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Facial pose variation is one of the major factors making face recognition (FR) a challenging task. One popular solution is to convert non-frontal faces to frontal ones on which face recognition is performed. Rotating faces causes the facial pixel value changes. Therefore, existing CNN-based methods learn to synthesize frontal faces in color space. However, this learning problem in color space is highly non-linear, causing the synthetic frontal faces to lose fine facial textures. In this work, we take the view that the nonfrontal-frontal pixel changes are essentially caused by geometric transformations (rotation, translation, etc) in space. Therefore, we aim to learn the nonfrontal-frontal facial conversion in spatial domain rather than the color domain to ease the learning task. To this end, we propose an Appearance Flow based Face Frontalization Convolutional Neural Network (A3F-CNN). Specifically, A3F-CNN learns to establish the dense correspondence between the non-frontal and frontal faces. Once the correspondence is built, frontal faces are synthesized by explicitly 'moving' pixels from the non-frontal one. In this way, the synthetic frontal faces can preserve fine facial textures. To improve the convergence of training, an appearance flow guided learning strategy is proposed. In addition, GAN loss is applied to achieve a more photorealistic face and a face mirroring method is introduced to handle the self-occlusion problem. Extensive experiments are conducted on face synthesis and pose invariant face recognition. Results show that our method can synthesize more photorealistic faces than existing methods in both controlled and uncontrolled lighting environments. Moreover, we achieve very competitive face recognition performance on the Multi-PIE, LFW and IJB-A databases.

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On the basis of the vectorial Rayleigh-Sommerfeld formulas and by means of the relation between Hermite and Laguerre polynomials, the analytical expressions for the propagation of the Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) beams beyond the paraxial approximation are derived, with the corresponding far-field propagation expressions and that for the Gaussian beams being given as special cases of the results. Some detailed comparisons of our results with the expansion series and paraxial expressions are made, which show the advantages of our results over the expansion series. With the results obtained, some typical intensity patterns of nonparaxial HG and LG beams are shown.