Quaternion softmax classifier

International audienceFor the feature extraction of red-blue-green (RGB) colour images, researchers usually deal with R, G and B channels separately to obtain three feature vectors, and then combine them together to obtain a long real feature vector. This approach does not exploit the relationships between the three channels of the colour images. Recently, attention has been paid to quaternion features, which take the relationships between channels into consideration and seem to be more suitable for representing colour images. However, there are only a few quaternion classifiers for dealing with quaternion features. To meet this requirement, a new quaternion classifier, namely, the quaternion softmax classifier is proposed, which is an extended version of the conventional softmax classifier generally defined in the complex (or real) domain. The proposed quaternion softmax classifier is applied to two of the most common quaternion features, that is, the quaternion principal components analysis feature and the colour image pixel feature. The experimental results show that the proposed method performs better than the quaternion back propagation neural network in terms of accuracy and convergence rate

Color Image Analysis by Quaternion-Type Moments

International audienceIn this paper, by using the quaternion algebra, the conventional complex-type moments (CTMs) for gray-scale images are generalized to color images as quaternion-type moments (QTMs) in a holistic manner. We first provide a general formula of QTMs from which we derive a set of quaternion-valued QTM invariants (QTMIs) to image rotation, scale and translation transformations by eliminating the influence of transformation parameters. An efficient computation algorithm is also proposed so as to reduce computational complexity. The performance of the proposed QTMs and QTMIs are evaluated considering several application frameworks ranging from color image reconstruction, face recognition to image registration. We show they achieve better performance than CTMs and CTM invariants (CTMIs). We also discuss the choice of the unit pure quaternion influence with the help of experiments. appears to be an optimal choice

Surface Networks

We study data-driven representations for three-dimensional triangle meshes, which are one of the prevalent objects used to represent 3D geometry. Recent works have developed models that exploit the intrinsic geometry of manifolds and graphs, namely the Graph Neural Networks (GNNs) and its spectral variants, which learn from the local metric tensor via the Laplacian operator. Despite offering excellent sample complexity and built-in invariances, intrinsic geometry alone is invariant to isometric deformations, making it unsuitable for many applications. To overcome this limitation, we propose several upgrades to GNNs to leverage extrinsic differential geometry properties of three-dimensional surfaces, increasing its modeling power. In particular, we propose to exploit the Dirac operator, whose spectrum detects principal curvature directions --- this is in stark contrast with the classical Laplace operator, which directly measures mean curvature. We coin the resulting models \emph{Surface Networks (SN)}. We prove that these models define shape representations that are stable to deformation and to discretization, and we demonstrate the efficiency and versatility of SNs on two challenging tasks: temporal prediction of mesh deformations under non-linear dynamics and generative models using a variational autoencoder framework with encoders/decoders given by SNs