v.surf.bspline(1grass) GRASS GIS User's Manual v.surf.bspline(1grass)
NAME
v.surf.bspline - Performs bicubic or bilinear spline interpolation
with Tykhonov regularization.
KEYWORDS
vector, surface, interpolation, LIDAR
SYNOPSIS
v.surf.bspline
v.surf.bspline --help
v.surf.bspline [-ce] input=name [layer=string] [column=name]
[sparse_input=name] [output=name] [raster_output=name]
[mask=name] [ew_step=float] [ns_step=float] [method=string]
[lambda_i=float] [solver=name] [maxit=integer] [error=float]
[memory=memory in MB] [--overwrite] [--help] [--verbose] [--quiet]
[--ui]
Flags:
-c
Find the best Tykhonov regularizing parameter using a
"leave-one-out" cross validation method
-e
Estimate point density and distance
Estimate point density and distance in map units for the input vec-
tor points within the current region extents and quit
--overwrite
Allow output files to overwrite existing files
--help
Print usage summary
--verbose
Verbose module output
--quiet
Quiet module output
--ui
Force launching GUI dialog
Parameters:
input=name [required]
Name of input vector point map
Or data source for direct OGR access
layer=string
Layer number or name
Vector features can have category values in different layers. This
number determines which layer to use. When used with direct OGR ac-
cess this is the layer name.
Default: 1
column=name
Name of the attribute column with values to be used for approxima-
tion
If not given and input is 3D vector map then z-coordinates are
used.
sparse_input=name
Name of input vector map with sparse points
Or data source for direct OGR access
output=name
Name for output vector map
raster_output=name
Name for output raster map
mask=name
Raster map to use for masking (applies to raster output only)
Only cells that are not NULL and not zero are interpolated
ew_step=float
Length of each spline step in the east-west direction
Default: 4 * east-west resolution
ns_step=float
Length of each spline step in the north-south direction
Default: 4 * north-south resolution
method=string
Spline interpolation algorithm
Options: bilinear, bicubic
Default: bilinear
bilinear: Bilinear interpolation
bicubic: Bicubic interpolation
lambda_i=float
Tykhonov regularization parameter (affects smoothing)
Default: 0.01
solver=name
The type of solver which should solve the symmetric linear equation
system
Options: cholesky, cg
Default: cholesky
maxit=integer
Maximum number of iteration used to solve the linear equation sys-
tem
Default: 10000
error=float
Error break criteria for iterative solver
Default: 0.000001
memory=memory in MB
Maximum memory to be used (in MB)
Cache size for raster rows
Default: 300
DESCRIPTION
v.surf.bspline performs a bilinear/bicubic spline interpolation with
Tykhonov regularization. The input is a 2D or 3D vector points map.
Values to interpolate can be the z values of 3D points or the values in
a user-specified attribute column in a 2D or 3D vector map. Output can
be a raster (raster_output) or vector (output) map. Optionally, a
"sparse point" vector map can be input which indicates the location of
output vector points.
NOTES
From a theoretical perspective, the interpolating procedure takes place
in two parts: the first is an estimate of the linear coefficients of a
spline function is derived from the observation points using a least
squares regression; the second is the computation of the interpolated
surface (or interpolated vector points). As used here, the splines are
2D piece-wise non-zero polynomial functions calculated within a lim-
ited, 2D area. The length (in mapping units) of each spline step is de-
fined by ew_step for the east-west direction and ns_step for the
north-south direction. For optimal performance, the length of spline
step should be no less than the distance between observation points.
Each vector point observation is modeled as a linear function of the
non-zero splines in the area around the observation. The least squares
regression predicts the the coefficients of these linear functions.
Regularization, avoids the need to have one observation and one coeffi-
cient for each spline (in order to avoid instability).
With regularly distributed data points, a spline step corresponding to
the maximum distance between two points in both the east and north di-
rections is sufficient. But often data points are not regularly dis-
tributed and require statistial regularization or estimation. In such
cases, v.surf.bspline will attempt to minimize the gradient of bilinear
splines or the curvature of bicubic splines in areas lacking point ob-
servations. As a general rule, spline step length should be greater
than the mean distance between observation points (twice the distance
between points is a good starting point). Separate east-west and
north-south spline step length arguments allows the user to account for
some degree of anisotropy in the distribution of observation points.
Short spline step lengths - especially spline step lengths that are
less than the distance between observation points - can greatly in-
crease the processing time.
Moreover, the maximum number of splines for each direction at each time
is fixed, regardless of the spline step length. As the total number of
splines used increases (i.e., with small spline step lengths), the re-
gion is automatically split into subregions for interpolation. Each
subregion can contain no more than 150x150 splines. To avoid subregion
boundary problems, subregions are created to partially overlap each
other. A weighted mean of observations, based on point locations, is
calculated within each subregion.
The Tykhonov regularization parameter (lambda_i) acts to smooth the in-
terpolation. With a small lambda_i, the interpolated surface closely
follows observation points; a larger value will produce a smoother in-
terpolation.
The input can be a 2D or 3D vector points map. If input is 3D and col-
umn is not given than z-coordinates are used for interpolation. Parame-
ter column is required when input is 2D vector map.
v.surf.bspline can produce a raster_output OR a output (but NOT simul-
taneously). Note that topology is not build for output vector point
map. The topology can be built if required by v.build.
If output is a vector points map and a sparse vector points map is not
specified, the output vector map will contain points at the same loca-
tions as observation points in the input map, but the values of the
output points are interpolated values. If instead a sparse vector
points map is specified, the output vector map will contain points at
the same locations as the sparse vector map points, and values will be
those of the interpolated raster surface at those points.
A cross validation "leave-one-out" analysis is available to help to de-
termine the optimal lambda_i value that produces an interpolation that
best fits the original observation data. The more points used for
cross-validation, the longer the time needed for computation. Empirical
testing indicates a threshold of a maximum of 100 points is recom-
mended. Note that cross validation can run very slowly if more than 100
observations are used. The cross-validation output reports mean and rms
of the residuals from the true point value and the estimated from the
interpolation for a fixed series of lambda_i values. No vector nor
raster output will be created when cross-validation is selected.
EXAMPLES
Basic interpolation
v.surf.bspline input=point_vector output=interpolate_surface method=bicubic
A bicubic spline interpolation will be done and a vector points map
with estimated (i.e., interpolated) values will be created.
Basic interpolation and raster output with a longer spline step
v.surf.bspline input=point_vector raster=interpolate_surface ew_step=25 ns_step=25
A bilinear spline interpolation will be done with a spline step length
of 25 map units. An interpolated raster map will be created at the cur-
rent region resolution.
Estimation of lambda_i parameter with a cross validation process
v.surf.bspline -c input=point_vector
Estimation on sparse points
v.surf.bspline input=point_vector sparse=sparse_points output=interpolate_surface
An output map of vector points will be created, corresponding to the
sparse vector map, with interpolated values.
Using attribute values instead z-coordinates
v.surf.bspline input=point_vector raster=interpolate_surface layer=1 \
column=attrib_column
The interpolation will be done using the values in attrib_column, in
the table associated with layer 1.
North carolina location example using z-coordinates for interpolation
g.region region=rural_1m res=2 -p
v.surf.bspline input=elev_lid792_bepts raster=elev_lid792_rast \
ew_step=5 ns_step=5 method=bicubic lambda_i=0.1
KNOWN ISSUES
Known issues:
In order to avoid RAM memory problems, an auxiliary table is needed for
recording some intermediate calculations. This requires the GROUP BY
SQL function is used, which is not supported by the DBF driver. For
this reason, vector map output (output) is not permitted with the DBF
driver. There are no problems with the raster map output from the DBF
driver.
REFERENCES
• Brovelli M. A., Cannata M., and Longoni U.M., 2004, LIDAR Data
Filtering and DTM Interpolation Within GRASS, Transactions in
GIS, April 2004, vol. 8, iss. 2, pp. 155-174(20), Blackwell
Publishing Ltd
• Brovelli M. A. and Cannata M., 2004, Digital Terrain model re-
construction in urban areas from airborne laser scanning data:
the method and an example for Pavia (Northern Italy). Computers
and Geosciences 30, pp.325-331
• Brovelli M. A e Longoni U.M., 2003, Software per il filtraggio
di dati LIDAR, Rivista dell’Agenzia del Territorio, n. 3-2003,
pp. 11-22 (ISSN 1593-2192)
• Antolin R. and Brovelli M.A., 2007, LiDAR data Filtering with
GRASS GIS for the Determination of Digital Terrain Models. Pro-
ceedings of Jornadas de SIG Libre, Girona, España. CD ISBN:
978-84-690-3886-9
SEE ALSO
v.surf.idw, v.surf.rst
Overview: Interpolation and Resampling in GRASS GIS
AUTHORS
Original version (s.bspline.reg) in GRASS 5.4: Maria Antonia Brovelli,
Massimiliano Cannata, Ulisse Longoni, Mirko Reguzzoni
Update for GRASS 6 and improvements: Roberto Antolin
SOURCE CODE
Available at: v.surf.bspline source code (history)
Accessed: unknown
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GRASS 7.8.7 v.surf.bspline(1grass)
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