def sample_image_dribinski(n=361, origin=None):
"""
Sample test image used in the BASEX paper Rev. Sci. Instrum. 73, 2634 (2002)
9x Gaussian functions of half-width 4 pixel + 1 background width 3600
- anisotropy - ß = -1 for cosθ term
- ß = +2 for sinθ term
- ß = 0 isotropic, no angular variation
(there are some missing negative exponents in the publication)
Parameters
----------
- n: integer (square) image width (height)
- origin: tuple, (row, col) center of image
Returns
-------
- IM: 2D n x n numpy array image
"""
def Gauss(r, r0, sigma2):
return np.exp(-(r - r0)**2 / sigma2)
def I(r, theta): # intensity function Eq. (16)
t0 = 7 * Gauss(r, 10, 4) * np.sin(theta)**2
t1 = 3 * Gauss(r, 15, 4)
t2 = 5 * Gauss(r, 20, 4) * np.cos(theta)**2
t3 = Gauss(r, 70, 4)
t4 = 2 * Gauss(r, 85, 4) * np.cos(theta)**2
t5 = Gauss(r, 100, 4) * np.sin(theta)**2
t6 = 2 * Gauss(r, 145, 4) * np.sin(theta)**2
t7 = Gauss(r, 150, 4)
t8 = 3 * Gauss(r, 155, 4) * np.cos(theta)**2
t9 = 20 * Gauss(r, 45, 3600) # background under t3 to t5
return 2000 * (t0 + t1 + t2) + 200 * (t3 + t4 + t5) + 50 * (t6 + t7 +
t8) + t9
# meshgrid @DanHickstein issue #67
x = np.arange(n)
n2 = n // 2 + n % 2
X, Y = np.meshgrid(x - n2, x - n2)
R, THETA = cart2polar(X, Y)
IM = I(R, THETA)
return IM
def sample_image_dribinski(n=361, origin=None):
"""
Sample test image used in the BASEX paper Rev. Sci. Instrum. 73, 2634 (2002)
9x Gaussian functions of half-width 4 pixel + 1 background width 3600
- anisotropy - ß = -1 for cosθ term
- ß = +2 for sinθ term
- ß = 0 isotropic, no angular variation
(there are some missing negative exponents in the publication)
Parameters
----------
- n: integer (square) image width (height)
- origin: tuple, (row, col) center of image
Returns
-------
- IM: 2D n x n numpy array image
"""
def Gauss (r, r0, sigma2):
return np.exp(-(r-r0)**2/sigma2)
def I(r, theta): # intensity function Eq. (16)
t0 = 7*Gauss(r, 10, 4)*np.sin(theta)**2
t1 = 3*Gauss(r, 15, 4)
t2 = 5*Gauss(r, 20, 4)*np.cos(theta)**2
t3 = Gauss(r, 70, 4)
t4 = 2*Gauss(r, 85, 4)*np.cos(theta)**2
t5 = Gauss(r, 100, 4)*np.sin(theta)**2
t6 = 2*Gauss(r, 145, 4)*np.sin(theta)**2
t7 = Gauss(r, 150, 4)
t8 = 3*Gauss(r, 155, 4)*np.cos(theta)**2
t9 = 20*Gauss(r, 45, 3600) # background under t3 to t5
return 2000*(t0+t1+t2) + 200*(t3+t4+t5) + 50*(t6+t7+t8) + t9
# meshgrid @DanHickstein issue #67
x = np.arange(n)
n2 = n//2 + n%2
X, Y = np.meshgrid(x-n2, x-n2)
R, THETA = cart2polar(X, Y)
IM = I(R, THETA)
return IM
def sample_image(n=361, name="dribinski", sigma=2, temperature=200):
"""
Sample images, made up of Gaussian functions
Parameters
----------
n : integer
image size n rows x n cols
name : str
one of "dribinski" or "Ominus"
sigma : float
Gaussian 1/e width (pixels)
temperature : float
for 'Ominus' only
anion levels have Boltzmann population weight
(2J+1) exp(-177.1 h c 100/k/temperature)
Returns
-------
IM : 2D np.array
image
"""
def gauss(r, r0, sigma):
return np.exp(-(r-r0)**2/sigma**2)
def dribinski(r, theta, sigma): # intensity function Eq. (16)
"""
Sample test image used in the BASEX paper
Rev. Sci. Instrum. 73, 2634 (2002)
9x Gaussian functions of half-width 4 pixel + 1 background width 3600
anisotropy - ß = -1 for cosθ term
- ß = +2 for sinθ term
- ß = 0 isotropic, no angular variation
(there are some missing negative exponents in the publication)
"""
sinetheta2 = np.sin(theta)**2
cosinetheta2 = np.cos(theta)**2
t0 = 7*gauss(r, 10, sigma)*sinetheta2
t1 = 3*gauss(r, 15, sigma)
t2 = 5*gauss(r, 20, sigma)*cosinetheta2
t3 = gauss(r, 70, sigma)
t4 = 2*gauss(r, 85, sigma)*cosinetheta2
t5 = gauss(r, 100, sigma)*sinetheta2
t6 = 2*gauss(r, 145, sigma)*sinetheta2
t7 = gauss(r, 150, sigma)
t8 = 3*gauss(r, 155, sigma)*cosinetheta2
t9 = 20*gauss(r, 45, 3600) # background under t3 to t5
return 2000*(t0+t1+t2) + 200*(t3+t4+t5) + 50*(t6+t7+t8) + t9
def Ominus(r, theta, sigma, boltzmann, rfact=1):
"""
Simulate the photoelectron spectrum of O- photodetachment
3PJ <- 2P3/2,1/2
6 transitions, triplet neutral, and doublet anion
"""
# positions based on 812.5 nm ANU O- PES
t1 = gauss(r, 341*rfact, sigma) # 3P2 <- 2P3/2
t2 = gauss(r, 285*rfact, sigma) # 3P1 <- 2P3/2
t3 = gauss(r, 257*rfact, sigma) # 3P0 <- 2P3/2
t4 = gauss(r, 394*rfact, sigma) # 3P2 <- 2P1/2
t5 = gauss(r, 348*rfact, sigma) # 3P1 <- 2P1/2
t6 = gauss(r, 324*rfact, sigma) # 3P0 <- 2P1/2
# intensities = fine-structure known ratios
# 2P1/2 transtions scaled by temperature dependent Boltzmann factor
t = t1 + 0.8*t2 + 0.36*t3 + (0.2*t4 + 0.36*t5 + 0.16*t6)*boltzmann
# anisotropy
sinetheta2 = np.sin(theta)**2*0.2 + 0.8
return t*sinetheta2
n2 = n//2
x = np.linspace(-n2, n2, n)
if name=='dribinski':
x = x * 180/n2
elif name=='Ominus':
x = x * 501/n2
X, Y = np.meshgrid(x, x)
R, THETA = cart2polar(X, Y)
if name == "dribinski":
IM = dribinski(R, THETA, sigma=sigma)
elif name == "Ominus":
boltzmann = 0.5*np.exp(-177.1*const.h*const.c*100/const.k/temperature)
IM = Ominus(R, THETA, sigma=sigma, boltzmann=boltzmann)
else:
raise ValueError('sample image name not recognized')
return IM
def reproject_image_into_polar(data,
origin=None,
Jacobian=False,
dr=1,
dt=None,
log=False):
"""
Reprojects a 2D numpy array (``data``) into a polar coordinate system.
"origin" is a tuple of (x0, y0) relative to the bottom-left image corner,
and defaults to the center of the image.
Parameters
----------
data : 2D np.array
origin : tuple
The coordinate of the image center, relative to bottom-left
Jacobian : boolean
Include ``r`` intensity scaling in the coordinate transform.
This should be included to account for the changing pixel size that
occurs during the transform.
dr : float
Radial coordinate spacing for the grid interpolation
tests show that there is not much point in going below 0.5
dt : float
Angular coordinate spacing (in radians)
if ``dt=None``, dt will be set such that the number of theta values
is equal to the maximum value between the height or the width of
the image.
Returns
-------
output : 2D np.array
The polar image (r, theta)
r_grid : 2D np.array
meshgrid of radial coordinates
theta_grid : 2D np.array
meshgrid of theta coordinates
Notes
-----
Adapted from:
http://stackoverflow.com/questions/3798333/image-information-along-a-polar-coordinate-system
"""
# bottom-left coordinate system requires numpy image to be np.flipud
data = np.flipud(data)
ny, nx = data.shape[:2]
if origin is None:
origin = (nx // 2, ny // 2)
# Determine that the min and max r and theta coords will be...
x, y = index_coords(data, origin=origin) # (x,y) coordinates of each pixel
r, theta = cart2polar(x, y) # convert (x,y) -> (r,θ), note θ=0 is vertical
nr = np.int(np.ceil((r.max() - r.min()) / dr))
if dt is None:
nt = max(nx, ny)
else:
# dt in radians
nt = np.int(np.ceil((theta.max() - theta.min()) / dt))
# Make a regular (in polar space) grid based on the min and max r & theta
if log:
r_i = np.logspace(np.log10(r.min() + 1),
np.log10(r.max()),
nr,
endpoint=False)
else:
r_i = np.linspace(r.min(), r.max(), nr, endpoint=False)
theta_i = np.linspace(theta.min(), theta.max(), nt, endpoint=False)
theta_grid, r_grid = np.meshgrid(theta_i, r_i)
# Project the r and theta grid back into pixel coordinates
X, Y = polar2cart(r_grid, theta_grid)
X += origin[0] # We need to shift the origin
Y += origin[1] # back to the bottom-left corner...
xi, yi = X.flatten(), Y.flatten()
coords = np.vstack((yi, xi)) # (map_coordinates requires a 2xn array)
zi = map_coordinates(data, coords)
output = zi.reshape((nr, nt))
if Jacobian:
output = output * r_i[:, np.newaxis]
return output, r_grid, theta_grid