\v Ww 
Regine Brügelmann 
  
  
Jump Edge Crease Edge Curvature Edge 
  
  
  
Figure 2: Different edge types 
  
  
edge type discontinuity in continuity in 
jump edge range 
crease edge surface normal range 
curvature edge curvature range and surface normal 
  
  
  
  
  
Table 1: Characteristics of edge types 
can partly be accredited to the noise strengthening effect of the second derivatives. Avoiding this effect (Chakreyavanich, 
1991) uses the Laplacian of Gaussian operator (LoG) for the detection of breakpoints. The Laplacian of Gaussian operator 
is a combination of a Laplacian operator (V?) and a Gaussian function G(c,r) with c = column, r = row: 
pi 5) 
e* 4- ^ — 98^ 
s4 
LoG z V? G(c,r) z G(c,or) with 5= 0Gausstan - (1) 
According to (Chakreyavanich, 1991), breakpoints (curvature edges) correspond to the points having positive maximum 
and negative minimum convoluted value of the signal between zero-crossings of LoG. Zero-crossings of the LoG represent 
jump edges or inflection points. 
2.3 Edge preserving filtering 
(Weidner, 1994) proposes an algorithm for parameterfree information-preserving surface restoration. The basic idea is to 
extract the data's signal and noise properties simultanously by variance-component estimation and use this information 
for the filtering of the data. This way discontinuities in the data are maintained. A similar idea is used by (Wild and 
Krzystek, 1996) in the course of an automatic DEM generation process from stereo-photographs. 
2.4 Second derivatives and hypothesis testing 
(Fórstner, 1998) proposes to treat breakline detection in range images on principle as finding edges in intensity images 
by means of performing hypothesis testing. Edge pixels are meant to be borders of homogeneous regions. Therefore the 
following two properties are inherent to them: 
1. A further to be defined homogeneity measure is significantly larger at edges than in homogeneous regions. 
2. This homogeneity measure is locally maximal across the edge. 
Following these principles a pixel with a significant large homogeneity with respect to noise will be called "edge pixel’. 
Thus edge pixels — actually being pixels which are non-homogeneous — indicate where there may be a signal, e.g. an 
edge. 
For intensity images g(c, r) often the squared gradient magnitude is used as homogeneity measure: 
  
  
  
h, z|Vgl! 2 934 o? Q) 
which becomes — being weighted assuming constant noise c? — the x2-distributed test statistic 
hy Vgl? 9° g2 
uz LN 2% LE (3) 
On! On! On! Op 
That means that pixels with 
2] > X2 «a 4) 
are significantly non-homogeneous, thus likely to be edge pixels. In multichannel images such as color images this test 
statistics can be extended to 
K K 9 
hi. V qi. |* j ; 
= = = > | I with A — number of channels , (5) 
k=1 n, kl n' 
  
“1 multi 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 111