code: 
  
  
  '''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()
  
  
  
  
style_context: 
  
  
  '''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()
  
  
  
  