Some Examples

Figure: Examples of polygons with their rectilinearity measured as proposed in this paper. Polygons are rotated to the orientations that maximised ${\cal Q}(P,\alpha)$.

1.000

0.849

0.334

0.041

0.038

0.860

0.813

0.603

0.508

0.277

0.668

0.516

0.425

0.380

0.351

0.748

0.457

0.287

0.204

0.134

The rectilinearity measure is applied to a (perfect) rectilinear polygon in the top left hand polygon in Fig. 3 which is then degraded in various ways. All examples show that the rectilinearity measure is well behaved; increasing distortion consistently decreases the computed value. Note also that the orientations that maximised ${\cal Q}(P,\alpha)=\frac{{\cal P}_2(P)}{{\cal P}_1(P, \alpha)}$ match our expectations except at high noise levels when the rectilinearity measure has dropped close to zero.

For each of the maximally degraded polygons (i.e. the rightmost examples in each row) Fig. 4 plots ${\cal Q}(P,\alpha)$. It can be seen that it is well behaved and, despite the effects of noise and other distortions which introduce local maxima, the main peak remains distinct.

Figure: Plots of normalised ${\cal Q}(P,\alpha)$ for each of the rightmost examples in Fig. 3.

The rectilinearity measure is now applied to a wide range of shapes, which are then ranked in order of decreasing rectilinearity (Fig. 5). Comparison against human rankings show that a similar ordering has been generated.

Figure: Shapes ranked by ${\cal R}$

Another example is shown, working this time with real data from a Digital Elevation Model (DEM). Some simple noise filtering and segmentation techniques were applied to produce a set of polygons. These are further processed using Ramer's line simplification algorithm to reduce the effects of quantisation. Fig 6b plots the regions filled with intensities proportional to their rectilinearity; thus rectilinear shapes generally appear bright.

Figure: (a) A DEM of Hanover; (b) segmented regions with filled intensities proportional to rectilinearity

(a)

(b)

A final example shows the deskewing of the outlines of the buildings extracted from the aerial photograph (see Fig. 7a&b). Assuming orthographic projection the simple affine mapping $(x',y') = (x+\beta y,y)$ is used, and applied at all orientations $\theta = 0 \ldots \frac{\pi}{4}$. Rectilinearity (${\cal R}$) is maximised over $\beta$ and $\theta$, and the deprojected contours are shown in Fig. 7c.

Figure: (a) Aerial photograph of Cardiff, (b) Extracted contours of buildings, (c) Contour set globally deskewed to maximise rectilinearity.

(a)

(b)

(c)