Orientation (computer vision)

In computer vision and image processing it is a common assumption that sufficiently small image regions can be characterized as locally one-dimensional, e.g., in terms of lines or edges. For natural images this assumption is usually correct except at specific points, e.g., corners or line junctions or crossings, or in regions of high frequency textures. However, what size the regions have to be in order to appear as one-dimensional varies both between images and within an image. Also, in practice a local region is never exactly one-dimensional but can be so to a sufficient degree of approximation.

Image regions which are one-dimensional are also refered to as simple or intrinsic one-dimensional (i1D).

Given an image of dimension d (d is usually = 2), a mathematical representation of a local i1D image region is

$f(\mathbf {x} )=g(\mathbf {x} \cdot {\hat {\mathbf {n} }})$

where $f$ is the image intensity function which varies over a local image coordinate $\mathbf {x}$ (a d-dimensional vector), $g$ is a one-variable function, and ${\hat {\mathbf {n} }}$ is a unit vector.

The intensity function $f$ is constant in all directions which are perpendicular to ${\hat {\mathbf {n} }}$. Intuitivly, the orientation of an i1D-region is therefore represented by the vector ${\hat {\mathbf {n} }}$. However, for a given $f$, ${\hat {\mathbf {n} }}$ is not uniquely determined. If

${\tilde {\mathbf {n} }}=-{\hat {\mathbf {n} }}$

${\tilde {g}}(x)=g(-x)$

then $f$ can be written as

$f(\mathbf {x} )={\tilde {g}}(\mathbf {x} \cdot {\tilde {\mathbf {n} }})$

which implies that ${\tilde {\mathbf {n} }}=-{\hat {\mathbf {n} }}$ also is a valid representation of the local orientation.

In order to avoid this ambiguity in the representation of local orientation two representations have been proposed

• The double angle representation
• The tensor representation