Exponential Image Enhancement in Daytime Fog Conditions Mihai Negru, Sergiu Nedevschi and Radu Ioan Peter Abstract— The images captured in fog conditions have degraded contrast, that makes current image processing applications sensitive and error prone. We propose in this paper an efficient single image enhancement algorithm suitable for daytime fog conditions and based on an original mathematical model, for computing the atmospheric veil, that takes into account the variation in fog density to the distance. This model is inspired by the functions that appear in partition of unity in the differential geometry field. When observing images captured in fog conditions, usually the fog has a very low density in front of the camera and this density has a non-linear increase with the distance, such that objects are no longer visible at greater distances. By using our mathematical model we are able to obtain superior reconstructions of the original fog-free image, when comparing to traditional methods. Another advantage of our method is the ability to adapt the model in accordance to the density of the fog. A quantitative and qualitative evaluation is performed on both synthetic and real camera images. This evaluation proves that our mathematical model is more suitable for image enhancement in both homogeneous and heterogeneous fog conditions. Our algorithm is able to perform image enhancement in real time for both color and gray scale images. I. INTRODUCTION Different natural phenomena can reduce the quality of im- ages and diminish the visibility. Such natural phenomena are haze, fog, mist, rain, etc. In these situations the visibility dis- tance is decreased because of the absorption and scattering of light by the atmospheric particles. The light that is reflected from objects in the captured scene is attenuated by scattering along the line of sight of the camera. Images of outdoor scenes, captured during fog conditions, are drastically de- graded. This weather phenomenon is especially dangerous in driving situations, because drivers tend to overestimate the visibility distance while traveling in fog conditions and drive with excessive speeds [1]. Due to the presence of fog, the visibility distance decreases exponentially, thus making fog one of the most dangerous weather condition for driving. Some of the negative effects of fog on the quality of the image are the loss of contrast and the alteration of the natural colors in the captured image. In addition the scattering effect of the transmitted light causes additional lightness in parts of the image [2]. These effect is called air-light or atmospheric Mihai Negru is with the Image Processing and Pattern Recognition Group, Computer Science Department, Technical University of Cluj-Napoca, Ro- mania. Mihai.Negru@cs.utcluj.ro Sergiu Nedevschi is the head of the Image Processing and Pattern Recognition Group, Computer Science Department, Technical University of Cluj-Napoca, Romania. Sergiu.Nedevschi@cs.utcluj.ro Radu Ioan Peter is with the Mathematics Depart- ment, Technical University of Cluj-Napoca, Romania. Ioan.Radu.Peter@math.utcluj.ro veil. In order to overcome these impediments we must either adapt the operating parameters of the camera or try to detect the presence of fog and remove its effects from the images. In this work we focus on the second approach, namely we are dealing with restoring the contrast and enhancing the quality of the original foggy image. Several algorithms were proposed in literature for restor- ing the contrast of foggy images. These methods can be categorized in two groups: model and non-model based enhancement techniques. Non-model based methods perform image enhancement relying only on the information obtained from the image; such as histogram equalization or adaptive histogram equalization [3], approaches based on Retinex theory [4]. Unfortunately, these methods do not maintain color fidelity and are not suitable for real time computer vision. Model based contrast restoration techniques can be further divided in two categories: with given depth and unknown depth. When the depth is supposed to be known, this information can be used to restore the original contrast of the image. The authors in [5], [6] and [7] studied different haze removal approaches based on given depth information. The depth is inferred by using the altitude, tilt and position of the camera [5], through the manual approximation of the sky area and vanishing point in the captured image [6] or by approximating the geometrical model of the analyzed image scene [7]. Because the depth information is provided by the user in all these above mentioned approaches and because the obtained depth information is erroneous and unreliable, these methods are not feasible for real world applications. Methods for restoring contrast without depth information are presented in [2], [8], [9] and [10]. They all use a single image for performing image enhancement and a mathemat- ical model that describes the fog in the image. Oakley [2] assumes that the distance between camera and the points in the scene is approximately constant, such that the air-light on the whole image is uniform. He then estimates the air-light by minimizing a cost function on the whole image. This cost function is a scaled version of the standard deviation of the normalized brightness in the image. This approach is only suitable for simple contrast loss correction of broadcast images, and fails in scenes where the distance to the scene points is not constant, such as driving scenarios. The method proposed by Tan in [8] restores the contrast of the original image by using a cost function in Markov Ran- dom Fields setting for estimating the air-light. The proposed method can produce halos at depth discontinuities. In [9] the authors introduce the dark channel prior (DCP), which states that in most of non-sky scenes at least one 2014 IEEE 17th International Conference on Intelligent Transportation Systems (ITSC) October 8-11, 2014. Qingdao, China 978-1-4799-6078-1/14/$31.00 ©2014 IEEE 1675