Steger Carsten
f)T), where f denotes the image, is used to classify the image into homogeneous and non-homogeneous regions. For
multispectral images, the average squared gradients in the individual channels are weighted by their noise variances and
are summed. The non-homogeneous regions are further classified into point and line regions. Through the use of the
average squared gradient, the algorithm is able to extract lines and edges simultaneously within the line regions with
the same approach. Edges and lines are simply given by the maxima of the average squared gradient in the direction
perpendicular to the edge and line. This direction corresponds to the larger eigenvalue of I'z. To extract lines of a
certain width, o; must be suitably chosen. This algorithm is able to extract bar-shaped lines as well as staircase lines.
Unfortunately, the squaring operation leads to strongly biased line positions even for lines with only moderately different
lateral contrast.
Edge detection approaches are too numerous to mention them all. See, for example, (Dougherty and Bowyer, 1998, Shin
et al., 1998) for a performance evaluation of several pixel precise edge detectors. For subpixel edge extraction, one can fit
primitives, e.g., planes or extruded 1D tanh curves as in (Nalwa and Binford, 1986) or an appropriate line or edge model
which is selected from the output of a quadrature filter pair (Kisworo et al., 1994), to windows in the image data, which
is very computationally expensive. Alternatively, the edge positions can be interpolated to subpixel precision, e.g., from
the Laplacian of the image (Huertas and Medioni, 1986) or the second derivative in the direction of the gradient (Busch,
1996, Cumani, 1991). All of the above approaches will often fail to extract edge junctions from the image although they
are clearly visible.
2 MODEL-BASED LINE AND EDGE EXTRACTION
2.1 Line and Edge Models
The features we are trying to detect, i.e., edges and lines, can be regarded as one-dimensional manifolds in the two-
dimensional space IR?. Both types of features therefore are curvilinear structures s(t) which can be characterized by their
typical ID gray value profile perpendicular to the feature. For edges, we will assume the traditional step edge profile,
which is given by
1, z 20
Ja) = | 0, 2<0. (1)
Edges of arbitrary contrast can be obtained by scaling this normalized profile by h, i.e., hfe(z).
Lines can be regarded as a combination of two step edges. It is useful to consider two different types of profiles, depending
on whether the two step edges have the same polarities or not. If the edges have equal polarities (when viewed from the
center of the line), we obtain the traditional bar-shaped line profile. Since the contrast on the two sides of the line is
usually different, we will use the following profile for lines with equal polarity:
0 r«-w
fat) — lh lvesw (2)
0, ZZ > WS
where a € [0, 1]. General lines of height ^ can be obtained by considering a scaled asymmetrical profile, i.e., h f, (x). In
this definition, w is half the total width of the line. For lines with different polarity we could simply use (2) with a > 1.
This definition would not be symmetrical, however. Therefore, a line with different polarity is better modeled by
0, r«-w
fir) m 4aj-]z|-s w (3)
1; *41»49;
where a € [0, 1]. Again, lines of arbitrary contrast can be obtained by scaling this profile.
2.2 Extraction of Lines with Equal Polarity
The extraction of lines with asymmetrical profiles usually results in biased line positions and widths, as was shown for
lines with equal polarity in (Steger, 1998b, Steger, 1998c) and was also considered in the context of edge detection in
(Shah et al., 1986, Chen and Medioni, 1989). In (Steger, 1997, Steger, 1998b) it was shown how the bias can be removed
from the extracted line positions and widths for lines with equal polarity. We will repeat the most important results here
so that we can extend the analysis to the case of lines with different polarity.
In order to detect lines that have the profile (2), we will convolve the image with the derivatives of the Gaussian kernel.
This leads to a scale-space description of the line profile:
ra(x,o,w,a) = (5 +wW) + (a — 1), (x — w) (4)
r(x,0,w,0). = go(Z+w)+ (a — 1)g0 (x — w) (5)
r'(x,0,w,0) = 9,(x+w)+(a—1)9,(x —w) , (6)
142 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.
