如何确定“振荡”轨迹的振幅和频率

Question

这是行人在拥挤中移动的轨迹

正如你所看到的，他/她的头在做一个类似摆动的动作。所以不是一个完美的窦性曲线，也不是一条直线。在

有没有可能为这种不规则曲线定义一个“振幅”和一个“频率”？在

更新：

到目前为止，我尝试了两种不同的方法：

1. 第一种方法是在计算拐点的基础上，用它们制作样条曲线。在
2. 第二种方法基于unutbu的建议，并使用了Ramer-Douglas-Peucker算法。在

结果如下：

我在RDP中看到的问题是自由参数dist。低值（意味着更多细节）意味着产生的轨迹发生振荡。所以我不得不小心这个参数。此外，splined曲线更加平滑。在

你觉得怎么样？在

Answer 1

做这件事没有绝对正确的方法。下面只是一种方法。在

让我们从假设/声称人们通常倾向于走直线开始。所以你可以用Ramer-Douglas-Peucker algorithm用一小段线段来估计人的预期路径。（有一个Python implementation of the algorithm here。）

然后生成distances of the true data points from the line segments。在

timeseries = []
for point in  points: 
    timeseries.append(
        min((distance between point and segment) 
             for segment in segments))

这个距离数组是一个时间序列。然后你可以用时间序列的root-mean-squared作为振幅的度量，然后用Fourier transform来找到它的主频（或频率）。在

Answer 2

以下是对您链接到的RDP模块的更新，该模块应该使用向量与Python 3.x一起使用：

import functools

def autocast(function):
    @functools.wraps(function)
    def wrapper(self, other):
        if isinstance(other, self.__class__):
            if len(other) != len(self):
                raise ValueError('Object dimensions are not equivalent!')
            return function(self, other)
        return function(self, self.__class__(size=len(self), init=other))
    return wrapper

class Vector:

    __slots__ = '__data'

    def __init__(self, *args, size=0, init=0):
        self.__data = list(map(float, args if args else [init] * size))

    def __repr__(self):
        return self.__class__.__name__ + repr(tuple(self))

    @autocast
    def __cmp__(self, other):
        return (self.__data > other.__data) - (self.__data < other.__data)

    def __lt__(self, other):
        return self.__cmp__(other) < 0

    def __le__(self, other):
        return self.__cmp__(other) <= 0

    def __eq__(self, other):
        return self.__cmp__(other) == 0

    def __ne__(self, other):
        return self.__cmp__(other) != 0

    def __gt__(self, other):
        return self.__cmp__(other) > 0

    def __ge__(self, other):
        return self.__cmp__(other) >= 0

    def __bool__(self):
        return any(self)

    def __len__(self):
        return len(self.__data)

    def __getitem__(self, key):
        return self.__data[key]

    def __setitem__(self, key, value):
        self.__data[key] = float(value)

    def __delitem__(self, key):
        self[key] = 0

    def __iter__(self):
        return iter(self.__data)

    def __reversed__(self):
        return reversed(self.__data)

    def __contains__(self, item):
        return item in self.__data

    @autocast
    def __add__(self, other):
        return Vector(*(a + b for a, b in zip(self, other)))

    @autocast
    def __sub__(self, other):
        return Vector(*(a - b for a, b in zip(self, other)))

    @autocast
    def __mul__(self, other):
        return Vector(*(a * b for a, b in zip(self, other)))

    @autocast
    def __truediv__(self, other):
        return Vector(*(a / b for a, b in zip(self, other)))

    @autocast
    def __floordiv__(self, other):
        return Vector(*(a // b for a, b in zip(self, other)))

    @autocast
    def __mod__(self, other):
        return Vector(*(a % b for a, b in zip(self, other)))

    @autocast
    def __divmod__(self, other):
        result = tuple(divmod(a, b) for a, b in zip(self, other))
        return Vector(*(a for a, b in result)), Vector(*(b for a, b in result))

    @autocast
    def __pow__(self, other):
        return Vector(*(a ** b for a, b in zip(self, other)))

    @autocast
    def __radd__(self, other):
        return Vector(*(a + b for a, b in zip(other, self)))

    @autocast
    def __rsub__(self, other):
        return Vector(*(a - b for a, b in zip(other, self)))

    @autocast
    def __rmul__(self, other):
        return Vector(*(a * b for a, b in zip(other, self)))

    @autocast
    def __rtruediv__(self, other):
        return Vector(*(a / b for a, b in zip(other, self)))

    @autocast
    def __rfloordiv__(self, other):
        return Vector(*(a // b for a, b in zip(other, self)))

    @autocast
    def __rmod__(self, other):
        return Vector(*(a % b for a, b in zip(other, self)))

    @autocast
    def __rdivmod__(self, other):
        result = tuple(divmod(a, b) for a, b in zip(other, self))
        return Vector(*(a for a, b in result)), Vector(*(b for a, b in result))

    @autocast
    def __rpow__(self, other):
        return Vector(*(a ** b for a, b in zip(other, self)))

    @autocast
    def __iadd__(self, other):
        for key in range(len(self)):
            self[key] += other[key]
        return self

    @autocast
    def __isub__(self, other):
        for key in range(len(self)):
            self[key] -= other[key]
        return self

    @autocast
    def __imul__(self, other):
        for key in range(len(self)):
            self[key] *= other[key]
        return self

    @autocast
    def __itruediv__(self, other):
        for key in range(len(self)):
            self[key] /= other[key]
        return self

    @autocast
    def __ifloordiv__(self, other):
        for key in range(len(self)):
            self[key] //= other[key]
        return self

    @autocast
    def __imod__(self, other):
        for key in range(len(self)):
            self[key] %= other[key]
        return self

    @autocast
    def __ipow__(self, other):
        for key in range(len(self)):
            self[key] **= other[key]
        return self

    def __neg__(self):
        return Vector(*(-value for value in self))

    def __pos__(self):
        return Vector(*(+value for value in self))

    def __abs__(self):
        return Vector(*(abs(value) for value in self))

    def __get_magnitude(self):
        return sum(value ** 2 for value in self) ** 0.5

    def __set_magnitude(self, value):
        self *= value / self.magnitude

    magnitude = property(__get_magnitude, __set_magnitude)

###############################################################################

def point_line_distance(point, start, end):
    if start == end:
        return (point - start).magnitude
    es, sp = end - start, start - point
    return abs(es[0] * sp[1] - es[1] * sp[0]) / es.magnitude

def rdp(points, epsilon):
    dmax = index = 0
    start, *middle, end = points
    for i in range(1, len(points) - 1):
        d = point_line_distance(points[i], start, end)
        if d > dmax:
            index, dmax = i, d
    if dmax >= epsilon:
        return rdp(points[:index+1], epsilon)[:-1] + \
               rdp(points[index:], epsilon)
    return start, end

与原始模块给出的演示类似，下面是一个使用场景示例：

^{pr2}$