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