Abstract

The parametric transformation is investigated to achieve invariance in optical pattern recognition. Two or more appropriately chosen feature vectors are parametrically combined to form a training vector. If chosen correctly, the training vector can be invariant under operations such as shift, scale, and transposition. As a simple and pedagogical shift-invariant example, consider a signal X(t) and its low- and high-pass-filtered versions, X_LP(t) and x_HP(t). Visualize a template formed by a parametric plot of these two functions on the (X_LP, x_hp) plane. Consider a shifted signal, $\widehat{X}\left(t\right)=X\left(t-t_0\right)$. We analogously form ${\widehat{X}}_{\text{LP}}\left(t\right)$ and ${\widehat{X}}_{\text{HP}}\left(t\right)$ on the (x_LP, x_HP) plane resulting in the same trajectory as that of the template. This concept can be extended to two dimensions to achieve rotation, shift, and scale invariance. To reduce the occurrence of false classification, the dimension of the parametric transformation can be increased. Indeed, for many cases of Importance, the parametric transformation becomes invertible to a member of the categorization class in the limit as the number of dimensions grows. When invertible, there are no false alarms due to the algorithm structure.

© 1988 Optical Society of America