


'''simple docstring'''





from typing import SupportsIndex

import numpy as np
from scipy.ndimage import convolve






def 	UpperCAmelCase__(							_SCREAMING_SNAKE_CASE				:   np.ndarray,_SCREAMING_SNAKE_CASE				:   np.ndarray,_SCREAMING_SNAKE_CASE				:   np.ndarray   ):






			"""simple docstring"""




			__A=  np.stack((horizontal_flow, vertical_flow),2   )

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

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

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

			# Set pixels at invalid locations to 0
			__A=  0

			return warped






def 	UpperCAmelCase__(							_SCREAMING_SNAKE_CASE				:   np.ndarray,_SCREAMING_SNAKE_CASE				:   np.ndarray,_SCREAMING_SNAKE_CASE				:   SupportsIndex,_SCREAMING_SNAKE_CASE				:   float | None = None,):






			"""simple docstring"""




			if alpha is None:
						__A=  0.1

			# Initialize flow
			__A=  np.zeros_like(_SCREAMING_SNAKE_CASE   )
			__A=  np.zeros_like(_SCREAMING_SNAKE_CASE   )

			# Prepare kernels for the calculation of the derivatives and the average velocity
			__A=  np.array([[-1, 1], [-1, 1]]   ) * 0.25
			__A=  np.array([[-1, -1], [1, 1]]   ) * 0.25
			__A=  np.array([[1, 1], [1, 1]]   ) * 0.25
			__A=  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(_SCREAMING_SNAKE_CASE   ):
						__A=  warp(_SCREAMING_SNAKE_CASE,_SCREAMING_SNAKE_CASE,_SCREAMING_SNAKE_CASE   )
						__A=  convolve(_SCREAMING_SNAKE_CASE,_SCREAMING_SNAKE_CASE   ) + convolve(_SCREAMING_SNAKE_CASE,_SCREAMING_SNAKE_CASE   )
						__A=  convolve(_SCREAMING_SNAKE_CASE,_SCREAMING_SNAKE_CASE   ) + convolve(_SCREAMING_SNAKE_CASE,_SCREAMING_SNAKE_CASE   )
						__A=  convolve(_SCREAMING_SNAKE_CASE,_SCREAMING_SNAKE_CASE   ) + convolve(_SCREAMING_SNAKE_CASE,-kernel_t   )

						__A=  convolve(_SCREAMING_SNAKE_CASE,_SCREAMING_SNAKE_CASE   )
						__A=  convolve(_SCREAMING_SNAKE_CASE,_SCREAMING_SNAKE_CASE   )

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

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

			return horizontal_flow, vertical_flow


if __name__ == "__main__":
					import doctest

					doctest.testmod()







'''simple docstring'''





UpperCAmelCase__      =							[[0, 1, 1], [0, 0, 1], [1, 0, 0]]
class 	a__				:





			'''simple docstring'''

			def __init__(     self       : Any ,						lowerCAmelCase_       : Optional[int]  )			->		Tuple:
						__A=  name
						__A=  []
						__A=  []
			def 				lowerCAmelCase    (     self       : Any ,						lowerCAmelCase_       : Dict  )			->		Any:
						self.inbound.append(lowerCAmelCase_  )
			def 				lowerCAmelCase    (     self       : str ,						lowerCAmelCase_       : Any  )			->		List[Any]:
						self.outbound.append(lowerCAmelCase_  )

			def __repr__(     self       : int  )			->		Union[str, Any]:
						return F"""<node={self.name} inbound={self.inbound} outbound={self.outbound}>"""






def 	UpperCAmelCase__(							_SCREAMING_SNAKE_CASE				:   Any,_SCREAMING_SNAKE_CASE				:   Optional[int]=3,_SCREAMING_SNAKE_CASE				:   Dict=0.85   ):






			"""simple docstring"""




			__A=  {}
			for node in nodes:
						__A=  1

			__A=  {}
			for node in nodes:
						__A=  len(node.outbound   )

			for i in range(_SCREAMING_SNAKE_CASE   ):
						print(f"""======= Iteration {i + 1} ======="""   )
						for _, node in enumerate(_SCREAMING_SNAKE_CASE   ):
									__A=  (1 - d) + d * sum(
									    ranks[ib] / outbounds[ib] for ib in node.inbound   )
						print(_SCREAMING_SNAKE_CASE   )






def 	UpperCAmelCase__(							):






			"""simple docstring"""




			__A=  list(input('Enter Names of the Nodes: '   ).split()   )

			__A=  [Node(_SCREAMING_SNAKE_CASE   ) for name in names]

			for ri, row in enumerate(_SCREAMING_SNAKE_CASE   ):
						for ci, col in enumerate(_SCREAMING_SNAKE_CASE   ):
									if col == 1:
												nodes[ci].add_inbound(names[ri]   )
												nodes[ri].add_outbound(names[ci]   )

			print('======= Nodes ======='   )
			for node in nodes:
						print(_SCREAMING_SNAKE_CASE   )

			page_rank(_SCREAMING_SNAKE_CASE   )


if __name__ == "__main__":
					main()



