Compute eigenvectors of image in python
Just a quick note, there are several tools to fit a gaussian to an image. The only thing I can think of off the top of my head is scikits.learn, which isn't completely image-oriented, but I know there are others.
To calculate the eigenvectors of the covariance matrix exactly as you had in mind is very computationally expensive. You have to associate each pixel (or a large-ish random sample) of image with an x,y point.
Basically, you do something like:
import numpy as np # grid is your image data, here... grid = np.random.random((10,10)) nrows, ncols = grid.shape i,j = np.mgrid[:nrows, :ncols] coords = np.vstack((i.reshape(-1), j.reshape(-1), grid.reshape(-1))).T cov = np.cov(coords) eigvals, eigvecs = np.linalg.eigh(cov)
You can instead make use of the fact that it's a regularly-sampled image and compute it's moments (or "intertial axes") instead. This will be considerably faster for large images.
As a quick example, (I'm using a part of one of my previous answers, in case you find it useful...)
import numpy as npimport matplotlib.pyplot as pltdef main(): data = generate_data() xbar, ybar, cov = intertial_axis(data) fig, ax = plt.subplots() ax.imshow(data) plot_bars(xbar, ybar, cov, ax) plt.show()def generate_data(): data = np.zeros((200, 200), dtype=np.float) cov = np.array([[200, 100], [100, 200]]) ij = np.random.multivariate_normal((100,100), cov, int(1e5)) for i,j in ij: data[int(i), int(j)] += 1 return data def raw_moment(data, iord, jord): nrows, ncols = data.shape y, x = np.mgrid[:nrows, :ncols] data = data * x**iord * y**jord return data.sum()def intertial_axis(data): """Calculate the x-mean, y-mean, and cov matrix of an image.""" data_sum = data.sum() m10 = raw_moment(data, 1, 0) m01 = raw_moment(data, 0, 1) x_bar = m10 / data_sum y_bar = m01 / data_sum u11 = (raw_moment(data, 1, 1) - x_bar * m01) / data_sum u20 = (raw_moment(data, 2, 0) - x_bar * m10) / data_sum u02 = (raw_moment(data, 0, 2) - y_bar * m01) / data_sum cov = np.array([[u20, u11], [u11, u02]]) return x_bar, y_bar, covdef plot_bars(x_bar, y_bar, cov, ax): """Plot bars with a length of 2 stddev along the principal axes.""" def make_lines(eigvals, eigvecs, mean, i): """Make lines a length of 2 stddev.""" std = np.sqrt(eigvals[i]) vec = 2 * std * eigvecs[:,i] / np.hypot(*eigvecs[:,i]) x, y = np.vstack((mean-vec, mean, mean+vec)).T return x, y mean = np.array([x_bar, y_bar]) eigvals, eigvecs = np.linalg.eigh(cov) ax.plot(*make_lines(eigvals, eigvecs, mean, 0), marker='o', color='white') ax.plot(*make_lines(eigvals, eigvecs, mean, -1), marker='o', color='red') ax.axis('image')if __name__ == '__main__': main()
Fitting a Gaussian robustly can be tricky. There was a fun article on this topic in the IEEE Signal Processing Magazine:
Hongwei Guo, "A Simple Algorithm for Fitting a Gaussian Function" IEEE Signal Processing Magazine, September 2011, pp. 134--137
I give an implementation of the 1D case here:
http://scipy-central.org/item/28/2/fitting-a-gaussian-to-noisy-data-points
(Scroll down to see the resulting fits)
Did you try Principal Component Analysis (PCA)? Maybe the MDP package could do the job with minimal effort.