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バイキュービック補間Python

私は、Pythonプログラミング言語を使用して学部生にデモするためのバイキュービック補間を開発しました。

方法論は wikipedia で説明されています。コードは正常に機能していますが、scipyライブラリを使用した場合に得られる結果とは少し異なります。

補間コードを以下の関数bicubic_interpolationに示します。

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits import mplot3d
from scipy import interpolate
import sympy as syp
import pandas as pd
pd.options.display.max_colwidth = 200
%matplotlib inline

def bicubic_interpolation(xi, yi, zi, xnew, ynew):

    # check sorting
    if np.any(np.diff(xi) < 0) and np.any(np.diff(yi) < 0) and\
    np.any(np.diff(xnew) < 0) and np.any(np.diff(ynew) < 0):
        raise ValueError('data are not sorted')

    if zi.shape != (xi.size, yi.size):
        raise ValueError('zi is not set properly use np.meshgrid(xi, yi)')

    z = np.zeros((xnew.size, ynew.size))

    deltax = xi[1] - xi[0]
    deltay = yi[1] - yi[0] 
    for n, x in enumerate(xnew):
        for m, y in enumerate(ynew):

            if xi.min() <= x <= xi.max() and yi.min() <= y <= yi.max():

                i = np.searchsorted(xi, x) - 1
                j = np.searchsorted(yi, y) - 1

                x0  = xi[i-1]
                x1  = xi[i]
                x2  = xi[i+1]
                x3  = x1+2*deltax

                y0  = yi[j-1]
                y1  = yi[j]
                y2  = yi[j+1]
                y3  = y1+2*deltay

                px = (x-x1)/(x2-x1)
                py = (y-y1)/(y2-y1)

                f00 = zi[i-1, j-1]      #row0 col0 >> x0,y0
                f01 = zi[i-1, j]        #row0 col1 >> x1,y0
                f02 = zi[i-1, j+1]      #row0 col2 >> x2,y0

                f10 = zi[i, j-1]        #row1 col0 >> x0,y1
                f11 = p00 = zi[i, j]    #row1 col1 >> x1,y1
                f12 = p01 = zi[i, j+1]  #row1 col2 >> x2,y1

                f20 = zi[i+1,j-1]       #row2 col0 >> x0,y2
                f21 = p10 = zi[i+1,j]   #row2 col1 >> x1,y2
                f22 = p11 = zi[i+1,j+1] #row2 col2 >> x2,y2

                if 0 < i < xi.size-2 and 0 < j < yi.size-2:

                    f03 = zi[i-1, j+2]      #row0 col3 >> x3,y0

                    f13 = zi[i,j+2]         #row1 col3 >> x3,y1

                    f23 = zi[i+1,j+2]       #row2 col3 >> x3,y2

                    f30 = zi[i+2,j-1]       #row3 col0 >> x0,y3
                    f31 = zi[i+2,j]         #row3 col1 >> x1,y3
                    f32 = zi[i+2,j+1]       #row3 col2 >> x2,y3
                    f33 = zi[i+2,j+2]       #row3 col3 >> x3,y3

                Elif i<=0: 

                    f03 = f02               #row0 col3 >> x3,y0

                    f13 = f12               #row1 col3 >> x3,y1

                    f23 = f22               #row2 col3 >> x3,y2

                    f30 = zi[i+2,j-1]       #row3 col0 >> x0,y3
                    f31 = zi[i+2,j]         #row3 col1 >> x1,y3
                    f32 = zi[i+2,j+1]       #row3 col2 >> x2,y3
                    f33 = f32               #row3 col3 >> x3,y3             

                Elif j<=0:

                    f03 = zi[i-1, j+2]      #row0 col3 >> x3,y0

                    f13 = zi[i,j+2]         #row1 col3 >> x3,y1

                    f23 = zi[i+1,j+2]       #row2 col3 >> x3,y2

                    f30 = f20               #row3 col0 >> x0,y3
                    f31 = f21               #row3 col1 >> x1,y3
                    f32 = f22               #row3 col2 >> x2,y3
                    f33 = f23               #row3 col3 >> x3,y3


                Elif i == xi.size-2 or j == yi.size-2:

                    f03 = f02               #row0 col3 >> x3,y0

                    f13 = f12               #row1 col3 >> x3,y1

                    f23 = f22               #row2 col3 >> x3,y2

                    f30 = f20               #row3 col0 >> x0,y3
                    f31 = f21               #row3 col1 >> x1,y3
                    f32 = f22               #row3 col2 >> x2,y3
                    f33 = f23               #row3 col3 >> x3,y3

                px00 = (f12 - f10)/2*deltax
                px01 = (f22 - f20)/2*deltax 
                px10 = (f13 - f11)/2*deltax 
                px11 = (f23 - f21)/2*deltax

                py00 = (f21 - f01)/2*deltay
                py01 = (f22 - f02)/2*deltay
                py10 = (f31 - f11)/2*deltay
                py11 = (f32 - f12)/2*deltay

                pxy00 = ((f22-f20) - (f02-f00))/4*deltax*deltay
                pxy01 = ((f32-f30) - (f12-f10))/4*deltax*deltay
                pxy10 = ((f23-f21) - (f03-f01))/4*deltax*deltay
                pxy11 = ((f33-f31) - (f13-f11))/4*deltax*deltay


                f = np.array([p00,  p01,  p10, p11,
                              px00,  px01,  px10, px11,
                              py00, py01,  py10,  py11,
                              pxy00,  pxy01, pxy10, pxy11])

                a = A@f

                a = a.reshape(4,4).transpose()
                z[n,m] = np.array([1, px, px**2, px**3]) @ a @ np.array([1, py, py**2, py**3])

    return z

関数bicubic_interpolationでは、入力はxi =古いxデータ範囲、yi =古いy範囲、zi =グリッドポイントの古い値(x、y )、xnew、およびynewは、新しい水平方向のデータ範囲です。 2D numpy配列であるziを除いて、すべての入力は1D numpy配列です。

関数をテストしているデータを以下に示します。結果をscipyおよび真のモデル(関数f)と比較しています。

def f(x,y):
    return np.sin(np.sqrt(x ** 2 + y ** 2))

x = np.linspace(-6, 6, 11)
y = np.linspace(-6, 6, 11)

xx, yy = np.meshgrid(x, y)

z = f(xx, yy)

x_new = np.linspace(-6, 6, 100)
y_new = np.linspace(-6, 6, 100)

xx_new, yy_new = np.meshgrid(x_new, y_new)

z_new = bicubic_interpolation(x, y, z, x_new, y_new)

z_true = f(xx_new, yy_new) 

f_scipy = interpolate.interp2d(x, y, z, kind='cubic')

z_scipy = f_scipy(x_new, y_new)

fig, ax = plt.subplots(2, 2, sharey=True, figsize=(16,12))

img0 = ax[0, 0].scatter(xx, yy, c=z, s=100)
ax[0, 0].set_title('original points')
fig.colorbar(img0, ax=ax[0, 0], orientation='vertical', shrink=1, pad=0.01)

img1 = ax[0, 1].imshow(z_new, vmin=z_new.min(), vmax=z_new.max(), Origin='lower',
           extent=[x_new.min(), x_new.max(), y_new.max(), y_new.min()])
ax[0, 1].set_title('bicubic our code')
fig.colorbar(img1, ax=ax[0, 1], orientation='vertical', shrink=1, pad=0.01)


img2 = ax[1, 0].imshow(z_scipy, vmin=z_scipy.min(), vmax=z_scipy.max(), Origin='lower',
           extent=[x_new.min(), x_new.max(), y_new.max(), y_new.min()])
ax[1, 0].set_title('bicubic scipy')
fig.colorbar(img2, ax=ax[1, 0], orientation='vertical', shrink=1, pad=0.01)


img3 = ax[1, 1].imshow(z_true, vmin=z_true.min(), vmax=z_true.max(), Origin='lower',
           extent=[x_new.min(), x_new.max(), y_new.max(), y_new.min()])
ax[1, 1].set_title('true model')
fig.colorbar(img3, ax=ax[1, 1], orientation='vertical', shrink=1, pad=0.01)

plt.subplots_adjust(wspace=0.05, hspace=0.15)

plt.show()

結果を以下に示します。

enter image description here

マトリックスA(関数内bicubic_interpolation)はWikipediaサイトで説明されているとおりで、次のコードを使用して簡単に取得できます。

x = syp.Symbol('x')
y = syp.Symbol('y')
a00, a01, a02, a03, a10, a11, a12, a13 = syp.symbols('a00 a01 a02 a03 a10 a11 a12 a13')
a20, a21, a22, a23, a30, a31, a32, a33 = syp.symbols('a20 a21 a22 a23 a30 a31 a32 a33')

p = a00 + a01*y + a02*y**2 + a03*y**3\
+ a10*x + a11*x*y + a12*x*y**2 + a13*x*y**3\
+ a20*x**2 + a21*x**2*y + a22*x**2*y**2 + a23*x**2*y**3\
+ a30*x**3 + a31*x**3*y + a32*x**3*y**2 + a33*x**3*y**3 

px = syp.diff(p, x)
py = syp.diff(p, y)
pxy = syp.diff(p, x, y)

df = pd.DataFrame(columns=['function', 'evaluation'])

for i in range(2):
    for j in range(2):
        function = 'p({}, {})'.format(j,i)
        df.loc[len(df)] = [function, p.subs({x:j, y:i})]
for i in range(2):
    for j in range(2):
        function = 'px({}, {})'.format(j,i)
        df.loc[len(df)] = [function, px.subs({x:j, y:i})]
for i in range(2):
    for j in range(2):
        function = 'py({}, {})'.format(j,i)
        df.loc[len(df)] = [function, py.subs({x:j, y:i})]
for i in range(2):
    for j in range(2):
        function = 'pxy({}, {})'.format(j,i)
        df.loc[len(df)] = [function, pxy.subs({x:j, y:i})]

eqns = df['evaluation'].tolist()
symbols = [a00,a01,a02,a03,a10,a11,a12,a13,a20,a21,a22,a23,a30,a31,a32,a33]
A = syp.linear_eq_to_matrix(eqns, *symbols)[0]
A = np.array(A.inv()).astype(np.float64)

print(df)

print(A) 

enter image description here

enter image description here

bicubic_interpolation関数のどこに問題があるのか​​、またscipyで得られた結果と少し異なるのはなぜですか?どんな助けでも大歓迎です!

7
Khalil Al Hooti

ウィキペディアの実装が期待どおりに機能しない理由がわかりません。おそらく、その理由は、これらの値がサイトで説明されているものとは異なる方法で概算される可能性があるためです。

px00 = (f12 - f10)/2*deltax
px01 = (f22 - f20)/2*deltax 
px10 = (f13 - f11)/2*deltax 
px11 = (f23 - f21)/2*deltax

py00 = (f21 - f01)/2*deltay
py01 = (f22 - f02)/2*deltay
py10 = (f31 - f11)/2*deltay
py11 = (f32 - f12)/2*deltay

pxy00 = ((f22-f20) - (f02-f00))/4*deltax*deltay
pxy01 = ((f32-f30) - (f12-f10))/4*deltax*deltay
pxy10 = ((f23-f21) - (f03-f01))/4*deltax*deltay
pxy11 = ((f33-f31) - (f13-f11))/4*deltax*deltay

ただし、実装が異なる this ドキュメントが見つかりました。ウィキペディアよりも説明がよく、理解しやすいです。この実装を使用して得られる結果は、SciPyによって得られる結果と非常に似ています。

def bicubic_interpolation2(xi, yi, zi, xnew, ynew):

    # check sorting
    if np.any(np.diff(xi) < 0) and np.any(np.diff(yi) < 0) and\
    np.any(np.diff(xnew) < 0) and np.any(np.diff(ynew) < 0):
        raise ValueError('data are not sorted')

    if zi.shape != (xi.size, yi.size):
        raise ValueError('zi is not set properly use np.meshgrid(xi, yi)')

    z = np.zeros((xnew.size, ynew.size))

    deltax = xi[1] - xi[0]
    deltay = yi[1] - yi[0] 
    for n, x in enumerate(xnew):
        for m, y in enumerate(ynew):

            if xi.min() <= x <= xi.max() and yi.min() <= y <= yi.max():

                i = np.searchsorted(xi, x) - 1
                j = np.searchsorted(yi, y) - 1

                x1  = xi[i]
                x2  = xi[i+1]

                y1  = yi[j]
                y2  = yi[j+1]

                px = (x-x1)/(x2-x1)
                py = (y-y1)/(y2-y1)

                f00 = zi[i-1, j-1]      #row0 col0 >> x0,y0
                f01 = zi[i-1, j]        #row0 col1 >> x1,y0
                f02 = zi[i-1, j+1]      #row0 col2 >> x2,y0

                f10 = zi[i, j-1]        #row1 col0 >> x0,y1
                f11 = p00 = zi[i, j]    #row1 col1 >> x1,y1
                f12 = p01 = zi[i, j+1]  #row1 col2 >> x2,y1

                f20 = zi[i+1,j-1]       #row2 col0 >> x0,y2
                f21 = p10 = zi[i+1,j]   #row2 col1 >> x1,y2
                f22 = p11 = zi[i+1,j+1] #row2 col2 >> x2,y2

                if 0 < i < xi.size-2 and 0 < j < yi.size-2:

                    f03 = zi[i-1, j+2]      #row0 col3 >> x3,y0

                    f13 = zi[i,j+2]         #row1 col3 >> x3,y1

                    f23 = zi[i+1,j+2]       #row2 col3 >> x3,y2

                    f30 = zi[i+2,j-1]       #row3 col0 >> x0,y3
                    f31 = zi[i+2,j]         #row3 col1 >> x1,y3
                    f32 = zi[i+2,j+1]       #row3 col2 >> x2,y3
                    f33 = zi[i+2,j+2]       #row3 col3 >> x3,y3

                Elif i<=0: 

                    f03 = f02               #row0 col3 >> x3,y0

                    f13 = f12               #row1 col3 >> x3,y1

                    f23 = f22               #row2 col3 >> x3,y2

                    f30 = zi[i+2,j-1]       #row3 col0 >> x0,y3
                    f31 = zi[i+2,j]         #row3 col1 >> x1,y3
                    f32 = zi[i+2,j+1]       #row3 col2 >> x2,y3
                    f33 = f32               #row3 col3 >> x3,y3             

                Elif j<=0:

                    f03 = zi[i-1, j+2]      #row0 col3 >> x3,y0

                    f13 = zi[i,j+2]         #row1 col3 >> x3,y1

                    f23 = zi[i+1,j+2]       #row2 col3 >> x3,y2

                    f30 = f20               #row3 col0 >> x0,y3
                    f31 = f21               #row3 col1 >> x1,y3
                    f32 = f22               #row3 col2 >> x2,y3
                    f33 = f23               #row3 col3 >> x3,y3


                Elif i == xi.size-2 or j == yi.size-2:

                    f03 = f02               #row0 col3 >> x3,y0

                    f13 = f12               #row1 col3 >> x3,y1

                    f23 = f22               #row2 col3 >> x3,y2

                    f30 = f20               #row3 col0 >> x0,y3
                    f31 = f21               #row3 col1 >> x1,y3
                    f32 = f22               #row3 col2 >> x2,y3
                    f33 = f23               #row3 col3 >> x3,y3

                Z = np.array([f00, f01, f02, f03,
                             f10, f11, f12, f13,
                             f20, f21, f22, f23,
                             f30, f31, f32, f33]).reshape(4,4).transpose()

                X = np.tile(np.array([-1, 0, 1, 2]), (4,1))
                X[0,:] = X[0,:]**3
                X[1,:] = X[1,:]**2
                X[-1,:] = 1

                Cr = [email protected](X)
                R = [email protected]([px**3, px**2, px, 1])

                Y = np.tile(np.array([-1, 0, 1, 2]), (4,1)).transpose()
                Y[:,0] = Y[:,0]**3
                Y[:,1] = Y[:,1]**2
                Y[:,-1] = 1

                Cc = np.linalg.inv(Y)@R

                z[n,m]=([email protected]([py**3, py**2, py, 1]))


    return z

def f(x,y):
    return np.sin(np.sqrt(x ** 2 + y ** 2))

x = np.linspace(-6, 6, 11)
y = np.linspace(-6, 6, 11)

xx, yy = np.meshgrid(x, y)

z = f(xx, yy)

x_new = np.linspace(-6, 6, 100)
y_new = np.linspace(-6, 6, 100)

xx_new, yy_new = np.meshgrid(x_new, y_new)

z_new = bicubic_interpolation2(x, y, z, x_new, y_new)

z_true = f(xx_new, yy_new) 

f_scipy = interpolate.interp2d(x, y, z, kind='cubic')

z_scipy = f_scipy(x_new, y_new)

fig, ax = plt.subplots(2, 2, sharey=True, figsize=(16,12))

img0 = ax[0, 0].scatter(xx, yy, c=z, s=100)
ax[0, 0].set_title('original points')
fig.colorbar(img0, ax=ax[0, 0], orientation='vertical', shrink=1, pad=0.01)

img1 = ax[0, 1].imshow(z_new, vmin=z_new.min(), vmax=z_new.max(), Origin='lower',
           extent=[x_new.min(), x_new.max(), y_new.max(), y_new.min()])
ax[0, 1].set_title('bicubic our code')
fig.colorbar(img1, ax=ax[0, 1], orientation='vertical', shrink=1, pad=0.01)


img2 = ax[1, 0].imshow(z_scipy, vmin=z_scipy.min(), vmax=z_scipy.max(), Origin='lower',
           extent=[x_new.min(), x_new.max(), y_new.max(), y_new.min()])
ax[1, 0].set_title('bicubic scipy')
fig.colorbar(img2, ax=ax[1, 0], orientation='vertical', shrink=1, pad=0.01)


img3 = ax[1, 1].imshow(z_true, vmin=z_true.min(), vmax=z_true.max(), Origin='lower',
           extent=[x_new.min(), x_new.max(), y_new.max(), y_new.min()])
ax[1, 1].set_title('true model')
fig.colorbar(img3, ax=ax[1, 1], orientation='vertical', shrink=1, pad=0.01)

plt.subplots_adjust(wspace=0.05, hspace=0.15)

plt.show()

enter image description here

fig, ax = plt.subplots(1, 2, sharey=True, figsize=(10, 6))

ax[0].plot(xx[0,:], z[5,:], 'or', label='original')
ax[0].plot(xx_new[0,:], z_true[int(100/10*5),:], label='true')
ax[0].plot(xx_new[0,:], z_new[int(100/10*5), :], label='our interpolation')

ax[1].plot(xx[0,:], z[5,:], 'or', label='original')
ax[1].plot(xx_new[0,:], z_true[int(100/10*5),:], label='true')
ax[1].plot(xx_new[0,:], z_scipy[int(100/10*5), :], label='scipy interpolation')


for axes in ax:
    axes.legend()
    axes.grid()


plt.show()

enter image description here

7
Khalil Al Hooti

将来の通知については、問題はウィキペディアで詳述されているアルゴリズムが単位正方形のバイキュービック補間用であることだと思います。代わりに直線グリッドで補間する場合は、ベクトルxを少し変更する必要があります。 Wikipediaのページに含まれる「直線グリッドへの拡張」セクションを参照してください。 https://en.wikipedia.org/wiki/Bicubic_interpolation#Extension_to_rectilinear_grids

0
Max Gains