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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
  
(a) (b) 
(c) (d) 
Figure 3: Orthogonal rectification: (a) and (c) original image regions inside Zwickel. (b) and (d) rectified image regions. 
We first calculate the edge orientation v and magnitude m 
at each pixel inside the rectified frame I: 
  
mr, qj) E fein + Lu = (I d zi Tori) (2) 
q(x, y) 7 atan((Iz—1,y * Los) s + Loti) (3) 
An orientation histogram is used as a region descriptor, the 
magnitude and the distance of the pixels from the origin 
are used as a weight. More formally the histogram is cal- 
culated as 
FO) =) 00.0) + wy, (4) 
QN 
where H (0) is the value for bin 0 (0 € [0?. 1^ .. .360?]) 
and q denotes angle values in a neighborhood N inside 
the Zwickel, w, is the weight of ç and ô(0, p) is the Kro- 
necker delta function. The angles ÿ are quantized in accor- 
dance with the histogram bins 0. The weight w, is com- 
puted from the magnitude of ¢ and a function decreasing 
with increasing radius r from the origin (zo, yo). We use 
a-Gaussian function thus w,(x, y) = mx, y) * g(r), with 
r— n9)? * (y — yo)? and g(r) = e? 
S] 
ii 
N 
  
  
The parameter c of the Gaussian function has to be adapted 
according to the detected scale. Due to the use of image 
derivatives illumination insensitivity is also achieved. 
3 MATCHING 
In the matching step we want to detect similar regions in 
an image pair. Using the Zwickel representation it is easy 
to implement several pre-selection criteria to speed up the 
matching by reducing the number of putative candidates. 
The pre-selection is preformed on the basis of geometric 
constraints as well as on image information. We only allow 
a maximal angle difference between corresponding lines of 
a Zwickel candidate pair. Furthermore we enforce the lines 
to have the same gradient direction. If a Zwickel encloses a 
darker region than the surrounding, the two lines have dif- 
ferent gradient directions and therefore different line types. 
Other pre-selection criteria for candidates e.g. by compar- 
ing the difference of the gray-value median for the Zwick- 
els can be easily implemented. For the remaining candi- 
dates we detect the most similar ones by comparing the 
descriptors. In order to accomplish this task we have to 
1121 
choose a proper distance function for the comparison of 
the orientation histograms. 
3.1 Distance functions 
Since the descriptors described in section 4 are histograms 
we use probabilistic distance measures to describe the sim- 
ilarity. Distance measures for histogram comparison are 
the L4 and L5 norm, the Bhattacharyya distance, and the 
Matusita distance. The earth movers distance is a more 
complex method for histogram comparison and is com- 
puted by solving the so called transportation problem, pro- 
posed for image indexing by Rubner et.al (Rubner et al., 
1998). Huet and Hancock (Huet and Hancock, 1996) give 
a comparison of the performance of this measures for his- 
togram comparison. Following the conclusions of Rubner 
we chose the Bhattacharyya distance which is defined as: 
  
D Bhatt (I4. Hp) = —In S^ 
1 
The Zwickel pair with the smallest distance is the most 
similar in terms of the histogram comparison. 
4 EXPERIMENTS 
We carried out several experiments to show the perfor- 
mance of the proposed method. In all experiments the re- 
gion size was 30 x 30 pixel. In order to increase the ro- 
bustness of the matching we also compute the normalized 
correlation coefficient cc for the rectified image patches. 
The distance function therefore modifies to: 
D = Drnau( Ha, Ha) * (L — cold, B)) where A and 
B denote the two rectified image patches and H 4 and Hp 
are the orientation histograms for the image patches. In the 
first experiment we assess the invariance of the descriptor 
against viewpoint changes. Sequences of several box-like 
objects were acquired by a turntable setup. The rotation 
between two subsequent images is five degrees resulting in 
a 72 image series. A key image is selected and we per- 
form the matching with all subsequent images. For evalu- 
ation purposes we keep thirty percent of the best matches 
(smallest D)and determined the number of correct matches 
by calculating the epipolar geometry. Figure 6(a) and Fig- 
ure 6(b) show the rate of correct matches versus the rota- 
tion angle between the camera of the key image and the 
camera of the second image used for the matching. The