点云处理－－－kd-tree

KD-tree在二叉树的基础上，实际上就是变成了多维度的分割，分割方法变为维度轮换分割：x-y-z-x-y-z…
1.kd-tree的构建
（1）节点定义
每个节点按我的理解其实就是单维度上的一片空间区域，该节点储存了该节点的分割维度axis,分割轴的坐标value,该节点区域内的点的索引point_indices,以分割轴分开的左右节点也是两个区域，被当做left、right两个节点存储起来。

from result_set import KNNResultsets, RadiusNNResultSet
import numpy as np
import math
import os
import random

# define the node


class Node:
    # init function
    def __init__(self, axis, value, left, right, point_indices):
        ''' pram: axis:划分的维度 ｖａｌｕｅ：节点的值 ｌｅｆｔ:左节点 ｒｉｇｈｔ:右节点 point_indices:点集中的序号 '''
        self.axis = axis
        self.value = value
        self.left = left
        self.right = right
        self.point_indices = point_indices
    # check if is leaf

    def is_leaf(self):
    #叶子节点的时候是不再对空间进行划分的，所以ｖａｌｕｅ是ｎｏｎe
        if self.value == None:
            return True
        else:
            return False

    # print function
    def __str__(self):
        output = ''
        output += 'axis %d, ' % self.axis
        if self.value is None:
            output += 'split value: leaf, '
        else:
            output += 'split value: %.2f, ' % self.value
        output += 'point_indices: '
        output += str(self.point_indices.tolist())
        return output

（2）kd树构建
kdtree_construction调用递归构造函数kdtree_recursive_builder构建ｋｄ-tree节点

def kdtree_construction(data, leaf_size):
    ''' input: data(numpy array) leaf_size(the smallest size of each node) '''
    N, dim = data.shape[0], data.shape[1]

    # build the kd-tree
    root = None
    root = kdtree_recursive_builder(
        root, data, np.arange(N), axis=0, leaf_size=leaf_size)
    return root

kdtree_recursive_builder

def kdtree_recursive_builder(root, db, point_indices, axis, leaf_size):
    """ param: root data:NxD point_indeces:N axis:scale leaf_size:scale """
 	#如果root不存在，先建立root
    if root is None:
        root = Node(axis, None, None, None, point_indices)

    
    #如果当前的节点数量大于叶子节点数，才进一步的进行分割，否则就不进行进一步分割，当前节点就作为叶子结点
    #叶子结点特点就是value==None
    if len(point_indices) > leaf_size:
        # --- get the split position ---
        #对当前传入的数据节点在划分维度上进行排序，选出当前维度的中间值数据点
        #划分点的value就等于中间值数据点的均值，注意此处划分的中间平面不穿过数据点
        point_indices_sorted, _ = sort_key_by_value(
            point_indices, db[point_indices, axis])  # 点的索引按照ｖａｌｕｅ的大小顺序进行排序
        #求出当前维度下中间值的点的索引位置
        middle_left_idx = math.ceil(point_indices_sorted.shape[0] / 2) - 1
        #中间点在原来点集合中的索引
        middle_left_point_idx = point_indices_sorted[middle_left_idx]
        #中间点的ｖａｌｕｅ值
        middle_left_point_value = db[middle_left_point_idx, axis]
		#中间点后一个点也一样
        middle_right_idx = middle_left_idx + 1
        middle_right_point_idx = point_indices_sorted[middle_right_idx]
        middle_right_point_value = db[middle_right_point_idx, axis]

        root.value = (middle_left_point_value + middle_right_point_value) * 0.5#取中间两个数据点value的平均值，不穿过数据点
        # === get the split position ===
        root.left = kdtree_recursive_builder(root.left,
                                             db,
                                             point_indices_sorted[0:middle_right_idx],
                                             axis_round_robin(
                                                 axis, dim=db.shape[1]),
                                             leaf_size)
        root.right = kdtree_recursive_builder(root.right,
                                              db,
                                              point_indices_sorted[middle_right_idx:],
                                              axis_round_robin(
                                                  axis, dim=db.shape[1]),
                                              leaf_size)
    return root

sort_key_by_value排序函数：

# sort_key_by_value
def sort_key_by_value(point_indeces, value):
    # 确保输入的点的索引和点的值维度相同
    assert(point_indeces.shape == value.shape)
    assert(len(point_indeces.shape) == 1)  # 1xN
    sort_idx = np.argsort(value)  # value是一个列表，不是ｎｕｍｐｙ
    point_indeces_sort = point_indeces[sort_idx]
    value_sort = value[sort_idx]
    return point_indeces_sort, value_sort

axis_round_robin函数，轮换确定分割维度：０－－＞１，１－－》２，２－－＞０

# axis_round_robin,改变分割维度
def axis_round_robin(axis, dim):
    if axis == dim-1:
        return 0
    else:
        return axis+1

2.kd-tree的KNN查找
和二叉树的查找很相似，不同之处在于初始时刻要判断当前节点是不是叶子节点，是的话就直接将各个节点计算和当前查询点的距离，把这些点插入到结果集合中。
不是叶子节点的时候：也是先比较查询点和当前节点的ｖａｌｕｅ的大小，选择从左边找还是从右边找。
是否查找一个节点区域的关键是：判断当前最坏距离与（查询点坐标—节点区域分割坐标之差）之间的大小关系。

def kdtree_knn_search(root: Node, data: np.ndarray, result_set: KNNResultsets, query_point: np.ndarray):
    if root == None:
        return
    # check if is a leaf
    #当当前节点是叶子节点的时候，由于不能再进一步的在当前节点空间上区分左右子空间了，
    #所以就暴力的把所有叶子节点中的节点都拿出来和查询点计算距离，把距离较近的点都插入到ｒｅｓｕｌｔ
    if root.is_leaf():
       # compare the contents of a leaf
        leaf_points = data[root.point_indices, :]#root.point_indeces是一个列表，存储的是点在元数据中的索引
        #print("leaf index:", root.point_indices)
        diff = np.linalg.norm(np.expand_dims(
            query_point, 0) - leaf_points, axis=1)
        for i in range(diff.shape[0]):
            result_set.add_point(diff[i], root.point_indices[i])
        return False
#距离小于ｒｏｏｔ　ｖａｌｕｅ，从左边开始找
    if query_point[root.axis] <= root.value:
        kdtree_knn_search(root.left, data, result_set, query_point)
        #如果左边没有找够，就继续从右边找
        if math.fabs(query_point[root.axis] - root.value) < result_set.worst_Dist():
            kdtree_knn_search(root.right, data, result_set, query_point)
    else:
    #从右边开始找
        kdtree_knn_search(root.right, data, result_set, query_point)
        if math.fabs(query_point[root.axis] - root.value) < result_set.worst_Dist():
            kdtree_knn_search(root.left, data, result_set, query_point)

    return False

3.kd-tree的radius查找

def kdtree_radius_search(root: Node, data: np.ndarray, result_set: RadiusNNResultSet, query_point: np.ndarray):
    if root == None:
        return
    # check if is a leaf
    if root.is_leaf():
       # compare the contents of a leaf
        leaf_points = data[root.point_indices, :]
        print("leaf index:", root.point_indices)
        diff = np.linalg.norm(np.expand_dims(
            query_point, 0) - leaf_points, axis=1)
        for i in range(diff.shape[0]):
            result_set.add_point(diff[i], root.point_indices[i])
        return False

    if query_point[root.axis] <= root.value:
        kdtree_radius_search(root.left, data, result_set, query_point)
        if math.fabs(query_point[root.axis] - root.value) < result_set.worst_Dist():
            kdtree_radius_search(root.right, data, result_set, query_point)
    else:
        kdtree_radius_search(root.right, data, result_set, query_point)
        if math.fabs(query_point[root.axis] - root.value) < result_set.worst_Dist():
            kdtree_radius_search(root.left, data, result_set, query_point)

    return False

main函数调用：

def main():
    # generate the data
    db = 64
    dim = 3
    leaf_size = 4
    k = 8
    print('runing')
    db_np = np.random.rand(db, dim)
    root = kdtree_construction(data=db_np, leaf_size=leaf_size)
    query = np.asarray([0, 0, 0])
    # result_set = KNNResultsets(capacity=k)
    # kdtree_knn_search(root, data=db_np,
    # result_set=result_set, query_point=query)

    # print(result_set)

    result_set = RadiusNNResultSet(radius=1)
    kdtree_radius_search(
        root, data=db_np, result_set=result_set, query_point=query)
    print(result_set)


if __name__ == '__main__':
    main()

python相关：

 diff = np.linalg.norm(np.expand_dims(
            query_point, 0) - leaf_points, axis=1)
  #计算范数：在列的方向上计算，也就是计算每个点和查询点之间的距离（空间距离）