active_contour_model.py

import numpy as np
import scipy.linalg
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
        Initialisation coordinates of snake. For periodic snakes, it should
        not include duplicate endpoints.
    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 to
        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).

    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)

    Initiliaze 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)
    xsave = np.empty((convergence_order, len(x)))
    ysave = np.empty((convergence_order, len(x)))

    # Build snake shape matrix for Euler equation
    n = len(x)
    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 = scipy.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 = np.dot(inv, gamma*x + fx)
        yn = np.dot(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