RANSAC算法与原理（一）

算法介绍

首先我们从Random sample consensus – Wikipedia上找到RANSAC原理的介绍。

RANSAC算法的中文名称是随机抽样一致算法(Random Sample Consenus)，简单的说，通过RANSAC算法，我们将数据分为inliers和outliers,inliers是对于模型拟合有效的点，也称之为内点；outliers是对于模型拟合无效的点，也就是错误的数据点，称之为外点。而我们在使用观测数据拟合模型的过程中，外点的存在对于使用数据拟合模型是有害的，那么我们该如何剔除这些外点呢？RANSAC算法就是能够剔除外点的一个迭代性的算法。

举例

下图所示就是RANSAC算法的作用：剔除外点，使模型估计更加准确，参考Robust line model estimation using RANSAC — skimage v0.19.2 docs (scikit-image.org)

算法的基本思想和流程

算法的实现流程：

1. 选择出估计模型的最小数据样本（对于二维和三维直线拟合来说，确定一条直线最少需要2个点；对于三维平面拟合来说，确定一个三维平面最少需要3个点）
2. 使用这个最小的数据样本，算出拟合的模型。（也就是直线方程或者平面方程）
3. 将所有的模型带入这个拟合的模型，计算出内点 的数量。（数据点和拟合的模型的误差在一定阈值范围内的数据点的数量）
4. 比较当前模型和之前迭代的得到的最好的模型的内点数量（内点数量越多，模型越好），记录最大的内点数的模型参数和内点数量。
5. 重复1-4步，直到达到迭代终止条件（例如达到最大迭代数、内点数量达到迭代终止条件）

迭代次数的推导

假设内点在数据中所占的比例为

如果我们每次迭代都需要个点，那么每次迭代至少有一个外点的概率是：

那么我们如果迭代次，所有的次迭代都至少有一个外点的概率为，那么这次迭代，能够采样到正确的个内点去计算模型的概率就是上述概率的补集。

通过上式，我们可以求得

注意：内点的概率是一个先验值，如果我们一开始不知道这个先验值，可以采用自适应迭代的方法，用当前的内点的比值来当成来估算迭代的次数。然后通过不断迭代，内点的比值也逐渐增大，再用新的更大的内点比值去代替的值；对于来说，一般会取一个定值0.99，等式（4）可以看出，当不变时，越大，越小，越小，越大。

算法的实现

Given:
    data – A set of observations.
    model – A model to explain observed data points.
    n – Minimum number of data points required to estimate model parameters.
    k – Maximum number of iterations allowed in the algorithm.
    t – Threshold value to determine data points that are fit well by model.
    d – Number of close data points required to assert that a model fits well to data.

Return:
    bestFit – model parameters which best fit the data (or null if no good model is found)

iterations = 0
bestFit = null
bestErr = something really large

while iterations < k do
    maybeInliers := n randomly selected values from data
    maybeModel := model parameters fitted to maybeInliers
    alsoInliers := empty set
    for every point in data not in maybeInliers do
        if point fits maybeModel with an error smaller than t
             add point to alsoInliers
        end if
    end for
    if the number of elements in alsoInliers is > d then
        // This implies that we may have found a good model
        // now test how good it is.
        betterModel := model parameters fitted to all points in maybeInliers and alsoInliers
        thisErr := a measure of how well betterModel fits these points
        if thisErr < bestErr then
            bestFit := betterModel
            bestErr := thisErr
        end if
    end if
    increment iterations
end while

return bestFit

python代码实现

以下代码参考scikit-image/fit.py at v0.19.2 · scikit-image/scikit-image (github.com)，也就是skimage的源码。

def _dynamic_max_trials(n_inliers, n_samples, min_samples, probability):
    
     if n_inliers == 0:
        return np.inf

    if probability == 1:
        return np.inf

    if n_inliers == n_samples:
        return 1

    nom = math.log(1 - probability)
    denom = math.log(1 - (n_inliers / n_samples) ** min_samples)

    return int(np.ceil(nom / denom))



def ransac(data, model_class, min_samples, residual_threshold,
   			is_data_valid=None, is_model_valid=None,
           max_trials=100, stop_sample_num=np.inf, stop_residuals_sum=0,
           stop_probability=1, random_state=None, initial_inliers=None):
    
    best_inlier_num = 0
    best_inlier_residuals_sum = np.inf
    best_inliers = []
    validate_model = is_model_valid is not None
    validate_data = is_data_valid is not None

    random_state = np.random.default_rng(random_state)

    # in case data is not pair of input and output, male it like it
    if not isinstance(data, (tuple, list)):
        data = (data, )
    num_samples = len(data[0])

    if not (0 < min_samples < num_samples):
        raise ValueError(f"`min_samples` must be in range (0, {num_samples})")

    if residual_threshold < 0:
        raise ValueError("`residual_threshold` must be greater than zero")

    if max_trials < 0:
        raise ValueError("`max_trials` must be greater than zero")

    if not (0 <= stop_probability <= 1):
        raise ValueError("`stop_probability` must be in range [0, 1]")

    if initial_inliers is not None and len(initial_inliers) != num_samples:
        raise ValueError(
            f"RANSAC received a vector of initial inliers (length "
            f"{len(initial_inliers)}) that didn't match the number of "
            f"samples ({num_samples}). The vector of initial inliers should "
            f"have the same length as the number of samples and contain only "
            f"True (this sample is an initial inlier) and False (this one "
            f"isn't) values.")

    # for the first run use initial guess of inliers
    spl_idxs = (initial_inliers if initial_inliers is not None
                else random_state.choice(num_samples, min_samples,
                                         replace=False))

    # estimate model for current random sample set
    model = model_class()

    for num_trials in range(max_trials):
        # do sample selection according data pairs
        samples = [d[spl_idxs] for d in data]

        # for next iteration choose random sample set and be sure that
        # no samples repeat
        spl_idxs = random_state.choice(num_samples, min_samples, replace=False)

        # optional check if random sample set is valid
        if validate_data and not is_data_valid(*samples):
            continue

        success = model.estimate(*samples)
        # backwards compatibility
        if success is not None and not success:
            continue

        # optional check if estimated model is valid
        if validate_model and not is_model_valid(model, *samples):
            continue

        residuals = np.abs(model.residuals(*data))
        # consensus set / inliers
        inliers = residuals < residual_threshold
        residuals_sum = residuals.dot(residuals)

        # choose as new best model if number of inliers is maximal
        inliers_count = np.count_nonzero(inliers)
        if (
            # more inliers
            inliers_count > best_inlier_num
            # same number of inliers but less "error" in terms of residuals
            or (inliers_count == best_inlier_num
                and residuals_sum < best_inlier_residuals_sum)):
            best_inlier_num = inliers_count
            best_inlier_residuals_sum = residuals_sum
            best_inliers = inliers
            dynamic_max_trials = _dynamic_max_trials(best_inlier_num,
                                                     num_samples,
                                                     min_samples,
                                                     stop_probability)
            if (best_inlier_num >= stop_sample_num
                    or best_inlier_residuals_sum <= stop_residuals_sum
                    or num_trials >= dynamic_max_trials):
                break

    # estimate final model using all inliers
    if any(best_inliers):
        # select inliers for each data array
        data_inliers = [d[best_inliers] for d in data]
        model.estimate(*data_inliers)
        if validate_model and not is_model_valid(model, *data_inliers):
            warn("Estimated model is not valid. Try increasing max_trials.")
    else:
        model = None
        best_inliers = None
        warn("No inliers found. Model not fitted")

    return model, best_inliers

References

Random sample consensus – Wikipedia

RANSAC算法详解(附Python拟合直线模型代码) – 知乎 (zhihu.com)

Robust line model estimation using RANSAC — skimage v0.19.2 docs (scikit-image.org)

scikit-image/fit.py at v0.19.2 · scikit-image/scikit-image (github.com)