forked from 170010011/fr
129 lines
4.3 KiB
Python
129 lines
4.3 KiB
Python
import numpy as np
|
|
|
|
|
|
def integral_image(image):
|
|
r"""Integral image / summed area table.
|
|
|
|
The integral image contains the sum of all elements above and to the
|
|
left of it, i.e.:
|
|
|
|
.. math::
|
|
|
|
S[m, n] = \sum_{i \leq m} \sum_{j \leq n} X[i, j]
|
|
|
|
Parameters
|
|
----------
|
|
image : ndarray
|
|
Input image.
|
|
|
|
Returns
|
|
-------
|
|
S : ndarray
|
|
Integral image/summed area table of same shape as input image.
|
|
|
|
References
|
|
----------
|
|
.. [1] F.C. Crow, "Summed-area tables for texture mapping,"
|
|
ACM SIGGRAPH Computer Graphics, vol. 18, 1984, pp. 207-212.
|
|
|
|
"""
|
|
S = image
|
|
for i in range(image.ndim):
|
|
S = S.cumsum(axis=i)
|
|
return S
|
|
|
|
|
|
def integrate(ii, start, end):
|
|
"""Use an integral image to integrate over a given window.
|
|
|
|
Parameters
|
|
----------
|
|
ii : ndarray
|
|
Integral image.
|
|
start : List of tuples, each tuple of length equal to dimension of `ii`
|
|
Coordinates of top left corner of window(s).
|
|
Each tuple in the list contains the starting row, col, ... index
|
|
i.e `[(row_win1, col_win1, ...), (row_win2, col_win2,...), ...]`.
|
|
end : List of tuples, each tuple of length equal to dimension of `ii`
|
|
Coordinates of bottom right corner of window(s).
|
|
Each tuple in the list containing the end row, col, ... index i.e
|
|
`[(row_win1, col_win1, ...), (row_win2, col_win2, ...), ...]`.
|
|
|
|
Returns
|
|
-------
|
|
S : scalar or ndarray
|
|
Integral (sum) over the given window(s).
|
|
|
|
|
|
Examples
|
|
--------
|
|
>>> arr = np.ones((5, 6), dtype=float)
|
|
>>> ii = integral_image(arr)
|
|
>>> integrate(ii, (1, 0), (1, 2)) # sum from (1, 0) to (1, 2)
|
|
array([3.])
|
|
>>> integrate(ii, [(3, 3)], [(4, 5)]) # sum from (3, 3) to (4, 5)
|
|
array([6.])
|
|
>>> # sum from (1, 0) to (1, 2) and from (3, 3) to (4, 5)
|
|
>>> integrate(ii, [(1, 0), (3, 3)], [(1, 2), (4, 5)])
|
|
array([3., 6.])
|
|
"""
|
|
start = np.atleast_2d(np.array(start))
|
|
end = np.atleast_2d(np.array(end))
|
|
rows = start.shape[0]
|
|
|
|
total_shape = ii.shape
|
|
total_shape = np.tile(total_shape, [rows, 1])
|
|
|
|
# convert negative indices into equivalent positive indices
|
|
start_negatives = start < 0
|
|
end_negatives = end < 0
|
|
start = (start + total_shape) * start_negatives + \
|
|
start * ~(start_negatives)
|
|
end = (end + total_shape) * end_negatives + \
|
|
end * ~(end_negatives)
|
|
|
|
if np.any((end - start) < 0):
|
|
raise IndexError('end coordinates must be greater or equal to start')
|
|
|
|
# bit_perm is the total number of terms in the expression
|
|
# of S. For example, in the case of a 4x4 2D image
|
|
# sum of image from (1,1) to (2,2) is given by
|
|
# S = + ii[2, 2]
|
|
# - ii[0, 2] - ii[2, 0]
|
|
# + ii[0, 0]
|
|
# The total terms = 4 = 2 ** 2(dims)
|
|
|
|
S = np.zeros(rows)
|
|
bit_perm = 2 ** ii.ndim
|
|
width = len(bin(bit_perm - 1)[2:])
|
|
|
|
# Sum of a (hyper)cube, from an integral image is computed using
|
|
# values at the corners of the cube. The corners of cube are
|
|
# selected using binary numbers as described in the following example.
|
|
# In a 3D cube there are 8 corners. The corners are selected using
|
|
# binary numbers 000 to 111. Each number is called a permutation, where
|
|
# perm(000) means, select end corner where none of the coordinates
|
|
# is replaced, i.e ii[end_row, end_col, end_depth]. Similarly, perm(001)
|
|
# means replace last coordinate by start - 1, i.e
|
|
# ii[end_row, end_col, start_depth - 1], and so on.
|
|
# Sign of even permutations is positive, while those of odd is negative.
|
|
# If 'start_coord - 1' is -ve it is labeled bad and not considered in
|
|
# the final sum.
|
|
|
|
for i in range(bit_perm): # for all permutations
|
|
# boolean permutation array eg [True, False] for '10'
|
|
binary = bin(i)[2:].zfill(width)
|
|
bool_mask = [bit == '1' for bit in binary]
|
|
|
|
sign = (-1)**sum(bool_mask) # determine sign of permutation
|
|
|
|
bad = [np.any(((start[r] - 1) * bool_mask) < 0)
|
|
for r in range(rows)] # find out bad start rows
|
|
|
|
corner_points = (end * (np.invert(bool_mask))) + \
|
|
((start - 1) * bool_mask) # find corner for each row
|
|
|
|
S += [sign * ii[tuple(corner_points[r])] if(not bad[r]) else 0
|
|
for r in range(rows)] # add only good rows
|
|
return S
|