python基于三阶贝塞尔曲线的数据平滑算法

# -*- coding: utf-8 -*-

import numpy as np

def bezier_curve(p0, p1, p2, p3, inserted):
 """
 三阶贝塞尔曲线
 
 p0, p1, p2, p3 - 点坐标，tuple、list或numpy.ndarray类型
 inserted  - p0和p3之间插值的数量
 """
 
 assert isinstance(p0, (tuple, list, np.ndarray)), u'点坐标不是期望的元组、列表或numpy数组类型'
 assert isinstance(p0, (tuple, list, np.ndarray)), u'点坐标不是期望的元组、列表或numpy数组类型'
 assert isinstance(p0, (tuple, list, np.ndarray)), u'点坐标不是期望的元组、列表或numpy数组类型'
 assert isinstance(p0, (tuple, list, np.ndarray)), u'点坐标不是期望的元组、列表或numpy数组类型'
 
 if isinstance(p0, (tuple, list)):
  p0 = np.array(p0)
 if isinstance(p1, (tuple, list)):
  p1 = np.array(p1)
 if isinstance(p2, (tuple, list)):
  p2 = np.array(p2)
 if isinstance(p3, (tuple, list)):
  p3 = np.array(p3)
 
 points = list()
 for t in np.linspace(0, 1, inserted+2):
  points.append(p0*np.power((1-t),3) + 3*p1*t*np.power((1-t),2) + 3*p2*(1-t)*np.power(t,2) + p3*np.power(t,3))
 
 return np.vstack(points)


def smoothing_base_bezier(date_x, date_y, k=0.5, inserted=10, closed=False):
 """
 基于三阶贝塞尔曲线的数据平滑算法
 
 date_x  - x维度数据集，list或numpy.ndarray类型
 date_y  - y维度数据集，list或numpy.ndarray类型
 k   - 调整平滑曲线形状的因子，取值一般在0.2~0.6之间。默认值为0.5
 inserted - 两个原始数据点之间插值的数量。默认值为10
 closed  - 曲线是否封闭，如是，则首尾相连。默认曲线不封闭
 """
 
 assert isinstance(date_x, (list, np.ndarray)), u'x数据集不是期望的列表或numpy数组类型'
 assert isinstance(date_y, (list, np.ndarray)), u'y数据集不是期望的列表或numpy数组类型'
 
 if isinstance(date_x, list) and isinstance(date_y, list):
  assert len(date_x)==len(date_y), u'x数据集和y数据集长度不匹配'
  date_x = np.array(date_x)
  date_y = np.array(date_y)
 elif isinstance(date_x, np.ndarray) and isinstance(date_y, np.ndarray):
  assert date_x.shape==date_y.shape, u'x数据集和y数据集长度不匹配'
 else:
  raise Exception(u'x数据集或y数据集类型错误')
 
 # 第1步：生成原始数据折线中点集
 mid_points = list()
 for i in range(1, date_x.shape[0]):
  mid_points.append({
   'start': (date_x[i-1], date_y[i-1]),
   'end':  (date_x[i], date_y[i]),
   'mid':  ((date_x[i]+date_x[i-1])/2.0, (date_y[i]+date_y[i-1])/2.0)
  })
 
 if closed:
  mid_points.append({
   'start': (date_x[-1], date_y[-1]),
   'end':  (date_x[0], date_y[0]),
   'mid':  ((date_x[0]+date_x[-1])/2.0, (date_y[0]+date_y[-1])/2.0)
  })
 
 # 第2步：找出中点连线及其分割点
 split_points = list()
 for i in range(len(mid_points)):
  if i < (len(mid_points)-1):
   j = i+1
  elif closed:
   j = 0
  else:
   continue
  
  x00, y00 = mid_points[i]['start']
  x01, y01 = mid_points[i]['end']
  x10, y10 = mid_points[j]['start']
  x11, y11 = mid_points[j]['end']
  d0 = np.sqrt(np.power((x00-x01), 2) + np.power((y00-y01), 2))
  d1 = np.sqrt(np.power((x10-x11), 2) + np.power((y10-y11), 2))
  k_split = 1.0*d0/(d0+d1)
  
  mx0, my0 = mid_points[i]['mid']
  mx1, my1 = mid_points[j]['mid']
  
  split_points.append({
   'start': (mx0, my0),
   'end':  (mx1, my1),
   'split': (mx0+(mx1-mx0)*k_split, my0+(my1-my0)*k_split)
  })
 
 # 第3步：平移中点连线，调整端点，生成控制点
 crt_points = list()
 for i in range(len(split_points)):
  vx, vy = mid_points[i]['end'] # 当前顶点的坐标
  dx = vx - split_points[i]['split'][0] # 平移线段x偏移量
  dy = vy - split_points[i]['split'][1] # 平移线段y偏移量
  
  sx, sy = split_points[i]['start'][0]+dx, split_points[i]['start'][1]+dy # 平移后线段起点坐标
  ex, ey = split_points[i]['end'][0]+dx, split_points[i]['end'][1]+dy # 平移后线段终点坐标
  
  cp0 = sx+(vx-sx)*k, sy+(vy-sy)*k # 控制点坐标
  cp1 = ex+(vx-ex)*k, ey+(vy-ey)*k # 控制点坐标
  
  if crt_points:
   crt_points[-1].insert(2, cp0)
  else:
   crt_points.append([mid_points[0]['start'], cp0, mid_points[0]['end']])
  
  if closed:
   if i < (len(mid_points)-1):
    crt_points.append([mid_points[i+1]['start'], cp1, mid_points[i+1]['end']])
   else:
    crt_points[0].insert(1, cp1)
  else:
   if i < (len(mid_points)-2):
    crt_points.append([mid_points[i+1]['start'], cp1, mid_points[i+1]['end']])
   else:
    crt_points.append([mid_points[i+1]['start'], cp1, mid_points[i+1]['end'], mid_points[i+1]['end']])
    crt_points[0].insert(1, mid_points[0]['start'])
 
 # 第4步：应用贝塞尔曲线方程插值
 out = list()
 for item in crt_points:
  group = bezier_curve(item[0], item[1], item[2], item[3], inserted)
  out.append(group[:-1])
 
 out.append(group[-1:])
 out = np.vstack(out)
 
 return out.T[0], out.T[1]


if __name__ == '__main__':
 import matplotlib.pyplot as plt
 
 x = np.array([2,4,4,3,2])
 y = np.array([2,2,4,3,4])
	
	plt.plot(x, y, 'ro')
 x_curve, y_curve = smoothing_base_bezier(x, y, k=0.3, closed=True)
 plt.plot(x_curve, y_curve, label='$k=0.3$')
 x_curve, y_curve = smoothing_base_bezier(x, y, k=0.4, closed=True)
 plt.plot(x_curve, y_curve, label='$k=0.4$')
 x_curve, y_curve = smoothing_base_bezier(x, y, k=0.5, closed=True)
 plt.plot(x_curve, y_curve, label='$k=0.5$')
 x_curve, y_curve = smoothing_base_bezier(x, y, k=0.6, closed=True)
 plt.plot(x_curve, y_curve, label='$k=0.6$')
 plt.legend(loc='best')
 
 plt.show()