foe’ NE wa 3.
€
»
Steger Carsten
r(5,0,1,0) — — 7x0,10) — e110) —
AN WS
NY
oooo
SS
TR
Figure 1: Scale-space behavior of the bar-shaped line f, with a = 0 and w = 1 when convolved with the derivatives of
Gaussian kernels for z € [—3,3] and o c [0.2, 2].
e
0.8 F True line position —
REM XN
0.6 | i | Line Position 4
s i |
j ||
04 | ; i»
i |
Edge Position i |
0.2 | m 1
|
ZZ
i Lp True line width
| à
3 -2 -1 0 1 2 3
X
Figure 2: Location of an asymmetrical line and its corresponding edges with width w = 1,0 = 1, anda € [0,1].
where g,(z) = or exp(— =) is the Gaussian kernel with standard deviation o and 4$, (x) is its integral. The scale-
space behavior of a line with w = 1 and a = 0 is displayed in Figure 1. The line position is given by the maxima (for
bright lines) or minima (for dark lines) of ra (x, 0, w, a), i.e., by the points where 7; (x, 0, w, a) = 0. We can see that with
this definition of the line position we can extract lines with any o > 0. However, in (Steger, 1998b) it was shown that for
small o the selection of salient lines will become difficult because the response 7 (x, o, w, a), on which the selection of
salient lines is based, will be small. To make the selection of salient lines easy, à > w/ V3 should be selected. From (5),
we can see that the line position is
d 2 tit — a) . (7)
2w
Hence, the line position will be biased if a # 0.
To extract the line width, we need to recall that a line is a combination of step edges, i.e., a line is bounded by an
edge on each side. The edge position can be obtained from the maxima of the absolute value of the first derivative of
the smoothed image, i.e., by the maxima of |r) (z,0,w,a)| or the zero crossings of r, (x, 0, w, a), where additionally
r"' (z, a, w, a)r (r,0,w,a) < 0 is required. From the center graph in Figure 1, it can be seen that the edges of the lines,
and consequently edges in general, can also be regarded as lines in the image of the first derivative of the input image.
In the 2D case, the image of the first derivative corresponds to the gradient image. We will use this interpretation of the
edges as lines in the gradient image throughout the rest of the paper.
Unlike the line position (7), the location of the edges can only be computed numerically. Figure 2 gives an example of
the line and edge positions for w — 1, c = 1, and a € [0, 1]. It can be seen that the position of the line and the edges is
greatly influenced by line asymmetry. As a gets larger the line and edge positions are pushed to the weak side, i.e., the
side that possesses the smaller edge gradient.
All these bias effects are not very encouraging if we want to extract lines and their width with high accuracy. Fortunately,
the definition of the line position by 7’ (x, 0, w,a) = 0 and of the edge position by 7, (T,0,w,a) = 0 can be used to
model the bias of the line position and width explicitly and hence to remove the bias, as was shown in (Steger, 1998b,
Steger, 1998¢). One of the properties that makes this analysis feasible is that the biased line positions and widths form
a scale-invariant system, which means that we can think in terms of scale-normalized line positions /, — //c and line
widths w, = w/o. This means that we save one parameter because we can set o to an arbitrary value, e.g., 0 = 1to
analyze the bias. With this, we can predict two features that we can extract from the image, i.e., the total line width v,
and the ratio of the gradients r, as a function of the true line width w, and the true asymmetry a. More formally, there is
a function f : (w,, a) + (vs, r). It can be shown that the inverse f ^! of this function exists and is unique. With this, it is
possible to remove the bias from the extracted line positions and widths. Figure 3 displays the bias removal function f "t
Note that f! cannot be described by an analytical formula. It must be computed by a multi-dimensional root finding
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 143
