AWRANGJEB AND LU: ROBUST IMAGE CORNER DETECTION BASED ON THE CHORD-TO-POINT DISTANCE ACCUMULATION TECHNIQUE 1061
Fig. 3. Effect of curve smoothing using Gaussian function with different
smoothing-scale
. The original curve as shown above is a high resolution
version of “curve 7” in Fig. 5(b).
when the curve point-locations are always approximated and,
therefore, may not be stable. Consequently, the derivative-based
curvature value defined in (4) may change considerably between
original and transformed curves. Such unstable and erroneous
estimated curvature not only affects the corner detection perfor-
mance under geometric transformations, but also may result in
poor corner matching performance in any of their later applica-
tions [4].
The second problem is related to the curve smoothing using
the Gaussian function. The aim of the smoothing is to reduce
the effect of local variation and noise so as to remove weak
and false corners. However, two direct effects of the Gaussian
smoothing are: first, it shrinks the curve and second, it smoothes
out the details if the smoothing-scale is high. On a smoothed
curve, corners are usually detected at those points where the
absolute curvature maxima values become higher than the
threshold. However, the estimated curvature value at a corner
point may decrease below the threshold with smoothing using
high
. The sharp (strong) corners are easily identifiable with
smoothing using large
, their estimated curvature is less accu-
rate and localization is worse than with smoothing using small
. However, using small may detect many weak and false
corners. Moreover, one smoothing-scale may not be suitable
for all curves. Since the local variation and noise on the curve
are unknown, curves of the same length may require different
smoothing-scales, even different segments of the same curve
may require different smoothing-scales. Therefore, choosing
an appropriate Gaussian smoothing-scale for a given curve is
a very difficult task.
This above problem is depicted in Fig. 3. The original curve
in Fig. 3 is a high resolution version of “curve 7” in Fig. 5(b),
where we see three strong corners (points 2, 3, and 4 in Fig. 3). In
the high resolution version of the curve (original curve in Fig. 3),
we see many weak corners and we choose points 1, 5, and 6
for illustration. Smoothing with both
and 4 can detect all
three strong corners; but while smoothing with
detects
all the weak corners, smoothing with
misses two weak
corners (points 1 and 5). However, corner localization is better
with lower smoothing-scale (see locations of point 3 in original
and smoothed curves in Fig. 3). Note that the detected corners
are shown with solid arrows and the missed corners are shown
with dotted arrows.
2) Existing CSS-Based Detectors: The first problem dis-
cussed above was an inherent problem with all the existing CSS
corner detectors [3], [5]–[11], since they used the same CSS
curvature estimation technique discussed in Section II-A. In the
rest of this section, we will discuss how the existing CSS-based
detectors suffer from the second problem.
The earliest CSS corner detector by Rattarangsi and Chin [5]
and its improved version [6] detected corners in the complete
scale-space by considering all possible smoothing-scales so as
to overcome the second problem. However, the tree construc-
tion and parsing using the complete scale-space resulted in high
computational complexity. The adaptive smoothing technique
in [7] reduced the computational complexity by eliminating the
construction of the complete scale-space map and by avoiding
the tree representation of the scale-space. However, the adaptive
smoothing technique itself incurred additional computational
cost which was supposed to be compensated by the slightly im-
proved robustness.
Mokhtarian and Suomela [8] detected corners at a high scale
and tracked them through multiple lower scales to the lowest
scale in order to improve localization. On the one hand, when
corners were detected in a very high scale, this detector showed
high robustness but lost many strong corners which might re-
quire lower smoothing-scales to be detected. On the other hand,
if corners were detected in a low scale it introduced many weak
corners which might require higher smoothing-scales to be re-
moved. As a result, this detector also suffered from the second
problem. Many of its improved versions (e.g., [3], [9]), which
selected the smoothing-scales based on the curve-length, also
failed to overcome this problem. Because even curves of the
same length may require different smoothing-scales depending
on the level of local variation and noise as discussed above.
The multi-scale curvature product (MSCP) detector [10]
and the single-scale detector [11] tried to overcome the second
problem to a great extent. They compensated the risk asso-
ciated with possible inappropriate smoothing-scale selection
by adopting different strategies. The MSCP detector used the
curvature product of three estimated curvature values at each
point using three smoothing-scales. As a result, in terms of cur-
vature product, the strong corners became more distinguishable
from the weak corners. He and Yung [11] used the adaptive
curvature-threshold and the dynamic region-of-support on both
sides of each curvature extremum point. Consequently, both of
them offered better corner detection performance than many of
the aforementioned detectors. In this paper, we will compare
most promising detectors [3], [8], [10], [11] with our proposed
corner detector (see Section IV).
C. CPDA Discrete Curvature Estimation
The previously proposed single chord-to-point distance mea-
surement technique [14] was not reliable, since such distance
depends upon the location of the chord [12]. In order to increase
the reliability, Han and Poston [12] proposed the chord-to-point
distance accumulation (CPDA) technique for measuring the dis-
crete curvature. A chord is moved along a curve and the perpen-
dicular distances from each point on the curve to the chord are
summed to represent the curvature. In contrast to the conven-
tional CSS curvature estimation technique adopted by the ex-
isting CSS corner detectors, the CPDA technique is completely
based on the Euclidean distance and does not involve any deriva-
tive of the curve-point locations at all.
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