CMSIS-DSP
Version 1.10.0
CMSIS DSP Software Library
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Functions | |
void | arm_spline_f32 (arm_spline_instance_f32 *S, const float32_t *xq, float32_t *pDst, uint32_t blockSize) |
Processing function for the floating-point cubic spline interpolation. More... | |
void | arm_spline_init_f32 (arm_spline_instance_f32 *S, arm_spline_type type, const float32_t *x, const float32_t *y, uint32_t n, float32_t *coeffs, float32_t *tempBuffer) |
Initialization function for the floating-point cubic spline interpolation. More... | |
Spline interpolation is a method of interpolation where the interpolant is a piecewise-defined polynomial called "spline".
Given a function f defined on the interval [a,b], a set of n nodes x(i) where a=x(1)<x(2)<...<x(n)=b and a set of n values y(i) = f(x(i)), a cubic spline interpolant S(x) is defined as:
S1(x) x(1) < x < x(2) S(x) = ... Sn-1(x) x(n-1) < x < x(n)
where
Si(x) = a_i+b_i(x-xi)+c_i(x-xi)^2+d_i(x-xi)^3 i=1, ..., n-1
Having defined h(i) = x(i+1) - x(i)
h(i-1)c(i-1)+2[h(i-1)+h(i)]c(i)+h(i)c(i+1) = 3/h(i)*[a(i+1)-a(i)]-3/h(i-1)*[a(i)-a(i-1)] i=2, ..., n-1
It is possible to write the previous conditions in matrix form (Ax=B). In order to solve the system two boundary conidtions are needed.
| 1 0 0 ... 0 0 0 || c(1) | | 0 | | h(0) 2[h(0)+h(1)] h(1) ... 0 0 0 || c(2) | | 3/h(2)*[a(3)-a(2)]-3/h(1)*[a(2)-a(1)] | | ... ... ... ... ... ... ... || ... |=| ... | | 0 0 0 ... h(n-2) 2[h(n-2)+h(n-1)] h(n-1) || c(n-1) | | 3/h(n-1)*[a(n)-a(n-1)]-3/h(n-2)*[a(n-1)-a(n-2)] | | 0 0 0 ... 0 0 1 || c(n) | | 0 |
| 1 -1 0 ... 0 0 0 || c(1) | | 0 | | h(0) 2[h(0)+h(1)] h(1) ... 0 0 0 || c(2) | | 3/h(2)*[a(3)-a(2)]-3/h(1)*[a(2)-a(1)] | | ... ... ... ... ... ... ... || ... |=| ... | | 0 0 0 ... h(n-2) 2[h(n-2)+h(n-1)] h(n-1) || c(n-1) | | 3/h(n-1)*[a(n)-a(n-1)]-3/h(n-2)*[a(n-1)-a(n-2)] | | 0 0 0 ... 0 -1 1 || c(n) | | 0 |
A is a tridiagonal matrix (a band matrix of bandwidth 3) of size N=n+1. The factorization algorithms (A=LU) can be simplified considerably because a large number of zeros appear in regular patterns. The Crout method has been used: 1) Solve LZ=B
u(1,2) = A(1,2)/A(1,1) z(1) = B(1)/l(11)
FOR i=2, ..., N-1 l(i,i) = A(i,i)-A(i,i-1)u(i-1,i) u(i,i+1) = a(i,i+1)/l(i,i) z(i) = [B(i)-A(i,i-1)z(i-1)]/l(i,i)
l(N,N) = A(N,N)-A(N,N-1)u(N-1,N) z(N) = [B(N)-A(N,N-1)z(N-1)]/l(N,N)
2) Solve UX=Z
c(N)=z(N)
FOR i=N-1, ..., 1 c(i)=z(i)-u(i,i+1)c(i+1)
c(i) for i=1, ..., n-1 are needed to compute the n-1 polynomials. b(i) and d(i) are computed as:
It is possible to compute the interpolated vector for x values outside the input range (xq<x(1); xq>x(n)). The coefficients used to compute the y values for xq<x(1) are going to be the ones used for the first interval, while for xq>x(n) the coefficients used for the last interval.
The initialization function takes as input two arrays that the user has to allocate: coeffs
will contain the b, c, and d coefficients for the (n-1) intervals (n is the number of known points), hence its size must be 3*(n-1); tempBuffer
is temporally used for internal computations and its size is n+n-1.
The x input array must be strictly sorted in ascending order and it must not contain twice the same value (x(i)<x(i+1)).
void arm_spline_f32 | ( | arm_spline_instance_f32 * | S, |
const float32_t * | xq, | ||
float32_t * | pDst, | ||
uint32_t | blockSize | ||
) |
[in] | S | points to an instance of the floating-point spline structure. |
[in] | xq | points to the x values ot the interpolated data points. |
[out] | pDst | points to the block of output data. |
[in] | blockSize | number of samples of output data. |
[in] | S | points to an instance of the floating-point spline structure. |
[in] | xq | points to the x values of the interpolated data points. |
[out] | pDst | points to the block of output data. |
[in] | blockSize | number of samples of output data. |
void arm_spline_init_f32 | ( | arm_spline_instance_f32 * | S, |
arm_spline_type | type, | ||
const float32_t * | x, | ||
const float32_t * | y, | ||
uint32_t | n, | ||
float32_t * | coeffs, | ||
float32_t * | tempBuffer | ||
) |
[in,out] | S | points to an instance of the floating-point spline structure. |
[in] | type | type of cubic spline interpolation (boundary conditions) |
[in] | x | points to the x values of the known data points. |
[in] | y | points to the y values of the known data points. |
[in] | n | number of known data points. |
[in] | coeffs | coefficients array for b, c, and d |
[in] | tempBuffer | buffer array for internal computations |