原：http://blog.csdn.net/liyuanbhu/article/details/11119081
仅供自己学习参考。
几个简单的数据点平滑处理算法
2013-09-05 13:29
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分类： 数值计算
（19）
版权声明：本文为博主原创文章，未经博主允许不得转载。
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。
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void
linearSmooth3
(
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
;
}
}
void
linearSmooth5
(
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
;
}
}
void
linearSmooth7
(
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
[
6
]
)
/
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
-
7
]
)
/
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
;
}
}
|
来自CODE的代码片 linearSmooth.c
然后是利用二次函数拟合平滑。
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void
quadraticSmooth5
(
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
;
}
}
void
quadraticSmooth7
(
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
]
=
(
32.0
*
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
[
6
]
)
/
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
-
7
]
)
/
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
;
}
}
|
来自CODE的代码片 quadraticSmooth.c
最后是三次函数拟合平滑。
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href="http://blog.csdn.net/liyuanbhu/article/details/11119081#L10" rel="#L10"> 10</a> <a target=_blank id="L11" href="http://blog.csdn.net/liyuanbhu/article/details/11119081#L11" rel="#L11"> 11</a> <a target=_blank id="L12" href="http://blog.csdn.net/liyuanbhu/article/details/11119081#L12" rel="#L12"> 12</a> <a target=_blank id="L13" href="http://blog.csdn.net/liyuanbhu/article/details/11119081#L13" rel="#L13"> 13</a> <a target=_blank id="L14" href="http://blog.csdn.net/liyuanbhu/article/details/11119081#L14" rel="#L14"> 14</a> <a target=_blank id="L15" href="http://blog.csdn.net/liyuanbhu/article/details/11119081#L15" rel="#L15"> 15</a> <a target=_blank id="L16" href="http://blog.csdn.net/liyuanbhu/article/details/11119081#L16" rel="#L16"> 16</a> <a target=_blank id="L17" href="http://blog.csdn.net/liyuanbhu/article/details/11119081#L17" rel="#L17"> 17</a> <a target=_blank id="L18" href="http://blog.csdn.net/liyuanbhu/article/details/11119081#L18" rel="#L18"> 18</a> <a target=_blank id="L19" href="http://blog.csdn.net/liyuanbhu/article/details/11119081#L19" rel="#L19"> 19</a> <a target=_blank id="L20" href="http://blog.csdn.net/liyuanbhu/article/details/11119081#L20" rel="#L20"> 20</a> <a target=_blank id="L21" href="http://blog.csdn.net/liyuanbhu/article/details/11119081#L21" rel="#L21"> 21</a> <a target=_blank id="L22" href="http://blog.csdn.net/liyuanbhu/article/details/11119081#L22" rel="#L22"> 22</a> <a target=_blank id="L23" href="http://blog.csdn.net/liyuanbhu/article/details/11119081#L23" rel="#L23"> 23</a> <a target=_blank id="L24" href="http://blog.csdn.net/liyuanbhu/article/details/11119081#L24" rel="#L24"> 24</a> <a target=_blank id="L25" href="http://blog.csdn.net/liyuanbhu/article/details/11119081#L25" rel="#L25"> 25</a> <a target=_blank id="L26" href="http://blog.csdn.net/liyuanbhu/article/details/11119081#L26" rel="#L26"> 26</a> <a target=_blank id="L27" href="http://blog.csdn.net/liyuanbhu/article/details/11119081#L27" rel="#L27"> 27</a> <a 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/**
* 五点三次平滑
*
*/
void
cubicSmooth5
(
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
;
}
void
cubicSmooth7
(
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
;
}
}
|
来自CODE的代码片 cubicSmooth.c
上面的代码经过了简单的测试，可以放心使用。
最近还有人向我要9点线性平滑和11点线性平滑的系数。我简单算了算，把系数列在了后面。不过我觉得这两组系数意义不大，与其这样做还不如设计一个FIR 滤波器或者 SG 滤波器。
9点线性平滑
yy[0]=(17*y[0] + 14*y[1] + 11*y[2] + 8*y[3] + 5*y[4] + 2*y[5] – y[6] – 4*y[7] – 7*y[8])/45;
yy[1]=(56*y[0] + 47*y[1] + 38*y[2] + 29*y[3] + 20*y[4] + 11*y[5] + 2*y[6] – 7*y[7] – 16*y[8])/180;
yy[2]=(22*y[0] + 19*y[1] + 16*y[2] + 13*y[3] + 10*y[4] + 7*y[5] + 4*y[6] + y[7] – 2*y[8])/90;
yy[3]=(32*y[0] + 29*y[1] + 26*y[2] + 23*y[3] + 20*y[4] + 17*y[5] + 14*y[6] + 11*y[7] + 8*y[8])/ 180;
yy[4]=(y[0] + y[1] + y[2] + y[3] + y[4] + y[5] + y[6] + y[7] + y[8])/9;
yy[5]=(8*y[0] + 11*y[1] + 14*y[2] + 17*y[3] + 20*y[4] + 23*y[5] + 26*y[6] + 29*y[7] + 32*y[8])/ 180;
yy[6]=(-2*y[0] + y[1] + 4*y[2] + 7*y[3] + 10*y[4] + 13*y[5] + 16*y[6] + 19*y[7] + 22*y[8])/90;
yy[7]=(-16*y[0] – 7*y[1] + 2*y[2] + 11*y[3] + 20*y[4] + 29*y[5] + 38*y[6] + 47*y[7] + 56*y[8])/ 180;
yy[8]=(-7*y[0] – 4*y[1] – y[2] + 2*y[3] + 5*y[4] + 8*y[5] + 11*y[6] + 14*y[7] + 17*y[8])/45;
11点线性平滑
yy[0]=0.3181818181818182*
y[0] + 0.2727272727272727*y[1] + 0.22727272727272727*y[2] +
0.18181818181818182*y[3] + 0.13636363636363635*y[4] +
0.09090909090909091*y[5] + 0.045454545454545456*y[6] +
1.6152909428804953*^-17*y[7] – 0.045454545454545456*y[8] –
0.09090909090909091*y[9] – 0.13636363636363635*y[10]
yy[1]=0.27272727272727276*
y[0] + 0.23636363636363633*y[1] + 0.19999999999999998*y[2] +
0.16363636363636364*y[3] + 0.12727272727272726*y[4] +
0.09090909090909091*y[5] + 0.05454545454545455*y[6] +
0.018181818181818202*y[7] – 0.01818181818181818*y[8] –
0.054545454545454536*y[9] – 0.09090909090909088*y[10]
yy[2]=0.2272727272727273*
y[0] + 0.19999999999999996*y[1] + 0.17272727272727267*y[2] +
0.14545454545454542*y[3] + 0.11818181818181815*y[4] +
0.09090909090909091*y[5] + 0.06363636363636363*y[6] +
0.03636363636363638*y[7] + 0.009090909090909094*y[8] –
0.01818181818181816*y[9] – 0.0454545454545454*y[10]
yy[3]=0.18181818181818188*
y[0] + 0.16363636363636355*y[1] + 0.1454545454545454*y[2] +
0.12727272727272723*y[3] + 0.10909090909090904*y[4] +
0.0909090909090909*y[5] + 0.07272727272727272*y[6] +
0.054545454545454564*y[7] + 0.036363636363636376*y[8] +
0.01818181818181823*y[9] + 8.326672684688674*^-17*y[10]
yy[4]=0.13636363636363644*
y[0] + 0.12727272727272718*y[1] + 0.11818181818181811*y[2] +
0.10909090909090903*y[3] + 0.09999999999999995*y[4] +
0.0909090909090909*y[5] + 0.08181818181818182*y[6] +
0.07272727272727275*y[7] + 0.06363636363636364*y[8] +
0.05454545454545459*y[9] + 0.04545454545454555*y[10]
yy[5]=0.090909090909091*
y[0] + 0.09090909090909077*y[1] + 0.09090909090909083*y[2] +
0.09090909090909083*y[3] + 0.09090909090909084*y[4] +
0.0909090909090909*y[5] + 0.09090909090909091*y[6] +
0.09090909090909094*y[7] + 0.09090909090909093*y[8] +
0.090909090909091*y[9] + 0.09090909090909102*y[10]
yy[6]=0.04545454545454558*
y[0] + 0.0545454545454544*y[1] + 0.06363636363636352*y[2] +
0.07272727272727264*y[3] + 0.08181818181818173*y[4] +
0.0909090909090909*y[5] + 0.1*y[6] + 0.10909090909090911*y[7] +
0.11818181818181821*y[8] + 0.12727272727272737*y[9] +
0.13636363636363652*y[10]
yy[7]=1.1102230246251565*^-\
16*y[0] + 0.01818181818181802*y[1] + 0.03636363636363624*y[2] +
0.05454545454545445*y[3] + 0.07272727272727264*y[4] +
0.0909090909090909*y[5] + 0.10909090909090909*y[6] +
0.12727272727272732*y[7] + 0.1454545454545455*y[8] +
0.16363636363636372*y[9] + 0.181818181818182*y[10]
yy[8]=-0.0454545454545453*
y[0] – 0.018181818181818354*y[1] + 0.009090909090908955*y[2] +
0.03636363636363624*y[3] + 0.06363636363636355*y[4] +
0.09090909090909088*y[5] + 0.11818181818181818*y[6] +
0.1454545454545455*y[7] + 0.17272727272727273*y[8] +
0.2000000000000001*y[9] + 0.22727272727272746*y[10]
yy[9]=-0.09090909090909072*
y[0] – 0.05454545454545473*y[1] – 0.018181818181818327*y[2] +
0.01818181818181805*y[3] + 0.05454545454545444*y[4] +
0.09090909090909088*y[5] + 0.12727272727272726*y[6] +
0.16363636363636366*y[7] + 0.2*y[8] + 0.23636363636363647*y[9] +
0.27272727272727293*y[10]
yy[10]=-0.1363636363636362*
y[0] – 0.09090909090909116*y[1] – 0.04545454545454561*y[2] –
1.6653345369377348*^-16*y[3] + 0.04545454545454533*y[4] +
0.09090909090909088*y[5] + 0.13636363636363635*y[6] +
0.18181818181818188*y[7] + 0.2272727272727273*y[8] +
0.27272727272727293*y[9] + 0.3181818181818184*y[10]
