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Related Work

From half a decade ago comes this long talk about http://videolectures.net/mlss05au_vidal_gpca/video/1/Generalized Principal Component Analysis (GPCA).

Further literature review shows work on GPCA in the context of face recognition. There is very limited amount of work on the subject which applies to 3-D sets however. In their 2005 paper, Kong et al. [19](Singaporean group) looked at the application of GPCA to 2-D face analysis, having sought to overcome the curse of dimensional while not accepting too much noise in their training set. ``In this work,'' they explain, a ``[k]ernel-based 2DPCA (K2DPCA) scheme is developed and the relationship between K2DPCA and KPCA (Scholkopf et al., 1998) is explored. Experimental results in face image representation and recognition show the excellent performance of G2DPCA.'' The acronym G2DPCA stands for Generalized 2D Principal Component Analysis (G2DPCA) and the ``K'' stands for kernel. There are comparative graphs there, supposedly showing the different performance of the different `families' of PCA-based algorithms as tested on various datasets, e.g. ORL, UMIST, and Yale databases (experimental results correspond only to the former in some of the cases, but all three datasets are used for experimental purposes). Quoting the short conclusions section: ``A framework of Generalized 2D Principal Component Analysis is proposed to extend the original 2DPCA in three ways: firstly, the essence of 2DPCA is clarified. Secondly, a bilateral 2DPCA scheme is introduced to remove the necessity of more coefficients in representing an image in 2DPCA than in PCA. Thirdly, a kernel-based 2DPCA scheme is introduced to remedy the shortage of 2DPCA in exploring the higher-order statistics among the rows/columns of the input data.''

In a later paper, this one from Wu et al. in 2006 [36], Generalised PCA is applied to virtual faces to achieve face recognition in 2-D. The Yale Database is used for experiments and a recognition rate of 81.5% is reported, compared to 59% for PCA on its own and 68.5% for Fisher faces.

3-D Application

After hours of searching and researching, we were unable to find any work which utilises GPCA for interpretation or separation between 3-D datasets, e.g. finding trajectories that distinguish between individuals, bar facial expressions variation. Search was not restricted to just face recognition, either.

One possibility we have is to implement GPCA, demonstrating its advantage over a purely PCA approach (which can be further refined implementation-wise). Later on, a GMDS-based measure can be introduced or embedded into the framework, hopefully showing in an empirical fashion what would be considered an analytical correlation.

In GPCA, vectors perpendicular to points that represent lines or other elongated distributions (whose principal axis at the given dimension defines this line's direction) are used to determine the separability between an unknown number of different clouds, e.g. a set of shapes belonging to a common person/expression. In GMDS, the relationship being exploited is that of distances between analogous points in surfaces. If each sample was to be incorporated and formally defined by the assemblage of aggregated distances, for example (alas, ordered meaningfully), then the dimensionality is defined consistently such that each dimension in hyperspace corresponds to an innately meaningful distance in 3-D space. Since the points move in harmony on a face, the real dimensionality (not that reached and formed by concatenation) is actually a lot lower. PCA allows us to automatically find an analogous set of axes that capture the variation and decompose this effectively, sorting everything in an ascending order. For instance, we expect to find that when the mouth is opened there is a particular expansion in several dimensions and if the training set exhibits this relationship, then the description length of the model (a la MDL) will remain small and similarity therefore accordingly high, which is exactly what we want as it may be the same person and consistent with a purely expression-imposed difference. The main relevance of the generality of PCA (the G in GPCA) is that it facilitates capturing and then clustering groups of faces matching similar criteria already inherent in the model, thus occupying less space. For that to work as a similarity measure we can adhere to calculation of the product of the eigenvalues, which is a fast approach already proven and tested. The true importance of geodesic distances in this context (and work with isometrics) is that they provide an anatomically meaningful set of measures on a given surface, even a partial one (which presents a challenge to more primitive sampling on a grid if there is limitation on dimensionality. We cannot sample every single point as an observation as it makes vast matrices of 480x640 dimensions. Only a subset of that is truly essential and the more compact the signal, the better.

Roy Schestowitz 2012-01-08