Figure 5.2: Illustration of the tangent space at the mean shape, where the co- variance matrix of a linearized data points are computed.
ance of a shape representation dataset as
cov(X) ∼1 q z X i=1 ( ¯f (˜c) − f (˜ci)2 1 q z X i=1 d( ¯f (˜c), f (˜ci)2, (5.14)
where f (·) is as given in (3.14) and d(·, ·) as (3.48) in Chapter 3. The above equation for the covariance matrix is the same as the (5.11) except that in (5.14) the mean is already estimated. Figure (5.2) shows an illustration of the covariance matrix computation at the tangent space of the mean.
5.4
Modelling shape distribution
There are several ways one could approach the estimation of a probability distri- bution for a random vector of type (5.4). The approaches are mainly guided by the assumptions imposed on the distributions:
1. Parametric: Given a sample dataset, parametric approach assumes the dis- tribution of a random vector X to take the form of a parametrized function, e.g., exponential distribution families. As such, in parametric distribution the effort is to correctly estimate the parameters of the chosen distribution form. We herein explore Gaussian and mixture of Gaussians for modelling shape distributions.
2. Non-parametric: An alternative to parametric modelling is to directly at- tempt to estimate the analytical form of the distribution from the dataset points. Such kind of approaches are called non-parametric estimation, e.g., histograms, Parzen-window (Kernel) density estimation, neural networks, etc. Non-parametric approaches are usually effective when the available dataset is large.
In this section, we will mainly cover parametric density estimation approaches for a data set of shape samples represented with the proposed method (3.14).
5.4.1
Parametric density estimation
Most existing statistical shape analysis approaches are based on parametric den- sity estimation or latent variable modelling [DM98]. In particular, in [DAO15] an inhomogeneous time Markov process is used to capture the statistical proper- ties of a deformable shape represented in a matrix Lie group. Similarly, here we consider a curve representation f (˜c) as a realization of a random vector X. Gaussian distribution: A Gaussian distribution of a random vector X is com- pletely determined by two parameters, mean µ and covariance Σ. Hence the evaluation of a D-dimensional observation X = x with Gaussian pdf is written as N(x; µ, Σ) = 1 (2π)D/2|Σ|1/2 exp(− 1 2(x − µ) TΣ−1 (x − µ)). (5.15) Refer to [GN99] for further details in matrix valued Gaussian random variables. In our case, however, if we assume that our shape dataset follows (5.15), the density estimation simply means the correct estimation of (5.11) and (5.14). Hence (5.15) amounts to the following in the defined representation space
1 (2π)D/2|Σ|1/2 exp(− 1 2v TΣ−1 v), (5.16)
where v is the vectorized form of an observation at tangent space of the mean. We write v, using vec(·) operator to vectorize a matrix, as follows
v = z Y i=1 veclog( ¯R T i Ri) R¯Tv¯i− vi 0 0 . (5.17)
5.4 Modelling shape distribution
Figure 5.3: Example dataset of dogs from Kimia 1070-shape dataset.
(a) (b)
(c) (d)
Figure 5.4: Mean shapes of the dogs dataset along with a randomly sampled shapes. (a) Mean shape based on area preserving parameter estimation, (b) ran- domly sampled shapes from a Gaussian distribution of shapes with area preserving parameter estimation. (c) Mean shape based on linear parameter estimation, (d) randomly sampled shapes from a Gaussian distribution of shapes with linear pa- rameter estimation.
One can further sample the estimated distribution using one of the well studied random samplers [Mad02]. However, since the sampled vector is going to be an element of the tangent vector it requires exponentiation to finally get the randomly sampled shape representation, see Appendix A. Hence, for a given Gaussian distribution sampler S(µ, Σ), we can generate a random shape as
fr = ¯f (˜c) expf (˜¯c)(S(0, Σ)), (5.18)
where fr represents the randomly sampled representation, see Figure 5.4.
random variables exists, a much more natural approach is to consider graphical models for modelling a distribution [Bes75]. As such, a given random vector X can be seen as spatially ordered set of random variables. Let the probability of a given random vector X = {X1, · · · , Xz} to take the value f (˜c) = {g1, · · · , gz} to
be expressed as
p(X1 = g1, · · · , Xz = gz), (5.19)
Regardless of what form p(·) takes, one can factor (5.19) using the chain rule, which follows directly from the axioms of a probability measure, as follows
p(g1, · · · , gz) = p(gz|gz−1, · · · , g1) · · · p(g2|g1)p(g1). (5.20)
Since the observed transformation matrices are indexed in an order, one can inter- pret (5.20) as a model of a stochastic process. Moreover, additional assumptions can be imposed on the dependency of the sequential transformation matrices. For instance, assume that (5.20) is a Markov chain1 then (5.20) is simplified as
p(g1, · · · , gz) = z−1
Y
i=1
p(gi+1|gi). (5.21)
In such a case, we model (5.21) as a non-stationary Markov process with the following transition rule
gi+1= ˆgi+1× gi. (5.22)
Note that theˆdenotes the observed transition matrix. Hence, the probability of a given shape representation f (˜cj) = {g1, · · · , gz} is as follows
p(g1, · · · , gz) = z−1 Y i=1 p(gi+1|gi) = z−1 Y i=1 p(ˆgi) (5.23)
Subsequently, it is possible to model the distribution of the transition matrices with a Gaussian distribution.