Single Image Haze Removal Using Dark Channel Prior

article2011TPAMI1,770 citationsBest Paper Award (CVPR 2009)

Introduces the dark channel prior, a simple statistical discovery that enables accurate estimation of haze thickness to recover clear outdoor scenes and generate depth maps from a single degraded photograph.

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Outdoor images frequently suffer from atmospheric haze, fog, and smoke, which scatter light, reduce contrast, and shift natural colors. This degradation impairs both human visual quality and downstream computer vision algorithms that assume clear scene radiance. Because haze density varies with unknown depth, restoring a clear image from a single picture has traditionally been an ill-posed problem requiring multiple exposures, polarization filters, or user-provided three-dimensional models.

The article demonstrates a novel, physically grounded method for haze removal from a single image using a newly identified statistical property called the dark channel prior. The core objective is to directly estimate haze thickness and restore clear scene radiance and depth information without requiring supplementary hardware or external 3D data.

The approach relies on an empirical observation across natural outdoor scenes: in local non-sky patches, at least one color channel typically exhibits very low intensity due to shadows, colorful surfaces, or dark objects. The authors validated this statistical prior on a dataset of 5,000 haze-free outdoor landscape and cityscape images. By integrating this prior into standard atmospheric scattering models, the method calculates haze transmission, refines edge boundaries using soft matting, and estimates atmospheric light from the most haze-opaque regions of the image.

The key findings confirm the effectiveness and robustness of this method. First, statistical analysis verified that approximately 75% of dark channel pixels in haze-free images have zero intensity, and 90% have intensities below 25 out of 255. Second, the algorithm effectively restores vivid colors and sharp structures even in heavily hazy scenes where previous methods fail due to faint color variation. Third, the process simultaneously extracts an accurate relative depth map as a direct by-product of dehazing. Fourth, the method prevents over-saturation and significantly suppresses halo artifacts around depth discontinuities, matching or exceeding the performance of techniques that rely on pre-existing 3D terrain models.

These findings provide immediate practical value for computer vision systems, aerial imaging, and photography by enhancing visibility and depth perception without expensive multi-sensor setups. However, the dark channel prior becomes invalid when scene objects are inherently bright, uniform, and match the atmospheric light without casting shadows (such as white marble buildings), which can lead to localized underestimations of transmission. Future work should integrate more complex atmospheric scattering models to handle non-uniform atmospheric effects, such as direct sunlight and horizon-specific color shifts.

Microsoft Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences The Chinese University of Hong Kong color images correlation analysis high resolution image dark channel prior single image haze removal image dehazing

He et al (2011).pdf

Paper: Single image dehazing, Raanan Fattal (2008). It formulates the core single-image dehazing problem using atmospheric scattering and surface shading statistics, establishing foundational concepts that the dark channel prior directly refines and improves upon.
Paper: A Closed-Form Solution to Natural Image Matting, Anat Levin et al. (2006). It introduces the matting Laplacian formulation used in the source paper for soft matting and refining coarse transmission maps.
Paper: Guided Image Filtering, Kaiming He et al. (2010). It introduces the guided filter, an edge-preserving smoothing filter designed as a fast alternative to the matting Laplacian for refining transmission maps in single image dehazing.

Paper: DehazeNet: An End-to-End System for Single Image Haze Removal, Bolun Cai et al. (2016). It advances single image dehazing from handcrafted heuristics like the dark channel prior to a learned, end-to-end convolutional neural network architecture for transmission map estimation.
Paper: AOD-Net: All-in-One Dehazing Network, Boyi Li et al. (2017). It re-formulates the atmospheric scattering model to bypass separate transmission and atmospheric light estimation in favor of an end-to-end unified restoration network.
Paper: Benchmarking Single-Image Dehazing and Beyond, Boyi Li et al. (2017). It builds a standardized, large-scale benchmark to systematically evaluate prior-based dehazing algorithms alongside deep learning alternatives across perceptual and task-driven criteria.
Paper: FFA-Net: Feature Fusion Attention Network for Single Image Dehazing, Xu Qin et al. (2019). It proposes an attention-driven deep feature fusion network to handle spatially uneven haze distributions that challenge classical statistical priors.

Cover for Single Image Haze Removal Using Dark Channel Prior

Abstract

In this paper, we propose a simple but effective image prior - dark channel prior to remove haze from a single input image. The dark channel prior is a kind of statistics of the haze-free outdoor images. It is based on a key observation - most local patches in haze-free outdoor images contain some pixels which have very low intensities in at least one color channel. Using this prior with the haze imaging model, we can directly estimate the thickness of the haze and recover a high quality haze-free image. Results on a variety of outdoor haze images demonstrate the power of the proposed prior. Moreover, a high quality depth map can also be obtained as a by-product of haze removal.

Topics

color images correlation analysis high resolution image single images digital image reference image

Keywords

dark channel prior single image haze removal image dehazing atmospheric light scene radiance soft matting depth map halo artifacts

1. Introduction
2. Background
3. Dark Channel Prior
4. Haze Removal Using Dark Channel Prior
4.1. Estimating the Transmission
4.2. Soft Matting
4.3. Recovering the Scene Radiance
4.4. Estimating the Atmospheric Light
5. Experimental Results
6. Discussions and Conclusions
References

Knowls

Knowl 1 — Dark Channel and Dark Channel Prior Definition
definition
For an arbitrary color image $J$ , the dark channel $J^{dark}(x)$ at pixel $x$ is defined as:

$J^{dark}(x) = \min_{c \in \{r, g, b\}} \left( \min_{y \in \Omega(x)} \left( J^c(y) \right) \right)$

where $J^c$ is the color channel $c \in \{r, g, b\}$ of $J$ , and $\Omega(x)$ is a local square patch centered at pixel $x$ .

The dark channel prior states that for natural, haze-free outdoor images, the intensity of $J^{dark}(x)$ is low and tends toward zero in non-sky regions:

$J^{dark}(x) \approx 0$

This low intensity stems from three primary image characteristics in outdoor scenes:
1. Shadows (such as cast shadows from buildings, vehicles, trees, and rocks).
2. Colorful objects or surfaces that have low reflectance in at least one color channel (such as green plants, red flowers, or blue water).
3. Inherently dark objects or surfaces with low reflectance across all channels (such as dark tree trunks and soil).
Knowl 2 — Empirical Statistics of Dark Channels in Outdoor Images
empirical result
The statistical validity of the dark channel prior was verified on a dataset of 5,000 haze-free outdoor cityscape and landscape images collected from image search engines using 150 popular user tags. The images were resized so that their maximum dimension was 500 pixels, sky regions were manually masked out, and dark channels were computed using a local patch size of $15 \times 15$ pixels.

The statistical findings are:
- Approximately 75% of the pixels in the dark channels have an intensity value equal to 0 (on an 8-bit scale $[0, 255]$ ).
- Approximately 90% of the pixels have an intensity value below 25.
- The histogram of the average dark channel intensity per image shows that the vast majority of haze-free outdoor images have near-zero average intensity, confirming that deviation from the prior is rare.
Knowl 3 — Coarse Transmission Estimation via Dark Channel Prior
equation
In the standard haze image formation model $I(x) = J(x)t(x) + A(1 - t(x))$ , $I$ is the observed hazy image, $J$ is the haze-free scene radiance, $A = (A^r, A^g, A^b)^T$ is the global atmospheric light vector, and $t(x)$ is the medium transmission. Under the assumption that transmission is locally constant $\tilde{t}(x)$ within patch $\Omega(x)$ , normalizing by $A^c$ and taking the spatial and channel minimum yields the coarse transmission estimate:

$\tilde{t}(x) = 1 - \omega \min_{c \in \{r, g, b\}} \left( \min_{y \in \Omega(x)} \left( \frac{I^c(y)}{A^c} \right) \right)$

where $\min_c (\min_{y \in \Omega(x)} (I^c(y)/A^c))$ is the dark channel of the normalized haze image.

The parameter $\omega \in (0, 1]$ is an aerial perspective constant (fixed to $\omega = 0.95$ ) that adaptively retains a small fraction of haze for distant objects to preserve depth cues.

In sky regions, $I^c(x) \approx A^c$ , so the normalized dark channel approaches 1 and $\tilde{t}(x) \to 0$ , naturally representing infinite scene depth without requiring prior sky segmentation.
Knowl 4 — Transmission Map Refinement Using Soft Matting Laplacian
model/method
Because the coarse transmission map $\tilde{t}(x)$ computed patch-wise contains block artifacts, it is refined to an edge-preserving continuous transmission map $t(x)$ by minimizing the soft matting cost function:

$E(t) = t^T L t + \lambda (t - \tilde{t})^T (t - \tilde{t})$

where $t$ and $\tilde{t}$ are the vectorized forms of the refined and coarse transmission maps, $\lambda = 10^{-4}$ is a regularization parameter, and $L$ is the Matting Laplacian matrix. The $(i, j)$ -th element of $L$ is defined as:

$L_{i,j} = \sum_{k \mid (i,j) \in w_k} \left( \delta_{i,j} - \frac{1}{|w_k|} \left( 1 + (I_i - \mu_k)^T \left( \Sigma_k + \frac{\varepsilon}{|w_k|} U_3 \right)^{-1} (I_j - \mu_k) \right) \right)$

where $I_i, I_j$ are the RGB colors at pixels $i$ and $j$ , $\delta_{i,j}$ is the Kronecker delta, $\mu_k$ is the $3 \times 1$ mean color vector in window $w_k$ , $\Sigma_k$ is the $3 \times 3$ color covariance matrix in window $w_k$ , $U_3$ is the $3 \times 3$ identity matrix, $|w_k|$ is the number of pixels in window $w_k$ , and $\varepsilon$ is a regularizing parameter.

The optimal transmission vector $t$ is computed by solving the linear system:

$(L + \lambda U)t = \lambda \tilde{t}$

where $U$ is an identity matrix of the same size as $L$ .
Knowl 5 — Atmospheric Light Estimation Using the Dark Channel
algorithm
Instead of selecting the globally brightest pixel in an image (which can be a white car, wall, or light source), the atmospheric light $A$ is estimated by finding the brightest pixel within the most haze-opaque regions identified by the dark channel:
Input: Haze image $I$ , dark channel $I^{dark}$
Output: Atmospheric light vector $A = (A^r, A^g, A^b)^T$
Identify the candidate set $P$ containing the top $0.1\%$ brightest pixels in $I^{dark}$
Find pixel $x^* = \arg\max_{x \in P} (\max_{c \in \{r,g,b\}} I^c(x))$
$A \leftarrow I(x^*)$
return $A$
Restricting the candidate set to the top 0.1% pixels in $I^{dark}$ ensures the search is localized to dense haze regions where airlight dominates scene radiance.
Knowl 6 — Scene Radiance Recovery and Lower Bounding
equation
Given the input hazy image $I(x)$ , the atmospheric light $A$ , and the refined transmission $t(x)$ , the scene radiance $J(x)$ is recovered by inverting the haze imaging equation:

$J(x) = \frac{I(x) - A}{\max(t(x), t_0)} + A$

where $t_0$ is a fixed transmission lower bound set to $t_0 = 0.1$ .

The threshold $t_0$ prevents division by values near zero in dense haze regions, protecting the recovered radiance from noise amplification while preserving a small amount of haze in extremely dense areas. Because haze-free scene radiance is typically dimmer than atmospheric light, the exposure of $J(x)$ is adjusted upward for display.
Knowl 7 — Single Image Haze Removal Procedure
algorithm
The complete end-to-end dehazing algorithm recovers the haze-free radiance $J$ and refined transmission map $t$ from a single RGB image $I$ :
Input: RGB haze image $I$ , patch size $15 \times 15$ , $\omega = 0.95$ , $\lambda = 10^{-4}$ , $t_0 = 0.1$
Output: Dehazed image $J$ , transmission map $t$
for each pixel $x$ in $I$ do
     $I^{dark}(x) \leftarrow \min_{c \in \{r,g,b\}} \min_{y \in \Omega(x)} I^c(y)$
$P \leftarrow \text{top } 0.1\% \text{ brightest pixels in } I^{dark}$
$x^* \leftarrow \arg\max_{x \in P} (\max_{c \in \{r,g,b\}} I^c(x))$
$A \leftarrow I(x^*)$
for each pixel $x$ in $I$ do
     $\tilde{t}(x) \leftarrow 1 - \omega \min_{c \in \{r,g,b\}} \min_{y \in \Omega(x)} (I^c(y) / A^c)$
Construct Matting Laplacian matrix $L$ from $I$
Solve $(L + \lambda U)t = \lambda \tilde{t}$ using Preconditioned Conjugate Gradient (PCG)
for each pixel $x$ and color channel $c$ do
     $J^c(x) \leftarrow (I^c(x) - A^c) / \max(t(x), t_0) + A^c$
return $J, t$
The local minimum operation is executed in $O(N)$ linear time with respect to image size $N$ using van Herk's fast algorithm. Solving the linear system with PCG takes 10–20 seconds for a $600 \times 400$ image on a 3.0 GHz processor.
Knowl 8 — Depth Map Reconstruction from Medium Transmission
model/method
In a homogeneous atmosphere with scattering coefficient $\beta$ , the medium transmission $t(x)$ decays exponentially with scene depth $d(x)$ :

$t(x) = e^{-\beta d(x)}$

Using the refined transmission map $t(x)$ , the relative scene depth $d(x)$ is reconstructed directly up to an unknown global scaling parameter $\beta$ :

$d(x) = -\frac{1}{\beta} \ln t(x)$

This produces a dense, depth-edge-preserving depth map of the scene as a direct byproduct of single-image haze removal.
Knowl 9 — Limitations and Failure Cases of the Dark Channel Prior
limitation
The dark channel prior fails under two principal scenarios:
1. Large, inherently bright objects without shadows: When a scene region contains objects whose intrinsic radiance is bright and uniform across all color channels (such as a large white marble surface or white car) with no cast shadows, $J^{dark}(x)$ is non-zero. The dark channel method interprets this high intensity as haze, underestimating transmission $\tilde{t}(x)$ and producing unnaturally dark, over-saturated output in those regions.
2. Invalidation of the atmospheric scattering model: The standard model $I(x) = J(x)t(x) + A(1 - t(x))$ assumes uniform atmospheric light and fails to capture non-homogeneous illumination effects, such as direct solar glare in the sky or wavelength-dependent Rayleigh scattering that introduces a bluish hue near the horizon.
Knowl 10 — Grayscale Image Dehazing Using Dark Channel Prior
model/method
The dark channel dehazing method applies to single-channel grayscale images $I(x)$ when sufficient local shadows or dark objects exist. Because color channels are absent, the cross-channel minimum is dropped, and the coarse transmission is given by:

$\tilde{t}(x) = 1 - \omega \min_{y \in \Omega(x)} \left( \frac{I(y)}{A} \right)$

where $A$ is the scalar atmospheric light estimated from the top 0.1% brightest pixels of the local minimum filtered image. Refinement uses the grayscale Matting Laplacian, and the grayscale scene radiance $J(x)$ is recovered via $J(x) = (I(x) - A)/\max(t(x), t_0) + A$ .

Coverage note — No substantial contributed material was omitted; all core components including prior definition, statistical verification, transmission estimation, matting refinement, atmospheric light estimation, radiance recovery, depth extraction, grayscale extension, and failure cases are included.

References

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Citation

MLA

Kaiming He, et al. “Single Image Haze Removal Using Dark Channel Prior”. IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 33, no. 12, 2011, pp. 2341–53, https://doi.org/10.1109/TPAMI.2010.168.

APA

Kaiming He, Jian Sun, & Xiaoou Tang. (2011). Single Image Haze Removal Using Dark Channel Prior. IEEE Transactions on Pattern Analysis and Machine Intelligence, 33(12), 2341–2353. https://doi.org/10.1109/TPAMI.2010.168

Chicago

Kaiming He, Jian Sun, and Xiaoou Tang. 2011. “Single Image Haze Removal Using Dark Channel Prior”. IEEE Transactions on Pattern Analysis and Machine Intelligence 33 (12): 2341–53. https://doi.org/10.1109/TPAMI.2010.168.

Harvard

Kaiming He, Jian Sun and Xiaoou Tang (2011) “Single Image Haze Removal Using Dark Channel Prior”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 33(12), pp. 2341–2353. Available at: https://doi.org/10.1109/TPAMI.2010.168.

Vancouver

1. Kaiming He, Jian Sun, Xiaoou Tang (2011) Single Image Haze Removal Using Dark Channel Prior. IEEE Transactions on Pattern Analysis and Machine Intelligence 33:2341–2353

BibTeX

@article{Kaiming_He_2011, title={Single Image Haze Removal Using Dark Channel Prior}, volume={33}, ISSN={2160-9292}, url={http://dx.doi.org/10.1109/TPAMI.2010.168}, DOI={10.1109/tpami.2010.168}, number={12}, journal={IEEE Transactions on Pattern Analysis and Machine Intelligence}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Kaiming He and Jian Sun and Xiaoou Tang}, year={2011}, month=Dec, pages={2341–2353} }

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Single Image Haze Removal Using Dark Channel Prior

Abstract

Topics

Keywords

Table of Contents

Knowls

Knowl 1 — Dark Channel and Dark Channel Prior Definition

Knowl 2 — Empirical Statistics of Dark Channels in Outdoor Images

Knowl 3 — Coarse Transmission Estimation via Dark Channel Prior

Knowl 4 — Transmission Map Refinement Using Soft Matting Laplacian

Knowl 5 — Atmospheric Light Estimation Using the Dark Channel

Knowl 6 — Scene Radiance Recovery and Lower Bounding

Knowl 7 — Single Image Haze Removal Procedure

Knowl 8 — Depth Map Reconstruction from Medium Transmission

Knowl 9 — Limitations and Failure Cases of the Dark Channel Prior

Knowl 10 — Grayscale Image Dehazing Using Dark Channel Prior

References

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