Multi-view Consensus CNN for 3D Facial Landmark Placement

The rapid increase in the availability of accurate 3D scanning devices has moved facial recognition and analysis into the 3D domain. 3D facial landmarks are often used as a simple measure of anatomy and it is crucial to have accurate algorithms for automatic landmark placement. The current state-of-the-art approaches have yet to gain from the dramatic increase in performance reported in human pose tracking and 2D facial landmark placement due to the use of deep convolutional neural networks (CNN). Development of deep learning approaches for 3D meshes has given rise to the new subfield called geometric deep learning, where one topic is the adaptation of meshes for the use of deep CNNs. In this work, we demonstrate how methods derived from geometric deep learning, namely multi-view CNNs, can be combined with recent advances in human pose tracking. The method finds 2D landmark estimates and propagates this information to 3D space, where a consensus method determines the accurate 3D face landmark position. We utilise the method on a standard 3D face dataset and show that it outperforms current methods by a large margin. Further, we demonstrate how models trained on 3D range scans can be used to accurately place anatomical landmarks in magnetic resonance images.Comment: This is a pre-print of an article published in proceedings of the asian conference on computer vision 2018 (LNCS 11361). The final authenticated version is available online at: https://doi.org/10.1007/978-3-030-20887-5_4

Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds

Motivated by the vast success of deep convolutional networks, there is a great interest in generalizing convolutions to non-Euclidean manifolds. A major complication in comparison to flat spaces is that it is unclear in which alignment a convolution kernel should be applied on a manifold. The underlying reason for this ambiguity is that general manifolds do not come with a canonical choice of reference frames (gauge). Kernels and features therefore have to be expressed relative to arbitrary coordinates. We argue that the particular choice of coordinatization should not affect a network's inference -- it should be coordinate independent. A simultaneous demand for coordinate independence and weight sharing is shown to result in a requirement on the network to be equivariant under local gauge transformations (changes of local reference frames). The ambiguity of reference frames depends thereby on the G-structure of the manifold, such that the necessary level of gauge equivariance is prescribed by the corresponding structure group G. Coordinate independent convolutions are proven to be equivariant w.r.t. those isometries that are symmetries of the G-structure. The resulting theory is formulated in a coordinate free fashion in terms of fiber bundles. To exemplify the design of coordinate independent convolutions, we implement a convolutional network on the M\"obius strip. The generality of our differential geometric formulation of convolutional networks is demonstrated by an extensive literature review which explains a large number of Euclidean CNNs, spherical CNNs and CNNs on general surfaces as specific instances of coordinate independent convolutions.Comment: The implementation of orientation independent M\"obius convolutions is publicly available at https://github.com/mauriceweiler/MobiusCNN