[GRASS-dev] [GRASS GIS] #3102: v.to.db compactness and fractal dimension

[GRASS GIS]

#3102: v.to.db compactness and fractal dimension
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 Reporter:  mmetz                  |      Owner:  grass-dev@…
     Type:  defect                 |     Status:  new
 Priority:  normal                 |  Milestone:  7.3.0
Component:  Vector                 |    Version:  7.0.1
 Keywords:  v.to.db,shape metrics  |        CPU:  All
 Platform:  All                    |
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 If several areas share the same category, v.to.db sums up area sizes and
 perimeters for all areas with the same category, but for compactness and
 fractal dimension, the values of the last area processed are used.
 Instead, for compactness and fractal dimension, area size and perimeter
 length should be summed up for each category and at the end compactness
 and fractal dimension should be calculated from the area size and
 perimeter length sums for each category value.

 Further on, compactness aka shape index is normalized to the area size of
 the most compact shape (a circle), but fractal dimension is not. This
 normalization is also done by e.g. FRAGSTATS.

 The formula for compactness is

 {{{
 perimeter / (2.0 * sqrt(M_PI * area))
 }}}

 which is for given area size and perimeter length of an object the area
 size of a cirlce with the same perimeter divided by the area size of the
 object.

 The formula for fractal dimension is

 {{{
 2.0 * log(perimeter) / log(area)
 }}}

 but should be

 {{{
 2.0 * log(perimeter / (2.0 * sqrt(M_PI)) / log(area)
 }}}

 which is for given area size and perimeter length of an object the log of
 the area size of a cirlce with the same perimeter divided by the log of
 the area size of the object.

 Further on, there are problems with using the log() function because
 log(1) is 0. That means that for area sizes smaller than 1, growing
 towards 1, the result approaches -infinity and for areas larger than 1 and
 growing further, the result decreases from +infinity. Thus for area sizes
 equal 1, the result is undefined, it could be + or -infinity. Results can
 also be negative or positive, depending on whether perimeter and/or area
 size are < 1. These problems could be avoided by using log(1 + x) instead
 of log(x).

 The fractal dimension is not only dependent of the shape (cirlce, square,
 star etc) but also on the size, e.g. larger squares have a smaller fractal
 dimension than smaller squares. This should be added to the manual to
 facilitate interpretation of the fractal dimension.

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Ticket URL: <https://trac.osgeo.org/grass/ticket/3102>
GRASS GIS <https://grass.osgeo.org>