| /* ---------------------------------------------------------------------- | |
| * 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. | |
| */ | |
| /** | |
| @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; | |
| 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; | |
| /* Create output for x(i)<x<x(i+1) */ | |
| for (i=0; i<n-1; i++) | |
| { | |
| 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; | |
| } | |
| 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)) */ | |
| /* 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; | |
| blkCnt2 = blkCnt; | |
| 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 | |
| */ | |