# Source code for dclab.features.volume

```
"""Volume computation based on contour revolution"""
import numpy as np
[docs]def get_volume(cont, pos_x, pos_y, pix, fix_orientation=False):
"""Calculate the volume of a polygon revolved around an axis
The volume estimation assumes rotational symmetry.
Parameters
----------
cont: ndarray or list of ndarrays of shape (N,2)
A 2D array that holds the contour of an event [px]
e.g. obtained using `mm.contour` where `mm` is an instance
of `RTDCBase`. The first and second columns of `cont`
correspond to the x- and y-coordinates of the contour.
pos_x: float or ndarray of length N
The x coordinate(s) of the centroid of the event(s) [µm]
e.g. obtained using `mm.pos_x`
pos_y: float or ndarray of length N
The y coordinate(s) of the centroid of the event(s) [µm]
e.g. obtained using `mm.pos_y`
pix: float
The detector pixel size in µm.
e.g. obtained using: `mm.config["imaging"]["pixel size"]`
fix_orientation: bool
If set to True, make sure that the orientation of the
contour is counter-clockwise in the r-z plane
(see :func:`vol_revolve`). This is False by default, because
(1) Shape-In always stores the contours in the correct
orientation and (2) there may be events with high porosity
where "fixing" the orientation makes things worse and a
negative volume is returned.
Returns
-------
volume: float or ndarray
volume in um^3
Notes
-----
The computation of the volume is based on a full rotation of the
upper and the lower halves of the contour from which the
average is then used.
The volume is computed radially from the the center position
given by (`pos_x`, `pos_y`). For sufficiently smooth contours,
such as densely sampled ellipses, the center position does not
play an important role. For contours that are given on a coarse
grid, as is the case for RT-DC, the center position must be
given.
References
----------
- https://de.wikipedia.org/wiki/Kegelstumpf#Formeln
- Yields identical results to the Matlab script by Geoff Olynyk
<http://de.mathworks.com/matlabcentral/fileexchange/36525-volrevolve>`_
"""
if np.isscalar(pos_x):
cont = [cont]
ret_list = False
else:
ret_list = True
# Convert input to 1D arrays
pos_x = np.atleast_1d(pos_x)
pos_y = np.atleast_1d(pos_y)
if pos_x.size != pos_y.size:
raise ValueError("Size of `pos_x` and `pos_y` must match!")
if pos_x.size > 1 and len(cont) <= 1:
raise ValueError("Number of given contours too small!")
# results are stored in a separate array initialized with nans
v_avg = np.zeros_like(pos_x, dtype=float)*np.nan
# v_avg has the shape of `pos_x`. We are iterating over the smallest
# length for `cont` and `pos_x`.
for ii in range(min(len(cont), pos_x.shape[0])):
# If the contour has less than 4 pixels, the computation will fail.
# In that case, the value np.nan is already assigned.
cc = cont[ii]
if cc.shape[0] >= 4:
# Center contour coordinates with given centroid
contour_x = cc[:, 0] - pos_x[ii] / pix
contour_y = cc[:, 1] - pos_y[ii] / pix
# Switch to r and z to follow notation of vol_revolve
# (In RT-DC the axis of rotation is x, but for vol_revolve
# we need the axis vertically)
contour_r = contour_y
contour_z = contour_x
if fix_orientation:
# Make sure the contour is counter-clockwise
contour_r, contour_z = counter_clockwise(contour_r, contour_z)
# Compute right volume
# Which points are at negative r-values (r<0)?
inx_neg = np.where(contour_r < 0)
# These points will be shifted up to r=0 directly on the z-axis
contour_right = np.copy(contour_r)
contour_right[inx_neg] = 0
vol_right = vol_revolve(contour_right, contour_x, pix)
# Compute left volume
# Which points are at positive r-values? (r>0)?
idx_pos = np.where(contour_r > 0)
# These points will be shifted down to y=0 to build an x-axis
contour_left = np.copy(contour_r)
contour_left[idx_pos] = 0
# Now we still have negative r values, but vol_revolve needs
# positive values, so we flip the sign...
contour_left[:] *= -1
# ... but in doing so, we have switched to clockwise rotation
# and we need to pass the array in reverse order
vol_left = vol_revolve(contour_left[::-1], contour_x[::-1], pix)
# Compute the average
v_avg[ii] = (vol_right + vol_left) / 2
if not ret_list:
# Do not return a list if the input contour was not in a list
v_avg = v_avg[0]
return v_avg
[docs]def counter_clockwise(cx, cy):
"""Put contour coordinates into counter-clockwise order
Parameters
----------
cx, cy: 1d ndarrays
The x- and y-coordinates of the contour
Returns
-------
cx_cc, cy_cc:
The x- and y-coordinates of the contour in
counter-clockwise orientation.
Notes
-----
The contour must be centered around (0, 0).
"""
# test orientation
angles = np.unwrap(np.arctan2(cy, cx))
grad = np.diff(angles)
if np.average(grad) < 0:
return cx[::-1], cy[::-1]
else:
return cx, cy
[docs]def vol_revolve(r, z, point_scale=1.):
r"""Calculate the volume of a polygon revolved around the Z-axis
This implementation yields the same results as the volRevolve
Matlab function by Geoff Olynyk (from 2012-05-03)
https://de.mathworks.com/matlabcentral/fileexchange/36525-volrevolve.
The difference here is that the volume is computed using (a much
more approachable) implementation using the volume of a truncated
cone (https://de.wikipedia.org/wiki/Kegelstumpf).
.. math::
V = \frac{h \cdot \pi}{3} \cdot (R^2 + R \cdot r + r^2)
Where :math:`h` is the height of the cone and :math:`r` and
`R` are the smaller and larger radii of the truncated cone.
Each line segment of the contour resembles one truncated cone. If
the z-step is positive (counter-clockwise contour), then the
truncated cone volume is added to the total volume. If the z-step
is negative (e.g. inclusion), then the truncated cone volume is
removed from the total volume.
.. versionchanged:: 0.37.0
The volume in previous versions was overestimated by on average
2µm³.
Parameters
----------
r: 1d np.ndarray
radial coordinates (perpendicular to the z axis)
z: 1d np.ndarray
coordinate along the axis of rotation
point_scale: float
point size in your preferred units; The volume is multiplied
by a factor of `point_scale**3`.
Notes
-----
The coordinates must be given in counter-clockwise order,
otherwise the volume will be negative.
"""
r = np.array(r).flatten()
z = np.array(z).flatten()
# sanity checks
assert len(r) == len(z)
assert len(r) >= 3
assert len(r.shape) == len(z.shape) == 1
assert np.all(r >= 0)
# make sure we have a closed contour
if (r[-1] != r[0]) or (z[-1] != z[0]):
# We have an open contour - close it.
r = np.resize(r, len(r) + 1)
z = np.resize(z, len(z) + 1)
rp = r[:-1]
# array of radii differences: R - r
dr = np.diff(r)
# array of height differences: h
dz = np.diff(z)
# If we expand the function in the doc string with
# dr = R - r and dz = h, then we get three terms for the volume
# (as opposed to four terms in Olynyk's script). Those three terms
# all resemble area slices multiplied by the z-distance dz.
a1 = 3 * rp**2
a2 = 3 * rp*dr
a3 = dr**2
# Note that the formula for computing the volume is symmetric
# with respect to r and R. This means that it does not matter
# which sign dr has (R and r are always positive). Since this
# algorithm assumes that the contour is ordered counter-clockwise,
# positive dz means adding to the contour while negative dz means
# subtracting from the contour (see test functions for more details).
# Conveniently so, dz only appears one time in this formula, so
# we can take the sign of dz as it is (Otherwise, we would have
# to take the absolute value of every truncated cone volume and
# multiply it by np.sign(dz)).
v = np.pi / 3 * dz * np.abs(a1 + a2 + a3)
vol = np.sum(v) * point_scale ** 3
return vol
```