spatialmath-python/spatialmath/base/numeric.py at master · Derek-TH-Wang/spatialmath-python · GitHub

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import numpy as np

from spatialmath import base

# this is a collection of useful algorithms, not otherwise categorized

def numjac(f, x, dx=1e-8, SO=0, SE=0):

r"""

Numerically compute Jacobian of function

:param f: the function, returns an m-vector

:type f: callable

:param x: function argument

:type x: ndarray(n)

:param dx: the numerical perturbation, defaults to 1e-8

:type dx: float, optional

:param SO: function returns SO(N) matrix, defaults to 0

:type SO: int, optional

:param SE: function returns SE(N) matrix, defaults to 0

:type SE: int, optional

:return: Jacobian matrix

:rtype: ndarray(m,n)

Computes a numerical approximation to the Jacobian for ``f(x)`` where

:math:`f: \mathbb{R}^n \mapsto \mathbb{R}^m`.

Uses first-order difference :math:`J[:,i] = (f(x + dx) - f(x)) / dx`.

If ``SO`` is 2 or 3, then it is assumed that the function returns

an SO(N) matrix and the derivative is converted to a column vector

.. math:

\vex \dmat{R} \mat{R}^T

If ``SE`` is 2 or 3, then it is assumed that the function returns

an SE(N) matrix and the derivative is converted to a colun vector.

"""

x = np.array(x)

Jcol = []

J0 = f(x)

I = np.eye(len(x))

f0 = np.array(f(x))

for i in range(len(x)):

fi = np.array(f(x + I[:, i] * dx))

Ji = (fi - f0) / dx

if SE > 0:

t = Ji[:SE, SE]

r = base.vex(Ji[:SE, :SE] @ J0[:SE, :SE].T)

Jcol.append(np.r_[t, r])

elif SO > 0:

R = Ji[:SO, :SO]

r = base.vex(R @ J0[:SO, :SO].T)

Jcol.append(r)

else:

Jcol.append(Ji)

# print(Ji)

return np.c_[Jcol].T

def numhess(J, x, dx=1e-8):

r"""

Numerically compute Hessian given Jacobian function

:param J: the Jacobian function, returns an ndarray(m,n)

:type J: callable

:param x: function argument

:type x: ndarray(n)

:param dx: the numerical perturbation, defaults to 1e-8

:type dx: float, optional

:return: Hessian matrix

:rtype: ndarray(m,n,n)

Computes a numerical approximation to the Hessian for ``J(x)`` where

:math:`f: \mathbb{R}^n \mapsto \mathbb{R}^{m \times n}`.

The result is a 3D array where

.. math::

H_{i,j,k} = \frac{\partial J_{j,k}}{\partial x_i}

Uses first-order difference :math:`H[:,:,i] = (J(x + dx) - J(x)) / dx`.

"""

I = np.eye(len(x))

Hcol = []

J0 = J(x)

for i in range(len(x)):

Ji = J(x + I[:,i] * dx)

Hi = (Ji - J0) / dx

Hcol.append(Hi)

return np.stack(Hcol, axis=0)

def array2str(X, valuesep=", ", rowsep=" | ", fmt="{:.3g}",

brackets=("[ ", " ]"), suppress_small=True):

"""

Convert array to single line string

:param X: 1D or 2D array to convert

:type X: ndarray(N,M), array_like(N)

:param valuesep: separator between numbers, defaults to ", "

:type valuesep: str, optional

:param rowsep: separator between rows, defaults to " | "

:type rowsep: str, optional

:param format: format string, defaults to "{:.3g}"

:type precision: str, optional

:param brackets: strings to be added to start and end of the string,

defaults to ("[ ", " ]"). Set to None to suppress brackets.

:type brackets: list, tuple of str

:param suppress_small: small values (:math:`|x| < 10^{-12}` are converted

to zero, defaults to True

:type suppress_small: bool, optional

:return: compact string representation of array

:rtype: str

Converts a small array to a compact single line representation.

"""

# convert to ndarray if not already

if isinstance(X, (list, tuple)):

X = base.getvector(X)

def format_row(x):

s = ""

for j, e in enumerate(x):

if abs(e) < 1e-12:

e = 0

if j > 0:

s += valuesep

s += fmt.format(e)

return s

if X.ndim == 1:

# 1D case

s = format_row(X)

else:

# 2D case

s = ""

for i, row in enumerate(X):

if i > 0:

s += rowsep

s += format_row(row)

if brackets is not None and len(brackets) == 2:

s = brackets[0] + s + brackets[1]

return s

def bresenham(p0, p1, array=None):

"""

Line drawing in a grid

:param p0: initial point

:type p0: array_like(2) of int

:param p1: end point

:type p1: array_like(2) of int

:return: arrays of x and y coordinates for points along the line

:rtype: ndarray(N), ndarray(N) of int

Return x and y coordinate vectors for points in a grid that lie on

a line from ``p0`` to ``p1`` inclusive.

The end points, and all points along the line are integers.

.. note:: The API is similar to the Bresenham algorithm but this

implementation uses NumPy vectorised arithmetic which makes it

faster than the Bresenham algorithm in Python.

"""

x0, y0 = p0

x1, y1 = p1

if array is not None:

_ = array[y0, x0] + array[y1, x1]

dx = x1 - x0

dy = y1 - y0

if abs(dx) >= abs(dy):

# shallow line -45° <= θ <= 45°

# y = mx + c

if dx == 0:

# case p0 == p1

x = np.r_[x0]

y = np.r_[y0]

else:

m = dy / dx

c = y0 - m * x0

if dx > 0:

# line to the right

x = np.arange(x0, x1 + 1)

elif dx < 0:

# line to the left

x = np.arange(x0, x1 - 1, -1)

y = np.round(x * m + c)

else:

# steep line θ < -45°, θ > 45°

# x = my + c

m = dx / dy

c = x0 - m * y0

if dy > 0:

# line to the right

y = np.arange(y0, y1 + 1)

elif dy < 0:

# line to the left

y = np.arange(y0, y1 - 1, -1)

x = np.round(y * m + c)

return x.astype(int), y.astype(int)

def mpq_point(data, p, q):

r"""

Moments of polygon

:param p: moment order x

:type p: int

:param q: moment order y

:type q: int

Returns the pq'th moment of the polygon

.. math::

M(p, q) = \sum_{i=0}^{n-1} x_i^p y_i^q

Example:

.. runblock:: pycon

>>> from spatialmath import Polygon2

>>> p = Polygon2([[1, 3, 2], [2, 2, 4]])

>>> p.moment(0, 0) # area

>>> p.moment(3, 0)

Note is negative for clockwise perimeter.

"""

x = data[0, :]

y = data[1, :]

return np.sum(x**p * y**q)

def gauss1d(mu, var, x):

"""

Gaussian function in 1D

:param mu: mean

:type mu: float

:param var: variance

:type var: float

:param x: x-coordinate values

:type x: array_like(n)

:return: Gaussian :math:`G(x)`

:rtype: ndarray(n)

:seealso: :func:`gauss2d`

"""

sigma = np.sqrt(var)

x = base.getvector(x)

return 1.0 / np.sqrt(sigma**2 * 2 * np.pi) * np.exp(-(x-mu)**2/2/sigma**2)

def gauss2d(mu, P, X, Y):

"""

Gaussian function in 2D

:param mu: mean

:type mu: array_like(2)

:param P: covariance matrix

:type P: ndarray(2,2)

:param X: array of x-coordinates

:type X: ndarray(n,m)

:param Y: array of y-coordinates

:type Y: ndarray(n,m)

:return: Gaussian :math:`g(x,y)`

:rtype: ndarray(n,m)

Computed :math:`g_{i,j} = G(x_{i,j}, y_{i,j})`

:seealso: :func:`gauss1d`

"""

x = X.ravel() - mu[0]

y = Y.ravel() - mu[1]

Pi = np.linalg.inv(P);

g = 1/(2*np.pi*np.sqrt(np.linalg.det(P))) * np.exp(

-0.5*(x**2 * Pi[0, 0] + y**2 * Pi[1, 1] + 2 * x * y * Pi[0, 1]));

return g.reshape(X.shape)

if __name__ == "__main__":

r = np.linspace(-4, 4, 6)

x, y = np.meshgrid(r, r)

print(gauss2d([0, 0], np.diag([1,2]), x, y))

# print(bresenham([2,2], [2,4]))

# print(bresenham([2,2], [2,-4]))

# print(bresenham([2,2], [4,2]))

# print(bresenham([2,2], [-4,2]))

# print(bresenham([2,2], [2,2]))

# print(bresenham([2,2], [3,6])) # steep

# print(bresenham([2,2], [6,3])) # shallow

# print(bresenham([2,2], [3,6])) # steep

# print(bresenham([2,2], [6,3])) # shallow

spatialmath-python/spatialmath/base/numeric.py at master · Derek-TH-Wang/spatialmath-python · GitHub

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