Inherited by BWSpline::BSpline< T >.
The base class for a spline object containing the nodes for a given domain, cutoff wavelength, and boundary condition.
To smooth a single curve, the BSpline interface contains a constructor which both sets up the domain and solves for the spline. Subsequent curves over the same domain can be created by apply()ing them to the BSpline object, where apply() is a BSplineBase method. [See apply().] New curves can also be smoothed within the same BSpline object by calling solve() with the new set of y values. [See BSpline.] A BSplineBase can be created on its own, in which case all of the computations dependent on the x values, boundary conditions, and cutoff wavelength have already been completed.
The solution of the cubic b-spline is divided into two parts. The first is the setup of the domain given the x values, boundary conditions, and wavelength. The second is the solution of the spline for a set of y values corresponding to the x values in the domain. The first part is done in the creation of the BSplineBase object (or when calling the setDomain method). The second part is done when creating a BSpline object (or calling solve() on a BSpline object).
A BSpline object can be created with either one of its constructors, or by calling apply() on an existing BSplineBase object. Once a spline has been solved, it can be evaluated at any x value. The following example creates a spline curve and evaluates it over the domain:
vector<float> x;
vector<float> y;
{ ... }
float wl = cutoff_wavelength( ... );
BSpline<float> spline (x.begin(), x.size(), y.begin(), wl, BSplineBase<float>::BC_ZERO_SECOND );
if (spline.ok())
{
ostream_iterator<float> of(cout, "\t ");
float xi = spline.Xmin();
float xs = (spline.Xmax() - xi) / 2000.0;
for (; xi <= spline.Xmax(); xi += xs)
{
*of++ = spline.evaluate (xi);
}
}
In the usual usage, the BSplineBase can compute a reasonable number of nodes for the spline, balancing between a few desirable factors. There needs to be at least 2 nodes per cutoff wavelength (preferably 4 or more) for the derivative constraint to reliably approximate a lo-pass filter. There should be at least 1 and preferably about 2 data points per node (measured just by their number and not by any check of the density of points across the domain). Lastly, of course, the fewer the nodes then the faster the computation of the spline. The computation of the number of nodes happens in the Setup() method during BSplineBase construction and when setDomain() is called. If the setup fails to find a desirable number of nodes, then the BSplineBase object will return false from ok().
The ok() method returns false when a BSplineBase or BSpline could not complete any operation successfully. In particular, as mentioned above, ok() will return false if some problem was detected with the domain values or if no reasonable number of nodes could be found for the given cutoff wavelength. Also, ok() on a BSpline object will return false if the matrix equation could not be solved, such as after BSpline construction or after a call to apply().
If letting Setup() determine the number of nodes is not acceptable, the constructors and setDomain() accept the parameter num_nodes. By default, num_nodes is passed as zero, forcing Setup() to calculate the number of nodes. However, if num_nodes is passed as 2 or greater, then Setup() will bypass its own algorithm and accept the given number of nodes instead. Obviously, it's up to the programmer to understand the affects of the number of nodes on the representation of the data and on the solution (or non-solution) of the spline. Remember to check the ok() method to detect when the spline solution has failed.
The algorithm is based on the cubic spline described by Katsuyuki Ooyama in Montly Weather Review, Vol 115, October 1987. This implementation has benefited from comparisons with a previous FORTRAN implementation by James L. Franklin, NOAA/Hurricane Research Division. In particular, the algorithm in the Setup() method is based mostly on his implementation (VICSETUP). The Setup() method finds a suitable default for the number of nodes given a domain and cutoff frequency. This implementation adopts most of the same constraints, including a constraint that the cutoff wavelength not be greater than the span of the domain values: wl < max(x) - min(x). If this is not an acceptable constraint, then use the num_nodes parameter to specify the number of nodes explicitly.
The cubic b-spline is formulated as the sum of some multiple of the basis function centered at each node in the domain. The number of nodes is determined by the desired cutoff wavelength and a desirable number of x values per node. The basis function is continuous and differentiable up to the second degree. A derivative constraint is included in the solution to achieve the effect of a low-pass frequency filter with the given cutoff wavelength. The derivative constraint can be disabled by specifying a wavelength value of zero, which reduces the analysis to a least squares fit to a cubic b-spline. The domain nodes, boundary constraints, and wavelength determine a linear system of equations, Qa=b, where a is the vector of basis function coefficients at each node. The coefficient vector is solved by first LU factoring along the diagonally banded matrix Q in BSplineBase. The BSpline object then computes the B vector for a set of y values and solves for the coefficient vector with the LU matrix. Only the diagonal bands are stored in memory and calculated during LU factoring and back substitution, and the basis function is evaluated as few times as possible in computing the diagonal matrix and B vector.
Copyright (c) 1998-2003 University Corporation for Atmospheric Research, UCAR
| typedef T BWSpline::BSplineBase< T >::datum_type |
typedef BSplineBaseP<T> BWSpline::BSplineBase< T >::Base [protected] |
| enum BWSpline::BSplineBase::BoundaryConditionTypes |
| BWSpline::BSplineBase< T >::BSplineBase | ( | const T * | x, | |
| int | nx, | |||
| const T * | w, | |||
| T | wl, | |||
| int | bc_type = BC_ZERO_SECOND, |
|||
| int | num_nodes = 0 | |||
| ) | [inline] |
Construct a spline domain for the given set of x values, weights, cutoff wavelength, and boundary condition type. The parameters are the same as for setDomain(). Call ok() to check whether domain setup succeeded after construction.
References BWSpline::BSplineBase< T >::setDomain().
| BWSpline::BSplineBase< T >::BSplineBase | ( | const BSplineBase< T > & | bb | ) | [inline] |
References BWSpline::BSplineBase< T >::alpha, BWSpline::BSplineBase< T >::base, BWSpline::BSplineBase< T >::DX, BWSpline::BSplineBase< T >::M, BWSpline::BSplineBase< T >::NX, BWSpline::BSplineBase< T >::waveLength, BWSpline::BSplineBaseP< T >::X, BWSpline::BSplineBase< T >::xmax, and BWSpline::BSplineBase< T >::xmin.
| BWSpline::BSplineBase< T >::~BSplineBase | ( | ) | [inline, virtual] |
References BWSpline::BSplineBase< T >::base.
| static const char* BWSpline::BSplineBase< T >::ImplVersion | ( | ) | [static] |
| static const char* BWSpline::BSplineBase< T >::IfaceVersion | ( | ) | [static] |
| bool BWSpline::BSplineBase< T >::setDomain | ( | const T * | x, | |
| int | nx, | |||
| const T * | w, | |||
| T | wl, | |||
| int | bc_type = BC_ZERO_SECOND, |
|||
| int | num_nodes = 0 | |||
| ) | [inline] |
Change the domain of this base. [If this is part of a BSpline object, this method {will not} change the existing curve or re-apply the smoothing to any set of y values.]
The x values can be in any order, but they must be of sufficient density to support the requested cutoff wavelength. The setup of the domain may fail because of either inconsistency between the x density and the cutoff wavelength, or because the resulting matrix could not be factored. If setup fails, the method returns false.
| x | The array of x values in the domain. | |
| nx | The number of values in the x array. | |
| w | The array of weights for the x values. | |
| wl | The cutoff wavelength, in the same units as the x values. A wavelength of zero disables the derivative constraint. | |
| bc_type | The enumerated boundary condition type. If omitted it defaults to BC_ZERO_SECOND. | |
| num_nodes | The number of nodes to use for the cubic b-spline. If less than 2 a reasonable number will be calculated automatically, if possible, taking into account the given cutoff wavelength. |
References BWSpline::BSplineBase< T >::addP(), BWSpline::BSplineBase< T >::Alpha(), BWSpline::BSplineBase< T >::alpha, BWSpline::BSplineBase< T >::base, BWSpline::BSplineBase< T >::BC, BWSpline::BSplineBase< T >::calculateQ(), BWSpline::BSplineBase< T >::factor(), BWSpline::BSplineBase< T >::NX, BWSpline::BSplineBase< T >::OK, BWSpline::BSplineBase< T >::Setup(), BWSpline::BSplineBase< T >::waveLength, and BWSpline::BSplineBaseP< T >::X.
Referenced by BWSpline::BSplineBase< T >::BSplineBase().
| BSpline< T > * BWSpline::BSplineBase< T >::apply | ( | const T * | y | ) | [inline] |
| const T * BWSpline::BSplineBase< T >::nodes | ( | int * | nnodes | ) | [inline] |
Return array of the node coordinates. Returns 0 if not ok(). The array of nodes returned by nodes() belongs to the object and should not be deleted; it will also be invalid if the object is destroyed.
References BWSpline::BSplineBase< T >::base, BWSpline::BSplineBase< T >::DX, BWSpline::BSplineBase< T >::M, BWSpline::BSplineBaseP< T >::Nodes, and BWSpline::BSplineBase< T >::xmin.
| int BWSpline::BSplineBase< T >::nNodes | ( | ) | [inline] |
Return the number of nodes (one more than the number of intervals).
References BWSpline::BSplineBase< T >::M.
| int BWSpline::BSplineBase< T >::nX | ( | ) | [inline] |
Number of original x values.
References BWSpline::BSplineBase< T >::NX.
| T BWSpline::BSplineBase< T >::Xmin | ( | ) | [inline] |
References BWSpline::BSplineBase< T >::xmin.
| T BWSpline::BSplineBase< T >::Xmax | ( | ) | [inline] |
| T BWSpline::BSplineBase< T >::Alpha | ( | T | wavelength | ) | [inline] |
Return the Alpha value for a given wavelength. Note that this depends on the current node interval length (DX).
References a, BWSpline::BSplineBase< T >::DX, and BWSpline::BSplineBase< T >::K.
| T BWSpline::BSplineBase< T >::Alpha | ( | ) | [inline] |
Return alpha currently in use by this domain.
References BWSpline::BSplineBase< T >::alpha.
Referenced by BWSpline::BSplineBase< T >::setDomain().
| bool BWSpline::BSplineBase< T >::ok | ( | ) | [inline] |
Return the current state of the object, either ok or not ok. Use this method to test for valid state after construction or after a call to setDomain(). ok() will return false if either fail, such as when an appropriate number of nodes and node interval cannot be found for a given wavelength, or when the linear equation for the coefficients cannot be solved.
References BWSpline::BSplineBase< T >::OK.
| bool BWSpline::BSplineBase< T >::Setup | ( | int | num_nodes = 0 |
) | [inline, protected] |
References BWSpline::BSplineBase< T >::base, BWSpline::BSplineBase< T >::DX, BWSpline::BSplineBase< T >::M, BWSpline::BSplineBase< T >::NX, BWSpline::BSplineBase< T >::Ratiod(), BWSpline::BSplineBase< T >::waveLength, BWSpline::BSplineBaseP< T >::X, BWSpline::BSplineBase< T >::xmax, and BWSpline::BSplineBase< T >::xmin.
Referenced by BWSpline::BSplineBase< T >::setDomain().
| void BWSpline::BSplineBase< T >::calculateQ | ( | ) | [inline, protected] |
References BWSpline::BSplineBase< T >::alpha, BWSpline::BSplineBase< T >::base, BWSpline::BSplineBase< T >::Beta(), BWSpline::BSplineBase< T >::M, BWSpline::BSplineBaseP< T >::Q, Q, BWSpline::BSplineBase< T >::qDelta(), and BWSpline::BandedMatrix< T >::setup().
Referenced by BWSpline::BSplineBase< T >::setDomain().
| T BWSpline::BSplineBase< T >::qDelta | ( | int | m1, | |
| int | m2 | |||
| ) | [inline, protected] |
| T BWSpline::BSplineBase< T >::Beta | ( | int | m | ) | [inline, protected] |
| void BWSpline::BSplineBase< T >::addP | ( | const T * | w | ) | [inline, protected] |
References BWSpline::BSplineBase< T >::base, BWSpline::BSplineBase< T >::Basis(), BWSpline::BSplineBase< T >::BasisInner(), BWSpline::BSplineBase< T >::DX, BWSpline::BSplineBase< T >::M, BWSpline::my::max(), BWSpline::my::min(), BWSpline::BSplineBase< T >::NX, P, BWSpline::BSplineBaseP< T >::Q, BWSpline::BSplineBaseP< T >::X, and BWSpline::BSplineBase< T >::xmin.
Referenced by BWSpline::BSplineBase< T >::setDomain().
| bool BWSpline::BSplineBase< T >::factor | ( | ) | [inline, protected] |
References BWSpline::BSplineBase< T >::base, BWSpline::LU_factor_banded(), and BWSpline::BSplineBaseP< T >::Q.
Referenced by BWSpline::BSplineBase< T >::setDomain().
| T BWSpline::BSplineBase< T >::Basis | ( | int | m, | |
| T | x | |||
| ) | [inline, protected] |
| T BWSpline::BSplineBase< T >::BasisInner | ( | int | m, | |
| T | x | |||
| ) | [inline, protected] |
References BWSpline::BSplineBase< T >::DX, and BWSpline::BSplineBase< T >::xmin.
Referenced by BWSpline::BSplineBase< T >::addP(), and BWSpline::BSpline< T >::solve().
| T BWSpline::BSplineBase< T >::DBasis | ( | int | m, | |
| T | x | |||
| ) | [inline, protected] |
| T BWSpline::BSplineBase< T >::Ratiod | ( | int & | ni, | |
| T & | deltax, | |||
| T & | ratiof | |||
| ) | [inline, protected] |
T BWSpline::BSplineBase< T >::waveLength [protected] |
int BWSpline::BSplineBase< T >::NX [protected] |
int BWSpline::BSplineBase< T >::K [protected] |
Referenced by BWSpline::BSplineBase< T >::Alpha(), and BWSpline::BSplineBase< T >::qDelta().
int BWSpline::BSplineBase< T >::BC [protected] |
Referenced by BWSpline::BSplineBase< T >::Beta(), and BWSpline::BSplineBase< T >::setDomain().
T BWSpline::BSplineBase< T >::xmax [protected] |
T BWSpline::BSplineBase< T >::xmin [protected] |
Referenced by BWSpline::BSplineBase< T >::addP(), BWSpline::BSplineBase< T >::Basis(), BWSpline::BSplineBase< T >::BasisInner(), BWSpline::BSplineBase< T >::BSplineBase(), BWSpline::BSpline< T >::curve(), BWSpline::BSplineBase< T >::DBasis(), BWSpline::BSpline< T >::evaluate(), BWSpline::BSplineBase< T >::nodes(), BWSpline::BSplineBase< T >::Ratiod(), BWSpline::BSplineBase< T >::Setup(), BWSpline::BSpline< T >::slope(), BWSpline::BSpline< T >::solve(), BWSpline::BSplineBase< T >::Xmax(), and BWSpline::BSplineBase< T >::Xmin().
int BWSpline::BSplineBase< T >::M [protected] |
Referenced by BWSpline::BSplineBase< T >::addP(), BWSpline::BSplineBase< T >::Basis(), BWSpline::BSplineBase< T >::Beta(), BWSpline::BSplineBase< T >::BSplineBase(), BWSpline::BSplineBase< T >::calculateQ(), BWSpline::BSpline< T >::coefficient(), BWSpline::BSpline< T >::curve(), BWSpline::BSplineBase< T >::DBasis(), BWSpline::BSpline< T >::evaluate(), BWSpline::BSplineBase< T >::nNodes(), BWSpline::BSplineBase< T >::nodes(), BWSpline::BSplineBase< T >::qDelta(), BWSpline::BSplineBase< T >::Setup(), BWSpline::BSpline< T >::slope(), BWSpline::BSpline< T >::solve(), and BWSpline::BSplineBase< T >::Xmax().
T BWSpline::BSplineBase< T >::DX [protected] |
Referenced by BWSpline::BSplineBase< T >::addP(), BWSpline::BSplineBase< T >::Alpha(), BWSpline::BSplineBase< T >::Basis(), BWSpline::BSplineBase< T >::BasisInner(), BWSpline::BSplineBase< T >::BSplineBase(), BWSpline::BSpline< T >::curve(), BWSpline::BSplineBase< T >::DBasis(), BWSpline::BSpline< T >::evaluate(), BWSpline::BSplineBase< T >::nodes(), BWSpline::BSplineBase< T >::Setup(), BWSpline::BSpline< T >::slope(), BWSpline::BSpline< T >::solve(), and BWSpline::BSplineBase< T >::Xmax().
T BWSpline::BSplineBase< T >::alpha [protected] |
bool BWSpline::BSplineBase< T >::OK [protected] |
Base* BWSpline::BSplineBase< T >::base [protected] |
Referenced by BWSpline::BSplineBase< T >::addP(), BWSpline::BSplineBase< T >::BSplineBase(), BWSpline::BSplineBase< T >::calculateQ(), BWSpline::BSplineBase< T >::factor(), BWSpline::BSplineBase< T >::nodes(), BWSpline::BSplineBase< T >::setDomain(), BWSpline::BSplineBase< T >::Setup(), BWSpline::BSpline< T >::solve(), and BWSpline::BSplineBase< T >::~BSplineBase().
const T BWSpline::BSplineBase< T >::BoundaryConditions [inline, static, protected] |
{
{ -4, -1, -1, -4 },
{ 0, 1, 1, 0 },
{ 2, -1, -1, 2 } }
Referenced by BWSpline::BSplineBase< T >::Beta().
1.6.1