v.surf.bspline - Bicubic or bilinear spline interpolation with Tykhonov regularization
v.surf.bspline [-c] input=name [sparse=name] [output=name] [raster=name] [sie=float] [sin=float] [type=string] [lambda_i=float] [layer=integer] [column=string] [--overwrite]
- Find best parameters using a cross validation method
- Force overwrite of output files
- Name of input vector map
- Name of input vector map of sparse points
- Name for output vector map
- Name for output raster map
- Interpolation spline step value in east direction
- Default: 4
- Interpolation spline step value in north direction
- Default: 4
- Spline type of interpolation
- Options: bilinear,bicubic
- Default: bilinear
- Thychonov regularization weigth
- Default: 1
- Field value. If set to 0, z coordinates are used. (3D vector only)
- Default: 0
- Attribute table column with values to interpolate (if layer>0)
v.surf.bspline makes a bilinear/bicubic spline interpolation
with Tykhonov regularization. The required input is an only 3d points
vector map that will be used to interpolate a reference surface.
Interpolation is carried out by adjusting a Least-Squares (LS) system
in which the parameters to estime are spline functions. The number of
splines doesn't depend on the resolution region, but it depends on the
spline steps values in the north-south and west-east directions. These
spline steps are set by "sin=" and "sie=",
respectively. If the number of splines is bigger than the number of
points, the LS system is bad conditioned because there are more unkowns
than observations. In that case the LS normal matrix can't be inverted.
To allow the inversion of the normal matrix a Tykhonov regularization
is done. The minimizing function is the gradient in the case of a bilinear
interpolation, and the curvature in the bicubic interpolation. The
lambda_i parameter associated with the regularization smooths the
interpolation. The higher the lambda_i parameter, the smoother the
The number of splines has a great influence on two things, mainly. The
first thing is the module's execution time. The second is the RAM use.
The higher the number of splines, the longer the time of execution and
the higher RAM use. A numerical example: 100 splines in each direction
imply 10e4 splines in total, that is, a square LS normal matrix of 10e4
size. Inverting this matrix means inverting 100 millions elements!
To improve this problems a Tcholebsky method with triangulars matrixes
is used in the normal matrix inversion. It has also fixed a maximum number
of splines for each direction. However, it is also possible running the
module with a higher number of splines. For a number of spline higher than
the fixed maximum, the whole region is divided into smaller regions. Each
subregion is 150x150 splines wide. To avoid contour problems, the subregions
are overlaped one to each other. To estimate a single value within the
overlaped zones, a weighted mean considering the point positions into each
subregion is carried out.
The required input is a 3d points vector. If nothing is specified z-coordinates
will be used in the interpolation. It could be also possible to consider
an attribute value by specifying "layer=" and "column="
parameters. If a vector map with another type of features is used, only
points will be considered. If the "sparse=" vector is
used, the "input=" vector map will be used to create a
reference surface. This surface will be used to make an estimation on the
points within the "sparse=". In this case a vector output
("output=") must be specify. If the "sparse="
is not supplied, the final interpolation output will be the interpolated
reference surface from the "input=" vector map. In this case,
one of both the raster or vector output format can be choosen. For raster
format ("raster="), the point estimation will be done
on a regular grid with a resolution equal to the GRASS region. For vector
format, the estimation will be done on the sparse points of the
"input=" vector supplied. Both, vector and raster output,
are not allowed simultaneously.
A cross validation method has been implemented. It helps to find the optimal
lambda_i value that fits the data. It shows the mean and rms
of the residuals from the true point value and the estimated from the
interpolation made with all the data without the point itself. This procedure
is done for fixed lambda_i values. The results of the cross validation will
appear in the stdout and no vector nor raster output will be created. The
external input ("sparse=") will be not considered. Due to
the nature of the algorithm, it is advised the user no to try the cross-
validation with more than 100 points at a time because it will take too long.
The execution time could be reduced by considering a lower number of splines.
Although, as seen, it is possible to use a high number of splines, more than
150x150 splines is not recommended.
In a raster map output ("raster="), region resolution implying
more than 2000x2000 (4 mill) cells are not allowed. If the user tries with a
more than those cells an error message will ask for a lower region resolution.
In this case, a bicubic spline interpolation will be done and an
estimation on the points of point_vector will be the output.
v.surf.bspline input=point_vector output=interpolate_surface type=bicubic
Basic interpolation and raster output with a long spline step
Now, a bilinear spline interpolation will be done on a grid. The spline steps
are set to 25. It doesn't mean that the grid will have a resolution equal to 25,
but that each 25 units there will be a spline.
v.surf.bspline input=point_vector raster=interpolate_surface sie=25 sin=25
Estimation of lambda_i parameter with a cross validation proccess
v.surf.bspline -c input=point_vector
Estimation on sparse points
In this last case, an estimation on the points of the sparse_points vector
will be done. The reference surface used for this estimation will be that
interpolated using the point_vector vector.
v.surf.bspline input=point_vector sparse=sparse_points output=interpolate_surface
Using attribute values instead Z-coordinates
This last case, the module uses the attribute values in attrib_column
in the table associated to layer 1.
v.surf.bspline input=point_vector raster=interpolate_surface layer=1 column=attrib_column
In order to avoid RAM memory problems, an auxiliar table will be needed for
recording some intermediate calculi. Since the "GROUP BY" SQL function is used,
which is not supported by the "dbf" driver, this driver is not
allowed with the vector map output "output=". There is no problem
with the raster map output.
At this time, using the external vector input ("sparse=") implies
interpoling with Z-coordinates. Updates to allow using attribute values
will be done in a near future (I hope).
Original version in GRASS 5.4: (s.bspline.reg)
Maria Antonia Brovelli, Massimiliano Cannata, Ulisse Longoni, Mirko Reguzzoni
Update for GRASS 6.X and improvements:
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 reconstruction 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)
Brovelli M. A., Cannata M. and Longoni U.M., 2002, DTM LIDAR in area urbana, Bollettino SIFET N.2, 2002, pp. 7-26
Last changed: $Date: 2007/04/16 21:36:04 $
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