Regine Brügelmann
which now is x2, -distributed. This way the information of the different channels can easily be combined to a single and
sensible homogeneity measure.
Transfering this principle to range images d(c, r) can be realized by starting with the gradient image g — Vd — (de, dr)!
as two-channel image (Förstner, 1998). This is due to the fact that edges in range images are pixels which do not lie on
flat surfaces but are expected to be pixels where the curvature — which is closely related to the second derivatives — is
significant compared to noise. Thus instead of deriving homogeneity hı from the first derivatives, the homogeneity hy is
derived from the second derivatives assuming constant noise variance:
h; — Addi) (D dh) (P rdi) (6)
z dUM.Ad, (7)
= irH(d)z XM(H);4X(H) (8)
= «2 + ne ; (9)
x1 and K+ are the principal curvatures which specify the curvature of surface curves in the directions of maximal and
minimal normal curvature at each point. They are a perfect pair of surface curvature descriptors, which are analytically
equivalent to the mean and Gaussian curvature pair (Besl, 1986). Eq.(9) only applies if the local metric is neglected.
Then the 2 x 2 Weingarten matrix (shape operator) transforms into the Hessian matrix H (Weidner, 1994). The so-called
‘quadratic variation’ ha is only Zero in case the pixel's surrounding is flat, that is in case both principle curvatures are
zero. Normalization of eq.(7) with the noise variances leads to the x3-distributed test statistics
déc + der 4 der
s) 2
0 a cr
Nec Mer Nyy
(10)
3 SUGGESTED APPROACH
The suggested approach is based on the latter described method of breakline detection by hypothesis testing. Fig.3 shows
an overview of the performed steps. Input is an greyvalue image representing the range data. The processing chain results
in smooth 3D vector breaklines. In the following we describe the single processing steps more detailed. The interim
results for the small dike testdata-set (fig. 1) are illustrated in fig. 4.
3.1 Detection of possible breakpixels
Since range data often is fair noisy and, moreover, noise is strongly strengthened by every derivation, it is sensible to
perform a smoothing before or during building the second derivatives which are required for calculating the homogeneity
measure (see eq.(7)). To perform smoothing and derivation in one step, the differentiation kernels for determining the
second derivatives could be discrete approximations of the corresponding Gaussian’s (Forstner, 1998)
0? mop g? er 0? r2 — 5°
aa Cs 7) m e Gsle,r), 35, 0^ r)= zi Gs(e,r), Ex r) = i Gs(e,r) (1D
with G,(c,r) z Gaussian and s = CGaussian - (12)
To build the test statistics z» in equation, the noise variances in equation (10) can explicitly be derived. For the case of the
Gaussian kernels they are (Fórstner, 1998):
GL. UL = t and d'u wee ; (13)
es rn 1675? er 16 s°
The choice of the appropriate filter size must carefully be done because noise has to be smoothed whereas significant
edges should still be detected. (Chakreyavanich, 1991) proposes to determine the correct filter size by the relationship
between spatial and frequency parameters of the Gaussian. The image noise c; can be measured in flat surface regions
or estimated from homogeneous image regions (see (Fórstner, 1998)). Fig. 4a shows the test statistics za per pixel. The
lighter the greyvalue the larger the test statistics, that is the larger the curvature in this point of the surface. Fig.4b denotes
possible breakpoint regions resulting from hypothesis testing (z2 > X3 a» With x3 0.99 = 11.34).
3.2 Nonmaxima-suppression
To reduce broad breakpoint regions to one pixel wide breaklines a simple thinning operation is not suitable because
neighbouring breakpoint regions sometimes melt (see left ascent in fig. 4b). Thinning would extract the middle axis of
the melted region, thus falsifying the position of the edge. In order to avoid this, a nonmaxima-suppression is performed
taking into account the direction of the maximum curvature which usually is the direction across the edge. The position
of the breakpixel is found as the local maximum of z». The direction of the maximum curvature corresponds with the
direction of the eigenvector of the maximum eigenvalue of the Hessian matrix. After the nonmaxima-suppression a
thinning operation is performed to eliminate cornerpixels. The pixels of the breakline then are 4-connected instead of
8-connected (fig. 4c). The heights of the pixels are taken from a slightly smoothed surface to eliminate the influence of
noise.
112 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.
