Manfred H. Günzl
e For technical reasons agricultural fields in most cases have got a preferably rectangular shape.
e Field boundaries are closed paths.
e Fields usually do not enclose other fields.
Using these assumptions the human viewer is able to "interpolate" broken boundaries caused by disturbing influences
like noise or speckle. Therefore, the consideration of the shape during an automatic approach of images segmentation is
important to implement fault tolerance.
2 SHAPES WITHIN GRID DATA
Segments in grid data are usually represented by sets of grid cells. If such segments are topologically simple, their shape
can be completely described by a single closed polygon. In case of grid data such polygons are chains of single grid cell
boarders. In an orthogonal isotropic grid these chain links are vertical or horizontal boarders of two neighboring cells.
Let P € NN be the number of enclosed grid cells (pixels) and E € N the number of single boarders (edges) that form the
surrounding polygon. Within a unit grid of 1 x 1 square cells, P is equal to the size of the enclosed area and E is equal to
the length of ine surrounding polygon. A common approach to Patameterize the shape is the measurement of compactness
according to i In the gener case this parameter is equal to 4 for a circle and smaller for all other shapes. In the case
of a grid the maximum of ic 5 is given by a square, if the Ski are parallel to the grid. Changing the squares orientation
towards the grid causes a Sampling error known as aliasing (Foley et al., 1997, 19.3.1). As a result E is greater than the
true length of the squares boarder. The length of the real boarder of a 45° tilted square is factor 4/2 times shorter than the
stairway like polygon. Therefore the aliasing causes an error in the measurement of compactness of up to factor 2.
2.1 Grid shape parameter
It is necessary to include the effects of grid sampling into the shape parameter to compensate this error. Let C € IN be the
number of changes in the direction of the polygon (corners).
A slant towards the grid obviously increases C. Considering two
P © squares one with parallel [J and one with 45° slanted edges © we
get the relations E2 = 16 - Pa, Co = 4, E2 = 32- Po — 16 and
Co — Eo. Out of these one can derive the general relation,
o € 2E? +16 — C?
P=9E=12 Tac (P, E,C) :— mp. . (1)
C=4rpgc=1 "e
o Ó P=13 E= 20 sum Tac (Pn, En, Cn) = rpec(Po, Eo, Co) = 1 for all 0? and
F S a : it ? slanted squares within a grid. The parameter 7, . reaches its
C 20 rPrOG —]1 a LUC
minimum of 1 for all squares with parallel or 45? diagonal edges.
For all other less compact shapes r,,,. derives to values greater than
1. Some examples are given in figure 1.
o © Within 7, the parameter C is reducing the influence of the orien-
P=12 E=14 5 tation by taking the aliasing into consideration. A polygon start-
C=A4Tpec=1.02 P=11 E=24 ing at point (x1, y1) approximating a straight line to point (z», y2)
C —12 rpgc = 2.9 within a grid contains Ax := |x1 — x2| horizontal and Ay :—
|y1 — y2| Vertical lines. The length of the polygon is Az + Ay
Figure 1: Some examples for the shape parameter whereas the = length of the approximated straight line derives to
Tor: + | — \/(Az)? + (Ay)?. Regarding the case Ay > Ax the number
of corners dû the polygon is 2Az. The angle of the line to-
wards the grid derives to w = arctan( az). Every square consists
of 4 parallel or perpendicular boarders that can be described in that way by their approximating polygon (— figure 2).
The parameters of such a slanted square results in
Ay : Ax 9 Ay - Ax >
P= 41 | + (Ay — Ar)“, P [33777] + (Ay — Az)”, (2)
- Ay - Ax 2 2 2 E
P= 1 + (Ay — Ar)“ = (Ay) + (Ar) for PI «P«P, (3)
E — A(Ax + Ay), C = 8Azx + 4. 4)
352 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.
