code: 
  '''simple docstring'''
  
  from typing import SupportsIndex
  
  import numpy as np
  from scipy.ndimage import convolve
  
  
  
  
  
  
  
  def 						snake_case__	(      UpperCamelCase			,UpperCamelCase			,UpperCamelCase	) ->      np.ndarray:
  		_UpperCamelCase       : Dict    =      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       : Optional[Any]    =      np.stack(
  		    np.meshgrid(np.arange(0			,image.shape[1]	)			,np.arange(0			,image.shape[0]	)	)			,2	)
  		_UpperCamelCase       : Tuple    =      np.round(grid - flow	).astype(np.intaa	)
  
  		# Find the locations outside of the original image
  		_UpperCamelCase       : str    =      (grid < 0) | (grid >= np.array([image.shape[1], image.shape[0]]	))
  		_UpperCamelCase       : Dict    =      0
  
  		_UpperCamelCase       : Union[str, Any]    =      image[grid[:, :, 1], grid[:, :, 0]]
  
  		# Set pixels at invalid locations to 0
  		_UpperCamelCase       : Tuple    =      0
  
  		return warped
  
  
  
  
  
  
  
  def 						snake_case__	(      UpperCamelCase			,UpperCamelCase			,UpperCamelCase			,UpperCamelCase = None			,) ->      tuple[np.ndarray, np.ndarray]:
  		if alpha is None:
  				_UpperCamelCase       : Optional[int]    =      0.1
  
  		# Initialize flow
  		_UpperCamelCase       : Tuple    =      np.zeros_like(UpperCamelCase	)
  		_UpperCamelCase       : Union[str, Any]    =      np.zeros_like(UpperCamelCase	)
  
  		# Prepare kernels for the calculation of the derivatives and the average velocity
  		_UpperCamelCase       : Tuple    =      np.array([[-1, 1], [-1, 1]]	) * 0.25
  		_UpperCamelCase       : Dict    =      np.array([[-1, -1], [1, 1]]	) * 0.25
  		_UpperCamelCase       : Tuple    =      np.array([[1, 1], [1, 1]]	) * 0.25
  		_UpperCamelCase       : Optional[Any]    =      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(UpperCamelCase	):
  				_UpperCamelCase       : Optional[int]    =      warp(UpperCamelCase			,UpperCamelCase			,UpperCamelCase	)
  				_UpperCamelCase       : List[Any]    =      convolve(UpperCamelCase			,UpperCamelCase	) + convolve(UpperCamelCase			,UpperCamelCase	)
  				_UpperCamelCase       : Optional[int]    =      convolve(UpperCamelCase			,UpperCamelCase	) + convolve(UpperCamelCase			,UpperCamelCase	)
  				_UpperCamelCase       : Union[str, Any]    =      convolve(UpperCamelCase			,UpperCamelCase	) + convolve(UpperCamelCase			,-kernel_t	)
  
  				_UpperCamelCase       : int    =      convolve(UpperCamelCase			,UpperCamelCase	)
  				_UpperCamelCase       : str    =      convolve(UpperCamelCase			,UpperCamelCase	)
  
  				# This updates the flow as proposed in the paper (Step 12)
  				_UpperCamelCase       : Union[str, Any]    =      (
  				    derivative_x * avg_horizontal_velocity
  				    + derivative_y * avg_vertical_velocity
  				    + derivative_t
  				)
  				_UpperCamelCase       : Tuple    =      update / (alpha**2 + derivative_x**2 + derivative_y**2)
  
  				_UpperCamelCase       : int    =      avg_horizontal_velocity - derivative_x * update
  				_UpperCamelCase       : int    =      avg_vertical_velocity - derivative_y * update
  
  		return horizontal_flow, vertical_flow
  
  
  if __name__ == "__main__":
  	import doctest
  
  	doctest.testmod()
  
  
  
  
  
  
style_context: 
  '''simple docstring'''
  
  from __future__ import annotations
  
  from collections.abc import Iterable
  from dataclasses import dataclass
  from typing import Any
  
  class 			UpperCAmelCase				:
  
  
  
  
  
  		"""simple docstring"""
  
  
  
  
  
  		A__  :   List[str]          =    ('_front', '_back', '_len')
  
  		@dataclass
  		class 			UpperCAmelCase				:
  
  
  
  
  
  				"""simple docstring"""
  
  
  
  
  
  				A__  :   Any          =    None
  				A__  :   Deque._Node | None          =    None
  				A__  :   Deque._Node | None          =    None
  
  
  
  
  
  
  
  		class 			UpperCAmelCase				:
  
  
  
  
  
  				"""simple docstring"""
  
  
  
  
  
  				A__  :   Any          =    ('_cur',)
  				def __init__(    self  ,					_snake_case     )				->    None:
  						_UpperCamelCase       : Any    =      cur
  				def __iter__(    self     )				->    Deque._Iterator:
  						return self
  
  
  
  
  				def _lowercase  (    self     )				->    Any:
  						if self._cur is None:
  								# finished iterating
  								raise StopIteration
  						_UpperCamelCase       : List[Any]    =      self._cur.val
  						_UpperCamelCase       : List[Any]    =      self._cur.next_node
  
  						return val
  
  
  
  
  
  
  
  		def __init__(    self  ,					_snake_case = None     )				->    None:
  				_UpperCamelCase       : Any    =      None
  				_UpperCamelCase       : Any    =      None
  				_UpperCamelCase       : int    =      0
  
  				if iterable is not None:
  						# append every value to the deque
  						for val in iterable:
  								self.append(_snake_case     )
  		def _lowercase  (    self  ,					_snake_case     )				->    None:
  				_UpperCamelCase       : List[str]    =      self._Node(_snake_case  ,					_snake_case  ,					_snake_case     )
  				if self.is_empty():
  						# front = back
  						_UpperCamelCase       : Union[str, Any]    =      node
  						_UpperCamelCase       : Tuple    =      1
  				else:
  						# connect nodes
  						_UpperCamelCase       : Union[str, Any]    =      node
  						_UpperCamelCase       : int    =      self._back
  						_UpperCamelCase       : List[Any]    =      node  # assign new back to the new node
  
  						self._len += 1
  
  						# make sure there were no errors
  						assert not self.is_empty(), "Error on appending value."
  		def _lowercase  (    self  ,					_snake_case     )				->    None:
  				_UpperCamelCase       : Optional[Any]    =      self._Node(_snake_case  ,					_snake_case  ,					_snake_case     )
  				if self.is_empty():
  						# front = back
  						_UpperCamelCase       : Union[str, Any]    =      node
  						_UpperCamelCase       : Optional[int]    =      1
  				else:
  						# connect nodes
  						_UpperCamelCase       : Tuple    =      self._front
  						_UpperCamelCase       : Optional[int]    =      node
  						_UpperCamelCase       : List[str]    =      node  # assign new front to the new node
  
  						self._len += 1
  
  						# make sure there were no errors
  						assert not self.is_empty(), "Error on appending value."
  		def _lowercase  (    self  ,					_snake_case     )				->    None:
  				for val in iterable:
  						self.append(_snake_case     )
  		def _lowercase  (    self  ,					_snake_case     )				->    None:
  				for val in iterable:
  						self.appendleft(_snake_case     )
  		def _lowercase  (    self     )				->    Any:
  				assert not self.is_empty(), "Deque is empty."
  
  				_UpperCamelCase       : int    =      self._back
  				_UpperCamelCase       : List[Any]    =      self._back.prev_node  # set new back
  				# drop the last node - python will deallocate memory automatically
  				_UpperCamelCase       : Union[str, Any]    =      None
  
  				self._len -= 1
  
  				return topop.val
  		def _lowercase  (    self     )				->    Any:
  				assert not self.is_empty(), "Deque is empty."
  
  				_UpperCamelCase       : List[Any]    =      self._front
  				_UpperCamelCase       : str    =      self._front.next_node  # set new front and drop the first node
  				_UpperCamelCase       : Union[str, Any]    =      None
  
  				self._len -= 1
  
  				return topop.val
  		def _lowercase  (    self     )				->    bool:
  				return self._front is None
  		def __len__(    self     )				->    int:
  				return self._len
  		def __eq__(    self  ,					_snake_case     )				->    bool:
  
  				if not isinstance(_snake_case  ,					_snake_case     ):
  						return NotImplemented
  
  				_UpperCamelCase       : Optional[int]    =      self._front
  				_UpperCamelCase       : List[Any]    =      other._front
  
  				# if the length of the dequeues are not the same, they are not equal
  				if len(self     ) != len(_snake_case     ):
  						return False
  
  				while me is not None and oth is not None:
  						# compare every value
  						if me.val != oth.val:
  								return False
  						_UpperCamelCase       : str    =      me.next_node
  						_UpperCamelCase       : List[str]    =      oth.next_node
  
  				return True
  		def __iter__(    self     )				->    Deque._Iterator:
  				return Deque._Iterator(self._front     )
  
  
  
  
  		def __repr__(    self     )				->    str:
  				_UpperCamelCase       : str    =      []
  				_UpperCamelCase       : Tuple    =      self._front
  				while aux is not None:
  						# append the values in a list to display
  						values_list.append(aux.val     )
  						_UpperCamelCase       : List[str]    =      aux.next_node
  
  				return F'''[{', '.join(repr(_snake_case     ) for val in values_list     )}]'''
  
  
  if __name__ == "__main__":
  	import doctest
  
  	doctest.testmod()
  
  
  
  
  
  