Test diffusion¶

This is an example using scopyon (https://scopyon.readthedocs.io/).

[1]:

import scopyon

Set the configuration first.

[2]:

config = scopyon.DefaultConfiguration()
config.default.magnification = 360.0
config.default.detector.exposure_time = 33.0e-3  # second

[3]:

# config.environ.processes = 20

A field of microscopic view could be calculated as follows:

[4]:

pixel_length = config.default.detector.pixel_length / config.default.magnification
L_2 = config.default.detector.image_size[0] * pixel_length * 0.5

Randomly generate positions of 1000 molecules on membrane. The diffusion rate D is 0.1e-12.

[5]:

import numpy
rng = numpy.random.RandomState(1)
num_frames = 50
duration = config.default.detector.exposure_time * 5
t = numpy.linspace(0, duration, num_frames + 1)

[6]:

N = 1000
D = 0.1e-12  # m ** 2 / s
inputs = scopyon.sample_inputs(t, N=N, lower=-L_2, upper=+L_2, ndim=2, D=D, rng=rng)

scopyon.form_image generates a single image from the given inputs.

[7]:

# import logging
# logging.basicConfig(level=logging.INFO)

[8]:

img1 = scopyon.form_image(inputs, config=config, rng=rng)

[9]:

spots1 = scopyon.analysis.spot_detection(
    img1.as_array(), min_sigma=1, max_sigma=4, threshold=40.0, overlap=0.5)

/home/kaizu/environ/anaconda3/lib/python3.7/site-packages/skimage/feature/blob.py:125: RuntimeWarning: invalid value encountered in double_scalars
  r1 = blob1[-1] / blob2[-1]
/home/kaizu/environ/anaconda3/lib/python3.7/site-packages/skimage/feature/blob.py:126: RuntimeWarning: divide by zero encountered in true_divide
  pos1 = blob1[:ndim] / (max_sigma * root_ndim)
/home/kaizu/environ/anaconda3/lib/python3.7/site-packages/skimage/feature/blob.py:127: RuntimeWarning: divide by zero encountered in true_divide
  pos2 = blob2[:ndim] / (max_sigma * root_ndim)
/home/kaizu/environ/anaconda3/lib/python3.7/site-packages/skimage/feature/blob.py:129: RuntimeWarning: invalid value encountered in subtract
  d = np.sqrt(np.sum((pos2 - pos1)**2))
/home/kaizu/environ/anaconda3/lib/python3.7/site-packages/skimage/feature/blob.py:126: RuntimeWarning: invalid value encountered in true_divide
  pos1 = blob1[:ndim] / (max_sigma * root_ndim)
/home/kaizu/environ/anaconda3/lib/python3.7/site-packages/skimage/feature/blob.py:127: RuntimeWarning: invalid value encountered in true_divide
  pos2 = blob2[:ndim] / (max_sigma * root_ndim)
/home/kaizu/.local/share/virtualenvs/latest-mxc9Lo-B/lib/python3.7/site-packages/scopyon-1.0.0a4-py3.7.egg/scopyon/analysis/spot_detection.py:81: RuntimeWarning: overflow encountered in exp
/home/kaizu/.local/share/virtualenvs/latest-mxc9Lo-B/lib/python3.7/site-packages/scopyon-1.0.0a4-py3.7.egg/scopyon/analysis/spot_detection.py:81: RuntimeWarning: overflow encountered in multiply

[10]:

r = 6
shapes = [dict(x=spot[0], y=spot[1], sigma=r, color='red')
        for spot in spots1]
img1.show(shapes=shapes)

Take another image at time ‘t=0.1’.

[11]:

img2 = scopyon.form_image(inputs, start_time=100e-3, config=config, rng=rng)

[12]:

spots2 = scopyon.analysis.spot_detection(
    img2.as_array(), min_sigma=1, max_sigma=4, threshold=40.0, overlap=0.5)

/home/kaizu/environ/anaconda3/lib/python3.7/site-packages/skimage/feature/blob.py:125: RuntimeWarning:

invalid value encountered in double_scalars

/home/kaizu/environ/anaconda3/lib/python3.7/site-packages/skimage/feature/blob.py:126: RuntimeWarning:

divide by zero encountered in true_divide

/home/kaizu/environ/anaconda3/lib/python3.7/site-packages/skimage/feature/blob.py:127: RuntimeWarning:

divide by zero encountered in true_divide

/home/kaizu/environ/anaconda3/lib/python3.7/site-packages/skimage/feature/blob.py:129: RuntimeWarning:

invalid value encountered in subtract

/home/kaizu/.local/share/virtualenvs/latest-mxc9Lo-B/lib/python3.7/site-packages/scopyon-1.0.0a4-py3.7.egg/scopyon/analysis/spot_detection.py:81: RuntimeWarning:

overflow encountered in exp

/home/kaizu/.local/share/virtualenvs/latest-mxc9Lo-B/lib/python3.7/site-packages/scopyon-1.0.0a4-py3.7.egg/scopyon/analysis/spot_detection.py:81: RuntimeWarning:

overflow encountered in multiply

Compare two images and choose the closest point for each points.

[13]:

closest = []
for spot in spots1:
    distance = spots2[:, : 2] - spot[: 2]
    idx = (distance ** 2).sum(axis=1).argmin()
    closest.extend(distance[idx])
closest = numpy.array(closest)

[14]:

closest

[14]:

array([ 0.81090703, -1.48551972, -0.58125853, ..., -3.13397873,
       14.1496484 , 13.72563499])

A diffusion rate is estimated based on the average of closest distances (displacements). The units below is a square of pixels per second.

[15]:

D_ = numpy.average(closest ** 2) / (2 * duration)
D_

[15]:

43.44057797332427

Rescaled into physical units (square meters per second).

[16]:

D_ * (pixel_length ** 2)

[16]:

8.580854908310966e-14

Plot data with probability densities.

[17]:

import plotly.graph_objects as go
from scipy.stats import norm

fig = go.Figure()

fig.add_trace(go.Histogram(x=closest, histnorm='probability density', name='Samples'))

x = numpy.linspace(closest.min(), closest.max(), 101)
y = norm.pdf(x, scale=numpy.sqrt(2 * D_ * duration))
fig.add_trace(go.Scatter(x=x, y=y, mode='lines', name='Estimated'))

y = norm.pdf(x, scale=numpy.sqrt(2 * (D / (pixel_length ** 2)) * duration))
fig.add_trace(go.Scatter(x=x, y=y, mode='lines', name='True'))

fig.show()