Fuzzy C means in python for brain tumor detection

Question

I found this code on GitHub and i try to understand the behavior of functions.I tried to compare this code with the formulas (from p. 6 of this paper):

I cannot find where this formulas are implemented into the code.Can anyone help me to explain the similarities between the formulas and the code?

class FCM() :
    def __init__(self, n_clusters=17, max_iter=100, m=2, error=1e-6):
        super().__init__()
        self.u, self.centers = None, None
        self.n_clusters = n_clusters
        self.max_iter = max_iter
        self.m = m
        self.error = error

def fit(self, X):
    N = X.shape[0]
    C = self.n_clusters
    centers = []

    u = np.random.dirichlet(np.ones(C), size=N)

    iteration = 0
    while iteration < self.max_iter:
        u2 = u.copy()

        centers = self.next_centers(X, u)
        u = self.next_u(X, centers)
        iteration += 1

        # Stopping rule
        if norm(u - u2) < self.error:
            break

    self.u = u
    self.centers = centers
    return centers

def next_centers(self, X, u):
    um = u ** self.m
    return (X.T @ um / np.sum(um, axis=0)).transpose()  #Vi

def next_u(self, X, centers):
    return self._predict(X, centers)

def _predict(self, X, centers):
    power = float(2 / (self.m - 1))
    temp = cdist(X, centers) ** power
    denominator_ = temp.reshape((X.shape[0], 1, -1)).repeat(temp.shape[-1], axis=1)
    denominator_ = temp[:, :, np.newaxis] / denominator_

    return 1 / denominator_.sum(2)

def predict(self, X):
    if len(X.shape) == 1:
        X = np.expand_dims(X, axis=0)

    u = self._predict(X, self.centers)
    return np.argmax(u, axis=-1)

img2 = ret.reshape(x * y, z)
    algorithm = FCM()
    cluster_centers = algorithm.fit(img2)
    output = algorithm.predict(img2)
    img = cluster_centers[output].astype(np.int16).reshape(x, y, 3)

Answer 1

I believe the most relevant parts are the following:

• The function fit describes the general steps of the algorithm:
  • a few initializations
  • the updates to the centers and labels (which are delegated to other functions)
  • a verification of the stopping criteria after each iteration (norm(u - u2) < self.error).
• The fuzzy membership from equation (4) is implemented inside the function _predict:
  • The exponent is stored in power
  • The norms of differences are taken care of by cdist from scipy.spatial.distance. A single call to this function is used to compute the distances between all combinations of a data point x_j and a cluster center v_i. Each of the results is raised to the power and the result is stored in a temp array.
  • The last 3 lines of _predict juggle around the entries of the temp array in such a way that, without loops, it computes the divisions between each "distance" and the appropriate sum of "distances" (actually, "power of distance(s)", but it is easier to ignore the exponent at first). For this, the code uses a few tricks available in numpy, such as reshape, repeat, and indexing.
• The fuzzy centers from equation (5) are computed by next_centers:
  • Precompute a matrix um containing the m-th power of each u_ij (for later use in both the numerator and denominator)
  • Use np.sum(um, axis=0) to create an array whose i-th entry is the sum which is used in the denominator when computing the i-th cluster
  • For the i-th cluster, the numerator of (5) will be computed in the i-th column of X.T @ um (which becomes a row once we .transpose()the resulting matrix)