"""simple docstring"""






def   a_						( lowerCamelCase	):
				return " ".join(sorted(set(sentence.split()	)	)	)


if __name__ == "__main__":
						import doctest

						doctest.testmod()




"""simple docstring"""






from typing import SupportsIndex

import numpy as np
from scipy.ndimage import convolve



def   a_						( lowerCamelCase ,      lowerCamelCase ,      lowerCamelCase	):
				UpperCAmelCase__					=							np.stack((horizontal_flow, vertical_flow) ,      2	)

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

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

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

				# Set pixels at invalid locations to 0
				UpperCAmelCase__					=							0

				return warped



def   a_						( lowerCamelCase ,      lowerCamelCase ,      lowerCamelCase ,      lowerCamelCase = None ,      ):
				if alpha is None:
								UpperCAmelCase__					=							0.1

				# Initialize flow
				UpperCAmelCase__					=							np.zeros_like(lowerCamelCase	)
				UpperCAmelCase__					=							np.zeros_like(lowerCamelCase	)

				# Prepare kernels for the calculation of the derivatives and the average velocity
				UpperCAmelCase__					=							np.array([[-1, 1], [-1, 1]]	) * 0.25
				UpperCAmelCase__					=							np.array([[-1, -1], [1, 1]]	) * 0.25
				UpperCAmelCase__					=							np.array([[1, 1], [1, 1]]	) * 0.25
				UpperCAmelCase__					=							np.array(
				    [[1 / 1_2, 1 / 6, 1 / 1_2], [1 / 6, 0, 1 / 6], [1 / 1_2, 1 / 6, 1 / 1_2]]	)

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

								UpperCAmelCase__					=							convolve(lowerCamelCase ,      lowerCamelCase	)
								UpperCAmelCase__					=							convolve(lowerCamelCase ,      lowerCamelCase	)

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

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

				return horizontal_flow, vertical_flow


if __name__ == "__main__":
						import doctest

						doctest.testmod()



