Theory and Replication of Work

As the first step of our work we wish to test the aforementioned approach for ourselves and replicate those results. Unfortunately, however, the training with PCA required a very large database of faces that are proprietary and cannot be accessed. Instead, we used a collection of many faces from hundreds of different individuals and used those for training. Given that prior groups working on this sort of challenge were unable to get high verification rates (e.g. Russ at al. [12], Mena-chalco at al. [10], and Gervei at al. [7] with rates of 83.3% for 540 3-D images from 60 individuals) we were hoping to reach success rates of at least 90% but were not overly optimistic. At our disposal, initially, we have PCA. Disney works on PCA for animation these days [13] and other groups working on verification in 3-D reported to have approached rates of about 85% (verification) several years ago, using piece-by-piece PCA alone.

PCA and GMDS have some clear similarities at some lower level. In MDS and GMDS we treat shapes in a metric space and assume that shape similarity can be reliably measured in terms of the distance between metric spaces. The duality of this problem can be outlined visually. To conceptualise this, it might look akin to Figure $[*]$ .

**Figure:** The proposed framework for GMDS improvement

It is possible to treat it somewhat differently, e.g. use ICP for rough alignment, GMDS for intrinsic fine alignment. If the alignment is onto a generalised face, then spectral decomposition takes place in the refined space - an Eigen functions of the generalised face. We need to address those issues, as shown in Figure $[*]$ .

**Figure:** A closer look at the GMDS approach

ICP could be considered Gromov-Hausdorff (G-H) in Euclidean space. The G-H-inspired method strives to identify and then calculate minimal distances for a group of geometric points with commonality in a more rigid space, wherein harmonic variation occurs in inherently non-orthogonal spaces. One way to model this type of variation and then explain its nature would be high-dimensional decomposition, which evidently requires that data be represented in a high-dimensional form such as vector of coordinates, intensities, energy, or discrete/quantised G-H distances (geometric terms). Figure $[*]$ provides an example of that subtle point.

**Figure:** Crude visual example of how typical PCA and GMDS relate to one another, approach-wise

Depending on the circumstance, different measurable attributes can be added to the space, even a hybrid of them (e.g. shape and texture, so as to reconstruct/recover the relationship between image intensity and the image shape in 2- and 3-D). For synthesis of images belonging to a particular class/subspace, e.g. a canonical form (bar embedding error), one requires that the model should be specific and generic. Specific - for the fact that it need preferably not be confused with similar images belonging to another class, and generic - for the fact that it must span a sufficiently large cloud in hyperspace in order to capture the variation of all images of the same class.

When it comes to sampling geodesic distances for PCA, http://www.ceremade.dauphine.fr/ peyre/numerical-tour/tours/fastmarching_5_sampling_2d/furthest point sampling would do the job of meaningful sampling[11]. It is 2-optimal in sense of sampling. $d_{GH(S,Q)}$ where distances are Euclidean is like ICP.