random-walker

random_walker

skimage.segmentation.random_walker(data, labels, beta=130, mode='bf', tol=0.001, copy=True, multichannel=False, return_full_prob=False, spacing=None) [source]

Random walker algorithm for segmentation from markers.

Random walker algorithm is implemented for gray-level or multichannel images.

Parameters:

data : array_like Image to be segmented in phases. Gray-level data can be two- or three-dimensional; multichannel data can be three- or four- dimensional (multichannel=True) with the highest dimension denoting channels. Data spacing is assumed isotropic unless the spacing keyword argument is used. labels : array of ints, of same shape as data without channels dimension Array of seed markers labeled with different positive integers for different phases. Zero-labeled pixels are unlabeled pixels. Negative labels correspond to inactive pixels that are not taken into account (they are removed from the graph). If labels are not consecutive integers, the labels array will be transformed so that labels are consecutive. In the multichannel case, labels should have the same shape as a single channel of data, i.e. without the final dimension denoting channels. beta : float Penalization coefficient for the random walker motion (the greater beta, the more difficult the diffusion). mode : string, available options {‘cg_mg’, ‘cg’, ‘bf’} Mode for solving the linear system in the random walker algorithm. If no preference given, automatically attempt to use the fastest option available (‘cg_mg’ from pyamg >> ‘cg’ with UMFPACK > ‘bf’). ‘bf’ (brute force): an LU factorization of the Laplacian is computed. This is fast for small images (<1024x1024), but very slow and memory-intensive for large images (e.g., 3-D volumes). ‘cg’ (conjugate gradient): the linear system is solved iteratively using the Conjugate Gradient method from scipy.sparse.linalg. This is less memory-consuming than the brute force method for large images, but it is quite slow. ‘cg_mg’ (conjugate gradient with multigrid preconditioner): a preconditioner is computed using a multigrid solver, then the solution is computed with the Conjugate Gradient method. This mode requires that the pyamg module (http://pyamg.org/) is installed. For images of size > 512x512, this is the recommended (fastest) mode. tol : float tolerance to achieve when solving the linear system, in cg’ and ‘cg_mg’ modes. copy : bool If copy is False, the labels array will be overwritten with the result of the segmentation. Use copy=False if you want to save on memory. multichannel : bool, default False If True, input data is parsed as multichannel data (see ‘data’ above for proper input format in this case) return_full_prob : bool, default False If True, the probability that a pixel belongs to each of the labels will be returned, instead of only the most likely label. spacing : iterable of floats Spacing between voxels in each spatial dimension. If None, then the spacing between pixels/voxels in each dimension is assumed 1.

Returns:

output : ndarray If return_full_prob is False, array of ints of same shape as data, in which each pixel has been labeled according to the marker that reached the pixel first by anisotropic diffusion. If return_full_prob is True, array of floats of shape (nlabels, data.shape). output[label_nb, i, j] is the probability that label label_nb reaches the pixel (i, j) first.

See also

skimage.morphology.watershed

watershed segmentation A segmentation algorithm based on mathematical morphology and “flooding” of regions from markers.

Notes

Multichannel inputs are scaled with all channel data combined. Ensure all channels are separately normalized prior to running this algorithm.

The spacing argument is specifically for anisotropic datasets, where data points are spaced differently in one or more spatial dimensions. Anisotropic data is commonly encountered in medical imaging.

The algorithm was first proposed in Random walks for image segmentation, Leo Grady, IEEE Trans Pattern Anal Mach Intell. 2006 Nov;28(11):1768-83.

The algorithm solves the diffusion equation at infinite times for sources placed on markers of each phase in turn. A pixel is labeled with the phase that has the greatest probability to diffuse first to the pixel.

The diffusion equation is solved by minimizing x.T L x for each phase, where L is the Laplacian of the weighted graph of the image, and x is the probability that a marker of the given phase arrives first at a pixel by diffusion (x=1 on markers of the phase, x=0 on the other markers, and the other coefficients are looked for). Each pixel is attributed the label for which it has a maximal value of x. The Laplacian L of the image is defined as:

• L_ii = d_i, the number of neighbors of pixel i (the degree of i)
• L_ij = -w_ij if i and j are adjacent pixels

The weight w_ij is a decreasing function of the norm of the local gradient. This ensures that diffusion is easier between pixels of similar values.

When the Laplacian is decomposed into blocks of marked and unmarked pixels:

L = M B.T
    B A

with first indices corresponding to marked pixels, and then to unmarked pixels, minimizing x.T L x for one phase amount to solving:

A x = - B x_m

where x_m = 1 on markers of the given phase, and 0 on other markers. This linear system is solved in the algorithm using a direct method for small images, and an iterative method for larger images.

Examples

>>> np.random.seed(0)
>>> a = np.zeros((10, 10)) + 0.2 * np.random.rand(10, 10)
>>> a[5:8, 5:8] += 1
>>> b = np.zeros_like(a)
>>> b[3, 3] = 1  # Marker for first phase
>>> b[6, 6] = 2  # Marker for second phase
>>> random_walker(a, b)
array([[1, 1, 1, 1, 1, 1, 1, 1, 1, 1],
       [1, 1, 1, 1, 1, 1, 1, 1, 1, 1],
       [1, 1, 1, 1, 1, 1, 1, 1, 1, 1],
       [1, 1, 1, 1, 1, 1, 1, 1, 1, 1],
       [1, 1, 1, 1, 1, 1, 1, 1, 1, 1],
       [1, 1, 1, 1, 1, 2, 2, 2, 1, 1],
       [1, 1, 1, 1, 1, 2, 2, 2, 1, 1],
       [1, 1, 1, 1, 1, 2, 2, 2, 1, 1],
       [1, 1, 1, 1, 1, 1, 1, 1, 1, 1],
       [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]], dtype=int32)