B_Spline_Approximation

B样条曲线的拟合主要是一个LSQ（least squares) 拟合问题，主要思想也是最小二乘法的思想，这与B-Spline曲线插值不同，拟合的曲线是尽量接近数据点，而不是完全通过。主要的方法可以参考cs3621

这里我定义了一个BS_curve类，类中的方法包括数据的参数化(parameterization)，节点(knots)的生成，计算系数$N_{i,p}$，De_Boor算法以及最小二乘拟合(approximation),完整代码如下：

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import math

class BS_curve(object):

    def __init__(self,n,p,cp=None,knots=None):
        self.n = n # n+1 control points >>> p0,p1,,,pn
        self.p = p
        if cp:
            self.cp = cp
            self.u = knots
            self.m = knots.shape[0]-1 # m+1 knots >>> u0,u1,,,nm
        else:
            self.cp = None
            self.u = None
            self.m = None

        self.paras = None


    def check(self):
        if self.m == self.n + self.p + 1:
            return 1
        else:
            return 0


    def coeffs(self,uq):
        # n+1 control points >>> p0,p1,,,pn
        # m+1 knots >>> u0,u1,,,nm
        # algorithm is from https://pages.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/B-spline/bspline-curve-coef.html
    
        #N[] holds all intermediate and the final results
        # in fact N is longer than control points,this is just to hold the intermediate value
        # at last, we juest extract a part of N,that is N[0:n+1]
        N = np.zeros(self.m+1,dtype=np.float64) 

        # rule out special cases Important Properties of clamped B-spline curve
        if uq == self.u[0]:
            N[0] = 1.0
            return N[0:self.n+1]
        elif uq == self.u[self.m]:
            N[self.n] = 1.0
            return N[0:self.n+1]

        # now u is between u0 and um
        # first find k uq in span [uk,uk+1)
        check = uq - self.u
        ind = check >=0
        k = np.max(np.nonzero(ind))
        # sk >>> multiplicity of u[k]
        sk = np.sum(self.u==self.u[k])

        N[k] = 1.0 # degree 0
        # degree d goes from 1 to p
        for d in range(1,self.p+1):
            r_max = self.m - d - 1 # the maximum subscript value of N in degree d,the minimum is 0
            if k-d >=0:
                if self.u[k+1]-self.u[k-d+1]:
                    N[k-d] = (self.u[k+1]-uq)/(self.u[k+1]-self.u[k-d+1])*N[k-d+1] #right (south-west corner) term only
                else:
                    N[k-d] = (self.u[k+1]-uq)/1*N[k-d+1] #right (south-west corner) term only

            for i in range(k-d+1,(k-1)+1):
                if i>=0 and i<=r_max:
                    Denominator1 = self.u[i+d]-self.u[i]
                    Denominator2 = self.u[i+d+1]-self.u[i+1]
                    # 0/0=0
                    if Denominator1 == 0:
                        Denominator1 = 1
                    if Denominator2 == 0:
                        Denominator2 = 1

                    N[i] = (uq-self.u[i])/(Denominator1)*N[i]+(self.u[i+d+1]-uq)/(Denominator2)*N[i+1]

            if k <= r_max:
                if self.u[k+d]-self.u[k]:
                    N[k] = (uq-self.u[k])/(self.u[k+d]-self.u[k])*N[k]
                else:
                    N[k] = (uq-self.u[k])/1*N[k]

        return N[0:self.n+1]


    def De_Boor(self,uq):
        # Input: a value u
        # Output: the point on the curve, C(u)

        # first find k uq in span [uk,uk+1)
        check = uq - self.u
        ind = check >=0
        k = np.max(np.nonzero(ind))
        
        # inserting uq h times
        if uq in self.u:
            # sk >>> multiplicity of u[k]
            sk = np.sum(self.u==self.u[k])
            h = self.p - sk
        else:
            sk = 0
            h = self.p

        # rule out special cases
        if h == -1:
            if k == self.p:
                return np.array(self.cp[0])
            elif k == self.m:
                return np.array(self.cp[-1])


        # initial values of P(affected control points) >>> Pk-s,0 Pk-s-1,0 ... Pk-p+1,0
        P = self.cp[k-self.p:k-sk+1]
        P = P.copy()
        dis = k-self.p # the index distance between storage loaction and varibale i
        # 1-h
        
        for r in range(1,h+1):
            # k-p >> k-sk
            temp = [] # uesd for Storing variables of the current stage
            for i in range(k-self.p+r,k-sk+1):
                a_ir = (uq-self.u[i])/(self.u[i+self.p-r+1]-self.u[i])
                temp.append((1-a_ir)*P[i-dis-1]+a_ir*P[i-dis])
            P[k-self.p+r-dis:k-sk+1-dis] = np.array(temp)
        # the last value is what we want
        return P[-1]


    def bs(self,us):
        y = []
        for x in us:
            y.append(self.De_Boor(x))
        y = np.array(y)
        return y


    def estimate_parameters(self,data_points,method="centripetal"):
        pts = data_points.copy()
        N = pts.shape[0]
        w = pts.shape[1]
        Li = []
        for i in range(1,N):
            Li.append(np.sum([pts[i,j]**2 for j in range(w)])**0.5)
        L = np.sum(Li)

        t= [0]
        for i in range(len(Li)):
            Lki = 0
            for j in range(i+1):
                Lki += Li[j]
            t.append(Lki/L)
        t = np.array(t)
        self.paras = t
        ind = t>1.0
        t[ind] = 1.0
        return t


    def get_knots(self,method="average"):

        knots = np.zeros(self.p+1).tolist()

        paras_temp = self.paras.copy()
        # m = n+p+1
        self.m = self.n + self.p + 1
        # we only need m+1 knots
        # so we just select m+1-(p+1)-(p+1)+(p-1)+1+1  paras to average
        num = self.m - self.p  # select n+1 paras

        ind = np.linspace(0,paras_temp.shape[0]-1,num)
        ind = ind.astype(int)
        paras_knots = paras_temp[ind]

        for j in range(1,self.n-self.p+1):
            k_temp = 0
            # the maximun of variable i is n-1
            for i in range(j,j+self.p-1+1):
                k_temp += paras_knots[i]
            k_temp /= self.p
            knots.append(k_temp)

        add = np.ones(self.p+1).tolist()
        knots = knots + add
        knots = np.array(knots)
        self.u = knots
        self.m = knots.shape[0]-1
        return knots


    def set_paras(self,parameters):
        self.paras = parameters


    def set_knots(self,knots):
        self.u = knots


    def approximation(self,pts):
        ## Obtain a set of parameters t0, ..., tn
        #pts_paras = self.estimate_parameters(pts)
        ## knot vector U;
        #knots = self.get_knots()
        num = pts.shape[0]-1 # (num+1) is the number of data points

        P = np.zeros((self.n+1,pts.shape[1]),dtype=np.float64) # n+1 control points
        P[0] = pts[0]
        P[-1] = pts[-1]

        # compute N
        N = []
        for uq in self.paras:
            N_temp = self.coeffs(uq)
            N.append(N_temp)
        N = np.array(N)

        Q = [0] # hold the location
        for k in range(1,num-1+1):
            Q_temp = pts[k] - N[k,0]*pts[0] - N[k,self.n]*pts[-1]
            Q.append(Q_temp)

        b = [0]
        for i in range(1,self.n-1+1):
            b_temp = 0
            for k in range(1,num-1+1):
                b_temp += N[k,i]*Q[k]
            b.append(b_temp)

        b = b[1::]
        b = np.array(b)

        N = N[:,1:(self.n-1)+1]
        A = np.dot(N.T,N)
        cpm = np.linalg.solve(A,b)
        P[1:self.n] = cpm
        self.cp = P
        return P


if __name__ =="__main__":
    bs = BS_curve(8,3)
    xx = np.linspace(0,4*np.pi,101)
    yy = np.sin(xx)+0.6*np.random.random(101)
    fig = plt.figure(figsize=(10,5))
    ax = fig.add_subplot(111)
    ax.scatter(xx,yy)

    data = np.array([xx,yy]).T
    paras = bs.estimate_parameters(data)
    knots = bs.get_knots()
    if bs.check():
        cp = bs.approximation(data)

    uq = np.linspace(0,1,101)
    y = bs.bs(uq)
    ax.plot(y[:,0],y[:,1],'-r')
    ax.plot(cp[:,0],cp[:,1],'-b*')
    plt.show()

上面代码的结果如下图所示：