# Low-light image restoration using bright channel prior-based variational Retinex model

## Abstract

This paper presents a low-light image restoration method based on the variational Retinex model using the bright channel prior (BCP) and total-variation minimization. The proposed method first estimates the bright channel to control the amount of brightness enhancement. Next, the variational Retinex-based energy function is iteratively minimized to estimate the improved illumination and reflectance using the BCP. Contrast of the estimated illumination is enhanced using the gamma correction and histogram equalization to reduce a color distortion and noise amplification. Experimental results show that the proposed method can provide the better restored result than the existing methods without unnatural artifacts such as noise amplification and halo effects near edges.

### Keywords

Low-light image enhancement Retinex Bright channel prior## 1 Introduction

Various imaging systems that consist of an optical system and imaging sensor have been widely used in various industrial and consumer application fields such as advanced driver assistance systems (ADAS), surveillance systems, robot vision, and medical imaging [1]. To acquire the high-quality images, sophisticated but compact imaging systems are particularly useful by reducing the size of a sensor and increasing the pixel density. However, in low-light condition, it is hard to obtain the high-quality input image since the sensor cannot sufficiently react to the very small amount of incoming light. In addition, the interference of the light between the reduced pixels leads to the chromatic noise. As a result, the low-light artifacts make the post-processing step difficult such as object recognition, detection, and tracking. To solve this problem, various image brightness enhancement methods have been proposed in the literature.

Histogram-based methods enhance the contrast of an input image by redistributing the intensity bins or modifying the cumulative distribution function (CDF) at the low-computational cost [2, 3, 4, 5, 6, 7]. However, since the low-light image provides the narrow histogram distribution, the CDF has many abrupt increases, which results in the brightness saturation and color distortion.

Recently, the haze removal method is adapted to the low-light image enhancement because the inversion of the low-light input image shows the visually similar property to the hazy image [8]. This method is based on the statistical prior knowledge of the haze-free natural image called dark channel prior (DCP) [9]. The DCP is estimated by finding the minimum value among the R, G, and B pixels, and it is used to estimate the transmission map of the non-haze region. Although the haze removal-based method can better preserve the bright region than the histogram-based methods without brightness saturation, it cannot avoid the noise amplification and color distortion in the resulting image.

On the other hand, Retinex-based methods are based on the human color perception system [10, 11]. Provenzi et al. mathematically analyzed the Retinex algorithm and demonstrated the performance according to various parameters such as threshold and the number of path of light paths to a pixel [11]. The Retinex methods enhance the input image by eliminating the illumination component using the low-pass filtering and logarithmic transformation [12, 13, 14]. However, the resulting image shows the halo effect near the edges.

To solve this problem, the state-of-the-art Retinex methods incorporate with the variational optimization method using *l* _{1}- and *l* _{2}-norm minimization [15, 16]. Li *et al* proposed a variational Retinex method using the constraint term that minimizes the combined reflectance component and the image gradient to reduce the halo effect [17]. Ma et al. performed the variational optimization by minimizing the constraint term on the reflectance component using *l* _{1}-norm [18]. Fu et al. proposed the bright channel prior (BCP) to reduce the halo effect and color distortion using *l* _{2}-norm minimization on the illumination and reflectance components [19]. However, since the BCP is estimated in the image patch, it results in the blocking and halo artifacts in the resulting image.

In this paper, we present the low-light image restoration method using the variational optimization-based Retinex model and BCP. In order to reduce the blocking and halo artifacts, the proposed method estimates the pixel-wise bright channel. In addition, the *l* _{1}-norm minimization of the reflectance component provides the edge-preserving noise reduction without the halo effect near the edges. In order to reduce color distortion and brightness over-enhancement, the proposed method performs the local histogram equalization on the illumination component. Finally, the restored result is generated by combining the optimized reflectance and enhanced illumination components. Experimental results show that the proposed method can provide significantly enhanced result without the halo effect, noise amplification, and color distortion.

This paper is organized as follows. Section 2 briefly describes variational Retinex model and total variation as theoretical background. Section 3 presents the proposed variational Retinex model. Experimental results are shown in Section 4, and Section 5 concludes the paper.

## 2 Theoretical background

### 2.1 Variational Retinex model using bright channel prior

where *g* represents the observed image, *f* _{ L } is the illumination component, which can be regarded as the light directly entered into an imaging sensor, and *f* _{ R } is the ratio of the reflected light by the object.

where ∗ represents the convolution operator, *G* is the Gaussian low-pass filter, and log*f* _{ R } is the reflectance image [12]. This method assumes that the slowly changing illumination component can be estimated by the Gaussian low-pass filtered version of the input image. Next, the reflectance component is computed by subtracting the estimated illumination component from the input image. However, the halo effect around the edges is generated according to the size of the Gaussian low-pass filter.

To solve this problem, the multi-scale Retinex (MSR) algorithm estimates several illumination components using multiple, different Gaussian low-pass filters [13]. The resulting enhanced image is reconstructed using the weighted sum of multiple reflectance components. However, the MSR method cannot completely remove the halo effect near the edges.

*l*

_{1}- and

*l*

_{2}-norm minimization on the illumination and reflectance components was recently proposed [15]. Specifically, Fu et al. restored the low-light image using the bright channel prior on the reflectance component in the variational Retinex method without logarithmic transform and Gaussian low-pass filtering [19]. The energy function is defined as

where *g* and *g* _{ b } represent the input low-light image and its bright channel, respectively. \(\left \| f_{R}f_{L}-g \right \|^{2}_{2}\) represents the data fidelity term, and \(\left \| \nabla f_{L} \right \|^{2}_{2}\) and \(\left \| \nabla f_{R} \right \|^{2}_{2}\) are the smoothness constraint term on the illumination and reflectance components. \(\left \| f_{L}-g_{b} \right \|^{2}_{2}\) penalizes the brightness of illumination component. *α*, *β*, and *γ* represent positive regularization parameters.

*g*

_{ b }is defined as

where *ω*(*x*) represents the image patch whose center is located at *x* and *c*∈{*R,G,B*} [19].

*f*

_{ R }≤1. Therefore, the constraint term \(\left \| f_{L}-g_{b} \right \|^{2}_{2}\) in (3) can be derived by taking the maximum operation on both sides of (1) as

where \(\tilde {f_{L}}\) represents the illumination component of the image patch. The constraint term \(\left \| f_{L}-g_{b} \right \|^{2}_{2}\) implies that the illumination component of an image is close to the bright channel to prevent the over-enhanced result.

### 2.2 Edge-preserving noise reduction using total variation

where *g* and *f* respectively represent the noisy and noise-free images and *η* is the additive white Gaussian noise.

*l*

_{1}-norm minimization [20]. The total variation-based noise reduction is defined as

*f*∥

_{1}the total variation constraint term on the smoothness of the resulting image, and

*λ*represents the regularization parameter. The energy function in (8) can be solved using the Euler-Lagrange equation as

The edges are preserved according to the magnitude of gradient of *f*. In the edge region, since the magnitude becomes lager than the flat region, the strong edges in the input noisy image are preserved in the resulting image.

## 3 Low-light image restoration method using the bright channel prior

In this paper, we present a variational retinex model using *l* _{1}- and *l* _{2}-norm minimization to enhance a low-light image. The reflectance component consists of rapidly changing high-frequency components such as edge and noise. For this reason, the proposed method estimates the illumination and reflectance components using *l* _{2}- and *l* _{1}-norm minimization, respectively, to suppress noise amplification while preserving the edge. Next, in order to prevent over-enhancement of the reflectance component, brightness of the estimated illumination component is corrected using histogram equalization and the sigmoid function.

where *λ* _{1}, *λ* _{2}, and *λ* _{3} represent the regularization parameters, \(\left \| f_{R}f_{L}-g \right \|^{2}_{2}\) the data-fidelity term, \(\left \| \nabla f_{L} \right \|^{2}_{2}\) and ∥∇*f* _{ R }∥_{1}, respectively, the smoothness constraints on the illumination and reflectance, and \(\left \| f_{L}-\hat {g}_{b} \right \|^{2}_{2}\) the data fidelity term between the illumination component and the bright channel.

### 3.1 Pixel-wise bright channel estimation

Existing Retinex-based methods estimate the illumination component using a Gaussian low-pass filter to extract the spatially smoothness component [15]. However, the estimated illumination component does not match the human visual system (HVS) near an edge since the estimated illumination component is continuous near the edge. The incorrectly estimated illumination component results in the halo effect.

In order to apply the discontinuity to the illumination component, Fu et al. proposed the bright channel which can suppress the halo effect [19]. However, this method cannot generate the optimal illumination component because of the blocking artifact caused by the patch-wise bright channel.

*Ω* represents the image patch whose center is located at *x*, *x* _{ i } is the ith pixel in the patch, *G* _{ r } and *G* _{ s } respectively are the range and spatial filters, and \(\hat {g}_{b}\) is the bright channel. The bilateral filtering process can effectively reduce the noise while preserving the edge. The enhanced bright channel enables to estimate the optimal illumination component as a constraint of the regularized minimization in (10).

### 3.2 Optimal reflectance and illumination components estimation

*f*

_{ R }is defined as

*l*

_{2}-norm minimization can reduce the noise with a large regularization parameter at the cost of blurred edge [19]. In order to perform edge-preserving denoising, the proposed method estimates the reflectance component, which contains high-frequency components, using

*l*

_{1}-norm minimization. The energy function in (15) can be solved using Euler-Lagrange equation as

*f*

_{ R }in (15) can be solved using the gradient descent method as

where *τ* represents the parameter that controls the converge speed.

At each iteration, \(f_{R}^{k}\) is forced to be in the range [0,1].

*f*

_{ R }, the illumination component is estimated by minimizing the energy function related to

*f*

_{ L }defined as

where *F* and *F* ^{−1} respectively represent the forward and backward FFT operators. Since \(f_{R}^{k}\) is forced to be in the range [0,1], \(f_{L}^{k}\) is forced to be larger value than *g* at each iteration [19].

*,*22]. Finally, the resulting image is reconstructed by multiplying the enhanced illumination and estimated reflectance as

## 4 Experimental results and discussion

In this section, to evaluate the performance of the proposed low-light enhancement method, the resulting image is compared with those of histogram-based [2,3], transmission map-based [8], variational optimization-based [23], and Retinex-based methods [14,16]. The regularization parameters *λ* _{1}, *λ* _{2}, and *λ* _{3} are determined to have the visually best enhancement result. The objective comparison of image enhancement performance is evaluated using the peak signal-to-noise ratio (PSNR) and structural similarity index measure (SSIM) [24]. The simulated low-light image were generated by decreasing 70% of the brightness and adding Gaussian noise of various standard deviations, such as *σ*=5,10,15, and 20.

### 4.1 Analysis on the regularization effect

*σ*=5. Figure 3 shows the results of proposed method using various different regularization parameters to analyze the effect of each regularization constraint. Figure 3 a, b shows the input and simulated low-light images, respectively. Figure 3 c shows the experimentally best enhancement result.

Figure 3 d shows a set of results with *λ* _{2}=0.1 and *λ* _{3}=0.9 and the three different values of *λ* _{1}, 1, 300, and 2000 from top to bottom. Since *λ* _{1} is a parameter related to the smoothness of the illumination component, the low- and high-frequency components are not sufficiently separated with a small *λ* _{1}. Therefore, both the illumination component and the resulting image contain noise. Figure 3 e shows a set of results with *λ* _{1}=300, *λ* _{3}=0.9, and three different values of *λ* _{2}, 0.01, 0.1, and 0.2 from top to bottom. Since *λ* _{2} is a parameter related to the smoothness of reflectance component, noise amplification cannot be suppressed with a small *λ* _{2}. However, when this parameter is set to too high, blurring artifacts occur. Figure 3 f shows a set of results with *λ* _{1}=300, *λ* _{2}=0.1, and three different values of *λ* _{3}, 0.05, 0.9, and 10 from top to bottom. Since *λ* _{3} is the parameter related to the data fidelity term which controls the brightness of the illumination component, noise amplification and color distortion are unavoidable with a small *λ* _{3}. Based on the observation, the optimal parameters are experimentally determined to produce satisfactory result.

### 4.2 Objective performance evaluation using simulated low-light images

Objective performance evaluation using PSNR and SSIM values with *σ*=5

[2] | [3] | [8] | [23] | [14] | [16] | Proposed | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | |

15.57 | 0.646 | 17.50 | 0.683 | 19.06 | 0.783 | 21.35 | 0.933 | 16.86 | 0.710 | 22.58 | 0.894 | | | |

16.57 | 0.662 | 18.92 | 0.676 | 18.36 | 0.722 | 20.25 | 0.889 | 17.10 | 0.645 | 21.21 | 0.819 | | | |

16.68 | 0.636 | 21.50 | 0.711 | 18.66 | 0.697 | 20.18 | 0.780 | 18.23 | 0.620 | 21.35 | 0.774 | | | |

16.87 | 0.720 | 18.69 | 0.735 | 19.40 | 0.821 | 20.49 | 0.876 | 16.68 | 0.706 | 22.44 | 0.896 | | | |

17.40 | 0.645 | 21.40 | 0.721 | 16.98 | 0.701 | 18.40 | 0.718 | 16.40 | 0.550 | 21.27 | 0.769 | | | |

16.48 | 0.590 | 21.69 | 0.715 | 19.09 | 0.671 | 21.18 | 0.806 | 17.50 | 0.566 | 21.26 | 0.741 | | |

Figure 5 a, b shows the ideal image and simulated low-light image with *σ*=5, respectively. Figure 5 c, d shows the results of histogram-based methods with saturation in the bright region. Figure 5 e shows the result of transmission map-based method that produces less saturation than the histogram-based methods. However, this method cannot avoid color distortion. Ravi et al. proposed variational optimization-based method using *l* _{1}-norm minimization that can suppress noise while preserving the sharp edge [23]. However, as shown in Fig. 5 f, it loses the dynamic range because of the brightness constraint term using gamma correction. Figure 5 g, h shows the result of Retinex-based methods with undesired artifacts. Although the Retinex-based variational optimization method provides a better enhanced result, it cannot suppress noise amplification. On the other hand, the result of proposed method, as shown in Fig. 5 i, shows the significantly enhanced result with minimum color distortion, saturation, and noise amplification than conventional enhancement methods.

Objective performance evaluation using PSNR and SSIM values with *σ*=10

[2] | [3] | [8] | [23] | [14] | [16] | Proposed | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | |

13.67 | 0.489 | 15.30 | 0.548 | 14.45 | 0.554 | 20.03 | 0.871 | 14.85 | 0.585 | 17.59 | 0.711 | | | |

14.52 | 0.486 | 16.59 | 0.557 | 15.72 | 0.567 | 19.04 | 0.775 | 15.52 | 0.546 | 17.12 | 0.623 | | | |

14.50 | 0.448 | 17.47 | 0.523 | 16.12 | 0.500 | 19.06 | 0.664 | 16.10 | 0.479 | 17.21 | 0.539 | | | |

14.83 | 0.596 | 16.17 | 0.629 | 15.76 | 0.650 | 19.12 | 0.818 | 14.83 | 0.599 | 17.72 | 0.733 | | | |

15.49 | 0.523 | 17.60 | 0.571 | 14.97 | 0.538 | 17.58 | 0.609 | 14.88 | 0.456 | 17.26 | 0.585 | | | |

14.31 | 0.396 | 17.48 | 0.496 | 15.27 | 0.423 | 19.73 | 0.667 | 15.65 | 0.426 | 17.26 | 0.500 | | |

Objective performance evaluation using PSNR and SSIM values with *σ*=15

[2] | [3] | [8] | [23] | [14] | [16] | Proposed | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | |

12.21 | 0.390 | 13.66 | 0.457 | 11.01 | 0.314 | 17.94 | 0.766 | 13.50 | 0.496 | 14.61 | 0.554 | | | |

12.94 | 0.391 | 14.73 | 0.471 | 13.25 | 0.418 | 17.28 | 0.651 | 14.22 | 0.468 | 14.52 | 0.484 | | | |

12.77 | 0.327 | 14.78 | 0.387 | 12.62 | 0.322 | 17.32 | 0.530 | 14.48 | 0.372 | 14.46 | 0.389 | | | |

13.26 | 0.504 | 14.31 | 0.549 | 14.53 | 0.595 | 17.33 | 0.729 | 13.56 | 0.521 | 14.90 | 0.597 | | | |

13.69 | 0.429 | 15.05 | 0.463 | 13.31 | 0.439 | 16.22 | 0.513 | 13.56 | 0.381 | 14.61 | 0.462 | | | |

12.62 | 0.284 | 14.78 | 0.360 | 13.94 | 0.354 | 17.69 | 0.522 | 14.19 | 0.335 | 14.47 | 0.355 | | |

Objective performance evaluation using PSNR and SSIM values with *σ*=20

[2] | [3] | [8] | [23] | [14] | [16] | Proposed | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | |

11.23 | 0.329 | 12.35 | 0.385 | 10.36 | 0.217 | 15.94 | 0.651 | 12.50 | 0.429 | 12.67 | 0.441 | | | |

11.94 | 0.333 | 13.23 | 0.398 | 10.23 | 0.289 | 15.67 | 0.549 | 13.20 | 0.409 | 12.78 | 0.392 | | | |

11.59 | 0.254 | 13.03 | 0.300 | 10.15 | 0.257 | 15.61 | 0.418 | 13.23 | 0.300 | 12.61 | 0.296 | | | |

12.11 | 0.436 | 12.93 | 0.481 | 10.93 | 0.361 | 15.67 | 0.637 | 12.62 | 0.464 | 13.09 | 0.500 | | | |

12.28 | 0.351 | 13.22 | 0.379 | 11.86 | 0.369 | 14.81 | 0.431 | 12.43 | 0.316 | 12.74 | 0.373 | | | |

11.58 | 0.225 | 13.07 | 0.280 | 10.95 | 0.202 | 15.84 | 0.410 | 13.10 | 0.276 | 12.65 | 0.269 | | |

### 4.3 Subjective evaluation using real low-light images

*f*

_{ R }using the gradient descent method, the processing time is longer than existing methods.

## 5 Conclusions

In this paper, a bright channel prior (BCP)-based variational Retinex model is presented to enhance the low-light image restoration. The existing low-light image enhancement methods cannot avoid undesired artifacts such as noise amplification and halo effect. To solve this problem, the proposed method simultaneously estimates the optimal illumination and reflectance components by minimizing the Retinex-based regularized energy functional to suppress noise amplification during brightness enhancement process. The constraint term related to the smoothness of reflectance component suppresses the noise while preserving the edge using *l* _{1}-norm minimization. In addition, the data-fidelity term on the illumination component prevents the halo effect near the edge. Experimental results show that the proposed method can provide better enhanced result than conventional low-light enhancement methods in the sense of both better brightness enhancement and less undesired artifact.

## Notes

### Funding

This research was supported by the Institute for Information & Communications Technology Promotion (IITP) grant funded by the Korean government (MSIP) (B0101-15-0525, Development of global multi-target tracking and event prediction techniques based on real-time large-scale video analysis) and Korea Aerospace Research Institute.

### Authors’ contributions

SP initiated the research and designed the experiments. BM, SK, and SY performed the experiments and analyzed the data. JP wrote the paper. All authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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