Polynomial interpolation fitting (I)

Use polynomial interpolation to fit between points .

Polynomial equation ：

Unfolding form ：f(x)=a0x+a1x^2+a2x^3+...

Given n Plane coordinate points , It can be solved by polynomial equation n Parametric solutions of equations , And the solution is unique （ Self evidence ）.

The solution adopts vandermund determinant .

Segment region fitting is sometimes used in image processing , Therefore, the interpolation method will n Add points between points .

First order equation ：y=kx+b Two points are needed (x1,y1),(x0,y0)

    Parameters     k=(y1-y0)/(x1-x0),b=y0-k*x0

have access to scipy Interpolation classes in the library

for instance ： Yes 5 First order interpolation between points

import numpy as np
from scipy.interpolate import interp1d,interp2d
import pylab as pl


x=np.array([1,2,3,4,5])
y=np.array([1,4,2,4,1])



fun=interp1d(x,y,kind='linear')# function  y Is a series of points , according to x,y Relation derivation function 
xint=np.linspace(x.min(),x.max(),20)
print(" library x:",xint)
print(" library y:",fun(xint))
pl.scatter(x,y,c='b',marker='o')
pl.plot(xint,fun(xint),color='red')
pl.show()

Interpolated image

  Output interpolation results

library x: [1.         1.21052632 1.42105263 1.63157895 1.84210526 2.05263158
 2.26315789 2.47368421 2.68421053 2.89473684 3.10526316 3.31578947
 3.52631579 3.73684211 3.94736842 4.15789474 4.36842105 4.57894737
 4.78947368 5.        ]
library y: [1.         1.63157895 2.26315789 2.89473684 3.52631579 3.89473684
 3.47368421 3.05263158 2.63157895 2.21052632 2.21052632 2.63157895
 3.05263158 3.47368421 3.89473684 3.52631579 2.89473684 2.26315789
 1.63157895 1.        ]

Custom interpolation

import numpy as np
from scipy.interpolate import interp1d,interp2d
import pylab as pl





#------------------------
x=np.array([1,2,3,4,5])
y=np.array([1,4,2,4,1])




xint=np.linspace(x.min(),x.max(),20)



# 1、 linear 
def linear_inter(x0,y0,x1,y1,nx):
    k=(y1-y0)/(x1-x0)
    b=y0-k*x0
    ny=k*nx+b
    return ny



for i in range(1,len(x)):
    print("----- The first ",i," Interval data -----")
    # print(ny)
    nx=[]
    for val in xint:
        if val>=x[i-1] and val<=x[i]:
            nx.append(val)

    print(nx)
    nx=np.array(nx)
    ny = linear_inter(x[i - 1], y[i - 1], x[i], y[i], nx)
    print(" result ",ny)

    pl.scatter(x,y,c='b',marker='o')
    pl.plot(nx,ny,color='red')
pl.show()

Because the segmented connection is different from the results in the above figure .

Obtained from each two-point interval y All values are equal .

----- The first 1 Interval data -----
[1.0, 1.2105263157894737, 1.4210526315789473, 1.631578947368421, 1.8421052631578947]
result [1.         1.63157895 2.26315789 2.89473684 3.52631579]
----- The first 2 Interval data -----
[2.052631578947368, 2.263157894736842, 2.473684210526316, 2.6842105263157894, 2.894736842105263]
result [3.89473684 3.47368421 3.05263158 2.63157895 2.21052632]
----- The first 3 Interval data -----
[3.1052631578947367, 3.3157894736842106, 3.526315789473684, 3.7368421052631575, 3.9473684210526314]
result [2.21052632 2.63157895 3.05263158 3.47368421 3.89473684]
----- The first 4 Interval data -----
[4.157894736842105, 4.368421052631579, 4.578947368421052, 4.789473684210526, 5.0]
result [3.52631579 2.89473684 2.26315789 1.63157895 1.        ]