2D Gaussian Fit for intensities at specific coordinates in Python

Question

I have a set of coordinates (x, y, z (x, y)) that describe the intensities (z) in x, y coordinates. For a given number of these intensities in different coordinates, I need to install a 2D Gaussian, which minimizes the mean square error. The data is in numpy matrices, and for each fitting session I will have either 4, 9, 16, or 25 coordinates. Ultimately, I just need to get the center position of the Gaussian (x_0, y_0) having the smallest MSE. All the examples I found use scipy.optimize.curve_fit, but the input they have is a whole grid, not a few coordinates. Any help would be greatly appreciated.

Answer 1

Introduction

There are several ways to approach this. You can use non-linear methods (e.g. scipy.optimize.curve_fit), but they will be slow and not guaranteed by convergence. You can linearize the problem (a quick, unique solution), but any noise in the “tails” of the propagation will cause problems. There are actually a few tricks that you can apply to this particular case to avoid the last problem. I will give some examples, but I don’t have time to demonstrate all the “tricks” right now.

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import numpy as np
import matplotlib.pyplot as plt

def gauss2d(x, y, amp, x0, y0, a, b, c):
    inner = a * (x - x0)**2 
    inner += 2 * b * (x - x0)**2 * (y - y0)**2
    inner += c * (y - y0)**2
    return amp * np.exp(-inner)

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np.random.seed(1977) # For consistency
x, y = np.random.random((2, 10))
x0, y0 = 0.3, 0.7
amp, a, b, c = 1, 2, 3, 4

zobs = gauss2d(x, y, amp, x0, y0, a, b, c)

fig, ax = plt.subplots()
scat = ax.scatter(x, y, c=zobs, s=200)
fig.colorbar(scat)
plt.show()

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scipy.optimize , , . , gauss2d:

def gauss2d(xy, amp, x0, y0, a, b, c):
    x, y = xy
    inner = a * (x - x0)**2
    inner += 2 * b * (x - x0)**2 * (y - y0)**2
    inner += c * (y - y0)**2
    return amp * np.exp(-inner)

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import numpy as np
import scipy.optimize as opt
import matplotlib.pyplot as plt

def main():
    x0, y0 = 0.3, 0.7
    amp, a, b, c = 1, 2, 3, 4
    true_params = [amp, x0, y0, a, b, c]
    xy, zobs = generate_example_data(10, true_params)
    x, y = xy

    i = zobs.argmax()
    guess = [1, x[i], y[i], 1, 1, 1]
    pred_params, uncert_cov = opt.curve_fit(gauss2d, xy, zobs, p0=guess)

    zpred = gauss2d(xy, *pred_params)
    print 'True parameters: ', true_params
    print 'Predicted params:', pred_params
    print 'Residual, RMS(obs - pred):', np.sqrt(np.mean((zobs - zpred)**2))

    plot(xy, zobs, pred_params)
    plt.show()

def gauss2d(xy, amp, x0, y0, a, b, c):
    x, y = xy
    inner = a * (x - x0)**2
    inner += 2 * b * (x - x0)**2 * (y - y0)**2
    inner += c * (y - y0)**2
    return amp * np.exp(-inner)

def generate_example_data(num, params):
    np.random.seed(1977) # For consistency
    xy = np.random.random((2, num))

    zobs = gauss2d(xy, *params)
    return xy, zobs

def plot(xy, zobs, pred_params):
    x, y = xy
    yi, xi = np.mgrid[:1:30j, -.2:1.2:30j]
    xyi = np.vstack([xi.ravel(), yi.ravel()])

    zpred = gauss2d(xyi, *pred_params)
    zpred.shape = xi.shape

    fig, ax = plt.subplots()
    ax.scatter(x, y, c=zobs, s=200, vmin=zpred.min(), vmax=zpred.max())
    im = ax.imshow(zpred, extent=[xi.min(), xi.max(), yi.max(), yi.min()],
                   aspect='auto')
    fig.colorbar(im)
    ax.invert_yaxis()
    return fig

main()

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True parameters:  [1, 0.3, 0.7, 2, 3, 4]
Predicted params: [ 1.   0.3  0.7  2.   3.   4. ]
Residual, RMS(obs - pred): 1.01560615193e-16

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. , , generate_example_data:

def generate_example_data(num, params):
    np.random.seed(1977) # For consistency
    xy = np.random.random((2, num))

    noise = np.random.normal(0, 0.3, num)
    zobs = gauss2d(xy, *params) + noise
    return xy, zobs

-:

:

True parameters:  [1, 0.3, 0.7, 2, 3, 4]
Predicted params: [  1.129    0.263   0.750   1.280   32.333   10.103  ]
Residual, RMS(obs - pred): 0.152444640098

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x0, y0 = -0.3, 1.1

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True parameters:  [1, -0.3, 1.1, 2, 3, 4]
Predicted params: [  0.546  -0.939   0.857  -0.488  44.069  -4.136]
Residual, RMS(obs - pred): 0.235664449826

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