Polynomial classifier

Polynomial classifiers for pattern recognition were developed from statistical decision theory and have the key function in text recognition (OCR), a branch of pattern recognition.

Their main advantage lies in the possibility of solving the adaptation task directly. Building on this basic technology , complex and high-performance classifier structures (trees, networks) were developed, which were able to demonstrate their performance in very different fields of application, for example for address readers, postal automation or form readers.

Mathematical definition

A polynomial classifier is a mapping of vectors from the -dimensional real feature space onto a set of classes: ${\displaystyle d}$ ${\displaystyle \left\{1,\ldots ,k\right\}}$

${\displaystyle f\colon \mathbb {R} ^{d}\to \left\{1,\ldots ,k\right\}}$

It is defined as the most significant component of the following vector ${\displaystyle f}$

${\displaystyle \mathbf {y} =\mathbf {p} (\mathbf {v} )={\begin{pmatrix}p_{1}(\mathbf {v} )\\\vdots \\p_{k}(\mathbf {v} )\end{pmatrix}}}$

The multivariate polynomials can be interpreted as probability functions that a given feature vector belongs to the class . Overall, the following applies . ${\displaystyle p_{i}(\mathbf {v} )=y_{i}\in [0,1]}$ ${\displaystyle \mathbf {v} }$ ${\displaystyle i}$ ${\displaystyle \|\mathbf {y} \|_{1}=1}$

The above notation can be simplified by calculating only one polynomial instead of many polynomials . Then it holds with a real-valued coefficient vector. Overall follows . ${\displaystyle p_{i}(\mathbf {v} )}$ ${\displaystyle \mathbf {x} (\mathbf {v} )}$ ${\displaystyle p_{i}(\mathbf {v} )=\mathbf {a} \cdot \mathbf {x} (\mathbf {v} )}$ ${\displaystyle \mathbf {p} (\mathbf {v} )=A^{T}\mathbf {x} (\mathbf {v} )}$

A new feature vector is thus classified by . ${\displaystyle \mathbf {\hat {v}} }$ ${\displaystyle \arg \max \mathbf {p} (\mathbf {\hat {v}} )=f(\mathbf {\hat {v}} )}$

literature

• J. Schürmann: Pattern Classification: A Unified View of Statistical and Neural Approaches . Wiley & Sons, 1996, ISBN 0471135348 .
• H. Niemann: Classification of Patterns . Springer, Berlin 1983, ISBN 3-540-12642-2 . (2nd edition, without publisher, 2003: PDF file; 6.5 MB ).