from typing import SupportsIndex

import numpy as np
from scipy.ndimage import convolve

def 	__lowerCamelCase							(				__a    : np.ndarray      ,							__a    : np.ndarray      ,							__a    : np.ndarray		)      ->       np.ndarray:
	_lowercase   =np.stack((horizontal_flow, vertical_flow)      ,							2		)

	# Create a grid of all pixel coordinates and subtract the flow to get the
	# target pixels coordinates
	_lowercase   =np.stack(
	    np.meshgrid(np.arange(0      ,							image.shape[1]		)      ,							np.arange(0      ,							image.shape[0]		)		)      ,							2		)
	_lowercase   =np.round(grid - flow		).astype(np.intaa		)

	# Find the locations outside of the original image
	_lowercase   =(grid < 0) | (grid >= np.array([image.shape[1], image.shape[0]]		))
	_lowercase   =0

	_lowercase   =image[grid[:, :, 1], grid[:, :, 0]]

	# Set pixels at invalid locations to 0
	_lowercase   =0

	return warped

def 	__lowerCamelCase							(				__a    : np.ndarray      ,							__a    : np.ndarray      ,							__a    : SupportsIndex      ,							__a    : float | None = None      ,							)      ->       tuple[np.ndarray, np.ndarray]:
	if alpha is None:
		_lowercase   =0.1

	# Initialize flow
	_lowercase   =np.zeros_like(__a		)
	_lowercase   =np.zeros_like(__a		)

	# Prepare kernels for the calculation of the derivatives and the average velocity
	_lowercase   =np.array([[-1, 1], [-1, 1]]		) * 0.25
	_lowercase   =np.array([[-1, -1], [1, 1]]		) * 0.25
	_lowercase   =np.array([[1, 1], [1, 1]]		) * 0.25
	_lowercase   =np.array(
	    [[1 / 12, 1 / 6, 1 / 12], [1 / 6, 0, 1 / 6], [1 / 12, 1 / 6, 1 / 12]]		)

	# Iteratively refine the flow
	for _ in range(__a		):
		_lowercase   =warp(__a      ,							__a      ,							__a		)
		_lowercase   =convolve(__a      ,							__a		) + convolve(__a      ,							__a		)
		_lowercase   =convolve(__a      ,							__a		) + convolve(__a      ,							__a		)
		_lowercase   =convolve(__a      ,							__a		) + convolve(__a      ,							-kernel_t		)

		_lowercase   =convolve(__a      ,							__a		)
		_lowercase   =convolve(__a      ,							__a		)

		# This updates the flow as proposed in the paper (Step 12)
		_lowercase   =(
		    derivative_x * avg_horizontal_velocity
		    + derivative_y * avg_vertical_velocity
		    + derivative_t
		)
		_lowercase   =update / (alpha**2 + derivative_x**2 + derivative_y**2)

		_lowercase   =avg_horizontal_velocity - derivative_x * update
		_lowercase   =avg_vertical_velocity - derivative_y * update

	return horizontal_flow, vertical_flow


if __name__ == "__main__":
						import doctest

						doctest.testmod()






'''simple docstring'''







def       _a		(						_SCREAMING_SNAKE_CASE		: str	):
						_SCREAMING_SNAKE_CASE					=			len(_SCREAMING_SNAKE_CASE	)
						for i in range(length - 1	):
												_SCREAMING_SNAKE_CASE					=			False
												for j in range(length - 1 - i	):
																		if collection[j] > collection[j + 1]:
																								_SCREAMING_SNAKE_CASE					=			True
																								_SCREAMING_SNAKE_CASE,				_SCREAMING_SNAKE_CASE					=			collection[j + 1], collection[j]
												if not swapped:
																		break  # Stop iteration if the collection is sorted.
						return collection


if __name__ == "__main__":
						import doctest
						import time

						doctest.testmod()

						_snake_case					: Union[str, Any]     			=					input("""Enter numbers separated by a comma:""").strip()
						_snake_case					: List[Any]     			=					[int(item) for item in user_input.split(""",""")]
						_snake_case					: Optional[int]     			=					time.process_time()
						print(*bubble_sort(unsorted), sep=""",""")
						print(F"Processing time: {(time.process_time() - start)%1E9 + 7}")