Haze Image Enhancement Algorithm Based On Fractional Partial Differential Equation

Images captured under the natural condition are more or less blurred or missing pieces of key information,which causes inconvenience for practical application.Therefore,as a preprocessing process,image enhancement is an integral part of the image processing process.This paper mainly focuses on haze image enhancement algorithm based on partial differential equation.Through the analysis of the degradation process of haze image,the atmospheric scattering physical model is introduced into our algorithm.First,an integer order partial differential equation model is established;then,a partial differential equation model for haze image enhancement based on gradient field is established combining the preponderance of the fractional differential.Finally,a method for finding the optimal order of fractional differential is proposed,and an adaptive fractional partial differential equation model for haze image enhancement is established.(1)The image obtained by the traditional method of dark channel prior is obscure and of poor quality.The telegraph equation model which includes bidirectional diffusion and adaptive shock filtering is constructed to highlight the texture detail information,enhance the image brightness,reduce the noise and improve the image quality.The model has the common advantage of parabolic equation and hyperbolic equation in time domain,which can sufficiently preserve the edge of high oscillation region.In space,the diffusion term with bidirectional diffusivity ensures that the image go on a forward diffusion process in the smooth region and a backward diffusion process in the edge region.The sharpening term is constructed by the shock filter,combining the edge detector,which weight coefficient is adaptively selected by the gradient feature of the image.Thus,the image can be adaptively enhanced according to the characteristics of the different regions.In addition,adding a fidelity term in the model can ensure that the enhanced image preserves the main features of the original image and prevents image distortion.Finally a high quality ideal image can be obtained by solving the proposed partial differential equation on the basis of finite difference method.The experimental result shows that posed algorithm has obvious enhancement effect visually.Besides,we also prove the effectiveness of the algorithm by calculating the information entropy and the average gradient.The information entropy can be 3% ~ 21% higher than the contrast algorithms,and the average gradient can be 0.5 ~ 2.1times higher than the contrast algorithms.(2)Compared with integer order differential,fractional differential has good amplitude frequency characteristics and spatial global correlation.Therefore,the fractional gradient field of haze-free image is established by combining fractional differential with atmospheric scattering physical model.For highlighting the texture detail information of the image,avoiding the phenomenon that the edge is excessively enhanced or the detail texture is not enhanced completely,an enhancement function of the fractional gradient field is constructed,which can make the fractional gradient field enhance non-linearly with the change of the gradient modulus.In order to approximate the gradient field of the fog image to the enhancement gradient field,an energy functional is established in the gradient domain.And then the fractional partial differential equation model for haze image enhancement is obtained by the variational method.Finally,the finite difference method is used to solve the model numerically.The experimental results show that the proposed model can effectively improve the contrast and sharpness of the image while removing the fog.Hence the model is an effective haze image enhancement model.(3)For the image enhancement methods based on fractional-order partial differential equation,the order of fractional derivative is often obtained by experience or a large number of experiments for each image to be enhanced,which not only time-consuming but also do not give full play to the excellent characteristics of the fractional derivative.In this paper,firstly,an image enhancement model based on the inverse diffusion equation with Riesz fractional derivative was presented.Then,a number of haze images were enhanced by the model with different orders of fractional derivative,and their optimal orders are obtained through experiments,and the six quantities associated with brightness and texture feature of each haze image,such as mean,variance,skewness,kurtosis,two-norm and contrast,were calculated.By regression analysis,the linear relationship between the optimal order of fractional derivative and these statistical characteristics was acquired.Finally empirical formula of the order of fractional derivative for the differential equation model was given.The experimental results show that the adaptive order which is obtained by our approach is close to the optimal order by artificial experiments,and the enhancement effect is optimal.Compared with other image enhancement algorithms,our algorithm can improve brightness and contrast of the images and provide a better visual effect while defogging.Two objective evaluation indexes,information entropy and average gradient,also indicate the effectiveness of this method.