Manfred H. Günzl
This indicates that 77, is independent of the size and only depends on the aspect ratio. Analogous to the square one can
assemble a rectangle out of 4 parallel or perpendicular boarders approximated within the grid (— figure 3). Distinguishing
the two cases (D) Az, — Ay,, Az, = Aye (D and Axa +1 = Aya, Axp + 1 = Ayp (ID Tor derives to
mn Ease werio-Iy (8)
ee 4v Av((Azzp 4- Ayp)v -- v + 1)’
(v 4- 1)? (sip
II meis t !
qD — m 4v 4v((Aas - Ayy)?v + 1) er
Both cases converge to the result of the parallel case as show in equation 7 for large rectangles. For twisted rectangles
with an orientation of w and Ayg > Axa + 1, Ayp > Axp + 1 the general case can be derived as
Ayay = AxaÂT; = AzaÂy tan(w), (10)
(Axa: Aya) = la(sin(w),cos(w)), (11)
(Axe, Aye) = l(cos(w),sin(w)), (12)
; 1)? 7
Yor. F (ot 17 ne) + cos(2w)) — A sin(w), (13)
4v vl
4 1 2
Tu m tal - (sin(2w) + cos(2w)) for ly — oo. (14)
U mme mme
pro (W)
fg; (0)
As a result the influence of the aspect ratio v and the orientation w can be splited into two independent factors Top (v) and
Tope (w). The fact that this is only possible for large enough rectangles with /,, and /j > 1 is the mathematical analogy to
the fact that objects with a size of the scale of the sampling cannot be recognized very well.
2.3 Error compensation
Equation 14 enables the compensation of the influence of the aspect ratio v and the orientation w independently by
multiplication of 7, with the reciprocal of 7,,.(v) or r,,.(w). Applied to land use segmentation there is no a priori
knowledge about these parameters.
A major advantage of r,,. is that P, E and C can be derived with very little computational effort. Applied to region
growing by merging (Schachter et al., 1979, Tilton and Cox, 1983, Beaulieu and Goldberg, 1989) it is possible to derive the
parameters of a merged region out of the parameters of the initial regions with a computational effort that is independent
of size and shape of the regions. This is possible, because P, E and C are all numbers of objects that can easily be added.
In the case of a region merging the sum of E and C only need a correction with regard to the border that was removed.
To keep this advantage the goal of error compensation is to derive v or w out of simple object counts. This can be realized
using totals of enclosed grid cell coordinates. Let (1, yj.) be the coordinate of the enclosed grid cells with 1 < k < P.
Further let qo fan}, ÿ = {yr}, YZ = E r, and Xy :— $7, , yy. Considering x; and y, as realizations of
random variables x and y the variance and covariance are defined as
Óm fx V fg
V. Vigra ll. (22) Wie) UL e IED) (15)
TL TL ; n n
i'j (Xz)(Xy)
AV:-V,-V, und C,,:-Cov(zy)- —- z (16)
= n n?
The main direction of a set of point (x, yx) is equal to the eigenvector € with the largest eigenvalue €.
V(x Tr, !
(ND Cm Y C mt deRzeR (17)
Cov(z,y) V(y)
As both eigenvectors are orthogonal and the error of orientation r,,,. (w) is 45? cyclic, it is insignificant whichever is used.
n. d uns, {ave AVE a0
1/2 = po Rum (18)
1/ €1/2,y 2C ry
w -— arctan(e1,)/61,;) — arctan(—e»2,,/e» ,), (19)
AV| -- 2|C,,
Tc (W) = sin(2w) + cos(2w) = BVI +90. (20)
«HAV + 402,
354 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.
