. Feature detection (blobs, lines, points) and feature descrim-
ination (junctions, symmetric features, edges, and lines)
4. Feature localization with subpixel accuracy, where features
are expected to lie inside the classified pixels.
In (Steger, 1996) lines are extracted as primitives. The image i$
regarded as a function g(z, y) and lines are detected as ridges and
ravines in this function by locally approximating the image func-
tion by its second order Taylor polynomial (not the facet model
like in (Busch, 1994)). The coefficients of the Taylor polynomial
are determined by convolving the image with the derivatives of
a Gaussian smoothing kernel. In contrast to (Fórstner, 1994) the
Hessian matrix
H zm ( [E Izy ) (2)
Jzy Ivy
is used to extract the local features.
Curvilinear structures in 2D are modeled as curves s(t) that
exhibit a characteristic 1D line profile in the direction perpen-
dicular to the line, i.e., perpendicular to s’(¢). Let this direction
be n(t). This means that the first directional derivative in the
direction n(t) should vanish and the second directional derivative
should be of large absolute value. To compute the direction of
the line locally for each image point the partial derivatives gs, gy.
gaz, Yay, and gyy of the image are estimated. This is done by
convolving the image with the appropriate 2D Gaussian kernels.
The direction in which the second directional derivative of g(z, y)
takes on its maximum absolute value is used as the direction n(t).
This direction is determined by calculating the eigenvalues and
eigenvectors of the Hessian matrix.
The use of the Taylor polynomial leads to a single response of
the filter to each line. Furthermore, the line position are deter-
mined with sub-pixel accuracy and the algorithm scales to lines
of arbitrary width
original image extracted lines
Figure 21: Extraction of lines using the appraoch of Steger
Other articles on the extraction of lines are: (Blaszka and
Deriche, 1994a), (Gruen and Agouris, 1994), (Koller et al., 1994),
(Koller et al., 1995), (Monga et al., 1995). Further articles on the
extraction of image primitives: (Reynolds and Beveridge, 1987),
(Blaszka and Deriche, 1994b), (Filbois and Gemmerlé, 1994).
6.3 Texture
A great variety of operators for texture segmentation have been
developed. The first class analyses the local frequency distribution
based on the idea that every texture has a specific spectrum. The
next class extracts local features (texture elements) by which the
global texture can be defined. Other approaches use stochastic
models for segmentation (Geman and Geman, 1984), (Kato et al.,
1991), (Nguyen and Cohen, 1993). Finally, local features like
the co occurrence matrix are used to describe textures (Lohmann,
1994).
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
The texture analysis using spectral decomposition will be ex-
plained in more detail. The idea is very simple: a set of n filters
has to be defined which extracts the amount of a specific frequency
range for the neighborhood of every pixel. Thus we have a vector
t(g) of length n desribing the pixel g and its neighborhood with
n values. This vector can be used as input for n-dimensional
(un)-supervised classification (see section 6.1).
For the implementation of this approach appropriate filters have
to be found. A simple set of filters was proposed by (Laws, 1980).
He defined 25 filters fi; of size 5 x 5 constructed from vectors
v € (l, e, 5, r, w) by convolution: f;; — vl xvj.
l = (1 4 4 à |;
e - (-152319 2$ 1)
8 = — 02 01)
r = (.1 — 10 —4 1)
w = (-1 23 0. —2 1)
Another popular set are the Gabor filters (Shao and Fórstner,
1994). They are defined in frequency space and have some nice
features: They have orientation selectivity, multiscale property,
linear phase and good localization both in spatial and frequency
domains.
As a last example for texture filters simple gauss shaped filters
can be used. They are invariant with respect to rotation and are
defined via center frequency and the deviation. Typical filters of
all three classes can be found in figure 22.
An example for the application of two laws filters is given in
figure 23. At first the filters f.. and f,; are calculated from the
gray image. The so called texture energy is calculated using a
lowpass filter (e.g., average) with a large filtermask to generalize
the texture. In this case a median filter with circular mask (diam-
eter 50 pixel) was used. These texture energy images can be used
as input to pixel classification.
6.4 Specialized Operations
Besides more general segmentation procedures like those of sec-
tion 6.2 and 6.3 are the specialized filters which emphasize special
structures in a gray image like points, lines, or corners.
The corner resonce operator, for example, is defined by (Harris
and Stephans, 1988):
c
g = Ga * 95 * Go * gy — Go * (9294) — (3)
k (Go * 92 + Go x92)
where g is the gray value and G, ist the Gaussian filter with
deviation c. The corner response function is invariant with respect
to rotation. A typical value for the factor k is 0.04. In this case
corners result in a positive g^ while edges have negative values.
An extension of the corner response function is given in formula
(4).
9° = Ga+(g2?-Go+(95)*-Ga+(g295)— @
k (Go + (98)? + Go + (98°)
© = olga
Here the filter is applied to the gradient and not to the original
gray values. The response is maximal for highly curved edges.
In figure 24 two examples are given. All maximums of the filter
above a given threshold are marked with a cross. Most of the
dominant points of the buildings are found as well as corners
caused by the shadows.
Besides the corner response filter a lot of other filters for corners
or "promiment" points have been defined. Some of these can
172
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