How to measure the perceptual quality of natural images is an important problem in low level vision. It is known that the Mean Squared Error (MSE) is not an effective index to describe the perceptual fidelity of images. Numerous perceptual fidelity indices have been developed, while the representatives include the Structural SIMilarity (SSIM) index and its variants. However, most of those perceptual measures are nonlinear, and they cannot be easily dopted as an objective function to minimize in various low level vision tasks. Can MSE be perceptual fidelity aware after some minor adaptation? In this paper we propose a simple framework to enhance the perceptual fidelity awareness of MSE by introducing an l2-norm structural error term to it. Such a Structural MSE (SMSE) can lead to very competitive image quality assessment (IQA) results. More surprisingly, we show that by using certain structure extractors, SMSE can be further turned into a Gaussian smoothed MSE (i.e., the Euclidean distance between the original and distorted images after Gaussian smooth filtering), which is much simpler to calculate but achieves rather better IQA performance than SSIM. The so called Perceptual-fidelity Aware MSE (PAMSE) can have great potentials in applications such as perceptual image coding and perceptual image restoration.