2009年CVPR最佳论文基于暗原色优先的去雾技术论文.pdf

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IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 33, NO. 12, DECEMBER 2011 2341 Single Image Haze Removal Using Dark Channel Prior Kaiming He, Jian Sun, and Xiaoou Tang, Fellow, IEEE 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 outdoor haze-free images. It is based on a key observation—most local patches in outdoor haze-free images contain some pixels whose intensity is very low 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 hazy images demonstrate the power of the proposed prior. Moreover, a high-quality depth map can also be obtained as a byproduct of haze removal. Index Terms—Dehaze, defog, image restoration, depth estimation. Ç 1 INTRODUCTION Haze removal1 (or dehazing) IMAGES of outdoor scenes are usually degraded by the turbid medium (e.g., particles and water droplets) in the atmosphere. Haze, fog, and smoke are such phenomena due to atmospheric absorption and scattering. The irradiance received by the camera from the scene point is attenuated along the line of sight. Furthermore, the incoming light is blended with the airlight [1]—ambient light reflected into the line of sight by atmospheric particles. The degraded images lose contrast and color fidelity, as shown in Fig. 1a. Since the amount of scattering depends on the distance of the scene points from the camera, the degradation is spatially variant. is highly desired in consumer/computational photography and computer vision applications. First, removing haze can significantly increase the visibility of the scene and correct the color shift caused by the airlight. In general, the haze-free image is more visually pleasing. Second, most computer vision algorithms, from low-level image analysis to high-level object recognition, usually assume that the input image (after radiometric calibration) is the scene radiance. The performance of many vision algorithms (e.g., feature detection, filtering, and photometric analysis) will inevita- bly suffer from the biased and low-contrast scene radiance. Last, haze removal can provide depth information and benefit many vision algorithms and advanced image 1. Haze, fog, and smoke differ mainly in the material, size, shape, and concentration of the atmospheric particles. See [2] for more details. In this paper, we do not distinguish these similar phenomena and use the term haze removal for simplicity. . K. He and X. Tang are with the Department of Information Engineering, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China. E-mail: {hkm007, xtang}@ie.cuhk.edu.hk. . J. Sun is with Microsoft Research Asia, F5, #49, Zhichun Road, Haldian District, Beijing, China. E-mail: jiansun@microsoft.com. Manuscript received 17 Dec. 2009; revised 24 June 2010; accepted 26 June 2010; published online 31 Aug. 2010. Recommended for acceptance by S.B. Kang. For information on obtaining reprints of this article, please send e-mail to: tpami@computer.org, and reference IEEECS Log Number TPAMISI-2009-12-0832. Digital Object Identifier no. 10.1109/TPAMI.2010.168. editing. Haze or fog can be a useful depth clue for scene understanding. A bad hazy image can be put to good use. However, haze removal is a challenging problem because the haze is dependent on the unknown depth. The problem is underconstrained if the input is only a single hazy image. Therefore, many methods have been proposed by using multiple images or additional information. Polarization- based methods [3], [4] remove the haze effect through two or more images taken with different degrees of polarization. In [5], [6], [7], more constraints are obtained from multiple images of the same scene under different weather conditions. Depth-based methods [8], [9] require some depth informa- tion from user inputs or known 3D models. Recently, single image haze removal has made significant progresses [10], [11]. The success of these methods lies on using stronger priors or assumptions. Tan [11] observes that a haze-free image must have higher contrast compared with the input hazy image and he removes haze by maximizing the local contrast of the restored image. The results are visually compelling but may not be physically valid. Fattal [10] estimates the albedo of the scene and the medium transmission under the assumption that the transmission and the surface shading are locally uncorrelated. This approach is physically sound and can produce impressive results. However, it cannot handle heavily hazy images well and may fail in the cases where the assumption is broken. In this paper, we propose a novel prior—dark channel prior—for single image haze removal. The dark channel prior is based on the statistics of outdoor haze-free images. We find that, in most of the local regions which do not cover the sky, some pixels (called dark pixels) very often have very low intensity in at least one color (RGB) channel. In hazy images, the intensity of these dark pixels in that channel is mainly contributed by the airlight. Therefore, these dark pixels can directly provide an accurate estimation of the haze transmis- sion. Combining a haze imaging model and a soft matting interpolation method, we can recover a high-quality haze- free image and produce a good depth map. Our approach is physically valid and is able to handle distant objects in heavily hazy images. We do not rely on 0162-8828/11/$26.00 ß 2011 IEEE Published by the IEEE Computer Society

2342 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 33, NO. 12, DECEMBER 2011 Fig. 1. Haze removal using a single image. (a) Input hazy image. (b) Image after haze removal by our approach. (c) Our recovered depth map. significant variance of transmission or surface shading. The result contains few halo artifacts. Like any approach using a strong assumption, our approach also has its own limitation. The dark channel prior may be invalid when the scene object is inherently similar to the airlight (e.g., snowy ground or a white wall) over a large local region and no shadow is cast on it. Although our approach works well for most outdoor hazy images, it may fail on some extreme cases. Fortunately, in such situations haze removal is not critical since haze is rarely visible. We believe that developing novel priors from different directions and combining them together will further advance the state of the art. 2 BACKGROUND In computer vision and computer graphics, the model widely used to describe the formation of a hazy image is [2], [5], [10], [11]: IðxÞ ¼ JðxÞtðxÞ þ Að1 tðxÞÞ; ð1Þ where I is the observed intensity, J is the scene radiance, A is the global atmospheric light, and t is the medium transmission describing the portion of the light that is not scattered and reaches the camera. The goal of haze removal is to recover J, A, and t from I. For an N-pixel there are 3N constraints and 4N þ 3 color image I, unknowns. This makes the problem of haze removal inherently ambiguous. In (1), the first term JðxÞtðxÞ on the right-hand side is called direct attenuation [11], and the second term Að1 tðxÞÞ is called airlight [11]. The direct attenuation describes the scene radiance and its decay in the medium, and the airlight results from previously scattered light and leads to the shift of the scene colors. While the direct [1], attenuation is a multiplicative distortion of radiance, the airlight is an additive one. the scene When the atmosphere is homogenous, the transmission t can be expressed as tðxÞ ¼ edðxÞ; ð2Þ where is the scattering coefficient of the atmosphere and d is the scene depth. This equation indicates that the scene radiance is attenuated exponentially with the depth. If we can recover the transmission, we can also recover the depth up to an unknown scale. Geometrically, the haze imaging equation (1) means that in RGB color space, the vectors A, IðxÞ, and JðxÞ are coplanar and their end points are collinear (see Fig. 2a). The transmission t is the ratio of two line segments: tðxÞ ¼ kA IðxÞk kA JðxÞk ¼ Ac IcðxÞ Ac J cðxÞ ; where c 2 fr; g; bg is the color channel index. ð3Þ Fig. 2. (a) Haze imaging model. (b) Constant albedo model used in Fattal’s work [10].

HE ET AL.: SINGLE IMAGE HAZE REMOVAL USING DARK CHANNEL PRIOR 2343 Fig. 3. Calculation of a dark channel. (a) An arbitrary image J. (b) For each pixel, we calculate the minimum of its (r, g, b) values. (c) A minimum filter is performed on (b). This is the dark channel of J. The image size is 800 551, and the patch size of is 15 15. Based on this model, Tan’s method [11] focuses on enhancing the visibility of the image. For a patch with uniform transmission t, the visibility (sum of gradient) of the input image is reduced by the haze since t < 1: krJðxÞk: krIðxÞk ¼ t krJðxÞk < X X X ð4Þ x x x The transmission t in a local patch is estimated by maximizing the visibility of the patch under a constraint that the intensity of JðxÞ is less than the intensity of A. An MRF model is used to further regularize the result. This approach is able to greatly unveil details and structures from hazy images. However, the output images usually tend to have larger saturation values because this method focuses solely on the enhancement of the visibility and does not intend to physically recover the scene radiance. Besides, the result may contain halo effects near the depth discontinuities. In [10], Fattal proposes an approach based on Indepen- dent Component Analysis (ICA). First, the albedo of a local patch is assumed to be a constant vector R. Thus, all vectors JðxÞ in the patch have the same direction R, as shown in Fig. 2b. Second, by assuming that the statistics of the surface shading kJðxÞk and the transmission tðxÞ are independent in the patch, the direction of R is estimated by ICA. Finally, an MRF model guided by the input color image is applied to extrapolate the solution to the whole image. This approach is physics-based and can produce a natural haze-free image together with a good depth map. But, due to the statistical this approach requires that the independent components vary significantly. Any lack of variation or low signal-to-noise ratio (often in dense haze region) will make the statistics unreliable. Moreover, as the statistic is based on color information, it is invalid for gray-scale images and it is difficult to handle dense haze that is colorless. independence assumption, In the next section, we present a new prior—dark channel prior—to estimate the transmission directly from an outdoor hazy image. 3 DARK CHANNEL PRIOR The dark channel prior is based on the following observa- tion on outdoor haze-free images: In most of the nonsky patches, at least one color channel has some pixels whose intensity are very low and close to zero. Equivalently, the minimum intensity in such a patch is close to zero. To formally describe this observation, we first define the concept of a dark channel. For an arbitrary image J, its dark channel J dark is given by ; min c2fr;g;bg J cðyÞ J darkðxÞ ¼ min y2 ðxÞ ð5Þ where J c is a color channel of J and ðxÞ is a local patch centered at x. A dark channel is the outcome of two minimum operators: minc2fr;g;bg is performed on each pixel (Fig. 3b), and miny2 ðxÞ is a minimum filter (Fig. 3c). The minimum operators are commutative. Using the concept of a dark channel, our observation says that if J is an outdoor haze-free image, except for the sky region, the intensity of J’s dark channel is low and tends to be zero: J dark ! 0: ð6Þ We call this observation dark channel prior. The low intensity in the dark channel is mainly due to three factors: a) shadows, e.g., the shadows of cars, buildings, and the inside of windows in cityscape images, or the shadows of leaves, trees, and rocks in landscape images; b) colorful objects or surfaces, e.g., any object with low reflectance in any color channel (for example, green grass/tree/plant, red or yellow flower/leaf, and blue water surface) will result in low values in the dark channel; c) dark objects or surfaces, e.g., dark tree trunks and stones. As the natural outdoor images are usually colorful and full of shadows, the dark channels of these images are really dark! To verify how good the dark channel prior is, we collect an outdoor image set from Flickr.com and several other image search engines using 150 most popular tags annotated by the Flickr users. Since haze usually occurs in outdoor landscape and cityscape scenes, we manually pick out the haze-free landscape and cityscape ones from the data set. Besides, we only focus on daytime images. Among them, we randomly select 5,000 images and manually cut out the sky regions. The images are resized so that the maximum of width and height is 500 pixels and their dark channels are computed using a patch size 15 15. Fig. 4 shows several outdoor images and the corresponding dark channels.

2344 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 33, NO. 12, DECEMBER 2011 Fig. 4. (a) Example images in our haze-free image database. (b) The corresponding dark channels. (c) A hazy image and its dark channel. Fig. 5a is the intensity histogram over all 5,000 dark channels and Fig. 5b is the corresponding cumulative distribution. We can see that about 75 percent of the pixels in the dark channels have zero values, and the intensity of 90 percent of the pixels is below 25. This statistic gives very strong support to our dark channel prior. We also compute the average intensity of each dark channel and plot the corresponding histogram in Fig. 5c. Again, most dark channels have very low average intensity, showing that only a small portion of outdoor haze-free images deviate from our prior. Due to the additive airlight, a hazy image is brighter than its haze-free version where the transmission t is low. So, the dark channel of a hazy image will have higher intensity in regions with denser haze (see the right-hand side of Fig. 4). Visually, the intensity of the dark channel is a rough approximation of the thickness of the haze. In the next section, we will use this property to estimate the transmis- sion and the atmospheric light. Note that we neglect the sky regions because the dark channel of a haze-free image may have high intensity here. Fortunately, we can gracefully handle the sky regions by using the haze imaging model (1) and our prior together. It is not necessary to cut out the sky regions explicitly. We discuss this issue in Section 4.1. Our dark channel prior is partially inspired by the well- known dark-object subtraction technique [12] widely used in multispectral remote sensing systems. In [12], spatially homogeneous haze is removed by subtracting a constant value corresponding to the darkest object in the scene. We generalize this idea and propose a novel prior for natural image dehazing. Fig. 5. Statistics of the dark channels. (a) Histogram of the intensity of the pixels in all of the 5,000 dark channels (each bin stands for 16 intensity levels). (b) Cumulative distribution. (c) Histogram of the average intensity of each dark channel.

HE ET AL.: SINGLE IMAGE HAZE REMOVAL USING DARK CHANNEL PRIOR 4 HAZE REMOVAL USING DARK CHANNEL PRIOR 4.1 Estimating the Transmission We assume that the atmospheric light A is given. An automatic method to estimate A is proposed in Section 4.3. We first normalize the haze imaging equation (1) by A: IcðxÞ Ac ¼ tðxÞ J cðxÞ Ac þ 1 tðxÞ: ð7Þ Note that we normalize each color channel independently. We further assume that the transmission in a local patch ðxÞ is constant. We denote this transmission as ~tðxÞ. Then, we calculate the dark channel on both sides of (7). Equivalently, we put the minimum operators on both sides: y2 ðxÞ min min IcðyÞ Ac þ 1 ~tðxÞ: c ¼ ~tðxÞ min y2 ðxÞ min c J cðyÞ Ac ð8Þ Since ~tðxÞ is a constant in the patch, it can be put on the outside of the min operators. As the scene radiance J is a haze-free image, the dark channel of J is close to zero due to the dark channel prior: ð9Þ ¼ 0: min J darkðxÞ ¼ min y2 ðxÞ As Ac is always positive, this leads to J cðyÞ c y2 ðxÞ min min c J cðyÞ Ac ¼ 0: ð10Þ Putting (10) into (8), we can eliminate the multiplicative term and estimate the transmission ~t simply by ~tðxÞ ¼ 1 min IcðyÞ Ac c : ð11Þ y2 ðxÞ min IcðyÞ Ac Þ is the dark channel of the It directly provides the . In fact, miny2 ðxÞðminc normalized hazy image IcðyÞ Ac estimation of the transmission. As we mentioned before, the dark channel prior is not a good prior for the sky regions. Fortunately, the color of the sky in a hazy image I is usually very similar to the atmospheric light A. So, in the sky region, we have IcðyÞ Ac ! 1; y2 ðxÞ min min c and (11) gives ~tðxÞ ! 0. Since the sky is infinitely distant and its transmission is indeed close to zero (see (2)), (11) gracefully handles both sky and nonsky regions. We do not need to separate the sky regions beforehand. In practice, even on clear days the atmosphere is not absolutely free of any particle. So the haze still exists when we look at distant objects. Moreover, the presence of haze is a fundamental cue for human to perceive depth [13], [14]. This phenomenon is called aerial perspective. If we remove the haze thoroughly, the image may seem unnatural and we may lose the feeling of depth. So, we can optionally keep a very small amount of haze for the distant objects by introducing a constant parameter ! (0 < ! 1) into (11): ~tðxÞ ¼ 1 ! min y2 ðxÞ min c IcðyÞ Ac : 2345 ð12Þ The nice property of this modification is that we adaptively keep more haze for the distant objects. The value of ! is application based. We fix it to 0.95 for all results reported in this paper. In the derivation of (11), the dark channel prior is essential for eliminating the multiplicative term (direct transmission) in the haze imaging model (1). Only the additive term (airlight) is left. This strategy is completely different from previous single image haze removal methods [10], [11], which rely heavily on the multiplicative term. These methods are driven by the fact that the multiplicative term changes the image contrast [11] and the color variance [10]. On the contrary, we notice that the additive term changes the intensity of the local dark pixels. With the help of the multiplicative term is discarded and the additive term is sufficient to estimate the transmission. We can further generalize (1) by the dark channel prior, IðxÞ ¼ JðxÞt1ðxÞ þ Að1 t2ðxÞÞ; ð13Þ where t1 and t2 are not necessarily the same. Using the method for deriving (11), we can estimate t2 and thus separate the additive term. The problem is reduced to a multiplicative form (JðxÞt1), and other constraints or priors can be used to further disentangle this term. In the literature of human vision research [15], the additive term is called a veiling luminance, and (13) can be used to describe a scene seen through a veil or glare of highlights. Fig. 6b shows the estimated transmission maps using (12). Fig. 6d shows the corresponding recovered images. As we can see, the dark channel prior is effective on recovering the vivid colors and unveiling low contrast objects. The transmission maps are reasonable. The main problems are some halos and block artifacts. This is because the transmission is not always constant in a patch. In the next section, we propose a soft matting method to refine the transmission maps. 4.2 Soft Matting We notice that the haze imaging equation (1) has a similar form as the image matting equation: I ¼ F þ Bð1 Þ; ð14Þ where F and B are foreground and background colors, respectively, and is the foreground opacity. A transmis- sion map in the haze imaging equation is exactly an alpha map. Therefore, we can apply a closed-form framework of matting [16] to refine the transmission. Denote the refined transmission map by tðxÞ. Rewriting tðxÞ and ~tðxÞ in their vector forms as t and ~t, we minimize the following cost function: EðtÞ ¼ tTLt þ ðt ~tÞTðt ~tÞ: ð15Þ Here, the first term is a smoothness term and the second term is a data term with a weight . The matrix L is called the matting Laplacian matrix [16]. Its ði; jÞ element is defined as

2346 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 33, NO. 12, DECEMBER 2011 Fig. 6. Haze removal. (a) Input hazy images. (b) Estimated transmission maps before soft matting. (c) Refined transmission maps after soft matting. (d), (e) Recovered images using (b) and (c), respectively. X ij 1 Fig. 6c shows the refined results using Fig. 6b as the constraint. As we can see, the halos and block artifacts are suppressed. The refined transmission maps manage to capture the sharp edge discontinuities and outline the profile of the objects. 4.3 Estimating the Atmospheric Light We have been assuming that the atmospheric light A is known. In this section, we propose a method to estimate A. In the previous works, the color of the most haze-opaque region is used as A [11] or as A’s initial guess [10]. However, little attention has been paid to the detection of the “most haze-opaque” region. In Tan’s work [11], the brightest pixels in the hazy image are considered to be the most haze-opaque. This is true only when the weather is overcast and the sunlight can be ignored. In this case, the atmospheric light is the only illumination source of the scene. So, the scene radiance of each color channel is given by ð18Þ where R 1 is the reflectance of the scene points. The haze imaging equation (1) can be written as JðxÞ ¼ RðxÞA; IðxÞ ¼ RðxÞAtðxÞ þ ð1 tðxÞÞA A: ð19Þ When pixels at infinite distance (t 0) exist in the image, the brightest I is the most haze-opaque and it approxi- mately equals A. Unfortunately, in practice we can rarely ignore the sunlight. Considering the sunlight S, we modify (18) by JðxÞ ¼ RðxÞðS þ AÞ; and (19) by IðxÞ ¼ RðxÞStðxÞ þ RðxÞAtðxÞ þ ð1 tðxÞÞA: ð20Þ ð21Þ kjði;jÞ2wk jwkj 1 þ ðIi kÞT k þ " ! ! 1ðIj kÞ ð16Þ where Ii and Ij are the colors of the input image I at pixels i and j, ij is the Kronecker delta, k and k are the mean and covariance matrix of the colors in window wk, U3 is a 3 3 identity matrix, " is a regularizing parameter, and jwkj is the number of pixels in the window wk. jwkj U3 ; The optimal t can be obtained by solving the following sparse linear system: ðL þ UÞt ¼ ~t; ð17Þ where U is an identity matrix of the same size as L. We set a small (104 in the experiments) so that t is softly constrained by ~t. The derivation of the matting Laplacian matrix in [16] is based on a color line assumption: The foreground/back- ground colors in a small local patch lie on a single line in the RGB color space. The color line assumption is also valid in the problem of haze removal. First, the scene radiance J is a natural image. According to [16], [17], the color line model holds for natural images. Second, the atmospheric light A is a constant, which of course satisfies the assumption. Therefore, it is valid to use the matting Laplacian matrix as the smoothness term in the haze removal problem. The closed-form matting framework has also been applied by Hsu et al. [18] to deal with the spatially variant white balance problem. In [16], [18], the constraint is known in some sparse regions, and this framework is mainly used to extrapolate the values into the unknown regions. In our application, the coarse constraint ~t has already filled the whole image. We consider this constraint as soft, and use the matting framework to refine the map. After solving the linear system (17), we perform a bilateral filter [19] on t to smooth its small scale textures.

HE ET AL.: SINGLE IMAGE HAZE REMOVAL USING DARK CHANNEL PRIOR 2347 Fig. 7. Estimating the atmospheric light. (a) Input image. (b) Dark channel and the most haze-opaque region. (c) The patch from where our method automatically obtains the atmospheric light. (d), (e) Two patches that contain pixels brighter than the atmospheric light. In this situation, the brightest pixel of the whole image can be brighter than the atmospheric light. They can be on a white car or a white building (Figs. 7d and 7e). As discussed in Section 3, the dark channel of a hazy image approximates the haze denseness (see Fig. 7b). So we can use the dark channel to detect the most haze-opaque region and improve the atmospheric light estimation. We first pick the top 0.1 percent brightest pixels in the dark channel. These pixels are usually most haze-opaque (bounded by yellow lines in Fig. 7b). Among these pixels, the pixels with highest intensity in the input image I are selected as the atmospheric light. These pixels are in the red rectangle in Fig. 7a. Note that these pixels may not be brightest ones in the whole input image. This method works well even when pixels at infinite distance do not exist in the image. In Fig. 8b, our method manages to detect the most haze-opaque regions. However, t is not close to zero here, so the colors of these regions may be different from A. Fortunately, t is small in these most haze-opaque regions, so the influence of sunlight is weak (see (21)). Therefore, these regions can still provide a good approximation of A. The haze removal result of this image is shown in Fig. 8c. This simple method based on the dark channel prior is more robust than the “brightest pixel” method. We use it to automatically estimate the atmospheric lights for all images shown in this paper. Fig. 8. (a) Input image. (b) Dark channel. The red pixels are the most haze-opaque regions detected by our method. (c) Our haze removal result. (d) Fattal’s haze removal result [10].

2348 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 33, NO. 12, DECEMBER 2011 Fig. 9. A haze-free image (600 400) and its dark channels using 3 3 and 15 15 patches, respectively. 4.4 Recovering the Scene Radiance With the atmospheric light and the transmission map, we can recover the scene radiance according to (1). But the direct attenuation term JðxÞtðxÞ can be very close to zero when the transmission tðxÞ is close to zero. The directly recovered scene radiance J is prone to noise. Therefore, we restrict the transmission tðxÞ by a lower bound t0, i.e., we preserve a small amount of haze in very dense haze regions. The final scene radiance JðxÞ is recovered by JðxÞ ¼ IðxÞ A maxðtðxÞ; t0Þ þ A: ð22Þ A typical value of t0 is 0.1. Since the scene radiance is usually not as bright as the atmospheric light, the image after haze removal looks dim. So we increase the exposure of JðxÞ for display. Some final recovered images are shown in Fig. 6e. 4.5 Patch Size A key parameter in our algorithm is the patch size in (11). On one hand, the dark channel prior becomes better for a larger patch size because the probability that a patch contains a dark pixel is increased. We can see this in Fig. 9: the larger the patch size, the darker the dark channel. Consequently, (9) is less accurate for a small patch, and the recovered scene radiance is oversaturated (Fig. 10b). On the other hand, the assumption that the transmission is constant in a patch becomes less appropriate. If the patch size is too large, halos near depth edges may become stronger (Fig. 10c). Fig. 11 shows the haze removal results using different patch sizes. The image sizes are 600 400. In Fig. 11b, the patch size is 3 3. The colors of some grayish surfaces look oversaturated (see the buildings in the first row and the subimages in the second and the third rows). In Figs. 11c and 11d, the patch sizes are 15 15 and 30 30, respectively. The results appear more natural than those in Fig. 11b. This shows that our method works well for sufficiently large patch sizes. The soft matting technique is able to reduce the artifacts introduced by large patches. We also notice that the images in Fig. 11d seem to be slightly hazier than those in Fig. 11c (typically in distant regions), but the differences are small. In the remainder of this paper, we use a patch size of 15 15. 5 EXPERIMENTAL RESULTS In our experiments, we compute the minimum filter by van Herk’s fast algorithm [20], whose complexity is linear to the image size. In the soft matting step, we use the Precondi- tioned Conjugate Gradient (PCG) algorithm as our solver. It takes about 10-20 seconds to process a 600 400 image on a PC with a 3.0 GHz Intel Pentium 4 Processor. Figs. 1 and 12 show the recovered images and the depth maps. The depth maps are computed according to (2) and are up to an unknown scaling parameter . The atmo- spheric lights in these images are automatically estimated using the method described in Section 4.3 (indicated by the red rectangles in Fig. 12). As can be seen, our approach can unveil the details and recover vivid colors even in heavily Fig. 10. (a) A 600 400 hazy image and the recovered scene radiance using (b) 3 3 and (c) 15 15 patches, respectively (without soft matting). The recovered scene radiance is oversaturated for a small patch size, while it contains apparent halos for a large patch size.

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