Face Recognition in Color Using Complex and Hypercomplex Representation

The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-540-72847-4_29Color has plenty of discriminative information that can be used to improve the performance of face recognition algorithms, although it is difficult to use it because of its high variability. In this paper we investigate the use of the quaternion representation of a color image for face recognition. We also propose a new representation for color images based on complex numbers. These two color representation methods are compared with the traditional grayscale and RGB representations using an eigenfaces based algorithm for identity verification. The experimental results show that the proposed method gives a very significant improvement when compared to using only the illuminance information.Work supported by the Spanish Project DPI2004-08279-C02-02 and the Generalitat Valenciana - Consellería d’Empresa, Universitat i Ciència under an FPI scholarship.Villegas, M.; Paredes Palacios, R. (2007). Face Recognition in Color Using Complex and Hypercomplex Representation. En Pattern Recognition and Image Analysis: Third Iberian Conference, IbPRIA 2007, Girona, Spain, June 6-8, 2007, Proceedings, Part I. Springer Verlag (Germany). 217-224. https://doi.org/10.1007/978-3-540-72847-4_29S217224Yip, A., Sinha, P.: Contribution of color to face recognition. Perception 31(5), 995–1003 (2002)Torres, L., Reutter, J.Y., Lorente, L.: The importance of the color information in face recognition. In: ICIP, vol. 3, pp. 627–631 (1999)Jones III, C., Abbott, A.L.: Color face recognition by hypercomplex gabor analysis. In: FG2006, University of Southampton, UK, pp. 126–131 (2006)Hamilton, W.R.: On a new species of imaginary quantities connected with a theory of quaternions. In: Proc. Royal Irish Academy, vol. 2, pp. 424–434 (1844)Zhang, F.: Quaternions and matrices of quaternions. Linear Algebra And Its Applications 251(1-3), 21–57 (1997)Pei, S., Cheng, C.: A novel block truncation coding of color images by using quaternion-moment preserving principle. In: ISCAS, Atlanta, USA, vol. 2, pp. 684–687 (1996)Sangwine, S., Ell, T.: Hypercomplex fourier transforms of color images. In: ICIP, Thessaloniki, Greece, vol. 1, pp. 137–140 (2001)Bihan, N.L., Sangwine, S.J.: Quaternion principal component analysis of color images. In: ICIP, Barcelona, Spain, vol. 1, pp. 809–812 (2003)Chang, J.-H., Pei, S.-C., Ding, J.J.: 2d quaternion fourier spectral analysis and its applications. In: ISCAS, Vancouver, Canada, vol. 3, pp. 241–244 (2004)Li, S.Z., Jain, A.K.: 6. In: Handbook of Face Recognition. Springer (2005)Gross, R., Brajovic, V.: An image preprocessing algorithm for illumination invariant face recognition. In: Kittler, J., Nixon, M.S. (eds.) AVBPA 2003. LNCS, vol. 2688, p. 1055. Springer, Heidelberg (2003)Lee, K., Ho, J., Kriegman, D.: Nine points of light: Acquiring subspaces for face recognition under variable lighting. In: CVPR, vol. 1, pp. 519–526 (2001)Zhang, L., Samaras, D.: Face recognition under variable lighting using harmonic image exemplars. In: CVPR, vol. 1, pp. 19–25 (2003)Villegas, M., Paredes, R.: Comparison of illumination normalization methods for face recognition. In: COST 275, University of Hertfordshire, UK, pp. 27–30 (2005)Turk, M., Pentland, A.: Face recognition using eigenfaces. In: CVPR, Hawaii, pp. 586–591 (1991)Bihan, N.L., Mars, J.: Subspace method for vector-sensor wave separation based on quaternion algebra. In: EUSIPCO, Toulouse, France (2002)XM2VTS (CDS00{1,6}), http://www.ee.surrey.ac.uk/Reseach/VSSP/xm2vtsdbLuettin, J., Maître, G.: Evaluation protocol for the extended M2VTS database (XM2VTSDB). IDIAP-COM 05, IDIAP (1998

Two-sided Clifford Fourier transform with two square roots of -1 in Cl(p,q)

We generalize quaternion and Clifford Fourier transforms to general two-sided Clifford Fourier transforms (CFT), and study their properties (from linearity to convolution). Two general \textit{multivector square roots} \in \cl{p,q} \textit{of} -1 are used to split multivector signals, and to construct the left and right CFT kernel factors. Keywords: Clifford Fourier transform, Clifford algebra, signal processing, square roots of -1 .Comment: 19 pages, 1 figur

Instantaneous frequency and amplitude of complex signals based on quaternion Fourier transform

The ideas of instantaneous amplitude and phase are well understood for signals with real-valued samples, based on the analytic signal which is a complex signal with one-sided Fourier transform. We extend these ideas to signals with complex-valued samples, using a quaternion-valued equivalent of the analytic signal obtained from a one-sided quaternion Fourier transform which we refer to as the hypercomplex representation of the complex signal. We present the necessary properties of the quaternion Fourier transform, particularly its symmetries in the frequency domain and formulae for convolution and the quaternion Fourier transform of the Hilbert transform. The hypercomplex representation may be interpreted as an ordered pair of complex signals or as a quaternion signal. We discuss its derivation and properties and show that its quaternion Fourier transform is one-sided. It is shown how to derive from the hypercomplex representation a complex envelope and a phase. A classical result in the case of real signals is that an amplitude modulated signal may be analysed into its envelope and carrier using the analytic signal provided that the modulating signal has frequency content not overlapping with that of the carrier. We show that this idea extends to the complex case, provided that the complex signal modulates an orthonormal complex exponential. Orthonormal complex modulation can be represented mathematically by a polar representation of quaternions previously derived by the authors. As in the classical case, there is a restriction of non-overlapping frequency content between the modulating complex signal and the orthonormal complex exponential. We show that, under these conditions, modulation in the time domain is equivalent to a frequency shift in the quaternion Fourier domain. Examples are presented to demonstrate these concepts

A unique polar representation of the hyperanalytic signal

The hyperanalytic signal is the straight forward generalization of the classical analytic signal. It is defined by a complexification of two canonical complex signals, which can be considered as an inverse operation of the Cayley-Dickson form of the quaternion. Inspired by the polar form of an analytic signal where the real instantaneous envelope and phase can be determined, this paper presents a novel method to generate a polar representation of the hyperanalytic signal, in which the continuously complex envelope and phase can be uniquely defined. Comparing to other existing methods, the proposed polar representation does not have sign ambiguity between the envelope and the phase, which makes the definition of the instantaneous complex frequency possible.Comment: 2014 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP

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