主动轮廓模型：Snake模型的python实现

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主要参考：Active Contour Model — skimage v0.16.dev0 docs - scikit-image

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skimage实现

函数声明

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.

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)

    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`

代码示例

import numpy as np
from matplotlib import pyplot as plt
from skimage.color import rgb2gray
from skimage import data
from skimage.filters import gaussian
from skimage.segmentation import active_contour

img = data.astronaut() # 读入图像
img = rgb2gray(img) # 灰度化

# 圆的参数方程：(220, 100) r=100
t = np.linspace(0, 2*np.pi, 400) # 参数t, [0,2π]
x = 220 + 100*np.cos(t)
y = 100 + 100*np.sin(t)

# 构造初始Snake
init = np.array([x, y]).T # shape=(400, 2)

# Snake模型迭代输出
snake = active_contour(gaussian(img,3), snake=init, alpha=0.1, beta=1, gamma=0.01, w_line=0, w_edge=10)

# 绘图显示
plt.figure(figsize=(5, 5))
plt.imshow(img, cmap="gray")
plt.plot(init[:, 0], init[:, 1], '--r', lw=3)
plt.plot(snake[:, 0], snake[:, 1], '-b', lw=3)
plt.xticks([]), plt.yticks([]), plt.axis("off")
plt.show()

结果显示

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Numpy实现

代码示例

import cv2 as cv
import numpy as np
from matplotlib import pyplot as plt


def getGaussianPE(src):
    """
    描述：计算负高斯势能(Negative Gaussian Potential Energy, NGPE)
    输入：单通道灰度图src
    输出：无符号的浮点型单通道，取值0.0 ~ 255.0
    """
    imblur = cv.GaussianBlur(src, ksize=(5, 5), sigmaX=3)
    dx = cv.Sobel(imblur, cv.CV_16S, 1, 0)  # X方向上取一阶导数，16位有符号数，卷积核3x3
    dy = cv.Sobel(imblur, cv.CV_16S, 0, 1)
    E = dx**2 + dy**2
    return E


def getDiagCycleMat(alpha, beta, n):
    """
    计算5对角循环矩阵
    """
    a = 2 * alpha + 6 * beta
    b = -(alpha + 4 * beta)
    c = beta
    diag_mat_a = a * np.eye(n)
    diag_mat_b = b * np.roll(np.eye(n), 1, 0) + b * np.roll(np.eye(n), -1, 0)
    diag_mat_c = c * np.roll(np.eye(n), 2, 0) + c * np.roll(np.eye(n), -2, 0)
    return diag_mat_a + diag_mat_b + diag_mat_c


def getCircleContour(centre=(0, 0), radius=(1, 1), N=200):
    """
    以参数方程的形式，获取n个离散点围成的圆形/椭圆形轮廓
    输入：中心centre=（x0, y0）, 半轴长radius=(a, b)， 离散点数N
    输出：由离散点坐标(x, y)组成的2xN矩阵
    """
    t = np.linspace(0, 2 * np.pi, N)
    x = centre[0] + radius[0] * np.cos(t)
    y = centre[1] + radius[1] * np.sin(t)
    return np.array([x, y])


def getRectContour(pt1=(0, 0), pt2=(50, 50)):
    """
    根据左上、右下两个顶点来计算矩形初始轮廓坐标
    由于Snake模型适用于光滑曲线，故这里用不到该函数
    """
    pt1, pt2 = np.array(pt1), np.array(pt2)
    r1, c1, r2, c2 = pt1[0], pt1[1], pt2[0], pt2[1]
    a, b = r2 - r1, c2 - c1
    length = (a + b) * 2 + 1
    x = np.ones((length), np.float)
    x[:b] = r1
    x[b:a + b] = np.arange(r1, r2)
    x[a + b:a + b + b] = r2
    x[a + b + b:] = np.arange(r2, r1 - 1, -1)
    y = np.ones((length), np.float)
    y[:b] = np.arange(c1, c2)
    y[b:a + b] = c2
    y[a + b:a + b + b] = np.arange(c2, c1, -1)
    y[a + b + b:] = c1
    return np.array([x, y])


def snake(img, snake, alpha=0.5, beta=0.1, gamma=0.1, max_iter=2500, convergence=0.01):
    """
    根据Snake模型的隐式格式进行迭代
    输入：弹力系数alpha，刚性系数beta，迭代步长gamma，最大迭代次数max_iter，收敛阈值convergence
    输出：由收敛轮廓坐标(x, y)组成的2xN矩阵， 历次迭代误差list
    """
    x, y, errs = snake[0].copy(), snake[1].copy(), []
    n = len(x)
    # 计算5对角循环矩阵A，及其相关逆阵
    A = getDiagCycleMat(alpha, beta, n)
    inv = np.linalg.inv(A + gamma * np.eye(n))
    # 初始化
    y_max, x_max = img.shape
    max_px_move = 1.0
    # 计算负高斯势能矩阵，及其梯度
    E_ext = -getGaussianPE(img)
    fx = cv.Sobel(E_ext, cv.CV_16S, 1, 0)
    fy = cv.Sobel(E_ext, cv.CV_16S, 0, 1)
    T = np.max([abs(fx), abs(fy)])
    fx, fy = fx / T, fy / T
    for g in range(max_iter):
        x_pre, y_pre = x.copy(), y.copy()
        i, j = np.uint8(y), np.uint8(x)
        try:
            xn = inv @ (gamma * x + fx[i, j])
            yn = inv @ (gamma * y + fy[i, j])
        except Exception as e:
            print("索引超出范围")
        # 判断收敛
        x, y = xn, yn
        err = np.mean(0.5 * np.abs(x_pre - x) + 0.5 * np.abs(y_pre - y))
        errs.append(err)
        if err < convergence:
            print(f"Snake迭代{g}次后，趋于收敛。\t err = {err:.3f}")
            break
    return x, y, errs


def main():
    src = cv.imread("circle.jpg", 0)
    img = cv.GaussianBlur(src, (3, 3), 5)

    # 构造初始轮廓线
    init = getCircleContour((140, 95), (110, 80), N=200)
    # Snake Model
    x, y, errs = snake(img, snake=init, alpha=0.1, beta=1, gamma=0.1)

    plt.figure() # 绘制轮廓图
    plt.imshow(img, cmap="gray")
    plt.plot(init[0], init[1], '--r', lw=1)
    plt.plot(x, y, 'g', lw=1)
    plt.xticks([]), plt.yticks([]), plt.axis("off")
    plt.figure() # 绘制收敛趋势图
    plt.plot(range(len(errs)), errs)
    plt.show()


if __name__ == '__main__':
    main()

结果显示

Snake迭代760次后，趋于收敛。	 err = 0.010

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