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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
2. Results of computer vision algorithms may contain scale-
dependent errors which can be estimated from the output
on images of different scales. Therefore, test images are
required realizing multiple levels of resolution.
Gaussian image pyramids. An image pyramid stores an image
I at multiple resolutions. For a high resolution image / we build
up a Gaussian image pyramid in the following way, cf. (Crowley,
JL et al; 2002):
We start from the full resolution image I9) — J of size N, x Ne
[pel ?] as the lowest level image. On each pyramid level rv, the
image // is smoothed by two-dimensional convolution
I(r, ¢) = Go (r, c) x I (v, c)
with an isotropic Gaussian kernel
- 1 à 22405
Gor, c) = rci 20°
7
of filter width à = 2 [pel]. The smoothed image I” is sub-
sampled using
JO C) md ar. ac)
relire) CL Na ix ums yield
ing the higher level image //"*
Noise characteristics of Gaussian image pyramids.
Smoothing with a Gaussian kernel of width c reduces image
noise. The relation between the noise variance oÿ of the orig-
inal image and the noise variance c? of the smoothed image is
given by (cf. (Fuchs, 1998), eq. 4.8
2 2 2 Lu à
7, =‘ Ga (r, c)drde = 74
di oc dro
Thus, if og denotes the noise variance of the level-0 image of an
image pyramid, the noise variance of the level k image is given
by
try! ol 2
(ot ) = nk Jo.
(4702)
Obviously, the amount of image noise decreases rapidly with in-
creasing pyramid levels.
Procedure. The method exploits the fact that the higher level
images of an image pyramid are practically noiseless but contain
the same image structure as the full resolution image — a fact that
qualifies the higher level pyramid images for the use as reference
data for the input signal of an algorithm. The natural image struc-
ture is widely prevented even in the higher level pyramid levels,
as the decimation step on each level of the image pyramid widely
compensates for smoothing the signal.
4 EXAMPLE: CHARACTERIZING FEATURE
EXTRACTION ALGORITHMS
To demonstrate the feasibility of the approaches to reference data
acquisition, we employ both methods for characterizing low level
feature extraction modules. We consider the following issues:
1063
1. Straight line and edge detection: Some modules for extract-
ing linear features, i. e. straight lines and edges from images,
for some reason provide line and edge segments which are
systematically too short. We investigate the shortening of
linear features exemplarily for the feature extraction mod-
ule presented in (Fuchs, 1998).
2. Point detection: Most investigations on the noise behavior
of point extraction algorithms are based on synthetic data
for the signal and for the noise. Unlike these. we investi-
gate the noise sensitivity of the point extraction proposed
by (Fórstner and Gülch, 1987) based on an almost noiseless
real signal with only the added noise being synthetic.
4.1 Basics
4.1.1 Notation. We use Euclidean and homogeneous repre-
sentations of points and straight lines in 2D and 3D. Euclidean
coordinates of a 2D point are denoted with lowercase slanted let-
ters x, its homogeneous coordinates are denoted with lowercase
upright letters x = (xl ; ai Homogeneous coordinates of a 3D
point are denoted with upright uppercase letters X. Segments of
straight lines and edges in 2D are represented by their bounding
points / : (xs, Xe). The line joining two end points x, and x, is
given homogeneous with 1 2 x, ^ Xe — X, X Xe.
Stochastic entities are underlined, e. g. x, their expectation values
are marked with a bar, e. g. x. Estimated entities wear a hat, e. g.
€ and reference values have a tilde, e. g. x.
4.1.2 Shortening of linear features at junctions. An exam-
ple for shortened lines and edges at junctions is depicted in fig.
3c). It is drawn from the output of the feature extraction software
FEX, which was applied to an intensity image of a polyhedral ob-
ject. The software expects a multi-channel intensity raster image
as input and provides a symbolic image description containing
lists of points, blobs and linear features, the linear features being
segments of straight lines or intensity edges.
a)
Figure 3: Situation for a single image: a) Rectified image b) Rec-
tified image with projected reference points c) Rectified image
with extracted edges d) Local situation for two junctions.
Given the true values x, and x« of the two end points x, and x, of
a straight line or edge, the expectation value d of the shortening
d is defined as
d= E(d) = E(| &. — & | - |, — x, D.
