曲线的平滑平滑处理

来源：互联网发布：淘宝皇冠买家编辑：程序博客网时间：2024/05/05 08:05

最近在写一些数据处理的程序。经常需要对数据进行平滑处理。直接用FIR滤波器或IIR滤波器都有一个启动问题，滤波完成后总要对数据掐头去尾。因此去找了些简单的数据平滑处理的方法。

在一本老版本的《数学手册》中找到了几个基于最小二乘法的数据平滑算法。将其写成了C 代码，测试了一下，效果还可以。这里简单的记录一下，算是给自己做个笔记。

算法的原理很简单，以五点三次平滑为例。取相邻的5个数据点，可以拟合出一条3次曲线来，然后用3次曲线上相应的位置的数据值作为滤波后结果。简单的说就是 Savitzky-Golay 滤波器。只不过Savitzky-Golay 滤波器并不特殊考虑边界的几个数据点，而这个算法还特意把边上的几个点的数据拟合结果给推导了出来。

不多说了，下面贴代码。首先是线性拟合平滑处理的代码. 分别为三点线性平滑、五点线性平滑和七点线性平滑。

voidlinearSmooth3 (double in[],double out[],int N )

{

int i;

if ( N <3 )

{

for ( i =0; i <= N -1; i++ )

{

out[i] = in[i];

}

else

{

out[0] = (5.0 * in[0] +2.0 * in[1] - in[2] ) /6.0;

for ( i =1; i <= N -2; i++ )

{

out[i] = ( in[i - 1] + in[i] + in[i +1] ) / 3.0;

}

out[N - 1] = (5.0 * in[N -1] + 2.0 * in[N -2] - in[N -3] ) / 6.0;

}

voidlinearSmooth5 (double in[],double out[],int N )

{

int i;

if ( N <5 )

{

for ( i =0; i <= N -1; i++ )

{

out[i] = in[i];

}

else

{

out[0] = (3.0 * in[0] +2.0 * in[1] + in[2] - in[4] ) / 5.0;

out[1] = (4.0 * in[0] +3.0 * in[1] +2 * in[2] + in[3] ) /10.0;

for ( i =2; i <= N -3; i++ )

{

out[i] = ( in[i - 2] + in[i - 1] + in[i] + in[i +1] + in[i +2] ) / 5.0;

}

out[N - 2] = (4.0 * in[N -1] + 3.0 * in[N -2] + 2 * in[N - 3] + in[N - 4] ) / 10.0;

out[N - 1] = (3.0 * in[N -1] + 2.0 * in[N -2] + in[N -3] - in[N -5] ) / 5.0;

}

voidlinearSmooth7 (double in[],double out[],int N )

{

int i;

if ( N <7 )

{

for ( i =0; i <= N -1; i++ )

{

out[i] = in[i];

}

else

{

out[0] = (13.0 * in[0] +10.0 * in[1] +7.0 * in[2] +4.0 * in[3] +

in[4] -2.0 * in[5] -5.0 * in[6] ) /28.0;

out[1] = (5.0 * in[0] +4.0 * in[1] +3 * in[2] +2 * in[3] +

in[4] - in[5] ) /14.0;

out[2] = (7.0 * in[0] +6.0 * in [1] +5.0 * in[2] +4.0 * in[3] +

3.0 * in[4] +2.0 * in[5] + in[6] ) /28.0;

for ( i =3; i <= N -4; i++ )

{

out[i] = ( in[i - 3] + in[i - 2] + in[i - 1] + in[i] + in[i +1] + in[i +2] + in[i +3] ) / 7.0;

}

out[N - 3] = (7.0 * in[N -1] + 6.0 * in [N -2] + 5.0 * in[N -3] +

4.0 * in[N -4] + 3.0 * in[N -5] + 2.0 * in[N -6] + in[N -7] ) / 28.0;

out[N - 2] = (5.0 * in[N -1] + 4.0 * in[N -2] + 3.0 * in[N -3] +

2.0 * in[N -4] + in[N -5] - in[N -6] ) / 14.0;

out[N - 1] = (13.0 * in[N -1] + 10.0 * in[N -2] + 7.0 * in[N -3] +

4 * in[N -4] + in[N -5] - 2 * in[N - 6] - 5 * in[N -7] ) / 28.0;

}

然后是利用二次函数拟合平滑。

voidquadraticSmooth5(double in[],double out[],int N)

{

int i;

if ( N <5 )

{

for ( i =0; i <= N -1; i++ )

{

out[i] = in[i];

}

else

{

out[0] = (31.0 * in[0] +9.0 * in[1] -3.0 * in[2] -5.0 * in[3] +3.0 * in[4] ) /35.0;

out[1] = (9.0 * in[0] +13.0 * in[1] +12 * in[2] +6.0 * in[3] -5.0 *in[4]) /35.0;

for ( i =2; i <= N -3; i++ )

{

out[i] = ( - 3.0 * (in[i -2] + in[i +2]) +

12.0 * (in[i -1] + in[i +1]) + 17 * in[i] ) / 35.0;

}

out[N - 2] = (9.0 * in[N -1] + 13.0 * in[N -2] + 12.0 * in[N -3] + 6.0 * in[N -4] - 5.0 * in[N -5] ) / 35.0;

out[N - 1] = (31.0 * in[N -1] + 9.0 * in[N -2] - 3.0 * in[N -3] - 5.0 * in[N -4] + 3.0 * in[N -5]) / 35.0;

}

voidquadraticSmooth7(double in[],double out[],int N)

{

int i;

if ( N <7 )

{

for ( i =0; i <= N -1; i++ )

{

out[i] = in[i];

}

else

{

out[0] = (5.0 * in[0] +15.0 * in[1] +3.0 * in[2] -4.0 * in[3] -

6.0 * in[4] -3.0 * in[5] +5.0 * in[6] ) /42.0;

out[1] = (5.0 * in[0] +4.0 * in[1] +3.0 * in[2] +2.0 * in[3] +

in[4] - in[5] ) /14.0;

out[2] = (1.0 * in[0] +3.0 * in [1] +4.0 * in[2] +4.0 * in[3] +

3.0 * in[4] +1.0 * in[5] -2.0 * in[6] ) /14.0;

for ( i =3; i <= N -4; i++ )

{

out[i] = ( -2.0 * (in[i -3] + in[i +3]) +

3.0 * (in[i -2] + in[i +2]) +

6.0 * (in[i -1] + in[i +1]) + 7.0 * in[i] ) /21.0;

}

out[N - 3] = (1.0 * in[N -1] + 3.0 * in [N -2] + 4.0 * in[N -3] +

4.0 * in[N -4] + 3.0 * in[N -5] + 1.0 * in[N -6] - 2.0 * in[N -7] ) / 14.0;

out[N - 2] = (5.0 * in[N -1] + 4.0 * in[N -2] + 3.0 * in[N -3] +

2.0 * in[N -4] + in[N -5] - in[N -6] ) / 14.0;

out[N - 1] = (32.0 * in[N -1] + 15.0 * in[N -2] + 3.0 * in[N -3] -

4.0 * in[N -4] - 6.0 * in[N -5] - 3.0 * in[N -6] + 5.0 * in[N -7] ) / 42.0;

}

最后是三次函数拟合平滑。

/**

* 五点三次平滑

voidcubicSmooth5 (double in[],double out[],int N )

{

int i;

if ( N <5 )

{

for ( i =0; i <= N -1; i++ )

out[i] = in[i];

}

else

{

out[0] = (69.0 * in[0] + 4.0 * in[1] -6.0 * in[2] +4.0 * in[3] - in[4]) /70.0;

out[1] = (2.0 * in[0] + 27.0 * in[1] +12.0 * in[2] -8.0 * in[3] +2.0 * in[4]) /35.0;

for ( i =2; i <= N -3; i++ )

{

out[i] = (-3.0 * (in[i -2] + in[i +2])+ 12.0 * (in[i -1] + in[i +1]) + 17.0 * in[i] ) /35.0;

}

out[N - 2] = (2.0 * in[N -5] - 8.0 * in[N -4] + 12.0 * in[N -3] + 27.0 * in[N -2] + 2.0 * in[N -1]) / 35.0;

out[N - 1] = (- in[N -5] + 4.0 * in[N -4] - 6.0 * in[N -3] + 4.0 * in[N -2] + 69.0 * in[N -1]) / 70.0;

}

return;

}

voidcubicSmooth7(double in[],double out[],int N)

{

int i;

if ( N <7 )

{

for ( i =0; i <= N -1; i++ )

{

out[i] = in[i];

}

else

{

out[0] = (39.0 * in[0] +8.0 * in[1] -4.0 * in[2] -4.0 * in[3] +

1.0 * in[4] +4.0 * in[5] -2.0 * in[6] ) /42.0;

out[1] = (8.0 * in[0] +19.0 * in[1] +16.0 * in[2] +6.0 * in[3] -

4.0 * in[4] -7.0* in[5] +4.0 * in[6] ) /42.0;

out[2] = ( -4.0 * in[0] + 16.0 * in [1] +19.0 * in[2] +12.0 * in[3] +

2.0 * in[4] -4.0 * in[5] +1.0 * in[6] ) /42.0;

for ( i =3; i <= N -4; i++ )

{

out[i] = ( -2.0 * (in[i -3] + in[i +3]) +

3.0 * (in[i -2] + in[i +2]) +

6.0 * (in[i -1] + in[i +1]) + 7.0 * in[i] ) /21.0;

}

out[N - 3] = ( -4.0 * in[N -1] + 16.0 * in [N -2] + 19.0 * in[N -3] +

12.0 * in[N -4] + 2.0 * in[N -5] - 4.0 * in[N -6] + 1.0 * in[N -7] ) / 42.0;

out[N - 2] = (8.0 * in[N -1] + 19.0 * in[N -2] + 16.0 * in[N -3] +

6.0 * in[N -4] - 4.0 * in[N -5] - 7.0 * in[N -6] + 4.0 * in[N -7] ) / 42.0;

out[N - 1] = (39.0 * in[N -1] + 8.0 * in[N -2] - 4.0 * in[N -3] -

4.0 * in[N -4] + 1.0 * in[N -5] + 4.0 * in[N -6] - 2.0 * in[N -7] ) / 42.0;

}

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