How to interpolate a 2D curve in Python

I have a set of x & y coordinate which is a curve / shape, I want the smooth the curve / sharp and plot a graph.

I tried different interpolation to smooth the curve / shape, But it still cannot fit my expectation. Using point to draw a smooth curve / shape.

Like the following, using x, y point to get a smooth circle / curve

However, I get something like

circle.jpg

curve.jpg

square.jpg

I also get trouble on spline interpolation, and rbf interpolation.

for cubic_spline_interpolation, I got

ValueError: Error on input data

for univariate_spline_interpolated, I got

ValueError: x must be strictly increasing

for rbf, I got

numpy.linalg.linalg.LinAlgError: Matrix is singular.

I have on idea to fix them and get correct sharp and curve. Many thanks for help.

Edit For those cannot download the source code and x, y coordinate file, I post the code and x, y coordinate in question.

The following is my code:

#!/usr/bin/env python3
from std_lib import *

import os
import numpy as np
import cv2

from scipy import interpolate
import matplotlib.pyplot as plt

CUR_DIR = os.getcwd()
CIRCLE_FILE = "circle.txt"
CURVE_FILE  = "curve.txt"
SQUARE_FILE = "square.txt"

#test
CIRCLE_NAME = "circle"
CURVE_NAME  = "curve"
SQUARE_NAME = "square"

SYS_TOKEN_CNT = 2   # x, y

total_pt_cnt = 0        # total no. of points      
x_arr = np.array([])    # x position set
y_arr = np.array([])    # y position set

def convert_coord_to_array(file_path):
    global total_pt_cnt
    global x_arr
    global y_arr

    if file_path == "":
        return FALSE

    with open(file_path) as f:
        content = f.readlines()

    content = [x.strip() for x in content] 

    total_pt_cnt = len(content)

    if (total_pt_cnt <= 0):
        return FALSE

    ##
    x_arr = np.empty((0, total_pt_cnt))
    y_arr = np.empty((0, total_pt_cnt))

    #compare the first and last x 
    # if ((content[0][0]) > (content[-1])):
        # is_reverse = TRUE

    for x in content:
        token_cnt = get_token_cnt(x, ',') 

        if (token_cnt != SYS_TOKEN_CNT):
            return FALSE

        for idx in range(token_cnt):
            token_string = get_token_string(x, ',', idx)
            token_string = token_string.strip()
            if (not token_string.isdigit()): 
                return FALSE

            # save x, y set
            if (idx == 0):
                x_arr = np.append(x_arr, int(token_string))
            else:
                y_arr = np.append(y_arr, int(token_string))

    return TRUE

def linear_interpolation(fig, axs):
    xnew = np.linspace(x_arr.min(), x_arr.max(), len(x_arr))
    f = interpolate.interp1d(xnew , y_arr)

    axs.plot(xnew, f(xnew))
    axs.set_title('linear')

def cubic_interpolation(fig, axs):
    xnew = np.linspace(x_arr.min(), x_arr.max(), len(x_arr))
    f = interpolate.interp1d(xnew , y_arr, kind='cubic')

    axs.plot(xnew, f(xnew))
    axs.set_title('cubic')

def cubic_spline_interpolation(fig, axs):
    xnew = np.linspace(x_arr.min(), x_arr.max(), len(x_arr))
    tck = interpolate.splrep(x_arr, y_arr, s=0) #always fail (ValueError: Error on input data)
    ynew = interpolate.splev(xnew, tck, der=0)

    axs.plot(xnew, ynew)
    axs.set_title('cubic spline')

def parametric_spline_interpolation(fig, axs):
    xnew = np.linspace(x_arr.min(), x_arr.max(), len(x_arr))
    tck, u = interpolate.splprep([x_arr, y_arr], s=0)
    out = interpolate.splev(xnew, tck)

    axs.plot(out[0], out[1])
    axs.set_title('parametric spline')

def univariate_spline_interpolated(fig, axs):   
    s = interpolate.InterpolatedUnivariateSpline(x_arr, y_arr)# ValueError: x must be strictly increasing
    xnew = np.linspace(x_arr.min(), x_arr.max(), len(x_arr))
    ynew = s(xnew)

    axs.plot(xnew, ynew)
    axs.set_title('univariate spline')

def rbf(fig, axs):
    xnew = np.linspace(x_arr.min(), x_arr.max(), len(x_arr))
    rbf = interpolate.Rbf(x_arr, y_arr) # numpy.linalg.linalg.LinAlgError: Matrix is singular.
    fi = rbf(xnew)

    axs.plot(xnew, fi)
    axs.set_title('rbf')

def interpolation():
    fig, axs = plt.subplots(nrows=4)
    axs[0].plot(x_arr, y_arr, 'r-')
    axs[0].set_title('org')

    cubic_interpolation(fig, axs[1])
    # cubic_spline_interpolation(fig, axs[2]) 
    parametric_spline_interpolation(fig, axs[2])
    # univariate_spline_interpolated(fig, axs[3])
    # rbf(fig, axs[3])        
    linear_interpolation(fig, axs[3])

    plt.show()

#------- main -------
if __name__ == "__main__":
    # np.seterr(divide='ignore', invalid='ignore')

    file_name = CUR_DIR + "/" + CIRCLE_FILE 
    convert_coord_to_array(file_name)  
    #file_name = CUR_DIR + "/" + CURVE_FILE 
    #convert_coord_to_array(file_name) 
    #file_name = CUR_DIR + "/" + SQUARE_FILE 
    #convert_coord_to_array(file_name) 
    #
    interpolation()

circle x, y coordinate

Solved

def linear_interpolateion(self, x, y):

        points = np.array([x, y]).T  # a (nbre_points x nbre_dim) array

        # Linear length along the line:
        distance = np.cumsum( np.sqrt(np.sum( np.diff(points, axis=0)**2, axis=1 )) )
        distance = np.insert(distance, 0, 0)

        alpha = np.linspace(distance.min(), int(distance.max()), len(x))
        interpolator =  interpolate.interp1d(distance, points, kind='slinear', axis=0)
        interpolated_points = interpolator(alpha)

        out_x = interpolated_points.T[0]
        out_y = interpolated_points.T[1]

        return out_x, out_y

Because the interpolation is wanted for generic 2d curve i.e. (x, y)=f(s) where s is the coordinates along the curve, rather than y = f(x), the distance along the line s have to be computed first. Then, the interpolation for each coordinates is performed relatively to s. (for instance, in the circle case y = f(x) have two solutions)

s (or distance in the code here) is calculated as the cumulative sum of the length of each segments between the given points.

import numpy as np
from scipy.interpolate import interp1d
import matplotlib.pyplot as plt

# Define some points:
points = np.array([[0, 1, 8, 2, 2],
                   [1, 0, 6, 7, 2]]).T  # a (nbre_points x nbre_dim) array

# Linear length along the line:
distance = np.cumsum( np.sqrt(np.sum( np.diff(points, axis=0)**2, axis=1 )) )
distance = np.insert(distance, 0, 0)/distance[-1]

# Interpolation for different methods:
interpolations_methods = ['slinear', 'quadratic', 'cubic']
alpha = np.linspace(0, 1, 75)

interpolated_points = {}
for method in interpolations_methods:
    interpolator =  interp1d(distance, points, kind=method, axis=0)
    interpolated_points[method] = interpolator(alpha)

# Graph:
plt.figure(figsize=(7,7))
for method_name, curve in interpolated_points.items():
    plt.plot(*curve.T, '-', label=method_name);

plt.plot(*points.T, 'ok', label='original points');
plt.axis('equal'); plt.legend(); plt.xlabel('x'); plt.ylabel('y');

which gives:

Regarding the graphs, it seems you are looking for a smoothing method rather than an interpolation of the points. Here, is a similar approach use to fit a spline separately on each coordinates of the given curve (see Scipy UnivariateSpline):

import numpy as np
import matplotlib.pyplot as plt

from scipy.interpolate import UnivariateSpline

# Define some points:
theta = np.linspace(-3, 2, 40)
points = np.vstack( (np.cos(theta), np.sin(theta)) ).T

# add some noise:
points = points + 0.05*np.random.randn(*points.shape)

# Linear length along the line:
distance = np.cumsum( np.sqrt(np.sum( np.diff(points, axis=0)**2, axis=1 )) )
distance = np.insert(distance, 0, 0)/distance[-1]

# Build a list of the spline function, one for each dimension:
splines = [UnivariateSpline(distance, coords, k=3, s=.2) for coords in points.T]

# Computed the spline for the asked distances:
alpha = np.linspace(0, 1, 75)
points_fitted = np.vstack( spl(alpha) for spl in splines ).T

# Graph:
plt.plot(*points.T, 'ok', label='original points');
plt.plot(*points_fitted.T, '-r', label='fitted spline k=3, s=.2');
plt.axis('equal'); plt.legend(); plt.xlabel('x'); plt.ylabel('y');

which gives:

来源：https://stackoverflow.com/questions/52014197/how-to-interpolate-a-2d-curve-in-python

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python

scipy

interpolation