I

have an image where I am trying to calculate the line integral (sum) along the circular path. My thoughts are:

1. Calculates the loop path sum
2. The image is masked according to the path, and everything is zeroed out except for the pixels that coincide with the path.
3. Sum all image pixels

I’m currently stuck between step one and step two, not knowing how to generate a circle on the same grid as the image.

In the code:

from scipy.stats import multivariate_normal

radius = 2

# Draw arbitrary image 
x, y = np.mgrid[-5:5:.1, -5:5:.1]
img = multivariate_normal.pdf(np.dstack((x, y)), cov=[[1, 0.7], [0.7, 1]])

# generate circle with desired radius 
circle = radius*np.exp(1j*np.linspace(-np.pi, np.pi, 100))

pyplot.pcolormesh(x, y, img)
pyplot.plot(np.real(circle), np.imag(circle), '-w')
pyplot.show()

Give:

Question:

How do I use a circle to mask image pixels that coincide with that circle?

This is another way to calculate integrals: it uses interpolation, makes the image a function defined on a rectangle, and then calculates the path integral using the standard integral solver.

from scipy.integrate import quad
from scipy.interpolate import RectBivariateSpline
from scipy.stats import multivariate_normal
import numpy as np

x, y = np.ogrid[-5:5:.1, -5:5:.1]
img = multivariate_normal.pdf(np.dstack(np.broadcast_arrays(x, y)),
                              cov=[[1, 0.7], [0.7, 1]])

f = RectBivariateSpline(x.ravel(), y.ravel(), img)

radius, centerx, centery = 3.0, 1.0, -1.5
def integrand(rad):
    return f(centerx+radius*np.cos(rad), centery+radius*np.sin(rad))

def true_integrand(rad):
    return multivariate_normal(cov=[[1, 0.7], [0.7, 1]]).pdf(
        (centerx+radius*np.cos(rad), centery+radius*np.sin(rad)))

print(quad(integrand, -np.pi, np.pi))
print(quad(true_integrand, -np.pi, np.pi))

Output:

(0.07985467350026378, 1.3411796499850778e-08)
(0.07985453947958436, 4.006916325573184e-11)