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

Question

我打算将 2D 高斯函数拟合到显示激光束的图像中，以获得其参数，例如

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，绘制了它，为其添加了一些噪声，然后尝试使用

curve_fit

来拟合它。除了我尝试将模型函数拟合到噪声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？这就是我将自变量传递给模型的方式吗

function/curve_fit

？

Answer 1

twoD_Gaussian

的输出需要是一维的。您可以做的就是在最后一行的末尾添加一个

.ravel()

，如下所示：

def twoD_Gaussian(xy, amplitude, xo, yo, sigma_x, sigma_y, theta, offset):
    x, y = xy
    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()

您显然需要重塑输出以进行绘图，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()

像以前一样进行安装：

# 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) For older versions. This has now been deprecated and later removed
ax.imshow(data_noisy.reshape(201, 201), cmap=plt.cm.jet, origin='lower',
    extent=(x.min(), x.max(), y.min(), y.max()))
ax.contour(x, y, data_fitted.reshape(201, 201), 8, colors='w')
plt.show()

Answer 2

为了稍微扩展 Dietrich 的答案，在使用 Python 3.4（在 Ubuntu 14.04 上）运行建议的解决方案时，我收到以下错误：

def twoD_Gaussian((x, y), amplitude, xo, yo, sigma_x, sigma_y, theta, offset):
                  ^
SyntaxError: invalid syntax

运行

2to3

建议进行以下简单修复：

def twoD_Gaussian(xdata_tuple, amplitude, xo, yo, sigma_x, sigma_y, theta, offset):
    (x, y) = xdata_tuple                                                        
    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()

其原因是，自 Python 3 起，将元组作为参数传递给函数时的自动解包已被删除。有关更多信息，请参阅此处：PEP 3113

Answer 3

curve_fit()

希望

xdata

的维度为

(2,n*m)

而不是

(2,n,m)

。

ydata

应该分别具有形状

(n*m)

而不是

(n,m)

。所以你使用

ravel()

来展平你的二维数组：

xdata = np.vstack((xx.ravel(),yy.ravel()))
ydata = data_noisy.ravel()
popt, pcov = opt.curve_fit(twoD_Gaussian, xdata, ydata, p0=initial_guess)

顺便说一句：我不确定三角项的参数化是否是最好的。例如，在数值方面和大偏差下，采用here描述的可能会更稳健一些。

Answer 4

为了补充 ali_m 的答案，这里是将 2D 高斯拟合到图像的代码片段，而不是假数据。

输入图像示例：

def twoD_Gaussian(xy, amplitude, xo, yo, sigma_x, sigma_y, theta, offset):
    x, y = xy
    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()

# use your favorite image processing library to load an image
im = cv2.imread(r"path_to_load\im.png", -1) 
h, w = im.shape
data = im.ravel()

x = np.linspace(0, w, w)
y = np.linspace(0, h, h)
x, y = np.meshgrid(x, y)

# initial guess of parameters
initial_guess = (1200, 120, 80, 20, 20, 0, 50)

# find the optimal Gaussian parameters
popt, pcov = curve_fit(twoD_Gaussian, (x, y), data, p0=initial_guess) 

# create new data with these parameters
data_fitted = twoD_Gaussian((x, y), *popt)

cv2.imwrite(r"path_to_save\data_fitted.png", data_fitted.reshape(h,w))