Source code for dclab.features.volume

# -*- coding: utf-8 -*-
"""Computation of volume for RT-DC measurements based on a rotation
of the contours"""
from __future__ import division, print_function, unicode_literals
import numpy as np

[docs]def get_volume(cont, pos_x, pos_y, pix): """Calculate the volume of a polygon revolved around an axis The volume estimation assumes rotational symmetry. Green`s theorem and the Gaussian divergence theorem allow to formulate the volume as a line integral. This is a translation from a Matlab script by Geoff Olynyk: 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` px_um: float The detector pixel size in µm. e.g. obtained using: `mm.config["image"]["pix size"]` Returns ------- volume: float or ndarray volume in um^3 Notes ----- The computation of the volume is based on a full rotation of the upper half of the contour to obtain the volume. Similarly, the lower part of the contour is rotated. Both volumes are then averaged. 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 ---------- Advanced Mathematics and Mechanics Applications with MATLAB 3rd ed. by H.B. Wilson, L.H. Turcotte, and D. Halpern, Chapman & Hall CRC Press, 2002, e-ISBN 978-1-4200-3544-5. See Chapter 5, Section 5.4, doi: 10.1201/9781420035445.ch5 """ 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 # Make sure contour is counter-clockwise contour_x, contour_y = counter_clockwise(contour_x, contour_y) # Which points are below the x-axis? (y<0)? ind_low = np.where(contour_y < 0) # These points will be shifted up to y=0 to build an x-axis # (wont contribute to lower volume). contour_y_low = np.copy(contour_y) contour_y_low[ind_low] = 0 # Which points are above the x-axis? (y>0)? ind_upp = np.where(contour_y > 0) # These points will be shifted down to y=0 to build an x-axis # (wont contribute to upper volume). contour_y_upp = np.copy(contour_y) contour_y_upp[ind_upp] = 0 # Move the contour to the left Z = contour_x # Last point of the contour has to overlap with the first point Z = np.hstack([Z, Z[0]]) Zp = Z[0:-1] dZ = Z[1:]-Zp # Last point of the contour has to overlap with the first point contour_y_low = np.hstack([contour_y_low, contour_y_low[0]]) contour_y_upp = np.hstack([contour_y_upp, contour_y_upp[0]]) vol_low = _vol_helper(contour_y_low, Z, Zp, dZ, pix) vol_upp = _vol_helper(contour_y_upp, Z, Zp, dZ, pix) v_avg[ii] = (vol_low + vol_upp) / 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
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. """ # test orientation angles = np.unwrap(np.arctan2(cy, cx)) grad = np.gradient(angles) if np.average(grad) > 0: return cx[::-1], cy[::-1] else: return cx, cy def _vol_helper(contour_y, Z, Zp, dZ, pix): # Instead of x and y, describe the contour by a Radius vector R and y # The Contour will be rotated around the x-axis. Therefore it is # Important that the Contour has been shifted onto the x-Axis R = np.sqrt(Z**2 + contour_y**2) Rp = R[0:-1] dR = R[1:] - Rp # 4 volume parts v1 = dR * dZ * Rp v2 = 2 * dZ * Rp**2 v3 = -1 * dR**2 * dZ v4 = -2 * dR * Rp * Zp V = (np.pi/3) * (v1 + v2 + v3 + v4) vol = np.sum(V) * pix**3 return abs(vol)