前言

随机样本一致性 (RANSAC) 是一种迭代方法，用于从一组包含异常值的观察数据中估计数学模型的参数，此时异常值不会对估计值产生影响。简言之，RANSAC是一种滤除异常值的常用算法。

算法流程

以直线拟合为例，通用RANSAC算法的流程如下

输入:data – 观测结果.model – 数学模型.n – 估计模型所需的最小样本.k – 最大迭代次数.t – 用于确定模型适合的数据点的阈值.输出:bestFit – 模型iterations = 0
bestFit = null
bestErr = something really largewhile iterations < k domaybeInliers := n randomly selected values from datamaybeModel := model parameters fitted to maybeInliersalsoInliers := empty setfor every point in data not in maybeInliers doif point fits maybeModel with an error smaller than tadd point to alsoInliersend ifend forif the number of elements in alsoInliers is > d thenbetterModel := model parameters fitted to all points in maybeInliers and alsoInliersthisErr := a measure of how well betterModel fits these pointsif thisErr < bestErr thenbestFit := betterModelbestErr := thisErrend ifend ifincrement iterations
end whilereturn bestFit

Python代码

"""_summary_
RANSAC直线拟合
"""
from copy import copy
import numpy as np
from numpy.random import default_rng
rng = default_rng()class RANSAC:def __init__(self, n=10, k=100, t=0.05, d=10, model=None, loss=None, metric=None):self.n = n              self.k = k              self.t = t              self.d = d              self.model = model     self.loss = loss        self.metric = metric  self.best_fit = Noneself.best_error = np.infdef fit(self, X, y):for _ in range(self.k):ids = rng.permutation(X.shape[0])maybe_inliers = ids[: self.n]maybe_model = copy(self.model).fit(X[maybe_inliers], y[maybe_inliers])thresholded = (self.loss(y[ids][self.n :], maybe_model.predict(X[ids][self.n :]))< self.t)inlier_ids = ids[self.n :][np.flatnonzero(thresholded).flatten()]if inlier_ids.size > self.d:inlier_points = np.hstack([maybe_inliers, inlier_ids])better_model = copy(self.model).fit(X[inlier_points], y[inlier_points])this_error = self.metric(y[inlier_points], better_model.predict(X[inlier_points]))if this_error < self.best_error:self.best_error = this_errorself.best_fit = maybe_modelreturn selfdef predict(self, X):return self.best_fit.predict(X)def square_error_loss(y_true, y_pred):return (y_true - y_pred) ** 2def mean_square_error(y_true, y_pred):return np.sum(square_error_loss(y_true, y_pred)) / y_true.shape[0]class LinearRegressor:def __init__(self):self.params = Nonedef fit(self, X: np.ndarray, y: np.ndarray):r, _ = X.shapeX = np.hstack([np.ones((r, 1)), X])self.params = np.linalg.inv(X.T @ X) @ X.T @ yreturn selfdef predict(self, X: np.ndarray):r, _ = X.shapeX = np.hstack([np.ones((r, 1)), X])return X @ self.paramsif __name__ == "__main__":regressor = RANSAC(model=LinearRegressor(), loss=square_error_loss, metric=mean_square_error)X = np.array([-0.848,-0.800,-0.704,-0.632,-0.488,-0.472,-0.368,-0.336,-0.280,-0.200,-0.00800,-0.0840,0.0240,0.100,0.124,0.148,0.232,0.236,0.324,0.356,0.368,0.440,0.512,0.548,0.660,0.640,0.712,0.752,0.776,0.880,0.920,0.944,-0.108,-0.168,-0.720,-0.784,-0.224,-0.604,-0.740,-0.0440,0.388,-0.0200,0.752,0.416,-0.0800,-0.348,0.988,0.776,0.680,0.880,-0.816,-0.424,-0.932,0.272,-0.556,-0.568,-0.600,-0.716,-0.796,-0.880,-0.972,-0.916,0.816,0.892,0.956,0.980,0.988,0.992,0.00400]).reshape(-1,1)y = np.array([-0.917,-0.833,-0.801,-0.665,-0.605,-0.545,-0.509,-0.433,-0.397,-0.281,-0.205,-0.169,-0.0531,-0.0651,0.0349,0.0829,0.0589,0.175,0.179,0.191,0.259,0.287,0.359,0.395,0.483,0.539,0.543,0.603,0.667,0.679,0.751,0.803,-0.265,-0.341,0.111,-0.113,0.547,0.791,0.551,0.347,0.975,0.943,-0.249,-0.769,-0.625,-0.861,-0.749,-0.945,-0.493,0.163,-0.469,0.0669,0.891,0.623,-0.609,-0.677,-0.721,-0.745,-0.885,-0.897,-0.969,-0.949,0.707,0.783,0.859,0.979,0.811,0.891,-0.137]).reshape(-1,1)regressor.fit(X, y)import matplotlib.pyplot as pltplt.style.use("seaborn-darkgrid")fig, ax = plt.subplots(1, 1)# 支持中文plt.rcParams['font.sans-serif'] = ['SimHei']  # 用来正常显示中文标签plt.rcParams['axes.unicode_minus'] = False  # 用来正常显示负号plt.scatter(X, y)plt.title("RANSAC拟合直线")line = np.linspace(-1, 1, num=100).reshape(-1, 1)plt.plot(line, regressor.predict(line), c="peru")plt.show()

RANSAC算法迭代参数的自适应

上面提到的RANSAC算法总要迭代足够的次数才能终止，这就留给设计者一个难题，迭代次数少了，找不到最优解，迭代次数多了，程序耗时增加。接下来给读者介绍一种自适应的迭代次数的方法。
当错误点所占比例较小时，则应该尽快停止迭代；反之，则需要足够的迭代次数再终止算法以获取更可靠的结果。
令 p:p:p: :RANSAC 算法在运行后提供至少一个有用结果的期望概率。
令w:w:w: 满足模型的点 / 检测到的所有点
令n:n:n:满足当前模型的点的个数
则1−p=(1−wn)k1-p = (1-{w}^n)^k1−p=(1−wn)k
即k=log(1−p)log(1−wn)k = \frac{log(1-p)}{log(1-w^{n})}k=log(1−wn)log(1−p)
代码实现过程如下

// 以下代码来自OpenCV
/*** @Method:    更新ransac的迭代次数* @Returns:   int 更新后的最大迭代次数* @Qualifier: * @Parameter: double p 信心分数* @Parameter: double ep 错误点所占的比例* @Parameter: int modelPoints 模型中点的数量* @Parameter: int maxIters 当前最大迭代次数*/
int RANSACUpdateNumIters( double p, double ep, int modelPoints, int maxIters )
{if( modelPoints <= 0 )CV_Error( Error::StsOutOfRange, "the number of model points should be positive" );p = MAX(p, 0.);p = MIN(p, 1.);ep = MAX(ep, 0.);ep = MIN(ep, 1.);// avoid inf's & nan'sdouble num = MAX(1. - p, DBL_MIN);double denom = 1. - std::pow(1. - ep, modelPoints);if( denom < DBL_MIN )return 0;num = std::log(num);denom = std::log(denom);return denom >= 0 || -num >= maxIters*(-denom) ? maxIters : cvRound(num/denom);
}

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RANSAC算法（原理及代码实现+迭代次数参数自适应）

RANSAC算法

前言

算法流程

Python代码

RANSAC算法迭代参数的自适应

RANSAC算法（原理及代码实现+迭代次数参数自适应）相关推荐

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