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import numpy as np
from scipy.interpolate import RectBivariateSpline
from ..util import img_as_float
from ..filters import sobel
def active_contour(image, snake, alpha=0.01, beta=0.1,
w_line=0, w_edge=1, gamma=0.01,
bc='periodic', max_px_move=1.0,
max_iterations=2500, convergence=0.1):
"""Active contour model.
Active contours by fitting snakes to features of images. Supports single
and multichannel 2D images. Snakes can be periodic (for segmentation) or
have fixed and/or free ends.
The output snake has the same length as the input boundary.
As the number of points is constant, make sure that the initial snake
has enough points to capture the details of the final contour.
Parameters
----------
image : (N, M) or (N, M, 3) ndarray
Input image.
snake : (N, 2) ndarray
Initial snake coordinates. For periodic boundary conditions, endpoints
must not be duplicated.
alpha : float, optional
Snake length shape parameter. Higher values makes snake contract
faster.
beta : float, optional
Snake smoothness shape parameter. Higher values makes snake smoother.
w_line : float, optional
Controls attraction to brightness. Use negative values to attract toward
dark regions.
w_edge : float, optional
Controls attraction to edges. Use negative values to repel snake from
edges.
gamma : float, optional
Explicit time stepping parameter.
bc : {'periodic', 'free', 'fixed'}, optional
Boundary conditions for worm. 'periodic' attaches the two ends of the
snake, 'fixed' holds the end-points in place, and 'free' allows free
movement of the ends. 'fixed' and 'free' can be combined by parsing
'fixed-free', 'free-fixed'. Parsing 'fixed-fixed' or 'free-free'
yields same behaviour as 'fixed' and 'free', respectively.
max_px_move : float, optional
Maximum pixel distance to move per iteration.
max_iterations : int, optional
Maximum iterations to optimize snake shape.
convergence: float, optional
Convergence criteria.
Returns
-------
snake : (N, 2) ndarray
Optimised snake, same shape as input parameter.
References
----------
.. [1] Kass, M.; Witkin, A.; Terzopoulos, D. "Snakes: Active contour
models". International Journal of Computer Vision 1 (4): 321
(1988). DOI:`10.1007/BF00133570`
Examples
--------
>>> from skimage.draw import circle_perimeter
>>> from skimage.filters import gaussian
Create and smooth image:
>>> img = np.zeros((100, 100))
>>> rr, cc = circle_perimeter(35, 45, 25)
>>> img[rr, cc] = 1
>>> img = gaussian(img, 2)
Initialize spline:
>>> s = np.linspace(0, 2*np.pi, 100)
>>> init = 50 * np.array([np.cos(s), np.sin(s)]).T + 50
Fit spline to image:
>>> snake = active_contour(img, init, w_edge=0, w_line=1) #doctest: +SKIP
>>> dist = np.sqrt((45-snake[:, 0])**2 + (35-snake[:, 1])**2) #doctest: +SKIP
>>> int(np.mean(dist)) #doctest: +SKIP
25
"""
max_iterations = int(max_iterations)
if max_iterations <= 0:
raise ValueError("max_iterations should be >0.")
convergence_order = 10
valid_bcs = ['periodic', 'free', 'fixed', 'free-fixed',
'fixed-free', 'fixed-fixed', 'free-free']
if bc not in valid_bcs:
raise ValueError("Invalid boundary condition.\n" +
"Should be one of: "+", ".join(valid_bcs)+'.')
img = img_as_float(image)
RGB = img.ndim == 3
# Find edges using sobel:
if w_edge != 0:
if RGB:
edge = [sobel(img[:, :, 0]), sobel(img[:, :, 1]),
sobel(img[:, :, 2])]
else:
edge = [sobel(img)]
for i in range(3 if RGB else 1):
edge[i][0, :] = edge[i][1, :]
edge[i][-1, :] = edge[i][-2, :]
edge[i][:, 0] = edge[i][:, 1]
edge[i][:, -1] = edge[i][:, -2]
else:
edge = [0]
# Superimpose intensity and edge images:
if RGB:
img = w_line*np.sum(img, axis=2) \
+ w_edge*sum(edge)
else:
img = w_line*img + w_edge*edge[0]
# Interpolate for smoothness:
intp = RectBivariateSpline(np.arange(img.shape[1]),
np.arange(img.shape[0]),
img.T, kx=2, ky=2, s=0)
x, y = snake[:, 0].astype(np.float), snake[:, 1].astype(np.float)
n = len(x)
xsave = np.empty((convergence_order, n))
ysave = np.empty((convergence_order, n))
# Build snake shape matrix for Euler equation
a = np.roll(np.eye(n), -1, axis=0) + \
np.roll(np.eye(n), -1, axis=1) - \
2*np.eye(n) # second order derivative, central difference
b = np.roll(np.eye(n), -2, axis=0) + \
np.roll(np.eye(n), -2, axis=1) - \
4*np.roll(np.eye(n), -1, axis=0) - \
4*np.roll(np.eye(n), -1, axis=1) + \
6*np.eye(n) # fourth order derivative, central difference
A = -alpha*a + beta*b
# Impose boundary conditions different from periodic:
sfixed = False
if bc.startswith('fixed'):
A[0, :] = 0
A[1, :] = 0
A[1, :3] = [1, -2, 1]
sfixed = True
efixed = False
if bc.endswith('fixed'):
A[-1, :] = 0
A[-2, :] = 0
A[-2, -3:] = [1, -2, 1]
efixed = True
sfree = False
if bc.startswith('free'):
A[0, :] = 0
A[0, :3] = [1, -2, 1]
A[1, :] = 0
A[1, :4] = [-1, 3, -3, 1]
sfree = True
efree = False
if bc.endswith('free'):
A[-1, :] = 0
A[-1, -3:] = [1, -2, 1]
A[-2, :] = 0
A[-2, -4:] = [-1, 3, -3, 1]
efree = True
# Only one inversion is needed for implicit spline energy minimization:
inv = np.linalg.inv(A + gamma*np.eye(n))
# Explicit time stepping for image energy minimization:
for i in range(max_iterations):
fx = intp(x, y, dx=1, grid=False)
fy = intp(x, y, dy=1, grid=False)
if sfixed:
fx[0] = 0
fy[0] = 0
if efixed:
fx[-1] = 0
fy[-1] = 0
if sfree:
fx[0] *= 2
fy[0] *= 2
if efree:
fx[-1] *= 2
fy[-1] *= 2
xn = inv @ (gamma*x + fx)
yn = inv @ (gamma*y + fy)
# Movements are capped to max_px_move per iteration:
dx = max_px_move*np.tanh(xn-x)
dy = max_px_move*np.tanh(yn-y)
if sfixed:
dx[0] = 0
dy[0] = 0
if efixed:
dx[-1] = 0
dy[-1] = 0
x += dx
y += dy
# Convergence criteria needs to compare to a number of previous
# configurations since oscillations can occur.
j = i % (convergence_order+1)
if j < convergence_order:
xsave[j, :] = x
ysave[j, :] = y
else:
dist = np.min(np.max(np.abs(xsave-x[None, :]) +
np.abs(ysave-y[None, :]), 1))
if dist < convergence:
break
return np.array([x, y]).T