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/* ----------------------------------------------------------------------
* Project: CMSIS DSP Library
* Title: arm_spline_interp_f32.c
* Description: Floating-point cubic spline interpolation
*
* $Date: 23 April 2021
* $Revision: V1.9.0
*
* Target Processor: Cortex-M and Cortex-A cores
* -------------------------------------------------------------------- */
/*
* Copyright (C) 2010-2021 ARM Limited or its affiliates. All rights reserved.
*
* SPDX-License-Identifier: Apache-2.0
*
* Licensed under the Apache License, Version 2.0 (the License); you may
* not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an AS IS BASIS, WITHOUT
* WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
#include "dsp/interpolation_functions.h"
/**
@ingroup groupInterpolation
*/
/**
@defgroup SplineInterpolate Cubic Spline Interpolation
Spline interpolation is a method of interpolation where the interpolant
is a piecewise-defined polynomial called "spline".
@par Introduction
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:
<pre>
S1(x) x(1) < x < x(2)
S(x) = ...
Sn-1(x) x(n-1) < x < x(n)
</pre>
where
<pre>
Si(x) = a_i+b_i(x-xi)+c_i(x-xi)^2+d_i(x-xi)^3 i=1, ..., n-1
</pre>
@par Algorithm
Having defined h(i) = x(i+1) - x(i)
<pre>
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
</pre>
It is possible to write the previous conditions in matrix form (Ax=B).
In order to solve the system two boundary conidtions are needed.
- Natural spline: S1''(x1)=2*c(1)=0 ; Sn''(xn)=2*c(n)=0
In matrix form:
<pre>
| 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 |
</pre>
- Parabolic runout spline: S1''(x1)=2*c(1)=S2''(x2)=2*c(2) ; Sn-1''(xn-1)=2*c(n-1)=Sn''(xn)=2*c(n)
In matrix form:
<pre>
| 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 |
</pre>
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
<pre>
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)
</pre>
2) Solve UX=Z
<pre>
c(N)=z(N)
FOR i=N-1, ..., 1
c(i)=z(i)-u(i,i+1)c(i+1)
</pre>
c(i) for i=1, ..., n-1 are needed to compute the n-1 polynomials.
b(i) and d(i) are computed as:
- b(i) = [y(i+1)-y(i)]/h(i)-h(i)*[c(i+1)+2*c(i)]/3
- d(i) = [c(i+1)-c(i)]/[3*h(i)]
Moreover, a(i)=y(i).
@par Behaviour outside the given intervals
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.
*/
/**
@addtogroup SplineInterpolate
@{
*/
/**
* @brief Processing function for the floating-point cubic spline interpolation.
* @param[in] S points to an instance of the floating-point spline structure.
* @param[in] xq points to the x values of the interpolated data points.
* @param[out] pDst points to the block of output data.
* @param[in] blockSize number of samples of output data.
*/
void arm_spline_f32(
arm_spline_instance_f32 * S,
const float32_t * xq,
float32_t * pDst,
uint32_t blockSize)
{
const float32_t * x = S->x;
const float32_t * y = S->y;
int32_t n = S->n_x;
/* Coefficients (a==y for i<=n-1) */
float32_t * b = (S->coeffs);
float32_t * c = (S->coeffs)+(n-1);
float32_t * d = (S->coeffs)+(2*(n-1));
const float32_t * pXq = xq;
int32_t blkCnt = (int32_t)blockSize;
int32_t blkCnt2;
int32_t i;
float32_t x_sc;
#ifdef ARM_MATH_NEON
float32x4_t xiv;
float32x4_t aiv;
float32x4_t biv;
float32x4_t civ;
float32x4_t div;
float32x4_t xqv;
float32x4_t temp;
float32x4_t diff;
float32x4_t yv;
#endif
/* Create output for x(i)<x<x(i+1) */
for (i=0; i<n-1; i++)
{
#ifdef ARM_MATH_NEON
xiv = vdupq_n_f32(x[i]);
aiv = vdupq_n_f32(y[i]);
biv = vdupq_n_f32(b[i]);
civ = vdupq_n_f32(c[i]);
div = vdupq_n_f32(d[i]);
while( *(pXq+4) <= x[i+1] && blkCnt > 4 )
{
/* Load [xq(k) xq(k+1) xq(k+2) xq(k+3)] */
xqv = vld1q_f32(pXq);
pXq+=4;
/* Compute [xq(k)-x(i) xq(k+1)-x(i) xq(k+2)-x(i) xq(k+3)-x(i)] */
diff = vsubq_f32(xqv, xiv);
temp = diff;
/* y(i) = a(i) + ... */
yv = aiv;
/* ... + b(i)*(x-x(i)) + ... */
yv = vmlaq_f32(yv, biv, temp);
/* ... + c(i)*(x-x(i))^2 + ... */
temp = vmulq_f32(temp, diff);
yv = vmlaq_f32(yv, civ, temp);
/* ... + d(i)*(x-x(i))^3 */
temp = vmulq_f32(temp, diff);
yv = vmlaq_f32(yv, div, temp);
/* Store [y(k) y(k+1) y(k+2) y(k+3)] */
vst1q_f32(pDst, yv);
pDst+=4;
blkCnt-=4;
}
#endif
while( *pXq <= x[i+1] && blkCnt > 0 )
{
x_sc = *pXq++;
*pDst = y[i]+b[i]*(x_sc-x[i])+c[i]*(x_sc-x[i])*(x_sc-x[i])+d[i]*(x_sc-x[i])*(x_sc-x[i])*(x_sc-x[i]);
pDst++;
blkCnt--;
}
}
/* Create output for remaining samples (x>=x(n)) */
#ifdef ARM_MATH_NEON
/* Compute 4 outputs at a time */
blkCnt2 = blkCnt >> 2;
while(blkCnt2 > 0)
{
/* Load [xq(k) xq(k+1) xq(k+2) xq(k+3)] */
xqv = vld1q_f32(pXq);
pXq+=4;
/* Compute [xq(k)-x(i) xq(k+1)-x(i) xq(k+2)-x(i) xq(k+3)-x(i)] */
diff = vsubq_f32(xqv, xiv);
temp = diff;
/* y(i) = a(i) + ... */
yv = aiv;
/* ... + b(i)*(x-x(i)) + ... */
yv = vmlaq_f32(yv, biv, temp);
/* ... + c(i)*(x-x(i))^2 + ... */
temp = vmulq_f32(temp, diff);
yv = vmlaq_f32(yv, civ, temp);
/* ... + d(i)*(x-x(i))^3 */
temp = vmulq_f32(temp, diff);
yv = vmlaq_f32(yv, div, temp);
/* Store [y(k) y(k+1) y(k+2) y(k+3)] */
vst1q_f32(pDst, yv);
pDst+=4;
blkCnt2--;
}
/* Tail */
blkCnt2 = blkCnt & 3;
#else
blkCnt2 = blkCnt;
#endif
while(blkCnt2 > 0)
{
x_sc = *pXq++;
*pDst = y[i-1]+b[i-1]*(x_sc-x[i-1])+c[i-1]*(x_sc-x[i-1])*(x_sc-x[i-1])+d[i-1]*(x_sc-x[i-1])*(x_sc-x[i-1])*(x_sc-x[i-1]);
pDst++;
blkCnt2--;
}
}
/**
@} end of SplineInterpolate group
*/
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