Our Neural Network Algorithm

The main idea of our algorithm [8] is based on the known feature of affine transformations: transformation of a given rectangle is completely conditioned by transformation of three of its corner points.

As previously, we work inside a unit square S^u of pixels for which we select three corner points (s₀, s_t, s_r), s₀ being left lower corner, s_t left top corner, s_r right lower corner. For every $f_k \in F$ ( $k = 1, \dots, \vert F\vert$) we only compute images of these three corner points, $(s_0^\prime=f_k(s_0), s_t^\prime=f_k(s_t), s_r^\prime=f_k(s_r))$. Then we define offsets $\Delta_x^t$ = $x(s_t^\prime)-x(s_0^\prime)$, $\Delta_y^t$ = $y(s_t^\prime)-y(s_0^\prime)$, $\Delta_x^r$ = $x(s_r^\prime)-x(s_0^\prime)$, and $\Delta_y^r$ = $y(s_r^\prime)-y(s_0^\prime)$ and use them in computing $\tan \alpha$ and $\tan \beta$, where $\alpha$ - the angle with x axis and $\beta$ - the angle with y axis. This computation gives us the set $f_k^{-1}(s^{\prime\prime})$ of points which are transformed by f_k onto $s^{\prime\prime}$ and, therefore, the synaptic weights $w_{s^{\prime\prime}s}=1$. Horizontal size $h(f_k^{-1}(s^{\prime\prime}))$ and vertical size $v(f_k^{-1}(s^{\prime\prime}))$ of the rectangle $f_k^{-1}(s^{\prime\prime})$ are determined by the contraction factors of f_k on x and y axes and calculated only once for each f_k. The inclination of each side of the rectangle $f_k^{-1}(s^{\prime\prime})$ is taken into account by $\tan \alpha$and $\tan \beta$, respectively.

The dynamics of the neural network is the same as in the Stark's algorithm.