In computer vision and image processing a common assumption is 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.
Contents
- Relation to direction
- Relation to gradients
- Estimation of local image orientation
- Application of local image orientation
- References
Image regions which are one-dimensional are also referred to as simple or intrinsic one-dimensional (i1D).
Given an image of dimension d (d = 2 for ordinary images), a mathematical representation of a local i1D image region is
where
The intensity function
then
which implies that
In order to avoid this ambiguity in the representation of local orientation two representations have been proposed
The double angle representation is only valid for 2D images (d=2), but the tensor representation can be defined for arbitrary dimensions d of the image data.
Relation to direction
A line between two points p1 and p2 has no given direction, but has a well-defined orientation. However, if one of the points p1 is used as a reference or origin, then the other point p2 can be described in terms of a vector which points in the direction to p2. Intuitively, orientation can be thought of as a direction without sign. Formally, this relates to projective spaces where the orientation of a vector corresponds to the equivalence class of vectors which are scaled versions of the vector.
For an image edge, we may talk of its direction which can be defined in terms of the gradient, pointing in the direction of maximum image intensity increase (from dark to bright). This implies that two edges can have the same orientation but the corresponding image gradients point in opposite directions if the edges go in different directions.
Relation to gradients
In image processing, the computation of the local image gradient is a common operation, e.g., for edge detection. If
Estimation of local image orientation
A number of methods have been proposed for computing or estimating an orientation representation from image data. These include
The first approach can be used both for the double angle representation (only 2D images) and the tensor representation, and the other methods compute a tensor representation of local orientation.
Application of local image orientation
Given that a local image orientation representation has been computed for some image data, this formation can be used for solving the following tasks: