python - 使用 scipy.optimize.curve_fit - ValueError 和 minpack.error 拟合 2D 高斯函数

Question

我打算将二维高斯函数拟合到显示激光束的图像中，以获取其参数，如 FWHM 和位置。到目前为止，我试图了解如何在 Python 中定义 2D 高斯函数以及如何将 x 和 y 变量传递给它。

我编写了一个小脚本，它定义了该函数，绘制它，为其添加一些噪声，然后尝试使用 curve_fit 对其进行拟合。除了我尝试将模型函数拟合到嘈杂数据的最后一步之外，一切似乎都有效。这是我的代码:

import scipy.optimize as opt
import numpy as np
import pylab as plt


#define model function and pass independant variables x and y as a list
def twoD_Gaussian((x,y), amplitude, xo, yo, sigma_x, sigma_y, theta, offset):
    xo = float(xo)
    yo = float(yo)    
    a = (np.cos(theta)**2)/(2*sigma_x**2) + (np.sin(theta)**2)/(2*sigma_y**2)
    b = -(np.sin(2*theta))/(4*sigma_x**2) + (np.sin(2*theta))/(4*sigma_y**2)
    c = (np.sin(theta)**2)/(2*sigma_x**2) + (np.cos(theta)**2)/(2*sigma_y**2)
    return offset + amplitude*np.exp( - (a*((x-xo)**2) + 2*b*(x-xo)*(y-yo) + c*((y-yo)**2)))

# Create x and y indices
x = np.linspace(0, 200, 201)
y = np.linspace(0, 200, 201)
x,y = np.meshgrid(x, y)

#create data
data = twoD_Gaussian((x, y), 3, 100, 100, 20, 40, 0, 10)

# plot twoD_Gaussian data generated above
plt.figure()
plt.imshow(data)
plt.colorbar()

# add some noise to the data and try to fit the data generated beforehand
initial_guess = (3,100,100,20,40,0,10)

data_noisy = data + 0.2*np.random.normal(size=len(x))

popt, pcov = opt.curve_fit(twoD_Gaussian, (x,y), data_noisy, p0 = initial_guess)

这是我使用 winpython 64-bit Python 2.7 运行脚本时收到的错误消息:

ValueError: object too deep for desired array
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "C:\Python\WinPython-64bit-2.7.6.2\python-2.7.6.amd64\lib\site-packages\spyderlib\widgets\externalshell\sitecustomize.py", line 540, in runfile
    execfile(filename, namespace)
  File "E:/Work Computer/Software/Python/Fitting scripts/2D Gaussian function fit/2D_Gaussian_LevMarq_v2.py", line 39, in <module>
    popt, pcov = opt.curve_fit(twoD_Gaussian, (x,y), data_noisy, p0 = initial_guess)
  File "C:\Python\WinPython-64bit-2.7.6.2\python-2.7.6.amd64\lib\site-packages\scipy\optimize\minpack.py", line 533, in curve_fit
    res = leastsq(func, p0, args=args, full_output=1, **kw)
  File "C:\Python\WinPython-64bit-2.7.6.2\python-2.7.6.amd64\lib\site-packages\scipy\optimize\minpack.py", line 378, in leastsq
    gtol, maxfev, epsfcn, factor, diag)
minpack.error: Result from function call is not a proper array of floats.

我做错了什么？是我如何将自变量传递给模型function/curve_fit？

Answer 1

twoD_Gaussian 的输出需要是一维的。您可以做的是在最后一行的末尾添加一个 .ravel()，如下所示:

def twoD_Gaussian((x, y), amplitude, xo, yo, sigma_x, sigma_y, theta, offset):
    xo = float(xo)
    yo = float(yo)    
    a = (np.cos(theta)**2)/(2*sigma_x**2) + (np.sin(theta)**2)/(2*sigma_y**2)
    b = -(np.sin(2*theta))/(4*sigma_x**2) + (np.sin(2*theta))/(4*sigma_y**2)
    c = (np.sin(theta)**2)/(2*sigma_x**2) + (np.cos(theta)**2)/(2*sigma_y**2)
    g = offset + amplitude*np.exp( - (a*((x-xo)**2) + 2*b*(x-xo)*(y-yo) 
                            + c*((y-yo)**2)))
    return g.ravel()

您显然需要 reshape 输出以进行绘图，例如:

# Create x and y indices
x = np.linspace(0, 200, 201)
y = np.linspace(0, 200, 201)
x, y = np.meshgrid(x, y)

#create data
data = twoD_Gaussian((x, y), 3, 100, 100, 20, 40, 0, 10)

# plot twoD_Gaussian data generated above
plt.figure()
plt.imshow(data.reshape(201, 201))
plt.colorbar()

像以前一样进行拟合:

# add some noise to the data and try to fit the data generated beforehand
initial_guess = (3,100,100,20,40,0,10)

data_noisy = data + 0.2*np.random.normal(size=data.shape)

popt, pcov = opt.curve_fit(twoD_Gaussian, (x, y), data_noisy, p0=initial_guess)

并绘制结果:

data_fitted = twoD_Gaussian((x, y), *popt)

fig, ax = plt.subplots(1, 1)
ax.hold(True)
ax.imshow(data_noisy.reshape(201, 201), cmap=plt.cm.jet, origin='bottom',
    extent=(x.min(), x.max(), y.min(), y.max()))
ax.contour(x, y, data_fitted.reshape(201, 201), 8, colors='w')
plt.show()