Direction in 2D - Vision with Direction - page 146

10
Direction in 2D
Directional processing of visual signals is the largest single analysis toolbox of mam-
malian visual system: it feeds other specialized visual processing areas [114, 173,
235], e.g., face recognition. Directional analysis is gaining increased traction even
in computer vision, as it moves from single-problem-solving systems towards multi-
problem-solving platforms. Nearly all applications of image analysis now have al-
ternatives using direction tensor fields. The necessary tools are more modern and
offer advanced low-level signal processing that was hitherto reserved to processing
of high-level tokens, such as binarized or skeletonized edge maps. In 2D, the earliest
solutions to the problem of finding the direction of an image patch, e.g., [51, 116],
consisted in projecting the image onto a number of fixed orthogonal functions. The
projection coefficients were then used to evaluate the orientation parameter of the
model. When the number of filters used is increased, the local image is described
better and better, but the inverse function, mapping the coefficients to the optimal
orientation, increase greatly in complexity. A generalization of the inverse projec-
tion approach to higher dimensions becomes therefore computationally prohibitive.
Here we will follow a different approach by modeling the shapes of isocurves via
tensors.
10.1 Linearly Symmetric Images
We will refer to a small 2D image patch around a point as an image , to the effect
that we will treat the local image patches in the same way as the global image. Let
the scalar function f , taking a two-dimensional vector r =( x, y ) T as argument,
represent an image. Assume that f r =( x, y ) T is a two-dimensional real vector that
represents the coordinates of a point in a plane on which an image f ( x, y ) is defined.
Furthermore, assume that k =( k x ,k y ) T is a two-dimensional unit vector represent-
ing a constant direction in the plane of the image.
Definition 10.1. The function f is called a linearly symmetric image if its isocurves
have a common direction, i.e., there exists a scalar function of one variable g such
that