A new Gaussian curvature of the image surface based variational model for haze or fog removal

PLOS ONE RESEARCH ARTICLE A new Gaussian curvature of the image surface based variational model for haze or fog removal Muhammad Arif1☯, Noor Badshah1☯, Tufail Ahmad Khan1☯, Asmat Ullah ID1☯*, Hena Rabbani1‡, Hadia Atta2‡, Nasra Begum3‡ a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 1 Department of Basic Sciences, University of Engineering and Technology, Peshawar, Pakistan, 2 Department of Mathematics, Islamia College Peshawar, Peshawar, Pakistan, 3 Department of Mathematics, Shaheed Benazir Bhutto Women University, Peshawar, Pakistan ☯ These authors contributed equally to this work. ‡ HR, HA and NB also contributed equally to this work. * Abstract OPEN ACCESS Citation: Arif M, Badshah N, Khan TA, Ullah A, Rabbani H, Atta H, et al. (2023) A new Gaussian curvature of the image surface based variational model for haze or fog removal. PLoS ONE 18(3): e0282568. https://doi.org/10.1371/journal. pone.0282568 Editor: Kapil Kumar Nagwanshi, Guru Ghasidas Vishwavidyalaya: Guru Ghasidas University, INDIA Received: June 20, 2022 Accepted: February 20, 2023 Published: March 23, 2023 Peer Review History: PLOS recognizes the benefits of transparency in the peer review process; therefore, we enable the publication of all of the content of peer review and author responses alongside final, published articles. The editorial history of this article is available here: https://doi.org/10.1371/journal.pone.0282568 Copyright: © 2023 Arif et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability Statement: All relevant data is available within the paper. Funding: All authors themselves will contribute for funding of the manuscript publication fee. Outdoor images are usually affected by haze which limits the visibility and reduces the contrast of the images. Removal of haze from real-world images is always a challenging task. Recently, many mathematical models have been proposed for the effective removal of haze from real-world images. However, these models may produce staircase effects or lower the image contrast or smooth the edges of the object. In this paper, we propose a model based on Gaussian curvature for the de-hazing of images. The atmospheric veil estimate is included based on dark channel prior (DCP), which can significantly reduce the artifacts on the edge of the image and increase the accuracy. The transmission map then changes to a high-quality map to reduce haze or fog from gray and color images. DCP combined with Gaussian curvature is done for the first time for image de-hazing/de-fogging. The augmented Lagrangian method is used to find the minimizer of the proposed functional, which will be a system of partial differential equations. To get fast convergence, fast Fourier transforms (FFT) is used to solve the system of PDEs. The performance of the proposed model is compared with other state-of-the-art models qualitatively and quantitatively. The proposed model is tested on various real and synthetic images which show better efficiency in staircase effects reduction, haze/fog removal, image contrast, corners, and sharp edges conservation respectively. 1 Introduction The atmospheric light is dispersed in different angles due to the presence of atmospheric particles (e.g. fog, haze, smog, smoke, and mist) in the air. The incoming light blended to the layer of ambient light (air-light), which reflects atmospheric particles in the line of sight, depending on the turbidity and distance from the scene to the observer (visual range), causing low contrast, reduced visibility, distortion of colour, deterioration of essential elements of the PLOS ONE | https://doi.org/10.1371/journal.pone.0282568 March 23, 2023 1 / 29 PLOS ONE Competing interests: The authors have declared that no competing interests exist. A new Gaussian curvature of the image surface based variational model for haze or fog removal photography scene and makes lots of trouble in the recognition and detection of a target in video surveillance system by blocking the direct scene transmission. Therefore, it is primal imperative to improve visibility and restore the essential features of the scene with a simple and effectual image restoration algorithm. The removal of haze or fog is considered to be an important procedure since haze or fogfree images are visually pleasing and can improve performance of computer vision tasks like object detection, classification, visual navigation, etc. According to Koschmieder [1], a degraded scene formed as displayed in Fig 1 can be formulated mathematically as follows IðxÞ ¼ TðxÞJðxÞ þ I1 ð TðxÞ þ 1Þ; ð1Þ where the observed hazy or foggy image intensity is I(x), the intensity of the scene is J(x), the intensity of the atmospheric light is I1 and T(x) = e−βd(x) is the transmission map (scene reflected light captured by the signal receiver depending upon the amount of haze) corresponds to values between 0 (no visibility) and 1 (clear visibility) with degradation coefficient β and distance d(x) from scene point to the observer. The first term T(x)J(x) in the R.H.S of Eq (1) is the direct attenuation, describes the radiance of the scene while the second term I1(−T (x) + 1) is the air-light (result of the scattered atmospheric light due to atmospheric particles), twists the radiance of the scene. The key task to get J from an observed hazy frame I is the T estimate and I1 atmospheric light. Many researchers have made significant progress in the estimation of transmission T using a single image [2–4] and took the average value from I or I as the atmospheric light I1 as its brightest pixel. It should be noted that inaccurate assumption or estimation of atmospheric light I1 leads towards imprecise results in the recuperation of J haze free image. Without the use of rational processes, several methods assume the pixels with dense fog to be pure white and take the transmissions in a direct relation with hazy colors [5–7]. Due to providing unsatisfactory transmission map they proceed towards Laplacian matting algorithm or guided filter algorithm for refinement. The inappropriate assumption of linearity between transmission map and haze colors applying by these two algorithms reflects every variance of hazy colors on transmission map without difference. Taking advantages of the physical properties of air-light map or atmospheric veil, Cho et al. [9] used a variational approach for the estimation of airlight map in order to restore a fog-free image. This approach can adequately remove the haze and satisfy the edge preserving property, but the restored images have very low contrast. For this purpose, a variety of histogram equalization methods were applied to make better the contrast of the restored images. During image de-hazing, it should be take into account that the acc (...truncated)