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import numpy as np
import scipy.sparse
from numba import njit
from ....utils.utils import get_h_w_c
@njit(
"void(f8[:, :, :], f8, i8, f8[:, :, :], i8[:], i8[:], b1[:, :])",
cache=True,
nogil=True,
)
def _cf_laplacian(image, epsilon, r, values, indices, indptr, is_known):
h, w, d = image.shape
assert d == 3
size = 2 * r + 1
window_area = size * size
for yi in range(h):
for xi in range(w):
i = xi + yi * w
k = i * (4 * r + 1) ** 2
for yj in range(yi - 2 * r, yi + 2 * r + 1):
for xj in range(xi - 2 * r, xi + 2 * r + 1):
j = xj + yj * w
if 0 <= xj < w and 0 <= yj < h:
indices[k] = j
k += 1
indptr[i + 1] = k
# Centered and normalized window colors
c = np.zeros((2 * r + 1, 2 * r + 1, 3))
# For each pixel of image
for y in range(r, h - r):
for x in range(r, w - r):
if np.all(is_known[y - r : y + r + 1, x - r : x + r + 1]):
continue
# For each color channel
for dc in range(3):
# Calculate sum of color channel in window
s = 0.0
for dy in range(size):
for dx in range(size):
s += image[y + dy - r, x + dx - r, dc]
# Calculate centered window color
for dy in range(2 * r + 1):
for dx in range(2 * r + 1):
c[dy, dx, dc] = (
image[y + dy - r, x + dx - r, dc] - s / window_area
)
# Calculate covariance matrix over color channels with epsilon regularization
a00 = epsilon
a01 = 0.0
a02 = 0.0
a11 = epsilon
a12 = 0.0
a22 = epsilon
for dy in range(size):
for dx in range(size):
a00 += c[dy, dx, 0] * c[dy, dx, 0]
a01 += c[dy, dx, 0] * c[dy, dx, 1]
a02 += c[dy, dx, 0] * c[dy, dx, 2]
a11 += c[dy, dx, 1] * c[dy, dx, 1]
a12 += c[dy, dx, 1] * c[dy, dx, 2]
a22 += c[dy, dx, 2] * c[dy, dx, 2]
a00 /= window_area
a01 /= window_area
a02 /= window_area
a11 /= window_area
a12 /= window_area
a22 /= window_area
det = (
a00 * a12 * a12
+ a01 * a01 * a22
+ a02 * a02 * a11
- a00 * a11 * a22
- 2 * a01 * a02 * a12
)
inv_det = 1.0 / det
# Calculate inverse covariance matrix
m00 = (a12 * a12 - a11 * a22) * inv_det
m01 = (a01 * a22 - a02 * a12) * inv_det
m02 = (a02 * a11 - a01 * a12) * inv_det
m11 = (a02 * a02 - a00 * a22) * inv_det
m12 = (a00 * a12 - a01 * a02) * inv_det
m22 = (a01 * a01 - a00 * a11) * inv_det
# For each pair ((xi, yi), (xj, yj)) in a (2 r + 1)x(2 r + 1) window
for dyi in range(2 * r + 1):
for dxi in range(2 * r + 1):
s = c[dyi, dxi, 0]
t = c[dyi, dxi, 1]
u = c[dyi, dxi, 2]
c0 = m00 * s + m01 * t + m02 * u
c1 = m01 * s + m11 * t + m12 * u
c2 = m02 * s + m12 * t + m22 * u
for dyj in range(2 * r + 1):
for dxj in range(2 * r + 1):
xi = x + dxi - r
yi = y + dyi - r
xj = x + dxj - r
yj = y + dyj - r
i = xi + yi * w
j = xj + yj * w
# Calculate contribution of pixel pair to L_ij
temp = (
c0 * c[dyj, dxj, 0]
+ c1 * c[dyj, dxj, 1]
+ c2 * c[dyj, dxj, 2]
)
value = (1.0 if (i == j) else 0.0) - (
1 + temp
) / window_area
dx = xj - xi + 2 * r
dy = yj - yi + 2 * r
values[i, dy, dx] += value
def cf_laplacian(image, epsilon=1e-7, radius=1, is_known=None):
"""
This function implements the alpha estimator for closed-form alpha matting as proposed by :cite:`levin2007closed`.
Parameters
------------
image: numpy.ndarray
Image with shape :math:`h\\times w \\times 3`
epsilon: float
Regularization strength, defaults to :math:`10^{-7}`. Strong regularization improves convergence but results in smoother alpha mattes.
radius: int
Radius of local window size, defaults to :math:`1`, i.e. only adjacent pixels are considered.
The size of the local window is given as :math:`(2 r + 1)^2`, where :math:`r` denotes the radius. A larger radius might lead to violated color line constraints, but also
favors further propagation of information within the image.
is_known: numpy.ndarray
Binary mask of pixels for which to compute the laplacian matrix.
Laplacian entries for known pixels will have undefined values.
Returns
-------
L: scipy.sparse.spmatrix
Matting Laplacian
"""
h, w, _ = get_h_w_c(image)
n = h * w
if is_known is None:
is_known = np.zeros((h, w), dtype=np.bool_)
is_known = is_known.reshape(h, w)
# Data for matting laplacian in csr format
indptr = np.zeros(n + 1, dtype=np.int64)
indices = np.zeros(n * (4 * radius + 1) ** 2, dtype=np.int64)
values = np.zeros((n, 4 * radius + 1, 4 * radius + 1), dtype=np.float64)
_cf_laplacian(image, epsilon, radius, values, indices, indptr, is_known)
L = scipy.sparse.csr_matrix((values.ravel(), indices, indptr), (n, n))
return L
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