





from typing import SupportsIndex

import numpy as np
from scipy.ndimage import convolve







def    UpperCAmelCase				(  _lowerCamelCase   :	np.ndarray      , _lowerCamelCase   :	np.ndarray      , _lowerCamelCase   :	np.ndarray  ):





						'''simple docstring'''


						SCREAMING_SNAKE_CASE__  :					List[str] =       np.stack((horizontal_flow, vertical_flow)      , 2  )

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

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

						SCREAMING_SNAKE_CASE__  :					Any =       image[grid[:, :, 1], grid[:, :, 0]]

						# Set pixels at invalid locations to 0
						SCREAMING_SNAKE_CASE__  :					Tuple =       0

						return warped







def    UpperCAmelCase				(  _lowerCamelCase   :	np.ndarray      , _lowerCamelCase   :	np.ndarray      , _lowerCamelCase   :	SupportsIndex      , _lowerCamelCase   :	float | None = None      , ):





						'''simple docstring'''


						if alpha is None:
												SCREAMING_SNAKE_CASE__  :					List[Any] =       0.1

						# Initialize flow
						SCREAMING_SNAKE_CASE__  :					Optional[int] =       np.zeros_like(_lowerCAmelCase  )
						SCREAMING_SNAKE_CASE__  :					List[str] =       np.zeros_like(_lowerCAmelCase  )

						# Prepare kernels for the calculation of the derivatives and the average velocity
						SCREAMING_SNAKE_CASE__  :					Optional[int] =       np.array([[-1, 1], [-1, 1]]  ) * 0.2_5
						SCREAMING_SNAKE_CASE__  :					str =       np.array([[-1, -1], [1, 1]]  ) * 0.2_5
						SCREAMING_SNAKE_CASE__  :					Dict =       np.array([[1, 1], [1, 1]]  ) * 0.2_5
						SCREAMING_SNAKE_CASE__  :					List[str] =       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(_lowerCAmelCase  ):
												SCREAMING_SNAKE_CASE__  :					Any =       warp(_lowerCAmelCase      , _lowerCAmelCase      , _lowerCAmelCase  )
												SCREAMING_SNAKE_CASE__  :					int =       convolve(_lowerCAmelCase      , _lowerCAmelCase  ) + convolve(_lowerCAmelCase      , _lowerCAmelCase  )
												SCREAMING_SNAKE_CASE__  :					Union[str, Any] =       convolve(_lowerCAmelCase      , _lowerCAmelCase  ) + convolve(_lowerCAmelCase      , _lowerCAmelCase  )
												SCREAMING_SNAKE_CASE__  :					Dict =       convolve(_lowerCAmelCase      , _lowerCAmelCase  ) + convolve(_lowerCAmelCase      , -kernel_t  )

												SCREAMING_SNAKE_CASE__  :					Dict =       convolve(_lowerCAmelCase      , _lowerCAmelCase  )
												SCREAMING_SNAKE_CASE__  :					List[str] =       convolve(_lowerCAmelCase      , _lowerCAmelCase  )

												# This updates the flow as proposed in the paper (Step 12)
												SCREAMING_SNAKE_CASE__  :					List[str] =       (
												    derivative_x * avg_horizontal_velocity
												    + derivative_y * avg_vertical_velocity
												    + derivative_t
												)
												SCREAMING_SNAKE_CASE__  :					Any =       update / (alpha**2 + derivative_x**2 + derivative_y**2)

												SCREAMING_SNAKE_CASE__  :					List[str] =       avg_horizontal_velocity - derivative_x * update
												SCREAMING_SNAKE_CASE__  :					Optional[Any] =       avg_vertical_velocity - derivative_y * update

						return horizontal_flow, vertical_flow


if __name__ == "__main__":
				import doctest

				doctest.testmod()





from math import cos, sin



def     lowerCAmelCase		(		_lowerCAmelCase				: float						):




 """simple docstring"""







 UpperCAmelCase__							=    float(_lowerCAmelCase						)
 return [[scaling_factor * int(x == y						) for x in range(2						)] for y in range(2						)]



def     lowerCAmelCase		(		_lowerCAmelCase				: float						):




 """simple docstring"""







 UpperCAmelCase__			,			UpperCAmelCase__							=    cos(_lowerCAmelCase						), sin(_lowerCAmelCase						)
 return [[c, -s], [s, c]]



def     lowerCAmelCase		(		_lowerCAmelCase				: float						):




 """simple docstring"""







 UpperCAmelCase__			,			UpperCAmelCase__							=    cos(_lowerCAmelCase						), sin(_lowerCAmelCase						)
 UpperCAmelCase__							=    c * s
 return [[c * c, cs], [cs, s * s]]



def     lowerCAmelCase		(		_lowerCAmelCase				: float						):




 """simple docstring"""







 UpperCAmelCase__			,			UpperCAmelCase__							=    cos(_lowerCAmelCase						), sin(_lowerCAmelCase						)
 UpperCAmelCase__							=    c * s
 return [[2 * c - 1, 2 * cs], [2 * cs, 2 * s - 1]]


print(F'''    {scaling(5) = }''')
print(F'''  {rotation(4_5) = }''')
print(F'''{projection(4_5) = }''')


print(F'''{reflection(4_5) = }''')





