Journal of Signal and Information Processing, 2013, 4, 158-163
doi:10.4236/jsip.2013.43B028 Published Online August 2013 (
Restricted Hysteresis Reduce Redundancy in Edge
Bo Li1, Ulrik Söderström1, Shafiq Ur Réhman1, Haibo Li2
1Department of Applied Physics and Electronics, Umeå University,Umeå, Sweden; 2Department of Media Technology and Interac-
tion Design, KTH Royal Institute of Technology, Stockholm, Sweden.
Received May 30, 2013.
In edge detection algorithms, there is a common redundancy problem, especially when the gradient direction is close to
-135°, -45°, 45°, and 135°. Double edge effect appears on the edges around these directions. This is caused by the dis-
crete calculation of non-maximum suppression. Many algorithms use edge points as feature for further task such as line
extraction, curve detection, matching and recognition. Redundancy is a very important factor of algorithm speed and
accuracy. We find that most edge detection algorithms have redundancy of 50% in the worst case and 0% in the best
case depending on the edge direction distribution. The common redundancy rate on natural images is approximately
between 15% and 20%. Based on Canny’s framework, we propose a restriction in the hysteresis step. Our experiment
shows that proposed restricted hysteresis reduce the redundancy successfully.
Keywords: Edge Detection; Hysteresis; Non-Maximum Suppression; Redundancy
1. Introduction
Edge detection is a fundamental image processing meth-
od that has been studied for decades [1-7]. In general, the
aim of edge detection is to significantly reduce the
amount of data in an image, while retaining the structural
properties to be used for further image processing.
Therefore, efforts have been spent in decades to reduce
noise and increase the detection rate in edge detection.
However, there is a problem that has not been fully ad-
dressed in the literature. As shown in Figure 1, edge
detection algorithms that follow Canny’s framework
usually have redundancy effect. On the diagonal direc-
tion, an edge detected with Canny edge detector has a
width of two pixels. This will in the worst case yield
50% redundancy to the resulting data. The redundancy
problem happens especially when the gradient direction
is close to -135°, -45°, 45°, and 135°. This is caused by
the discrete calculation of non-maximum suppression.
Our proposed method can successfully delete such re-
dundancy and its relative increased cost is less than 5%.
Generally, time consumption of an algorithm is be-
tween O(n) to O(nc) depending on its complexity. Here n
is the number of inputs and c is the power number. In
natural images, the number of detected edge points is
usually less than 5 % of the number of the original image
pixels. Our method is able to further reduce n by elimi-
nating the edge redundancy. Edge feature brings an op-
portunity of fast computing for high level computer vi-
sion tasks. Edge as an important image feature has been
used in many high level tasks [12-17]. For example,
straight line can be detected by Hough transform. Ballard
developed a generalized Hough transform to detect arbi-
trary shapes [12]. In both [13,15] they experimented with
edge matching method in their application scenario.
Borgefors developed a face recognition method using
line edge map [15]. One reason of using edge instead of
using pure image is because pure image is high dimen-
sional and it brings extraordinary computational cost. For
example [16] subclass discriminant analysis method is
used to detect object based on pure image as input. In
their algorithm, image transformation to a high dimen-
sional matrix is the main factor of time consumption. The
time consumption of above algorithms highly depends on
the number of detected edge points. Our proposed meth-
od is able to reduce time consumption caused by edge
Canny’s [1] edge detector has been developed as a
standard function in Matlab and OpenCV [8]. Many edge
detectors have been developed in the last two decades
after Canny [3-6]. Most edge detectors follow Canny’s
last two steps: non-maximum suppression and hysteresis
to get thin edges with one pixel width. However, no work
addresses redundancy problem of edge detectors.
In section 2, we explain the background of edge detec-
Copyright © 2013 SciRes. JSIP
Restricted Hysteresis Reduce Redundancy in Edge Detection 159
tion including non-maximum suppression and hysteresis.
We present our simple solution using restricted hysteresis
in section 3. Experiment result is presented in section 4.
(a) (b) (c) (d) (e)
Figure 1. (a) A synthetic image; (b) Part of enlarged image;
(c) Edge detected by Canny edge detector; (d) Redundant
edge point, denoted by red color; (e) Edge detected by our
proposed method.
2. Background
Edges are normally representations of changes in inten-
sity functions of an image; i.e., image intensity variations
such as steps, lines and junctions [3]. The widespread
edge detection methods detect edges by finding local
maxima of first-order derivative function or zero-cross-
ing of second-order derivative function of the intensity
profile of given image. In practice, image gradients are
estimated by convoluting images with first-order deriva-
tive operators (also known as kernels), such as Robert’s
cross operator, Prewitt operator and Sobel operator [9].
The kernel convolution finds the abrupt changes in inten-
sity of the image. In case of Sobel kernel both horizontal
and vertical changes are approximated. Formally if I(x, y)
is intensity level at a point in a given source image, and
Gx, Gy are horizontal and vertical derivative approxima-
tions, then the gradient magnitude is given as
GGG (1)
Once the gradient magnitude is computed, the next
step is to apply a threshold, to decide whether an edge is
present or not at an image point. Appropriate thresholds
filter out most noise and keep edge points. After this step,
the resulting edge is still thick. Canny [1] introduced the
notion of non-maximum suppression (NMS) to find
edges with one-pixel thickness by comparing gradient
value along gradient direction. The gradient direction is
computed as:
arctan( )
As shown in Figure 2, non-maximum suppression
procedure checks if the pixel is a local maximum along
gradient direction by comparing pixel with its interpo-
lated neighbor pixels p and r [10]. This interpolation is
implemented in Matlab.
In order to achieve a real-time algorithm, OpenCV do
not use interpolated pixels but its rounded neighbor pix-
els [8]. As shown in Figure 3, such approximation
rounds the angle θ to one of four directions: 0°, 45°, 90°,
or 135°. This means in the ranges [22.5°, -22.5°] and
[157.5°,‐202.5°] the algorithm round θ
to 0o. Since the
comparison is symmetrical, the absolute value of the
lower half of the circle is mirrored to the upper one. So,
three pixels around pixel (x, y) are examined:
If θ (x, y) = 0°, then the pixels at (x + 1, y), I(x, y),
and I(x 1, y) are examined.
If θ (x, y) = 90°, then the pixels at (x, y + 1), (x, y),
and (x, y 1) are examined.
If θ (x, y) = 45°, then the pixels at (x + 1, y + 1), (x,
y), and (x 1, y 1) are examined.
If θ (x, y) = 135°, then the pixels at (x + 1, y 1), (x,
y), and (x 1, y + 1) are examined.
If pixel (x, y) has the highest gradient magnitude of the
three pixels examined, it is kept as an edge. We have
evaluated Canny edge detection provided by Matlab and
OpenCV. Both of them yield the same result with a re-
dundancy problem.
(a) (b)
Figure 2. Non-maximum suppression. (a) Gradient image
and gradient direction; (b) Neighbor interpolation.
Figure 3. Gradient direction regions.
The redundancy problem is caused by the inherent na-
ture of discretization of image. As shown in Figure 4,
the pixels with gradient value of 8 are the local maxi-
mum compared with their neighbor along the gradient
direction. However, because there is a staggered effect
because of discretization, the local maximum pixels pre-
sent double edge effect and thus cause redundancy. If
any algorithm could link these redundant edge pixels as
one line with one pixel edge, the redundancy problem
can be solved. However, this is not a straightforward
method because most linking methods use gradient di-
rection to predict the next pixel and then the result would
be two parallel lines.
Copyright © 2013 SciRes. JSIP
Restricted Hysteresis Reduce Redundancy in Edge Detection
Canny edge detector’s final step is called hysteresis. It
keeps all the week edge pixels which are linked to the
strong edge pixels. Canny edge detector uses two thresh-
olds on the gradient image– a lower threshold decides if
a pixel is a weak edge and an upper threshold decides if a
pixel is a strong edge. Hysteresis step starts tracing from
a strong edge and stops when the gradient is below the
lower threshold. It improves noise reduction rate by
eliminating weak edges which are not connected to a
strong edge. However, the redundancy problem remains
in the detected edge. The tracing involves predicting the
next edge point. As shown in Figure 5, assume the
marked point is an edge point. Then we construct the
tangent to the edge curve and use this to predict the next
point. The tangent is normal to the gradient at the point.
The predicted next point is either point r or s. The
choosing of r or s can be similar to the non-maximum
suppression step by using rounded direction range.
Figure 4. Gradient image illustrating non-maximum sup-
Figure 5. Predicting the next edge point.
3. Restricted Hysteresis
Restricted hysteresis means that when one starts to link a
new edge part, a restriction is required to check if a pix-
el’s 8 neighbor pixels have already been linked in previ-
ous hysteresis. If anyone of the 8 neighbors has already
been linked as an edge point, we do not start new hys-
teresis from this pixel. As shown in Figure 6, P(x,y) is a
strong edge so hysteresis step can be started. However,
because of the restriction, we cannot start linking because
two of its neighbors are already determined as edge
points. Our restriction is only applied on each new start-
ing point of hysteresis step. Once the starting point pass
the restriction, the following linking is done as usual.
Otherwise, it will corrupt all the following linking. We
have tested our method on many images including syn-
thetic images and natural images. Restricted hysteresis
performs well both regarding redundancy reduction rate
and algorithm speed.
Figure 6. Restricted hysteresis. 2 denotes strong edge and 1
denotes week edge.
The above mentioned method works well in most cas-
es. However there are occasional exceptions according to
our observation. As shown in Figure 7 (b), a new hys-
teresis starts from point P(x,y) and stops at Point R(x,y)
by mistake. The pixels in red color are denoted as re-
dundant pixels. This happens when there is a T–junction
or X-junction in the edge. We can easily fix this excep-
tion by revising our restriction. In hysteresis linking step,
we check if less than three of a pixel’s eight neighbors
have already been linked as edge, then we continue the
hysteresis linking. Otherwise the hysteresis linking is
stopped and we start a new hysteresis from a new pixel.
New restricted hysteresis can successfully stop the link-
ing on the position of T-junction while not eliminating
the correct edge points.
Figure 7. (a)Synthetic image; (b) Occasional exception when
restricted hysteresis fail; (c) Revised restricted hysteresis.
4. Experiment Result
We find that most edge detection algorithms have re-
dundancy of 50% in the worst case and 0% in the best
case depending on the edge direction distribution. The
average redundancy on natural images is approximately
15% to 20%. The worst case happens when all the edge
direction is ±45° or ±135°. The best case happens when
all the edge direction is ±0° or ±90°. In Figure 8, we can
see that when edge direction is close to ±45°, the redun-
dancy is about 50% and reduce when edge become more
Copyright © 2013 SciRes. JSIP
Restricted Hysteresis Reduce Redundancy in Edge Detection 161
and more horizontal.
(a) (b)
(c) (d)
Figure 8. (a) Synthetic image; (b) Edge detected by Canny
edge detector; (c) Edge detected by using restricted hystere-
sis; (d) shows a comparison between (b) and (c), red pixels
are the redundancy edge points.
The time consumption of restricted hysteresis is minor
compared with gradient calculation and non-maximum
suppression. Gradient calculation and non-maximum
suppression need to operate N times on each pixel. And
hysteresis only operates on the edge detected by non-
maximum suppression. After non-maximum suppression,
the number of edge pixels is on average less than 5% of
image resolution. This means the time consumption of
hysteresis is less than 5% of gradient calculation and
non-maximum suppression.
In order to evaluate our proposed method, we compare
the number of detected edge points by Canny edge de-
tector and restricted hysteresis. The number of detected
edge points by Canny edge detector is denoted as Nc, and
restricted hysteresis as Nrh. Then the successful reduced
number of redundancy points is:
Redundancy reduction rate is:
Since there is no available dataset providing the
ground truth of redundant points, it is difficult to know
the true redundancy rate of an edge image, denoted as Rre
(R’re<= Rre). Based on our experiment, R’re is about 15%
to 20% on natural images. Table 1 presents our experi-
ment results of on natural image. The natural im-
ages are resized and then tested on Canny edge detector
and restricted hysteresis (Figure 9). Since the hysteresis
itself eliminates the edges which are not connected to
strong edge points, and we want to analyze the power of
redundancy reduction but not the power of noise cancel-
lation, we just set the lower threshold and upper thresh-
old to the same value in our calculation. This means
all the detected edges are strong edge and eliminated
edge points are due to redundancy. In applying restricted
hysteresis in real application, the principle of setting
threshold is the same as in Canny’s framework.
Table 1. Reduced redundancy R’re.
Worst Best
Figure 8(b) Figure 9(a2)
50% 0% 22.65% 19.21%
Figure 9(b2)Figure 9(c2)Figure 9(d2) Figure 9(e2)
15.31% 13.37% 16.43% 14.48%
5. Discussion
There is a concern about the accuracy of the detected
edge. As shown in Figure 10, is we use Hough transform
to detect straight lines on detected edge points, then the
detected line positions based on Canny edge detection
are different from restricted hysteresis (Figure 10(a) and
(b) ) . The difference is 0.3535 pixel. Since edge pixels
are presented in discrete locations, it is not sufficient to
present accurate sub pixel locations unless we implement
further sub-pixel analysis [7]. Therefore, even assuming
line position in Figure 10(a) is more accurate, the max-
imum error of 0.3535 pixel in Figure 10(b) is reasonable.
Compared with the accuracy issue, the advantage of re-
stricted hysteresis is obvious. For example, Hough trans-
form is time consuming. By reducing average redun-
dancy of 20%, the time consumption of Hough transform
will be reduced by 20%. Most high level object detection
and recognition tasks do not use Hough transform but
linking based line or curve detection because of speed
consideration. Linking based algorithms usually result
detected lines such as in Figure 10(c). This will result in
redundant lines. Redundant lines are a potential factor of
low speed and accuracy in high level computer vision
6. Conclusions
Non-maximum suppression and hysteresis as the funda-
mental thinning step in Canny’s edge detector has been
used widely in many other edge detection algorithms.
The procedure can be found in many literatures [1,3,5,7,
17]. However, there is no work that discusses the redun-
dancy effect of detected edge points. Our work highlights
that there is a common redundancy problem, especially
when the gradient direction is close to -135°, -45°, 45°,
and 135°. Double edge effec appears on the edge around t
Copyright © 2013 SciRes. JSIP
Restricted Hysteresis Reduce Redundancy in Edge Detection
Copyright © 2013 SciRes. JSIP
(a1) (a2) (a3)
(b1) (b2) (b3)
(c1) (c2) (c3)
(d1) (d2) (d3)
(e1) (e2) (e3)
Figure 9. (a1), (b1), (c1), (d1), (e1): Natural images; (a2), (b2), (c2), (d2), (e2): Edge detected by Canny edge detector; (a3),
b3), (c3), (d3), (e3): Edge detected by using restricted hysteresis. (
Restricted Hysteresis Reduce Redundancy in Edge Detection 163
Figure 10. Hough transform line detection based on (a)
Canny edge detector; (b) Restricted hysteresis; (c)Linking
based line detection.
these directions. This is caused by the discrete calcula-
tion of non-maximum suppression. Our proposed solu-
tion is to put a restriction in the hysteresis step to further
reduce the double edges to one pixel width. The in-
creased time consumption is far less than 5%. Our pro-
posed method successfully reduces redundant edge
points which are usually 15% to 20% of the detected
edge points. Restricted hysteresis is simple to implement
and can be used by any edge detection algorithm which
need a thinning step. Redundancy reduction helps to im-
prove algorithm speed when edge points are needed for
further computer vision task such as line extraction or
object recognition.
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