How to represent a face pattern? While it is presented in a continuous way in our visual system, computers often store and process the face image in a discrete manner with 2D arrays of pixels. In this study, we attempt to learn a continuous representation for face images with explicit functions. First, we propose an explicit model (EmFace) for human face representation in the form of a finite sum of mathematical terms, where each term is an analytic function element. Further, to estimate the unknown parameters of EmFace, a novel neural network, EmNet, is designed with an encoder-decoder structure and trained using the backpropagation algorithm, where the encoder is defined by a deep convolutional neural network and the decoder is an explicit mathematical expression of EmFace. Experimental results show that EmFace has a higher representation performance on faces with various expressions, postures, and other factors, compared to that of other methods. Furthermore, EmFace achieves reasonable performance on several face image processing tasks, including face image restoration, denoising, and transformation.

Scale variation is one of the most challenging problems in face detection. Modern face detectors employ feature pyramids to deal with scale variation. However, it might break the feature consistency across different scales of faces. In this paper, we propose a simple yet effective method named the receptive field pyramids (RFP) method to enhance the representation ability of feature pyramids. It can learn different receptive fields in each feature map adaptively based on the varying scales of detected faces. Empirical results on two face detection benchmark datasets, i.e., WIDER FACE and UFDD, demonstrate that our proposed method can accelerate the inference rate significantly while achieving state-of-the-art performance. The source code of our method is available at \url{https://github.com/emdata-ailab/EMface}.