Existing image dehazing algorithms have problems in restoring hazy images with bright areas, such as color distortion, color offset, and brightness reduction. Aiming at the shortcomings of existing methods, a single image dehazing algorithm is proposed based on exponential mapping and adaptive weight energy function. Firstly, according to statistical law of the dark channel prior, the attenuation characteristics of the exponential function are utilized to construct a dark channel mapping model between clear image and hazy image. Subsequentially, the estimated value of the transmission can be calculated based on that of obtained dark channel. Secondly, according to the Markov property of images, a Markov network-based adaptive weight energy function is constructed to optimize the transmission and the down-sampling method is used to reduce the algorithm complexity. Finally, the haze-free image is restored by using the optimized transmission and local atmospheric light. The experimental results show that the restored images of the proposed algorithm have clear visual effects and high color fidelity. And several objective evaluation parameters reach the highest values. The histogram correlation coefficient reaches 0.4521, which is 67.3% higher than those of the average performance of comparison algorithms. In summary, the proposed algorithm can effectively solve the recovery problems of hazy image with bright areas.