A Probabilistic Approach to Geometric Hashing using Line Features

CHAPTER 3. NOISE IN THE HOUGH TRANSFORM 37
Indeed, our experiments showed that most of the undetected lines are slanting lines with
small slopes.
While background subtraction works well to remove bias in favor of long line segments
over short line segments, it in fact implicitly applies the principle of the-winner-takes-all. In
particular, when a long line segment intersects a short line segment, their intersection pixel
will be classiæed as belonging to the long line segment and subtracted from the image after
the long line segment is detected. However, when these two line segments do not intersect
each other even though their extended lines do, the perceptual grouping technique we use
reduces this adverse eæect.
In regard to accuracy, we observe that line length is the key role aæecting the accuracy
of Hough transform results. The Hough transform works well to detect long line segments
even if the image is quite noisy. As line length decreases, noise aæects the accuracy of the
Hough transform even more seriously.
When the center of a segment is far away from the projection of the origin onto the
extended line of the segment, the average error of r increases. This phenomenon can be
rationalized as follows èsee Figure 3.6è: Since our algorithm loops through ç, we in fact
consider a line through the origin with slope angle ç èi.e., with line parameter èç +90 æ ; 0èè
and project all the image edgels onto this line. If bucket èç; rè in Hough space is detected
with n votes, roughly n edgels are projected onto the line at positions roughly r distant
from the origin. With this viewpoint in mind, due to the quantization of images, if the
center of a line segment, with line parameter èç; rè, is far away from the intersection point
of line èç; rè and line èç +90 æ ; 0è, the projections èonto line èç +90 æ ; 0èè of the points of
that line segment disperse èon line èç +90 æ ; 0èè around the position r distant from the
origin èexcept when the orientation of the line segment is0 æ ,45 æ ,90 æ or 135 æ .è