def gradient_points(
self, points: np.ndarray, params: Union[List[float], List[Interval]]
) -> Union[List[np.ndarray], List[Interval]]:
(alpha, ) = params
x = points[:, 0]
y = points[:, 1]
z = points[:, 2]
zero = iv.zeros_like(z)
one = iv.ones_like(z)
return [
iv.stack(
[iv.cos(alpha * z), iv.sin(alpha * z), zero],
axis=1,
convert=isinstance(alpha, Interval)),
iv.stack([-iv.sin(alpha * z),
iv.cos(alpha * z), zero],
axis=1,
convert=isinstance(alpha, Interval)),
iv.stack([
alpha * (-iv.sin(alpha * z) * x - iv.cos(alpha * z) * y),
alpha * (iv.cos(alpha * z) * x - iv.sin(alpha * z) * y), one
],
axis=1,
convert=isinstance(alpha, Interval)),
]
def hessian_points(self, points: np.ndarray, params: Union[List[float], List[Interval]]) -> Union[List[List[np.ndarray]], List[List[Interval]]]:
a, b, theta = params
z = points[:, 2]
zero = iv.zeros_like(z)
one = iv.ones_like(z)
return [
[
iv.stack([
zero,
zero,
zero
], axis=1, convert=isinstance(a, Interval)),
iv.stack([
zero,
zero,
zero
], axis=1, convert=isinstance(a, Interval)),
iv.stack([
(0.5 * iv.square(a) + b) * iv.cos(theta) * one,
(0.5 * iv.square(a) + b) * iv.sin(theta) * one,
zero
], axis=1, convert=isinstance(a, Interval)),
],
[
iv.stack([
zero,
zero,
zero
], axis=1, convert=isinstance(a, Interval)),
iv.stack([
zero,
zero,
zero
], axis=1, convert=isinstance(a, Interval)),
iv.stack([
-(0.5 * iv.square(a) + b) * iv.sin(theta) * one,
(0.5 * iv.square(a) + b) * iv.cos(theta) * one,
zero
], axis=1, convert=isinstance(a, Interval))
],
[
iv.stack([
(0.5 * iv.square(a) + b) * iv.cos(theta) * one,
(0.5 * iv.square(a) + b) * iv.sin(theta) * one,
zero
], axis=1, convert=isinstance(a, Interval)),
iv.stack([
-(0.5 * iv.square(a) + b) * iv.sin(theta) * one,
(0.5 * iv.square(a) + b) * iv.cos(theta) * one,
zero
], axis=1, convert=isinstance(a, Interval)),
iv.stack([
zero,
zero,
zero
], axis=1, convert=isinstance(a, Interval))
],
]
def gradient_points(self, points: np.ndarray, params: Union[List[float], List[Interval]]) -> Union[List[np.ndarray], List[Interval]]:
sx, sy = params
z = points[:, 2]
zero = iv.zeros_like(z)
one = iv.ones_like(z)
return [
iv.stack([one, zero, zero], axis=1, convert=isinstance(sx, Interval)),
iv.stack([zero, one, zero], axis=1, convert=isinstance(sx, Interval)),
iv.stack([sx * one, sy * one, one], axis=1, convert=isinstance(sx, Interval)),
]
def hessian_points_params(self, points: np.ndarray, params: Union[List[float], List[Interval]]) -> Union[List[List[np.ndarray]], List[List[Interval]]]:
a, b = params
x = points[:, 0]
zero = iv.zeros_like(x)
one = iv.ones_like(x)
return [
[
iv.stack([
-iv.sin(a) * one,
iv.cos(a) * one,
zero
], axis=1, convert=isinstance(a, Interval)),
iv.stack([
zero,
zero,
zero
], axis=1, convert=isinstance(a, Interval)),
],
[
iv.stack([
-iv.cos(a) * iv.cos(b) * one,
-iv.sin(a) * iv.cos(b) * one,
zero
], axis=1, convert=isinstance(a, Interval)),
iv.stack([
iv.sin(a) * iv.sin(b) * one,
-iv.cos(a) * iv.sin(b) * one,
iv.cos(b) * one
], axis=1, convert=isinstance(a, Interval)),
],
[
iv.stack([
iv.cos(a) * iv.sin(b) * one,
iv.sin(a) * iv.sin(b) * one,
zero
], axis=1, convert=isinstance(a, Interval)),
iv.stack([
iv.sin(a) * iv.cos(b) * one,
-iv.cos(a) * iv.cos(b) * one,
-iv.sin(b) * one
], axis=1, convert=isinstance(a, Interval)),
]
]
def hessian_points_params(self, points: np.ndarray, params: Union[List[float], List[Interval]]) -> Union[List[List[np.ndarray]], List[List[Interval]]]:
sx, sy = params
x = points[:, 0]
zero = iv.zeros_like(x)
one = iv.ones_like(x)
return [
[
iv.stack([zero, zero, zero], axis=1, convert=isinstance(sx, Interval)),
iv.stack([zero, zero, zero], axis=1, convert=isinstance(sx, Interval)),
],
[
iv.stack([zero, zero, zero], axis=1, convert=isinstance(sx, Interval)),
iv.stack([zero, zero, zero], axis=1, convert=isinstance(sx, Interval)),
],
[
iv.stack([one, zero, zero], axis=1, convert=isinstance(sx, Interval)),
iv.stack([zero, one, zero], axis=1, convert=isinstance(sx, Interval)),
],
]
def gradient_points(
self, points: np.ndarray, params: Union[List[float], List[Interval]]
) -> Union[List[np.ndarray], List[Interval]]:
a, b = params
x = points[:, 0]
y = points[:, 1]
z = points[:, 2]
zero = iv.zeros_like(z)
return [
iv.stack([(0.5 * iv.square(a) * z + b * z + 1), zero, zero],
axis=1,
convert=isinstance(a, Interval)),
iv.stack([zero, (0.5 * iv.square(a) * z + b * z + 1), zero],
axis=1,
convert=isinstance(a, Interval)),
iv.stack([(0.5 * iv.square(a) + b) * x,
(0.5 * iv.square(a) + b) * y,
iv.ones_like(z)],
axis=1,
convert=isinstance(a, Interval)),
]
def gradient_points(self, points: np.ndarray, params: Union[List[float], List[Interval]]) -> Union[List[np.ndarray], List[Interval]]:
a, b, theta = params
x = points[:, 0]
y = points[:, 1]
z = points[:, 2]
zero = iv.zeros_like(z)
one = iv.ones_like(z)
return [
iv.stack([
(0.5 * iv.square(a) * z + b * z + 1) * iv.cos(theta),
(0.5 * iv.square(a) * z + b * z + 1) * iv.sin(theta),
zero
], axis=1, convert=isinstance(a, Interval)),
iv.stack([
-(0.5 * iv.square(a) * z + b * z + 1) * iv.sin(theta),
(0.5 * iv.square(a) * z + b * z + 1) * iv.cos(theta),
zero
], axis=1, convert=isinstance(a, Interval)),
iv.stack([
(0.5 * iv.square(a) + b) * (iv.cos(theta) * x - iv.sin(theta) * y),
(0.5 * iv.square(a) + b) * (iv.sin(theta) * x + iv.cos(theta) * y),
one
], axis=1, convert=isinstance(a, Interval))
]