如何用NumPy计算欧氏距离?

在三维空间中有两个点

a = (ax, ay, az)
b = (bx, by, bz)

我想计算它们之间的距离:

dist = sqrt((ax-bx)^2 + (ay-by)^2 + (az-bz)^2)

我如何用NumPy做到这一点?我有:

import numpy
a = numpy.array((ax, ay, az))
b = numpy.array((bx, by, bz))

当前回答

我想用各种表演笔记来阐述这个简单的答案。Np.linalg.norm可能会做的比你需要的更多:

dist = numpy.linalg.norm(a-b)

首先，这个函数被设计用于处理一个列表并返回所有的值，例如比较pA到点集sP的距离:

sP = set(points)
pA = point
distances = np.linalg.norm(sP - pA, ord=2, axis=1.)  # 'distances' is a list

记住几件事:

Python函数调用的开销很大。 [常规]Python不缓存名称查找。

def distance(pointA, pointB):
    dist = np.linalg.norm(pointA - pointB)
    return dist

并不像看上去那么无辜。

>>> dis.dis(distance)
  2           0 LOAD_GLOBAL              0 (np)
              2 LOAD_ATTR                1 (linalg)
              4 LOAD_ATTR                2 (norm)
              6 LOAD_FAST                0 (pointA)
              8 LOAD_FAST                1 (pointB)
             10 BINARY_SUBTRACT
             12 CALL_FUNCTION            1
             14 STORE_FAST               2 (dist)

  3          16 LOAD_FAST                2 (dist)
             18 RETURN_VALUE

首先，每次我们调用它时，我们都必须对“np”进行全局查找，对“linalg”进行范围查找，对“norm”进行范围查找，而仅仅调用这个函数的开销就相当于几十条python指令。

最后，我们浪费了两个操作来存储结果并重新加载它以返回…

改进的第一步:使查找更快，跳过存储

def distance(pointA, pointB, _norm=np.linalg.norm):
    return _norm(pointA - pointB)

我们得到了更精简的:

>>> dis.dis(distance)
  2           0 LOAD_FAST                2 (_norm)
              2 LOAD_FAST                0 (pointA)
              4 LOAD_FAST                1 (pointB)
              6 BINARY_SUBTRACT
              8 CALL_FUNCTION            1
             10 RETURN_VALUE

不过，函数调用开销仍然需要一些工作。你会想要做基准测试，以确定你自己做数学是否会更好:

def distance(pointA, pointB):
    return (
        ((pointA.x - pointB.x) ** 2) +
        ((pointA.y - pointB.y) ** 2) +
        ((pointA.z - pointB.z) ** 2)
    ) ** 0.5  # fast sqrt

在某些平台上，**0.5比math.sqrt快。你的里程可能会有所不同。

****高级性能说明。

你为什么要计算距离?如果唯一的目的是展示它，

 print("The target is %.2fm away" % (distance(a, b)))

沿着。但是如果你在比较距离，进行范围检查等等，我想添加一些有用的性能观察。

让我们以两种情况为例:按距离排序或将列表剔除到满足范围约束的项。

# Ultra naive implementations. Hold onto your hat.

def sort_things_by_distance(origin, things):
    return things.sort(key=lambda thing: distance(origin, thing))

def in_range(origin, range, things):
    things_in_range = []
    for thing in things:
        if distance(origin, thing) <= range:
            things_in_range.append(thing)

我们需要记住的第一件事是，我们使用毕达哥拉斯来计算距离(dist =根号(x²+ y²+ z²))，所以我们做了很多根号调用。数学101:

dist = root ( x^2 + y^2 + z^2 )
:.
dist^2 = x^2 + y^2 + z^2
and
sq(N) < sq(M) iff M > N
and
sq(N) > sq(M) iff N > M
and
sq(N) = sq(M) iff N == M

简而言之:直到我们真正需要以X为单位的距离，而不是X^2，我们才能消除计算中最难的部分。

# Still naive, but much faster.

def distance_sq(left, right):
    """ Returns the square of the distance between left and right. """
    return (
        ((left.x - right.x) ** 2) +
        ((left.y - right.y) ** 2) +
        ((left.z - right.z) ** 2)
    )

def sort_things_by_distance(origin, things):
    return things.sort(key=lambda thing: distance_sq(origin, thing))

def in_range(origin, range, things):
    things_in_range = []

    # Remember that sqrt(N)**2 == N, so if we square
    # range, we don't need to root the distances.
    range_sq = range**2

    for thing in things:
        if distance_sq(origin, thing) <= range_sq:
            things_in_range.append(thing)

很好，这两个函数都不再做昂贵的平方根了。这样会快得多，但在进一步讨论之前，请检查自己:为什么sort_things_by_distance两次都需要一个“天真”的免责声明?在最下面回答(*a1)。

我们可以通过将in_range转换为生成器来改进它:

def in_range(origin, range, things):
    range_sq = range**2
    yield from (thing for thing in things
                if distance_sq(origin, thing) <= range_sq)

如果你在做以下事情，这尤其有好处:

if any(in_range(origin, max_dist, things)):
    ...

但如果你接下来要做的事需要一段距离，

for nearby in in_range(origin, walking_distance, hotdog_stands):
    print("%s %.2fm" % (nearby.name, distance(origin, nearby)))

考虑生成元组:

def in_range_with_dist_sq(origin, range, things):
    range_sq = range**2
    for thing in things:
        dist_sq = distance_sq(origin, thing)
        if dist_sq <= range_sq: yield (thing, dist_sq)

如果你可能要进行连锁范围检查(“找到X附近和Y的Nm范围内的东西”，因为你不需要再次计算距离)，这可能特别有用。

但如果我们在搜索一个很大的列表，我们预计其中有很多不值得考虑呢?

其实有一个很简单的优化:

def in_range_all_the_things(origin, range, things):
    range_sq = range**2
    for thing in things:
        dist_sq = (origin.x - thing.x) ** 2
        if dist_sq <= range_sq:
            dist_sq += (origin.y - thing.y) ** 2
            if dist_sq <= range_sq:
                dist_sq += (origin.z - thing.z) ** 2
                if dist_sq <= range_sq:
                    yield thing

这是否有用取决于“事物”的大小。

def in_range_all_the_things(origin, range, things):
    range_sq = range**2
    if len(things) >= 4096:
        for thing in things:
            dist_sq = (origin.x - thing.x) ** 2
            if dist_sq <= range_sq:
                dist_sq += (origin.y - thing.y) ** 2
                if dist_sq <= range_sq:
                    dist_sq += (origin.z - thing.z) ** 2
                    if dist_sq <= range_sq:
                        yield thing
    elif len(things) > 32:
        for things in things:
            dist_sq = (origin.x - thing.x) ** 2
            if dist_sq <= range_sq:
                dist_sq += (origin.y - thing.y) ** 2 + (origin.z - thing.z) ** 2
                if dist_sq <= range_sq:
                    yield thing
    else:
        ... just calculate distance and range-check it ...

同样，考虑生成dist_sq。热狗的例子就变成了:

# Chaining generators
info = in_range_with_dist_sq(origin, walking_distance, hotdog_stands)
info = (stand, dist_sq**0.5 for stand, dist_sq in info)
for stand, dist in info:
    print("%s %.2fm" % (stand, dist))

(*a1: sort_things_by_distance的排序键为每一项调用distance_sq，而那个看起来无辜的键是一个lambda，这是必须调用的第二个函数…)

2017-11-28 22:54:12

其他回答

使用scipy.spatial.distance.euclidean:

from scipy.spatial import distance
a = (1, 2, 3)
b = (4, 5, 6)
dst = distance.euclidean(a, b)

2014-02-24 11:32:29