### 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.

Original language | English (US) |
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Title of host publication | Advances in Neural Information Processing Systems 22 - Proceedings of the 2009 Conference |

Pages | 1033-1041 |

Number of pages | 9 |

State | Published - 2009 |

Event | 23rd Annual Conference on Neural Information Processing Systems, NIPS 2009 - Vancouver, BC, Canada Duration: Dec 7 2009 → Dec 10 2009 |

### Other

Other | 23rd Annual Conference on Neural Information Processing Systems, NIPS 2009 |
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Country | Canada |

City | Vancouver, BC |

Period | 12/7/09 → 12/10/09 |

### Fingerprint

### ASJC Scopus subject areas

- Information Systems

### Cite this

*Advances in Neural Information Processing Systems 22 - Proceedings of the 2009 Conference*(pp. 1033-1041)

**Fast image deconvolution using hyper-laplacian priors.** / Krishnan, Dilip; Fergus, Rob.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Advances in Neural Information Processing Systems 22 - Proceedings of the 2009 Conference.*pp. 1033-1041, 23rd Annual Conference on Neural Information Processing Systems, NIPS 2009, Vancouver, BC, Canada, 12/7/09.

}

TY - GEN

T1 - Fast image deconvolution using hyper-laplacian priors

AU - Krishnan, Dilip

AU - Fergus, Rob

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84858712496&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84858712496&partnerID=8YFLogxK

M3 - Conference contribution

SN - 9781615679119

SP - 1033

EP - 1041

BT - Advances in Neural Information Processing Systems 22 - Proceedings of the 2009 Conference

ER -