Linear feature modeling:

The premise of linear feature modeling is to extract rich descriptions of lower-level local image primitives and use these descriptions for subsequent grouping into higher-level features (linear line segments). The following illustrates the modeling of the perceptual grouping process described in [8] for the collection of edge segments $\omega_{k}$'s, to form a longer linear line $\omega_j$, (figure 1(a)). Let ${\bf {r}} = \{x,y\}$ denote the $x$- and $y$-coordinates of an end-point of an edge segment $\omega_k$, and $\phi \in {\cal S}^1$ represents the orientation of the edge segment. We treat $\bf r$ and $\phi$ as independent variables, so that all possible orientations for $\omega_k$ exist at each corresponding position $\bf r$. A certain collection ${\cal C}_i$ of $\omega_{k}$'s is collected, which will be replaced by $\omega_j$, that maximizes the energy $\lambda_i$ given as:

$$\lambda_i^{(n)} = \lambda_i^{(n-1)} \: + \sum_{k \in {\cal K... ...}} \xi_{kl},\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \lambda_i^{(0)} = 0$$ (5)

where the superscript $n$ is an iteration index, and (omitting the subscript $i$), the energy functional $\xi_{kl}: (\omega_b, \omega_k, \omega_l) \to \Re$ is expressed as:

$$\xi_{kl}(\omega_b \:\vert\: \omega_k, \omega_l) = \Lambda(q)... ...k-{\bf {{r}}}_l-s{\bf {e}}_{{kl}})\: \delta(\phi_b - {\phi}_l)$$ (6)

where $\omega_b$ is a certain base edge segment in the collection that is used to determine that all other edge segments are parallel to it, $\Lambda$ is a weighting function and $q$ is the maximum length of the orthogonal distance of any point of $\omega_l$ from $\omega_b$. In the above equation, ${\bf {r}}_k$ and ${\bf {{r}}}_l$ represent those end-points of two edge segments $\omega_{k}$ and $\omega_{{l}}$ respectively, (at the lower-level), that are closer to each other, and $\phi_b$ and $\phi_l$ are the orientations of $\omega_b$ and $\omega_l$, respectively. In addition, $\delta$ is the Dirac delta function, ${\bf e}_{{kl}}$ is a unit vector in the direction of ${\bf r}_k - {\bf r}_l$ and $s$ is a distance parameter along an axis parallel to the direction of ${\bf r}_k - {\bf r}_l$. The Boolean parameter $t$ is such that $t = 0$ if the length of the orthogonal projection of $\omega_l$ on $\omega_k$ is greater than zero, otherwise $t = 1$. In our system $\Lambda$ is represented by a constant function (not equal to zero) with compact support. Specifically, we have selected the constant as 1 and the support is equal to 5 units (pixels). Equation 5 indicates the iterative nature of the grouping. At the start ${\cal C}_i$ consists of only one segment $\omega_b$. At the end of each iteration those $\omega_l$'s for which $\xi _{kl}$ is non-zero are put into ${\cal C}_i$. The grouping is started again and continued until there is no increase in $\lambda_i$. The higher-level longer linear line $\omega_j$ is then obtained by a weighted average of the lengths and orientations of all edge segments in ${\cal C}_i$ [8].

The form of the energy functional expressed in equation 6 is similar to the one defined in [18], however, in their model ${\bf r}_k$ represents the V1 image of the center of the receptive field of a neuron, and ${\bf e}_{kl}$ represents the V1 image of the orientation preference of the neuron. Unlike their model, in our system ${\bf e}_{kl}$ points in the direction of ${\bf r}_k - {\bf r}_l$ and incorporates the non-collinearity of two edge segments to an arbitrary extent (e.g., figure 1). (To further emphasize closer points, unequal weights, as opposed to constant weights in the support of $\Lambda$, can be achieved by replacing $\Lambda$ with an appropriate weighting function, such as a Gaussian function.)