A review on dark channel prior based image dehazing algorithms <s> Soft matting <s> Interactive digital matting, the process of extracting a foreground object from an image based on limited user input, is an important task in image and video editing. From a computer vision perspective, this task is extremely challenging because it is massively ill-posed - at each pixel we must estimate the foreground and the background colors, as well as the foreground opacity ("alpha matte") from a single color measurement. Current approaches either restrict the estimation to a small part of the image, estimating foreground and background colors based on nearby pixels where they are known, or perform iterative nonlinear estimation by alternating foreground and background color estimation with alpha estimation. In this paper we present a closed form solution to natural image matting. We derive a cost function from local smoothness assumptions on foreground and background colors, and show that in the resulting expression it is possible to analytically eliminate the foreground and background colors to obtain a quadratic cost function in alpha. This allows us to find the globally optimal alpha matte by solving a sparse linear system of equations. Furthermore, the closed form formula allows us to predict the properties of the solution by analyzing the eigenvectors of a sparse matrix, closely related to matrices used in spectral image segmentation algorithms. We show that high quality mattes can be obtained on natural images from a surprisingly small amount of user input. <s> BIB001 </s> A review on dark channel prior based image dehazing algorithms <s> Soft matting <s> 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. <s> BIB002
We found that the Gaussian and bilateral filters are effective for removing false color Fig. 12 The result of the Gaussian filter. a Hazy images. b Transmission map obtained using the local patch with the size 3 × 3. c Gaussian filtered transmission map using σ s = 5. d Transmission map obtained using the local patch with the size 15 × 15. e Gaussian filtered transmission map using σ s = 5 Fig. 13 The average RMSE between ground-truth and Gaussian filtered transmission maps using the FRIDA [46] . The RMSE results with respect to σ s when the local patch size is (a) 3 × 3, (b) 11 × 11, and (c) 15 × 15 textures in the transmission map. However, the transmission map should have a similar level of sharpness to the color image for dehazing, which is impossible if the color image is not used in the transmission map refinement. To this end, the original DCP-based dehazing algorithm  adopted the soft matting to refine the transmission map. From the observation that the degradation model in Eq. (6) is similar to the matting equation BIB001 , the refined transmission mapt is obtained by minimizing the following energy function: wheret is the transmission map to be refined and a weight λ controls the importance of the data term. It is demonstrated in BIB002 that the solution of Eq. (21) is equivalent to that of the following sparse linear equation: where U represents an identity matrix. Note that in order to exploit sharp details in the hazy image, the Laplacian matrix L is determined from the hazy image. We refer the readers for more details about image matting to BIB002 BIB001 . Figure 16 shows the refined transmission maps obtained by the soft matting. As can be seen, blurry edges in the transmission maps have been sharpened due to the use of color images. It should be noted here that the bilateral filter was also applied to the result of the soft matting to further refine the transmission map  . To evaluate the performance of the soft matting only, the bilateral filter is not applied. Figure 17 shows the quantitative performance of the soft matting. Different values of λ and patch sizes were used to find out the dependency of the performance of the soft matting on the parameters. A large value of λ was preferred when a small local patch was used because the transmission map before the refinement tended to be inherently similar to the hazy image. When the local patch of the size 15 × 15 was used, a proper value of λ (=2 × 10 − 4 in our experiment) showed the best performance.