C# 线性插值的实现示例
作者：ou.cs
线性插值是针对一维数据的插值方法,本文主要介绍了C# 线性插值的实现示例,具有一定的参考价值,感兴趣的可以了解一下
线性插值是一种常用的插值算法，适用于许多实际场景。
传感器数据处理：在传感器数据处理中，可能会出现数据点不连续或不均匀的情况。使用线性插值可以根据已知的数据点来估算在两个数据点之间的数值，从而填补数据中的缺失或不连续之处。
图像处理：在图像处理中，可能需要调整图像的大小或分辨率。线性插值可以用来估算新像素点的数值，从而实现图像的缩放或放大。
信号处理：在数字信号处理中，线性插值可以用来估算信号在离散时间点之间的数值，从而平滑信号或填充信号中的缺失值。
数值模拟：在线性数值模拟中，线性插值可以用来逼近复杂函数的数值输出，从而简化数学模型的计算过程。
运动学模型：在线性运动学模型中，可能需要根据已知的位置数据来估算物体在不同时间点的轨迹或位置信息，线性插值可以帮助实现这一目标。
案例有一组数据，电压，温度会随着改变，电压和温度都是浮点型数据，快速获取（-40~120）整形温度对应的电压。
using System;
class Program
{
static void Main()
{
// 假设已知的电压-温度数据点
double[] temperatures = { -40, 0, 25, 50, 100 };
double[] voltages = { 1.0, 2.5, 3.0, 3.5, 4.0 };
// 需要获取的整数温度值
int targetTemperature = 75;
// 寻找相邻的温度数据点
int lowerIndex = Array.BinarySearch(temperatures, targetTemperature);
if (lowerIndex < 0)
{
lowerIndex = ~lowerIndex - 1;
}
int upperIndex = lowerIndex + 1;
// 线性插值计算电压值
double lowerTemperature = temperatures[lowerIndex];
double upperTemperature = temperatures[upperIndex];
double lowerVoltage = voltages[lowerIndex];
double upperVoltage = voltages[upperIndex];
double interpolatedVoltage = lowerVoltage + (upperVoltage - lowerVoltage) * (targetTemperature - lowerTemperature) / (upperTemperature - lowerTemperature);
Console.WriteLine($"在温度 {targetTemperature} 度时对应的电压值为: {interpolatedVoltage}");
}
}1、分段线性插值
/// <summary>
/// 分段线性插值，将一组数插值为所需点数
/// </summary>
/// <param name="dataIn">待插值的数据数组</param>
/// <param name="n">插值点数</param>
/// <returns>插值后的数据数组</returns>
public static float[] Interpolation(float[] dataIn, int n)
{
float[] dataOut = new float[n];
int lenIn = dataIn.Length;
float[] a = new float[lenIn];
float[] divIn = new float[lenIn];
float[] divOut = new float[n];
divIn[0] = 0;
for (int i = 1; i < lenIn; i++)
{
divIn[i] = divIn[i - 1] + 1;
}
divOut[0] = 0;
for (int i = 1; i < n; i++)
{
divOut[i] = divOut[i - 1] + lenIn / (float)n;
}
int k = 0;
for (int i = k; i < n; i++)
{
for (int j = 0; j < lenIn - 1; j++)
{
if (divOut[i] >= divIn[j] && divOut[i] < divIn[j + 1])
{
dataOut[i] = (dataIn[j + 1] - dataIn[j]) * (divOut[i] - divIn[j]) / (divIn[j + 1] - divIn[j]) + dataIn[j];
k = i;
}
}
}
return dataOut;
}2、三次样条插值
/// <summary>
/// 三次样条插值
/// </summary>
/// <param name="points">排序好的x、y点集合</param>
/// <param name="xs">输入x轴数据，插值计算出对应的y轴点</param>
/// <param name="chf">写1</param>
/// <returns>返回计算好的Y轴数值</returns>
public static double[] SplineInsertPoint(PointClass[] points, double[] xs, int chf)
{
int plength = points.Length;
double[] h = new double[plength];
double[] f = new double[plength];
double[] l = new double[plength];
double[] v = new double[plength];
double[] g = new double[plength];
for (int i = 0; i < plength - 1; i++)
{
h[i] = points[i + 1].x - points[i].x;
f[i] = (points[i + 1].y - points[i].y) / h[i];
}
for (int i = 1; i < plength - 1; i++)
{
l[i] = h[i] / (h[i - 1] + h[i]);
v[i] = h[i - 1] / (h[i - 1] + h[i]);
g[i] = 3 * (l[i] * f[i - 1] + v[i] * f[i]);
}
double[] b = new double[plength];
double[] tem = new double[plength];
double[] m = new double[plength];
double f0 = (points[0].y - points[1].y) / (points[0].x - points[1].x);
double fn = (points[plength - 1].y - points[plength - 2].y) / (points[plength - 1].x - points[plength - 2].x);
b[1] = v[1] / 2;
for (int i = 2; i < plength - 2; i++)
{
// Console.Write(" " + i);
b[i] = v[i] / (2 - b[i - 1] * l[i]);
}
tem[1] = g[1] / 2;
for (int i = 2; i < plength - 1; i++)
{
//Console.Write(" " + i);
tem[i] = (g[i] - l[i] * tem[i - 1]) / (2 - l[i] * b[i - 1]);
}
m[plength - 2] = tem[plength - 2];
for (int i = plength - 3; i > 0; i--)
{
//Console.Write(" " + i);
m[i] = tem[i] - b[i] * m[i + 1];
}
m[0] = 3 * f[0] / 2.0;
m[plength - 1] = fn;
int xlength = xs.Length;
double[] insertRes = new double[xlength];
for (int i = 0; i < xlength; i++)
{
int j = 0;
for (j = 0; j < plength; j++)
{
if (xs[i] < points[j].x)
break;
}
j = j - 1;
Console.WriteLine(j);
if (j == -1 || j == points.Length - 1)
{
if (j == -1)
throw new Exception("插值下边界超出");
if (j == points.Length - 1 && xs[i] == points[j].x)
insertRes[i] = points[j].y;
else
throw new Exception("插值下边界超出");
}
else
{
double p1;
p1 = (xs[i] - points[j + 1].x) / (points[j].x - points[j + 1].x);
p1 = p1 * p1;
double p2; p2 = (xs[i] - points[j].x) / (points[j + 1].x - points[j].x);
p2 = p2 * p2;
double p3; p3 = p1 * (1 + 2 * (xs[i] - points[j].x) / (points[j + 1].x - points[j].x)) * points[j].y + p2 * (1 + 2 * (xs[i] - points[j + 1].x) / (points[j].x - points[j + 1].x)) * points[j + 1].y;
double p4; p4 = p1 * (xs[i] - points[j].x) * m[j] + p2 * (xs[i] - points[j + 1].x) * m[j + 1];
// Console.WriteLine(m[j] + " " + m[j + 1] + " " + j);
p4 = p4 + p3;
insertRes[i] = p4;
//Console.WriteLine("f(" + xs[i] + ")= " + p4);
}
}
//Console.ReadLine();
return insertRes;
}排序计算
public class PointClass
{
public double x = 0;
public double y = 0;
public PointClass()
{
x = 0; y = 0;
}
//-------写一个排序函数，使得输入的点按顺序排列，是因为插值算法的要求是，x轴递增有序的---------
public static PointClass[] DeSortX(PointClass[] points)
{
int length = points.Length;
double temx, temy;
for (int i = 0; i < length - 1; i++)
{
for (int j = 0; j < length - i - 1; j++)
if (points[j].x > points[j + 1].x)
{
temx = points[j + 1].x;
points[j + 1].x = points[j].x;
points[j].x = temx;
temy = points[j + 1].y;
points[j + 1].y = points[j].y;
points[j].y = temy;
}
}
return points;
}
}3、拉格朗日插值
（1）一元全区间不等距插值
/// <summary>
/// 一元全区间不等距插值
/// 拉格朗日插值算法
/// </summary>
/// <param name="x">一维数组，长度为n，存放给定的n个结点的值x(i)，要求x(0)<x(1)<...<x(n-1)</param>
/// <param name="y">一维数组，长度为n，存放给定的n个结点的函数值y(i)，y(i) = f(x(i)), i=0,1,...,n-1</param>
/// <param name="t">存放指定的插值点的x值</param>
/// <returns>指定的查指点t的函数近似值y=f(t)</returns>
public static double Lagrange(double[] x, double[] y, double t)
{
// x,y点数
int n = x.Length;
double z = 0.0;
// 特例处理
if (n < 1)
{
return (z);
}
else if (n == 1)
{
z = y[0];
return (z);
}
else if (n == 2)
{
z = (y[0] * (t - x[1]) - y[1] * (t - x[0])) / (x[0] - x[1]);
return (z);
}
// 开始插值
int ik = 0;
while ((x[ik] < t) && (ik < n))
{
ik = ik + 1;
}
int k = ik - 4;
if (k < 0)
{
k = 0;
}
int m = ik + 3;
if (m > n - 1)
{
m = n - 1;
}
for (int i = k; i <= m; i++)
{
double s = 1.0;
for (int j = k; j <= m; j++)
{
if (j != i)
{
// 拉格朗日插值公式
s = s * (t - x[j]) / (x[i] - x[j]);
}
}
z = z + s * y[i];
}
return (z);
} /// <summary>
/// 一元全区间不等距插值
/// </summary>
/// <param name="points">点集（含XY坐标）</param>
/// <param name="t"></param>
/// <returns></returns>
public static double Lagrange(PointF[] points, double t)
{
double[] x = new double[points.Length];
double[] y = new double[points.Length];
for (int i = 0; i < points.Length; i++)
{
x[i] = points[i].X;
y[i] = points[i].Y;
}
return Lagrange(x, y, t);
} /// <summary>
/// 一元全区间不等距插值
/// </summary>
/// <param name="points">二元组类型的点集（含XY坐标）</param>
/// <param name="t"></param>
/// <returns></returns>
public static double Lagrange(List<Tuple<double, double>> points, double t)
{
double[] x = new double[points.Count];
double[] y = new double[points.Count];
for (int i = 0; i < points.Count; i++)
{
x[i] = points[i].Item1;
y[i] = points[i].Item2;
}
return Lagrange(x, y, t);
} /// <summary>
/// 一元全区间不等距插值，获得插值后的曲线（折线拟合）数据
/// </summary>
/// <param name="points">点集（含XY坐标）</param>
/// <param name="segment_count">每数据段的分割数</param>
/// <returns></returns>
public static PointF[] Lagrange_Curve(PointF[] points, int segment_count = 10)
{
int n = points.Length;
PointF[] segments = new PointF[n * segment_count + 1];
for (int i = 0; i < points.Length - 1; i++)
{
double dt = (points[i + 1].X - points[i].X) / segment_count;
double t = points[i].X;
for (int j = 0; j <= segment_count; j++, t += dt)
{
PointF p = new PointF(0.0F, 0.0F);
p.X = (float)t;
if (j == 0) p.Y = points[i].Y;
else if (j == segment_count) p.Y = points[i + 1].Y;
else p.Y = (float)(Lagrange(points, t));
segments[i] = p;
}
}
return segments;
} /// <summary>
/// 一元全区间等距插值
/// (使用非等距插值的方法)
/// </summary>
/// <param name="x0">存放等距n个结点中第一个结点的值</param>
/// <param name="step">等距结点的步长</param>
/// <param name="y">一维数组，长度为n，存放给定的n个结点的函数值y(i)，y(i) = f(x(i)), i=0,1,...,n-1</param>
/// <param name="t">存放指定的插值点的x值</param>
/// <returns>指定的查指点t的函数近似值y=f(t)</returns>
public static double Lagrange(double x0, double step, double[] y, double t)
{
double[] x = new double[y.Length];
for (int i = 0; i < y.Length; i++, x0 += step)
{
x[i] = x0;
}
return Lagrange(x, y, t);
}（2）一元全区间等距插值
/// <summary>
/// 一元全区间等距插值
/// </summary>
/// <param name="x0">存放等距n个结点中第一个结点的值</param>
/// <param name="step">等距结点的步长</param>
/// <param name="y">一维数组，长度为n，存放给定的n个结点的函数值y(i)，y(i) = f(x(i)), i=0,1,...,n-1</param>
/// <param name="t">存放指定的插值点的x值</param>
/// <returns>指定的查指点t的函数近似值y=f(t)</returns>
public static double Lagrange_Equidistant(double x0, double step, double[] y, double t)
{
int n = y.Length;
double z = 0.0;
// 特例处理
if (n < 1)
{
return (z);
}
else if (n == 1)
{
z = y[0];
return (z);
}
else if (n == 2)
{
z = (y[1] * (t - x0) - y[0] * (t - x0 - step)) / step;
return (z);
}
// 开始插值
int ik = 0;
if (t > x0)
{
double p = (t - x0) / step;
ik = (int)p;
double q = (float)ik;
if (p > q)
{
ik = ik + 1;
}
}
else
{
ik = 0;
}
int k = ik - 4;
if (k < 0)
{
k = 0;
}
int m = ik + 3;
if (m > n - 1)
{
m = n - 1;
}
for (int i = k; i <= m; i++)
{
double s = 1.0;
double xi = x0 + i * step;
for (int j = k; j <= m; j++)
{
if (j != i)
{
double xj = x0 + j * step;
// 拉格朗日插值公式
s = s * (t - xj) / (xi - xj);
}
}
z = z + s * y[i];
}
return (z);
}4、埃尔米特插值
（1）埃尔米特不等距插值
/// <summary>
/// 埃尔米特不等距插值
/// </summary>
/// <param name="x">一维数组，长度为n，存放给定的n个结点的值x(i)，要求x(0)<x(1)<...<x(n-1)</param>
/// <param name="y">一维数组，长度为n，存放给定的n个结点的函数值y(i)，y(i) = f(x(i)), i=0,1,...,n-1</param>
/// <param name="dy">一维数组，长度为n，存放给定的n个结点的函数导数值y'(i)，y'(i) = f'(x(i)), i=0,1,...,n-1</param>
/// <param name="t">存放指定的插值点的x值</param>
/// <returns>指定的查指点t的函数近似值y=f(t)</returns>
public static double Hermite(double[] x, double[] y, double[] dy, double t)
{
int n = x.Length;
double z = 0.0;
// 循环插值
for (int i = 1; i <= n; i++)
{
double s = 1.0;
for (int j = 1; j <= n; j++)
{
if (j != i)
{
s = s * (t - x[j - 1]) / (x[i - 1] - x[j - 1]);
}
}
s = s * s;
double p = 0.0;
for (int j = 1; j <= n; j++)
{
if (j != i)
{
p = p + 1.0 / (x[i - 1] - x[j - 1]);
}
}
double q = y[i - 1] + (t - x[i - 1]) * (dy[i - 1] - 2.0 * y[i - 1] * p);
z = z + q * s;
}
return (z);
} /// <summary>
/// 埃尔米特等距插值
/// (使用非等距插值的方法)
/// </summary>
/// <param name="x0">存放等距n个结点中第一个结点的值</param>
/// <param name="step">等距结点的步长</param>
/// <param name="y">一维数组，长度为n，存放给定的n个结点的函数值y(i)，y(i) = f(x(i)), i=0,1,...,n-1</param>
/// <param name="dy">一维数组，长度为n，存放给定的n个结点的函数导数值y'(i)，y'(i) = f'(x(i)), i=0,1,...,n-1</param>
/// <param name="t">存放指定的插值点的x值</param>
/// <returns>指定的查指点t的函数近似值y=f(t)</returns>
public static double Hermite(double x0, double step, double[] y, double[] dy, double t)
{
double[] x = new double[y.Length];
for (int i = 0; i < y.Length; i++, x0 += step)
{
x[i] = x0;
}
return Hermite(x, y, dy, t);
}（2）埃尔米特等距插值
/// <summary>
/// 埃尔米特等距插值
/// </summary>
/// <param name="x0">等距n个结点中第一个结点的值</param>
/// <param name="step">等距结点的步长</param>
/// <param name="y">一维数组，长度为n，存放给定的n个结点的函数值y(i)，y(i) = f(x(i)), i=0,1,...,n-1</param>
/// <param name="dy">一维数组，长度为n，存放给定的n个结点的函数导数值y'(i)，y'(i) = f'(x(i)), i=0,1,...,n-1</param>
/// <param name="t">存放指定的插值点的x值</param>
/// <returns>指定的查指点t的函数近似值y=f(t)</returns>
public static double Hermite(double x0, double step, double[] y, double[] dy, double t)
{
int n = y.Length;
double z = 0.0;
// 循环插值
for (int i = 1; i <= n; i++)
{
double s = 1.0;
double q = x0 + (i - 1) * step;
double p;
for (int j = 1; j <= n; j++)
{
p = x0 + (j - 1) * step;
if (j != i)
{
s = s * (t - p) / (q - p);
}
}
s = s * s;
p = 0.0;
for (int j = 1; j <= n; j++)
{
if (j != i)
{
p = p + 1.0 / (q - (x0 + (j - 1) * step));
}
}
q = y[i - 1] + (t - q) * (dy[i - 1] - 2.0 * y[i - 1] * p);
z = z + q * s;
}
return (z);
}到此这篇关于C# 线性插值的实现示例的文章就介绍到这了,更多相关C# 线性插值内容请搜索脚本之家以前的文章或继续浏览下面的相关文章希望大家以后多多支持脚本之家！