Viewing File: /home/ubuntu/combine_ai/combine/lib/python3.10/site-packages/skimage/measure/_moments_analytical.py
"""Analytical transformations from raw image moments to central moments.
The expressions for the 2D central moments of order <=2 are often given in
textbooks. Expressions for higher orders and dimensions were generated in SymPy
using ``tools/precompute/moments_sympy.py`` in the GitHub repository.
"""
import itertools
import math
import numpy as np
def _moments_raw_to_central_fast(moments_raw):
"""Analytical formulae for 2D and 3D central moments of order < 4.
`moments_raw_to_central` will automatically call this function when
ndim < 4 and order < 4.
Parameters
----------
moments_raw : ndarray
The raw moments.
Returns
-------
moments_central : ndarray
The central moments.
"""
ndim = moments_raw.ndim
order = moments_raw.shape[0] - 1
float_dtype = moments_raw.dtype
# convert to float64 during the computation for better accuracy
moments_raw = moments_raw.astype(np.float64, copy=False)
moments_central = np.zeros_like(moments_raw)
if order >= 4 or ndim not in [2, 3]:
raise ValueError(
"This function only supports 2D or 3D moments of order < 4."
)
m = moments_raw
if ndim == 2:
cx = m[1, 0] / m[0, 0]
cy = m[0, 1] / m[0, 0]
moments_central[0, 0] = m[0, 0]
# Note: 1st order moments are both 0
if order > 1:
# 2nd order moments
moments_central[1, 1] = m[1, 1] - cx*m[0, 1]
moments_central[2, 0] = m[2, 0] - cx*m[1, 0]
moments_central[0, 2] = m[0, 2] - cy*m[0, 1]
if order > 2:
# 3rd order moments
moments_central[2, 1] = (m[2, 1] - 2*cx*m[1, 1] - cy*m[2, 0]
+ cx**2*m[0, 1] + cy*cx*m[1, 0])
moments_central[1, 2] = (m[1, 2] - 2*cy*m[1, 1] - cx*m[0, 2]
+ 2*cy*cx*m[0, 1])
moments_central[3, 0] = m[3, 0] - 3*cx*m[2, 0] + 2*cx**2*m[1, 0]
moments_central[0, 3] = m[0, 3] - 3*cy*m[0, 2] + 2*cy**2*m[0, 1]
else:
# 3D case
cx = m[1, 0, 0] / m[0, 0, 0]
cy = m[0, 1, 0] / m[0, 0, 0]
cz = m[0, 0, 1] / m[0, 0, 0]
moments_central[0, 0, 0] = m[0, 0, 0]
# Note: all first order moments are 0
if order > 1:
# 2nd order moments
moments_central[0, 0, 2] = -cz*m[0, 0, 1] + m[0, 0, 2]
moments_central[0, 1, 1] = -cy*m[0, 0, 1] + m[0, 1, 1]
moments_central[0, 2, 0] = -cy*m[0, 1, 0] + m[0, 2, 0]
moments_central[1, 0, 1] = -cx*m[0, 0, 1] + m[1, 0, 1]
moments_central[1, 1, 0] = -cx*m[0, 1, 0] + m[1, 1, 0]
moments_central[2, 0, 0] = -cx*m[1, 0, 0] + m[2, 0, 0]
if order > 2:
# 3rd order moments
moments_central[0, 0, 3] = (2*cz**2*m[0, 0, 1]
- 3*cz*m[0, 0, 2]
+ m[0, 0, 3])
moments_central[0, 1, 2] = (-cy*m[0, 0, 2]
+ 2*cz*(cy*m[0, 0, 1] - m[0, 1, 1])
+ m[0, 1, 2])
moments_central[0, 2, 1] = (cy**2*m[0, 0, 1] - 2*cy*m[0, 1, 1]
+ cz*(cy*m[0, 1, 0] - m[0, 2, 0])
+ m[0, 2, 1])
moments_central[0, 3, 0] = (2*cy**2*m[0, 1, 0]
- 3*cy*m[0, 2, 0]
+ m[0, 3, 0])
moments_central[1, 0, 2] = (-cx*m[0, 0, 2]
+ 2*cz*(cx*m[0, 0, 1] - m[1, 0, 1])
+ m[1, 0, 2])
moments_central[1, 1, 1] = (-cx*m[0, 1, 1]
+ cy*(cx*m[0, 0, 1] - m[1, 0, 1])
+ cz*(cx*m[0, 1, 0] - m[1, 1, 0])
+ m[1, 1, 1])
moments_central[1, 2, 0] = (-cx*m[0, 2, 0]
- 2*cy*(-cx*m[0, 1, 0] + m[1, 1, 0])
+ m[1, 2, 0])
moments_central[2, 0, 1] = (cx**2*m[0, 0, 1]
- 2*cx*m[1, 0, 1]
+ cz*(cx*m[1, 0, 0] - m[2, 0, 0])
+ m[2, 0, 1])
moments_central[2, 1, 0] = (cx**2*m[0, 1, 0]
- 2*cx*m[1, 1, 0]
+ cy*(cx*m[1, 0, 0] - m[2, 0, 0])
+ m[2, 1, 0])
moments_central[3, 0, 0] = (2*cx**2*m[1, 0, 0]
- 3*cx*m[2, 0, 0]
+ m[3, 0, 0])
return moments_central.astype(float_dtype, copy=False)
def moments_raw_to_central(moments_raw):
ndim = moments_raw.ndim
order = moments_raw.shape[0] - 1
if ndim in [2, 3] and order < 4:
return _moments_raw_to_central_fast(moments_raw)
moments_central = np.zeros_like(moments_raw)
m = moments_raw
# centers as computed in centroid above
centers = tuple(m[tuple(np.eye(ndim, dtype=int))] / m[(0,)*ndim])
if ndim == 2:
# This is the general 2D formula from
# https://en.wikipedia.org/wiki/Image_moment#Central_moments
for p in range(order + 1):
for q in range(order + 1):
if p + q > order:
continue
for i in range(p + 1):
term1 = math.comb(p, i)
term1 *= (-centers[0]) ** (p - i)
for j in range(q + 1):
term2 = math.comb(q, j)
term2 *= (-centers[1]) ** (q - j)
moments_central[p, q] += term1*term2*m[i, j]
return moments_central
# The nested loops below are an n-dimensional extension of the 2D formula
# given at https://en.wikipedia.org/wiki/Image_moment#Central_moments
# iterate over all [0, order] (inclusive) on each axis
for orders in itertools.product(*((range(order + 1),) * ndim)):
# `orders` here is the index into the `moments_central` output array
if sum(orders) > order:
# skip any moment that is higher than the requested order
continue
# loop over terms from `m` contributing to `moments_central[orders]`
for idxs in itertools.product(*[range(o + 1) for o in orders]):
val = m[idxs]
for i_order, c, idx in zip(orders, centers, idxs):
val *= math.comb(i_order, idx)
val *= (-c) ** (i_order - idx)
moments_central[orders] += val
return moments_central
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