High efficiency video coding (HEVC) is the latest video coding standard, and it has the best performance among all the existing standards. HEVC main still picture profile (HEVC-MSP) also achieves top performance in image compr-ession. In this paper, we propose a closed-form bit allocation approach to optimize the saliency-guided PSNR (viewed as perceptual distortion) such that the coding efficiency of HEVC-based image compression can be significantly improved from a subjective perspective. Specifically, a bit allocation formulation is established to minimize perceptual distortion with a constraint on bit-rates. Then, this formulation is solved using the proposed recursive Taylor expansion method with a closed-form solution. On the basis of our solution, a bit allocation and re-allocation process is developed in our approach to minimize perceptual distortion, meanwhile accurately controlling bit-rates. In addition, we provide both theoretical and numerical analyses of the computational complexity, verifying the little extra time cost of our approach. The experimental results demonstrate the superior performance of our approach over the state-of-the-art HEVC-MSP, and the BD-rate savings are approximately 40% and 24% for face and generic images, respectively.