Robustness of Raw Images Classifiers Against the Class Imbalance – A Case Study

Abstract

Our aim is to investigate the robustness of classifiers against the class imbalance. From this point of view, we compare several most widely used classifiers as well as the one recently proposed, which is based on the assumption that the probability densities in classes have the matrix normal distribution. As the base for comparison we take a sequence of images from that laser based additive manufacturing process. It is important that the classifiers are fed by raw images. The classifiers are compared according to several criterions and the methodology of all pair-wise comparisons is used to rank them.

A Appendix

The densities of the matrix normal distribution are defined as follows:

$$\begin{aligned} f_j({\mathbf {X}})=\frac{1}{c_j}\,\exp \left[ -\frac{1}{2}\, {\text {tr}}[U_j^{-1}({\mathbf {X}} - {\mathbf {M}}_j)\,V_j^{-1}\, ({\mathbf {X}} - {\mathbf {M}}_j)^T ] \right] , \end{aligned}$$

(18)

where the normalization constants are given by:

$$\begin{aligned} c_j{\mathop {=}\limits ^{def}} (2\, \pi )^{0.5\,{n\, m}}\, {\text {det}}[U_j]^{0.5\,n}\, {\text {det}}[V_j]^{0.5\,m}\, , \end{aligned}$$

(19)

where $n\times m$ matrices $M_j$’s denote the class means matrices. The covariance structure of MND class densities is as follows

• 1. $n\times n$ matrix $U_j$ denotes the covariance matrix between rows of an image from j-th class,

• 2. $m\times m$ matrix $V_j$ stands for the covariance matrix between columns of an image from j-th class.

The above definitions are meaningful only when ${\text {det}}[U_j]>0$, ${\text {det}}[V_j]>0$.

The equivalent description of MND is the following:

$$\begin{aligned} \text {vec}{({\mathbf {X}})} \sim \mathcal {N}_{n\, m}(\text {vec}{({\mathbf {M}}_j}),\, \Sigma _j ), \; {\text {for }} j=1,\, 2,\ldots , J, \end{aligned}$$

(20)

where $\mathcal {N}_{K}$ stands for the classic (vector valued) normal distribution with K componentnts. In (20), $\text {vec}{({\mathbf {X}})}$ is the operation of stacking columns of matrix ${\mathbf {X}}$, while $\Sigma _j$ is a $n\, m\times n\, m$ covariance matrix of j-th class, which is the Kronecker product (denoted as $\otimes$) of $U_j$ and $V_j$, i.e.,

$$\begin{aligned} \Sigma _j {\mathop {=}\limits ^{def}} U_j\otimes V_j,\quad j=1,\, 2,\ldots , J. \end{aligned}$$

(21)

Formulas (20) and (21) show clearly that MND’s form a subclass of all normal distributions. Namely, MND’s have the special structure of the covariance matrix given by (21) (see [7]). Thus, in practice, it suffices to estimate two much smaller matrices $U_j$ and $V_j$ instead of a general covariance matrix which is $n\, m\times n\, m$. As the consequence, it suffices to have:

$$\begin{aligned} N_j \ge \max \left\{ \frac{n}{m}, \frac{m}{n} \right\} +1, \quad j=1,\, 2, \ldots ,\, J. \end{aligned}$$

(22)

(see [6] for the proof).