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Single Image Haze Removal Using Dark Channel Prior HE, Kaiming A Thesis Submitted in Partial Fulﬁlment of the Requirements for the Degree of Doctor of Philosophy in Information Engineering The Chinese University of Hong Kong August 2011

Abstract of thesis entitled: Single Image Haze Removal Using Dark Channel Prior Submitted by HE, Kaiming supervised by Prof. TANG, Xiaoou for the degree of Doctor of Philosophy at The Chinese University of Hong Kong in August 2011 Haze is a natural phenomenon that obscures scenes, reduces visibility, and changes colors. It is an annoying problem for photographers since it degrades image quality. It is also a threat to the reliability of many applications, like outdoor surveillance, object detection, and aerial imaging. So removing haze from images is important in computer vision/graphics. But haze removal is highly challenging due to its mathematical ambiguity, typically when the input is merely a single image. In this thesis, we propose a simple but eﬀective image prior, called dark channel prior, to remove haze from a single image. The dark channel prior is a statistical property of out- door haze-free images: most patches in these images should contain pixels which are dark in at least one color channel. Using this prior with a haze imaging model, we can easily recover high quality haze-free images. Exper- iments demonstrate that this simple prior is powerful in various situations and outperforms many previous approaches. Speed is an important issue in practice. Like many computer vision prob- lems, the time-consuming step in haze removal is to combine pixel-wise con- straints with spatial continuities. In this thesis, we propose two novel tech- niques to solve this problem eﬃciently. The ﬁrst one is an unconventional large-kernel-based linear solver. The second one is a generic edge-aware ﬁl- ter which enables real-time performance. This ﬁlter is superior in various applications including haze removal, in terms of speed and quality. The human visual system is able to perceive haze, but the underlying mechanism remains unknown. In this thesis, we present new illusions showing that the human visual system is possibly adopting a mechanism similar to the dark channel prior. Our discovery casts new insights into human vision research in psychology and physiology. It also reinforces the validity of the dark channel prior as a computer vision algorithm, because a good way for artiﬁcial intelligence is to mimic human brains. i

Contents Abstract Contents 1 Introduction 1.1 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Background 2.1 Haze Imaging Model . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Image Degradation . . . . . . . . . . . . . . . . . . . . i ii 1 4 4 6 6 8 9 2.1.3 Problem Formulation and Ambiguity . . . . . . . . . . 11 2.2 Related Works . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.1 Multiple-Image Haze Removal . . . . . . . . . . . . . . 12 2.2.2 Single Image Haze Removal . . . . . . . . . . . . . . . 17 3 Dark Channel Prior and Single Image Haze Removal 22 3.1 Dark Channel Prior . . . . . . . . . . . . . . . . . . . . . . . . 23 3.1.1 Observation . . . . . . . . . . . . . . . . . . . . . . . . 23 3.1.2 Mathematical Formulation . . . . . . . . . . . . . . . . 25 3.1.3 Experimental Veriﬁcation . . . . . . . . . . . . . . . . 29 ii

3.2 A Novel Algorithm for Single Image Haze Removal . . . . . . 32 3.2.1 Transmission Estimation . . . . . . . . . . . . . . . . . 33 3.2.2 Soft Matting . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2.3 Atmospheric Light Estimation . . . . . . . . . . . . . . 39 3.2.4 Scene Radiance Recovery . . . . . . . . . . . . . . . . . 43 3.2.5 Implementation . . . . . . . . . . . . . . . . . . . . . . 45 3.2.6 Relation to Previous Methods . . . . . . . . . . . . . . 46 3.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . 48 3.3.1 Patch Size . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3.2 Results of Our Method . . . . . . . . . . . . . . . . . . 49 3.3.3 Comparisons with Previous Methods . . . . . . . . . . 51 3.3.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . 57 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4 Eﬃcient Solutions to Reﬁnement 64 4.1 A Large-Kernel-Based Linear Solver . . . . . . . . . . . . . . . 65 4.1.1 Related Works: Linear Solvers . . . . . . . . . . . . . . 65 4.1.2 Matting Laplacian Matrix . . . . . . . . . . . . . . . . 66 4.1.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.1.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . 76 4.1.5 Conclusion and Discussion . . . . . . . . . . . . . . . . 77 4.2 Guided Image Filtering . . . . . . . . . . . . . . . . . . . . . . 78 4.2.1 Related Works: Edge-aware Filtering . . . . . . . . . . 80 4.2.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.2.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.2.4 Experiments and Applications . . . . . . . . . . . . . . 94 4.2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 101 5 Dark Channel Prior and Human Vision 103 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.2 Related Works . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 iii

5.3 Illusion Experiments . . . . . . . . . . . . . . . . . . . . . . . 107 5.4 Proposed Model . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.5 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . 116 6 Conclusion A Physical Model 118 120 A.1 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 A.2 Direct Attenuation . . . . . . . . . . . . . . . . . . . . . . . . 121 A.3 Airlight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 A.4 Colorimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 B Mathematical Derivations 126 B.1 Derivation 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 B.2 Derivation 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 B.3 Derivation 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Bibliography 131 iv

Chapter 1 Introduction Haze is an atmospheric phenomenon where turbid media obscure the scenes. Haze brings troubles to many computer vision/graphics applications. It re- duces the visibility of the scenes and lowers the reliability of outdoor surveil- lance systems; it reduces the clarity of the satellite images; it also changes the colors and decreases the contrast of daily photos, which is an annoying problem to photographers (see Fig. 1.1 left1). Therefore, removing haze from images is an important and widely demanded topic in computer vision and computer graphics areas. The main challenge lies in the ambiguity of the problem. Haze attenuates the light reﬂected from the scenes, and further blends it with some additive light in the atmosphere. The target of haze removal is to recover the re- ﬂected light (i.e., the scene colors) from the blended light. This problem is mathematically ambiguous: there are an inﬁnite number of solutions given the blended light. How can we know which solution is true? We need to answer this question in haze removal. Ambiguity is a common challenge for many computer vision problems. In terms of mathematics, ambiguity is because the number of equations is smaller than the number of unknowns. The methods in computer vision to solve the ambiguity can roughly categorized into two strategies. The ﬁrst one is to acquire more known variables, e.g., some haze removal algorithms capture multiple images of the same scene under diﬀerent settings (like po- larizers). But it is not easy to obtain extra images in practice. The second strategy is to impose extra constraints using some knowledge or assumptions 1All the images in this thesis are best viewed in the electronic version. 1

Figure 1.1: Haze removal from a single image. Left: input hazy image. Right: haze removal result of our approach. known beforehand, namely, some “priors”. This way is more practical since it requires as few as only one image. To this end, we focus on single image haze removal in this thesis. The key is to ﬁnd a suitable prior. Priors are important in many computer vision topics. A prior tells the al- gorithm “what can we know about the fact beforehand” when the fact is not directly available. In general, a prior can be some statistical/physical prop- erties, rules, or heuristic assumptions. The performance of the algorithms is often determined by the extent to which the prior is valid. Some widely used priors in computer vision are the smoothness prior, sparsity prior, and symmetry prior. In this thesis, we develop an eﬀective but very simple prior, called the dark channel prior, to remove haze from a single image. The dark channel prior is a statistical property of outdoor haze-free images: most patches in these images should contain pixels which are dark in at least one color channel. These dark pixels can be due to shadows, colorfulness, geometry, or other factors. This prior provides a constraint for each pixel, and thus solves the ambiguity of the problem. Combining this prior with a physical haze imaging model, we can easily recover high quality haze-free images. Experiments demonstrate that our method is very successful in various situations (e.g., Fig. 1.1 right) and outperforms many previous approaches. Besides quality, speed is another concern in practical applications, typi- cally in real-time video processing and interactive image editing. The time- 2

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