r.fill.stats is a module for fast gap filling and interpolation (with smoothing) of dense raster data.
The r.fill.stats module is capable of quickly filling smalldatagaps in large and high-resolution raster
maps. It’s primary purpose is to improve high-resolution, rasterized sensor data (such as Lidar data). As
a rule of thumb, there should be at least as many data cells as there are "no data" (NULL) cells in the
input raster map. Several interpolation modes are available. By default, all values of the input raster
map will be replaced with locally interpolated values in the output raster map. This is the equivalent of
running a low-pass smoothing filter on the interpolated data and is often desirable, owing to noisy
nature of high-resolution sensor data. With dense data and small gaps, this should result in only slight
deviations from the original data and produce smooth output. Alternatively, setting the -k flag will
disable the low-pass filter and preserve the original cell values.
Available gap filling modes:
• spatially weighted mean (default)
• simple mean
• simple median
• simple mode
The spatially weighted mean is equivalent to an Inverse Distance Weighting (IDW; see also r.surf.idw)
interpolation. The simple mean is equivalent to a simple low-pass filter. Median and mode replacements
can also be achieved using r.neighbors.
Note that r.fill.stats is highly optimized for fast processing of large raster datasets with a small
interpolation distance threshold (i.e. closing small gaps). As a trade-off for speed and a small memory
foot print, some spatial accuracy is lost due to the rounding of the distance threshold to the closest
approximation in input raster cells and the use of a matrix of precomputed weights at cell resolution
(see further down for details). Note also that processing time will increase exponentially with higher
distance settings. Cells outside the distance threshold will not be interpolated, preserving the edges of
the original data extent.
This module is not well suited for interpolating sparse input data and closing large gaps. For such
purposes more appropriate methods are available, such as v.surf.bspline, v.surf.idw or v.surf.rst.
Applications where the properties of r.fill.stats are advantageous include the processing of high
resolution, close range scanning and remote sensing datasets. Such datasets commonly feature densely
spaced measurements that have some gaps after rasterization, due to blind spots, instrument failures, and
misalignments between the GIS’ raster cell grids and the original measurement locations. In these cases,
r.fill.stats should typically be run using the "weighted mean" (default) method and with a small distance
setting (the default is to use a search radius of just three cells).
The images below show a gradiometer dataset with gaps and its interpolated equivalent, produced using the
spatially weighted mean operator (mode="wmean").
In addition, r.fill.stats can be useful for raster map generalization. Often, this involves removing
small clumps of categorized cells and then filling the resulting data gaps without "inventing" any new
values. In such cases, the "mode" or "median" interpolators should be used.
Usage
The most critical user-provided settings are the interpolation/gap filling method (mode) and the maximum
distance, up to which r.fill.stats will scan for cells with values (distance). The distance can be
expressed as a number of cells (default) or in the current coordinate reference system’s units (if the -m
flag is given). The latter are typically meters, but can be any other units of a planar coordinate
system.
Note that proper handling of geodetic coordinates (lat/lon) and distances is currently not implemented.
For lat/lon data, the distance should therefore be specified in cells and usage of r.fill.stats should be
restricted to small areas to avoid large inaccuracies that may arise from the different extents of cells
along the latitudinal and longitudinal axes.
Distances specified in map units (-m flag) will be approximated as accurately as the current region’s
cell resolution settings allow. The program will warn if the distance cannot be expressed as whole cells
at the current region’s resolution. In such case, the number of cells in the search window will be
rounded up. Due to the rounding effect introduced by using cells as spatial units, the actual maximum
distance considered by the interpolation may be up to half a cell diagonal larger than the one specified
by the user.
The interpolator type "wmean" uses a precomputed matrix of spatial weights to speed up computation. This
matrix can be examined (printed to the screen) before running the interpolation, by setting the -w flag.
In mode "wmean", the power option has the usual meaning in IDW: higher values mean that cell values in
the neighborhood lose their influence on the cell to be interpolated more rapidly with increasing
distance from the latter. Another way of explaining this effect is to state that larger "power" settings
result in more localized interpolation, smaller ones in more globalized interpolation. The default
setting is power=2.0.
The interpolators "mean", "median" and "mode" are calculated from all cell values within the search
radius. No spatial weighting is applied for these methods. The "mode" of the input data may not always be
unique. In such case, the mode will be the smallest value with the highest frequency.
Often, input data will contain spurious extreme measurements (spikes, outliers, noise) caused by the
limits of device sensitivity, hardware defects, environmental influences, etc. If the normal, valid range
of input data is known beforehand, then the minimum and maximum options can be used to exclude those
input cells that have values below or above that range, respectively. This will prevent the influence of
spikes and outliers from spreading through the interpolation.
Unless the -k (keep) flag is given, data cells of the input map will be replaced with interpolated values
instead of preserving them in the output. In modes "wmean" and "mean", this results in a smoothing effect
that includes the original data (see below)!
Besides the result of the interpolation/gap filling, a second output map can be specified via the
uncertainty option. The cell values in this map represent a simple measure of how much uncertainty was
involved in interpolating each cell value of the primary output map, with "0" being the lowest and "1"
being the theoretic highest uncertainty. Uncertainty is measured by summing up all cells in the
neighborhood (defined by the search radius distance) that contain a value in the input map, multiplied by
their weights, and dividing the result by the sum of all weights in the neighborhood. For mode=wmean,
this means that interpolated output cells that were computed from many nearby input cells have very low
uncertainty and vice versa. For all other modes, all weights in the neighborhood are constant "1" and the
uncertainty measure is a simple measure of how many input data cells were present in the search window.
Smoothing
The r.fill.stats module uses the interpolated values to adjust the original values and create a smooth
surface, which is akin to running a low-pass filter on the data. Since most high-resolution sensor data
is noisy, this is normally a desired effect and results in output that is more suitable for deriving
secondary products, such as slope, aspect and curvature maps. Larger settings for the search radius
(distance) will result in a stronger smoothing. In practice, some experimentation with different settings
for distance and power might be required to achieve good results. In some cases (e.g. when dealing with
low-noise or classified data), it might be desirable to turn off data smoothing by setting the -k (keep)
flag. This will ensure that the original cell data is copied through to the result map without
alteration.
Effectofsmoothingtheoriginaldata:Thetoprowshowsagap-filledsurfacecomputedfromarasterizedLidarpointcloud(usingmode=wmean and power=2), and the derived slope, aspect, and profile curvature
maps. The smoothing effect is clearly visible. The bottom row shows the effect of setting the -k flag:
Preserving the original cell values in the interpolated output produces and unsmoothed, noisy surface,
and likewise noisy derivative maps. The effect can be seen in the illustration above: Slope, aspect, and
profile curvature are computed using the r.slope.aspect module, which uses a window (kernel) for
computations that considers only the immediate neighborhood of each cell. When performed on noisy data,
such local operations result in equally noisy derivatives if the original data is preserved (by setting
the -k flag) and no smoothing is performed.
(Note that the effects of noisy data can also be avoided by using modules that are not restricted to
minimal kernel sizes. For example, aspect and other morphometric parameters can be computed using the
r.param.scale module which operates with variable-size cell neighborhoods.)
Spatialweightingscheme
The key to getting good gap filling results is to understand the spatial weighting scheme used in mode
"wmean". The weights are precomputed and assigned per cell within the search window centered on the
location at which to interpolate/gap fill all cells within the user-specified distance.
The illustration below shows the precomputed weights matrix for a search distance of four cells from the
center cell:
000.00 000.01 000.04 000.07 000.09 000.07 000.04 000.01 000.00
000.01 000.06 000.13 000.19 000.22 000.19 000.13 000.06 000.01
000.04 000.13 000.25 000.37 000.42 000.37 000.25 000.13 000.04
000.07 000.19 000.37 000.56 000.68 000.56 000.37 000.19 000.07
000.09 000.22 000.42 000.68 001.00 000.68 000.42 000.22 000.09
000.07 000.19 000.37 000.56 000.68 000.56 000.37 000.19 000.07
000.04 000.13 000.25 000.37 000.42 000.37 000.25 000.13 000.04
000.01 000.06 000.13 000.19 000.22 000.19 000.13 000.06 000.01
000.00 000.01 000.04 000.07 000.09 000.07 000.04 000.01 000.00
Note that the weights in such a small window drop rapidly for the default setting of power=2.
If the distance is given in map units (flag -m), then the search window can be modeled more accurately as
a circle. The illustration below shows the precomputed weights for a distance in map units that is
approximately equivalent to four cells from the center cell:
...... ...... ...... 000.00 000.00 000.00 ...... ...... ......
...... 000.00 000.02 000.06 000.09 000.06 000.02 000.00 ......
...... 000.02 000.11 000.22 000.28 000.22 000.11 000.02 ......
000.00 000.07 000.22 000.44 000.58 000.44 000.22 000.07 000.00
000.00 000.09 000.28 000.58 001.00 000.58 000.28 000.09 000.00
000.00 000.07 000.22 000.44 000.58 000.44 000.22 000.07 000.00
...... 000.02 000.11 000.22 000.28 000.22 000.11 000.02 ......
...... 000.00 000.02 000.06 000.09 000.06 000.02 000.00 ......
...... ...... ...... 000.00 000.00 000.00 ...... ...... ......
When using a small search radius, cells must also be set to a small value. Otherwise, there may not be
enough cells with data within the search radius to support interpolation.
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