| 1) - g(z; y)
iction a sec-
ins. rr =
; + Di Ka =
e parameters
least-squares
hler & Nagel
The method
] to compute
2-4.4) model
ntially based
“onsequently,
ot at all war-
ages of non-
‘eatures than
tion is much
! noise model
chemes. Fur-
done in the
d, truncated,
port, often as
situation may
rely different
ition filters.
Marr-Hildreth
ator, (4) Ed-
tors, and (6)
rily based on
f the human
les. We con-
image is first
lurring effect
he edges are
/ariant Lapla-
the Gaussian
an (LOG).
wide variety
ed at several
n. The edges
to form the
more general
g, introduced
developed in
Malik, 1990;
0: Liu et al”,
io embed the
e scale-space
age g(z, y; 0)
with a Gaussian kernel G(z,y;0) of standard deviation c,
the scale-space parameter, which describes the current level
of scale resolution: g(x,9; 0) = 9(x,9y,0) * G(1,9y;0). The
larger the value of o the coarser the resulting resolution and
the more the image is blurred. The choice of the Gaussian is
motivated by the fact that it is the only kernel in a broad class
of functions which satisfies adequate scale-space conditions
(Babaud et al. 1986): (1) causality: increase of a should not
generate spurious details and (2) homogeneity and isotropy:
the blurring is shift invariant and does not depend upon the
grey values. The Marr-Hildreth operator suffers from several
deficits:
e Theoretically the zero-crossing contours are closed.
However, due to noise, texture and quantization ef-
fects, breaks may occur since the magnitude of the
pixel differences on the two-sides of a zero-crossing do
not exceed an acceptance threshold.
e T-junctions or trihedral vertices are incorrectly de-
tected. Instead of 3 meeting zero-crossing lines 2 dis-
connected lines are detected, i.e. a spurious line is
detected. In general, at positions where the edges are
highly curves the zero crossings are located improperly.
The larger the width of the Gaussian, the larger this
effect.
Gaussian smoothing causes merging of closely spaced
edges, resulting in the detection of phantom edges. For
example, a set of parallel edges may be joined to one
edge after convolution with the Gaussian.
e A good method to combine the results at different
scales is lacking.
e Since the Laplacian is a second derivative operator, the
operator is sensitive to noise. (A nonlinear Laplacian
for use on noisy images has been developed by van
Vliet et al. (1989)).
5.2 Canny Edge Detector
Canny (1986) formulates edge detection as an optimization
problem and derives optimal filters for the detection of step
edges in the presence of Gaussian noise. The product of SNR
and the localization measure in one-dimension is optimized,
using as performance criteria: 1) good detection, 2) good
localization, and 3) only one response to a single edge should
appear. The steps of the scheme are:
1. Filtering of the image to smooth the effects of noise
and to produce a multiscale representation of the image
data. For computational reasons a suboptimal Gaus-
sian filter is chosen.
2. Differentiation by taking directional first derivatives us-
ing templates at an interval of 30°.
3. Non-maxima suppression by interpolating gradient vec-
tors in a 3 x 3 neighbourhood.
4. Multithreshold hysteresis linking which draws on in-
formation concerning edge-connectivity. Edge-pixels
are initially labeled if their response exceeds a high-
threshold value. Pixels lying above a weaker response
threshold are then admitted provided they belong to
edge-segments which are connected to the initially la-
beled pixels. Finally, unconnected high-response pixels
are deleted.
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
Although the Canny operator may remove genuine high-
frequency edge-features such as corners, its performance is
good. This is, however, not due to its optimal design for
step edges embedded in Gaussian noise, but is based on the
use of local context (generic geometric) information in step
4. Based on the Canny approach Petrou & Kittler (1991)
developed an optimal detector for ramp edges.
5.3 Forstner Operator
The Forstner (1986, 1993, 1994) operator, which is well-
known within the photogrammetric community as interest
operator for feature-based matching, is based on examination
of a small set of connected gradient components g. and gy,
enabling to distinguish isotropic structure such as blobs, cor-
ners, and texture from non-isotropic structure such as edges
and lines (e.g. roads). The operator requires two thresholds:
one on the strength of the averaged squared gradients to de-
cide upon the presence of a feature; if a feature is present, the
other threshold is to decide whether the feature is isotropic
or non-isotropic. Based on this scheme Forstner (1994) de-
veloped a framework for low level feature extraction.
5.4 Edgeness Operator
In the template approach (section 4.3), the maximum of the
responses is taken as edge measure. The relationships among
the directional template responses are not taken into account,
although they may give an essential point whether the re-
sponse is due to noise and texture or to the presence of a real
edge pixel. A scheme that is able to carry out such a coher-
ence examination is developed by Cheng (1990). The basic
notion of this edgeness operator is as follows. If an edge
is present the responses of the templates in subsequent di-
rections, starting from the template oriented along the edge,
will be monotonically decreasing and will show symmetry. For
noisy pixels this systematics will be absent. If the maximum of
the template responses exceeds a threshold and all responses
show sufficient systematics, the presence of an edge is ac-
cepted. The scheme is especially developed for use on radar
images. An extensive analysis carried out in Lemmens (1996)
shows that the operator, using 5 x 5 templates, performs very
well on images heavily corrupted by noise and/or texture.
5.5 Cascade of Local Edge Detectors
Usually, one out of the many local edge detectors is applied.
However, one may employ several edge detectors in sequence
to improve quality and/or to reduce the computational costs.
McLean & Jernigan (1988) developed in such away a fast
scheme for real time processing of large images. The basic
idea is to use in a first stage a time efficient operator that
indicates pixels of interest. A simple cross operator over the
diagonals (mask: [-1 0 1]) may suffice. This operator may
classify non-edge pixels as pixels of interest but the number of
edges that are wrongly not detected should be preferable zero,
since this type of misclassification can not be corrected in a
subsequent stage. Next the pixels of interest are considered
more thoroughly by a more sophisticated operator. Tan &
Loh (1993) make the above approach still more efficient by
using in addition multiple resolutions by establishing an image
pyramid, however at the cost of missing closely spaced edges.
It will be obvious that cascades of operators may be realized
in many ways.
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