## Single image recovery in scattering medium by propagating deconvolution |

Optics Express, Vol. 22, Issue 7, pp. 8114-8119 (2014)

http://dx.doi.org/10.1364/OE.22.008114

Acrobat PDF (2357 KB)

### Abstract

This paper proposed an effective method called *propagating deconvolution* to recover single image degraded in a scattering medium. The propagating deconvolution is in the manner of the calculus, which is based on multi-layered decomposition of the scattering volume and modeled blurring function of a single-layer scattering volume. Parameters of the deconvolution algorithm are estimated just from in situ measurement of the pure scattered background from one single image.

© 2014 Optical Society of America

## 1. Introduction

*g*(

*x, y*) is the blurred image,

*f*(

*x, y*) is the true image,

*h*(

*x, y*) is the point spread function (PSF),

*n*(

*x, y*) is the additive noise and

***denotes the convolution operation.

8. G. Wang, B. Zheng, and F. F. Sun, “Estimation-based approach for underwater image restoration,” Opt. Lett. **36**(13), 2384–2386 (2011). [CrossRef] [PubMed]

*propagating deconvolution*. Consequently, image decovolution can be realized flexibly by a back-tracking scheme without measure of the depth, approaching to the desired solution through an iterative routine in range dimension. The propagating nature of this method prevents computational instability and assures convergence. The physical model of the sub-systems and the implementation of the algorithm are described in details and the experimental results are given. What’s more, the experiments on color images show the advantage of the algorithm in color recovery.

## 2. Image formation by layered decomposition model

*m*layers along the symmetrical optical axis, denoted as

*z*. The model widely used to describe the formation of a hazy image is [2]

*E*is the observed intensity,

*J*is the scene radiance,

*A*is the air light,

*β*is the attenuation coefficient of the scattering medium and

*z*is the scene depth.

*h*

_{Δ}_{.}The whole blurring function is just the output of all the successive sub-systems in form

*h*

_{nΔ}= h_{Δ}^{(1)}*

*h*

_{Δ}^{(2)}*…*

*h*

_{Δ}^{(m)}, and

*h*

_{Δ}^{(1)}

*= h*

_{Δ}^{(2)}

*=*…

*= h*

_{Δ}^{(m)}

*= h*

_{Δ}..*Δ*(

*Δ*is sufficiently small); the single-layered blurring function is modeled as a Gaussian PSF [8

8. G. Wang, B. Zheng, and F. F. Sun, “Estimation-based approach for underwater image restoration,” Opt. Lett. **36**(13), 2384–2386 (2011). [CrossRef] [PubMed]

*z*is determined by the range between the target and the sensor and

*z*is

*k*(

*x, y*) as the unit-length scattering coefficient, the received back scattered light intensity generated by layer

*Δ = dz*at distance

*z*is

*h*(

_{z}*x, y*) is the PSF in the spatial domain associated with

*I*is the illumination intensity at the layer at distance

*z*. In the atmosphere, environmental illumination projects equal intensity on each layer. Thus, the illumination intensity at each layer is just

*I*=

*A*exp(-

*βz*) +

*A*(1-exp(-

*βz*)) =

*A*. In the frequency domain, we can write (6) as

*k*(

*x, y*) =

*k*

_{0}+

*γ*, where

*k*denotes the mean component and

_{0}*γ*denotes the random component respectively.

*γ*is allowed to be the white Gaussian noise with variance σ

^{2}, because the scattering generated by particles in a sufficiently thin layer is thought to be non-contributory. The mean component of the total back scattering intensity, associated to

*k*

_{0}is

*γ*can be represented as

## 3. Algorithm of propagating deconvolution

### 3.1 Principle of propagating deconvlution

*f*passes

*m*layers whose total length is

*z*, the received degraded image

*g*can be formulated as

*f*can be obtained through successive layer-based deconvolutions, each of which is a knowledge-based solution if the PSF of single layer has been predetermined. Without knowing the entire scattering volume, we can approach the final solution progressively, i.e., by the way of propagating deconvolution in range dimension.

*m*layer with a thickness of

^{th}*Δ*, according to (5), the PSF with respect to this layer is

*z*to

*z + Δ*, giving rise to the mean level

*S*and the power spectrum of the random component

_{d}^{(m)}*P*of this layer respectively:

_{n}^{(m)}*z*=

*mΔ*, then

*z*is thus

*I*=

*A*exp(-

*βz*). Without loss of essentiality, the formulation of (7) are alternatively formulated as (18).

*2 β*instead of

*β*in the formulation of (9), (10), (13) and (14).

*z*→∞ and

*m*→∞, there remains a pure scattering background. Fitting the observed scattering background to

*η*

_{1,}η_{2}and

*η*,.

_{3}### 3.2 Control of the propagating process

9. M. Pinchas and B. Z. Bobrovsky, “A maximum entropy approach for blind deconvolution,” Signal Process. **86**(10), 2913–2931 (2006). [CrossRef]

*f*is defined as [9

9. M. Pinchas and B. Z. Bobrovsky, “A maximum entropy approach for blind deconvolution,” Signal Process. **86**(10), 2913–2931 (2006). [CrossRef]

*p*is the frequency of occurrence of pixels at

_{i}*i*grey level,

^{th}*i*= 1, 2...

*L*, The iteration stopped when the updated image possessed the maximum of

*S*. It should be noted that the result of

*S*at each iteration relies on the selection of the step length, i.e., the preset

*α*.

## 4. Implementation and experimental results

*η*

_{1},

*η*

_{2}and

*η*

_{3}were evaluated and

*α*was preset, the original image can be regarded as

*f*

^{(}^{0}

*(*

^{)}*x, y*), and well developed deconvolution techniques can be applied through an iterative routine. The Wiener filter is widely used in image restoration and can be applied to implement the algorithm. The computational steps are as below and finally end by control criteria:

*S*in our experiments. However, there still exist uncertainties in the evaluation of the best recovery of the image, either caused by the disagreement between the objective metrics and the visual inspection, or due to the inappropriate quantification of the parameter

*α*. A sophisticated solution is to “slow down” the propagating steps when approaching to the best evaluation by the defined blur measure, to refine the recovered output to agree to visual inspection as much as possible. Thus we can change the parameter

*α*to smaller scales at the step before the maximum

*S*is being reached to obtain the final output image.

4. R. Fattal, “Single image dehazing,” J. ACM Siggraph **27**(3), 1–9 (2008). [CrossRef]

*S*.

*S*, while the method of He [5] performed the best for the sample on second row. It is noted that the propagating deconvolution algorithm runs by way of deblurring the scene at successive depth associated to the propagating step, i.e., the depth based local recovery because

*mα*corresponds to the volume of the scatter medium. Since the blur measure

*S*is in fact computed globally, for a scene with large depth of view, such as the second row in Fig. 3, the globally computed

*S*, but locally associated to a certain depth, is inevitably biased evaluated.

## 4. Conclusion

## Acknowledgments

## References and links

1. | S. G. Narasimhan and S. K. Nayar, “Chromatic framework for vision in bad weather,” in Proceedings of IEEE Conference on CVPR (IEEE, 2000), 598–605. [CrossRef] |

2. | Y. Schechner, S. G. Narasimhan, and S. K. Nayar, “Instant dehazing of images using polarization,” in Proceedings of IEEE Conference on CVPR (IEEE, 2001), |

3. | R. Tan, “Visibility in bad weather from a single image,” in Proceedings of IEEE Conference on CVPR (IEEE, 2008), 1–8. [CrossRef] |

4. | R. Fattal, “Single image dehazing,” J. ACM Siggraph |

5. | K. He, J. Sun, and X. Tang, “Single image haze removal using dark channel prior,” in Proceedings of IEEE Conference on CVPR (IEEE, 2009), 1956–1963. |

6. | W. Hou, D. J. Gray, A. D. Weidemann, and R. A. Arnone, “Comparison and validation of point spread models for imaging in natural waters,” Opt. Express |

7. | K. Iqbal, R. Abdul Salam, A. Osman, and A. Zawawi Talib, “Underwater image enhancement using an integrated color model” J. Comput. Sci. |

8. | G. Wang, B. Zheng, and F. F. Sun, “Estimation-based approach for underwater image restoration,” Opt. Lett. |

9. | M. Pinchas and B. Z. Bobrovsky, “A maximum entropy approach for blind deconvolution,” Signal Process. |

**OCIS Codes**

(100.0100) Image processing : Image processing

(100.1830) Image processing : Deconvolution

**ToC Category:**

Scattering

**History**

Original Manuscript: January 9, 2014

Revised Manuscript: February 27, 2014

Manuscript Accepted: March 20, 2014

Published: March 31, 2014

**Citation**

Rui Wang and Guoyu Wang, "Single image recovery in scattering medium by propagating deconvolution," Opt. Express **22**, 8114-8119 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-7-8114

Sort: Year | Journal | Reset

### References

- S. G. Narasimhan, S. K. Nayar, “Chromatic framework for vision in bad weather,” in Proceedings of IEEE Conference on CVPR (IEEE, 2000), 598–605. [CrossRef]
- Y. Schechner, S. G. Narasimhan, S. K. Nayar, “Instant dehazing of images using polarization,” in Proceedings of IEEE Conference on CVPR (IEEE, 2001), 1, 325–332.
- R. Tan, “Visibility in bad weather from a single image,” in Proceedings of IEEE Conference on CVPR (IEEE, 2008), 1–8. [CrossRef]
- R. Fattal, “Single image dehazing,” J. ACM Siggraph 27(3), 1–9 (2008). [CrossRef]
- K. He, J. Sun, X. Tang, “Single image haze removal using dark channel prior,” in Proceedings of IEEE Conference on CVPR (IEEE, 2009), 1956–1963.
- W. Hou, D. J. Gray, A. D. Weidemann, R. A. Arnone, “Comparison and validation of point spread models for imaging in natural waters,” Opt. Express 16(13), 9958–9965 (2008). [CrossRef] [PubMed]
- K. Iqbal, R. Abdul Salam, A. Osman, A. Zawawi Talib, “Underwater image enhancement using an integrated color model” J. Comput. Sci. 34, 2–12 (2007).
- G. Wang, B. Zheng, F. F. Sun, “Estimation-based approach for underwater image restoration,” Opt. Lett. 36(13), 2384–2386 (2011). [CrossRef] [PubMed]
- M. Pinchas, B. Z. Bobrovsky, “A maximum entropy approach for blind deconvolution,” Signal Process. 86(10), 2913–2931 (2006). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.