Bayesian-based iterative method of image restoration. 62:55–59, 1972.
16.S. Roth and M. J. Black. Fields of Experts: A Framework for Learning Image Priors. In CVPR, volume 2, pages 860–867, 2005.
17.L. Rudin, S. Osher, and E. Fatemi. Nonlinear total variation based noise removal algorithms. Physica D, 60:259–268, 1992.
18.E. Simoncelli and E. H. Adelson. Noise removal via bayesian wavelet coring. In ICIP, pages 379–382, 1996.
19.C. V. Stewart. Robust parameter estimation in computer vision. SIAM Reviews, 41(3):513–537, Sept. 1999.
20.M. F. Tappen, B. C. Russell, and W. T. Freeman. Exploiting the sparse derivative prior for super-resolution and image demosaicing. In SCTV, 2003.
21.M. Wainwright and S. Simoncelli. Scale mixtures of gaussians and teh statistics of natural images. In NIPS, pages 855–861, 1999.
22.Y. Wang, J. Yang, W. Yin, and Y. Zhang. A new alternating minimization algorithm for total variation image reconstruction. SIAM J. Imaging Sciences, 1(3):248–272, 2008.
23.E. W. Weisstein. Cubic formula. http://mathworld.wolfram.com/CubicFormula.html.
24.E. W. Weisstein. Quartic equation. http://mathworld.wolfram.com/QuarticEquation.html.
25.M. Welling, G. Hinton, and S. Osindero. Learning sparse topographic representations with products of student-t distributions. In NIPS, 2002.
26.S. Wright, R. Nowak, and M. Figueredo. Sparse reconstruction by separable approximation. IEEE Trans. Signal Processing, page To appear, 2009.

Citation

MLA

Krishnan, D., and R. Fergus. “Fast Image Deconvolution Using Hyper-Laplacian Priors”. Advances in Neural Information Processing Systems, vol. 22, 2009, https://proceedings.neurips.cc/paper_files/paper/2009/file/3dd48ab31d016ffcbf3314df2b3cb9ce-Paper.pdf.

APA

Krishnan, D., & Fergus, R. (2009). Fast Image Deconvolution using Hyper-Laplacian Priors. Advances in Neural Information Processing Systems, 22. https://proceedings.neurips.cc/paper_files/paper/2009/file/3dd48ab31d016ffcbf3314df2b3cb9ce-Paper.pdf

Chicago

Krishnan, D., and R. Fergus. 2009. “Fast Image Deconvolution Using Hyper-Laplacian Priors”. Advances in Neural Information Processing Systems 22. https://proceedings.neurips.cc/paper_files/paper/2009/file/3dd48ab31d016ffcbf3314df2b3cb9ce-Paper.pdf.

Harvard

Krishnan, D. and Fergus, R. (2009) “Fast Image Deconvolution using Hyper-Laplacian Priors”, Advances in Neural Information Processing Systems. Curran Associates, Inc. Available at: https://proceedings.neurips.cc/paper_files/paper/2009/file/3dd48ab31d016ffcbf3314df2b3cb9ce-Paper.pdf.

Vancouver

1. Krishnan D, Fergus R (2009) Fast Image Deconvolution using Hyper-Laplacian Priors. Advances in Neural Information Processing Systems 22:

BibTeX

@inproceedings{krishnan2009fast,
  title = {Fast Image Deconvolution using Hyper-Laplacian Priors},
  author = {Krishnan, Dilip and Fergus, Rob},
  year = {2009},
  booktitle = {Advances in Neural Information Processing Systems},
  publisher = {Curran Associates, Inc.},
  volume = {22},
  url = {https://proceedings.neurips.cc/paper_files/paper/2009/file/3dd48ab31d016ffcbf3314df2b3cb9ce-Paper.pdf}
}

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Fast Image Deconvolution using Hyper-Laplacian Priors

Abstract

Topics

Keywords

Table of Contents

Knowls

Knowl 1 — Alternating Minimization Algorithm for Fast Hyper-Laplacian Deconvolution

Knowl 2 — Half-Quadratic Splitting for Hyper-Laplacian Image Deconvolution

Knowl 3 — Closed-Form Image Subproblem Solution via Fast Fourier Transform

Knowl 4 — Separable 1D Auxiliary Variable Subproblem and Look-Up Table Solution

Knowl 5 — Optimal Root Selection Criterion for the 1D Hyper-Laplacian Subproblem

Knowl 6 — Analytic Solution of the 1D Subproblem for Exponent $\alpha = 1/2$

Knowl 7 — Analytic Solution of the 1D Subproblem for Exponent $\alpha = 2/3$