Image Processing Reference
In-Depth Information
Fig. 14.1. Top row illustrates images containing 1, 2, and 3 directions. Bottom row shows the
same, but with different group directions
π
2
within the same image, i.e., the constituent line directions differ by
. The amount
π
4
, 3 π
4
, or 5 π
4
of relative rotation between the images is
. Can we deduce the relative
image rotation from the contents? Can we represent the direction of the line constel-
lation in each of the two images separately and uniquely as we did in the linearly
symmetric images? Yes, we can use a “quadruple-angle” representation:
θ =4 ϕ
(14.2)
where ϕ is any of the four possible normal directions, to achieve this. We will call
θ the group direction . The mapping is continuous w.r.t. any rotation and preserves
uniqueness of the rotation in that it continuously maps four different angles to one
and the same angle, which in turn represents an equivalence class of four angles
continuously. Employing the normal directions in the second column of images, we
obtain θ =0and θ = π as the group directions for the top and the bottom images,
respectively. While representing the “direction” of the image, the notion of “group
direction” also represents the direction of a constellation of lines that is invariant to
specific (group) rotations, justifying its name. In the case of linear symmetries, it was
not urgent to “invent” the notion of group direction because images were assumed to
contain only one direction. In this case, it was fairly simple to use this as a reference
direction of the image itself, allowing us to deduce the relative rotation between two
linearly symmetric images.
When there are many directions, the correspondence between the normal direc-
tion angles of individual lines and the rotation angle of a line constellation is more
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