
from typing import SupportsIndex

import numpy as np
from scipy.ndimage import convolve




def 		A						(  SCREAMING_SNAKE_CASE     ,						SCREAMING_SNAKE_CASE     ,						SCREAMING_SNAKE_CASE							):




			"""simple docstring"""


			UpperCAmelCase__		: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
			UpperCAmelCase__		:Union[str, Any] 		=    np.stack(
			    np.meshgrid(np.arange(0     ,						image.shape[1]							)     ,						np.arange(0     ,						image.shape[0]							)							)     ,						2							)
			UpperCAmelCase__		:str 		=    np.round(grid - flow							).astype(np.intaa							)

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

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

			# Set pixels at invalid locations to 0
			UpperCAmelCase__		:List[Any] 		=    0

			return warped




def 		A						(  SCREAMING_SNAKE_CASE     ,						SCREAMING_SNAKE_CASE     ,						SCREAMING_SNAKE_CASE     ,						SCREAMING_SNAKE_CASE = None     ,						):




			"""simple docstring"""


			if alpha is None:
						UpperCAmelCase__		:int 		=    0.1

			# Initialize flow
			UpperCAmelCase__		:Optional[Any] 		=    np.zeros_like(SCREAMING_SNAKE_CASE							)
			UpperCAmelCase__		:Tuple 		=    np.zeros_like(SCREAMING_SNAKE_CASE							)

			# Prepare kernels for the calculation of the derivatives and the average velocity
			UpperCAmelCase__		:Optional[int] 		=    np.array([[-1, 1], [-1, 1]]							) * 0.25
			UpperCAmelCase__		:str 		=    np.array([[-1, -1], [1, 1]]							) * 0.25
			UpperCAmelCase__		:Optional[int] 		=    np.array([[1, 1], [1, 1]]							) * 0.25
			UpperCAmelCase__		: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(SCREAMING_SNAKE_CASE							):
						UpperCAmelCase__		:Union[str, Any] 		=    warp(SCREAMING_SNAKE_CASE     ,						SCREAMING_SNAKE_CASE     ,						SCREAMING_SNAKE_CASE							)
						UpperCAmelCase__		:List[Any] 		=    convolve(SCREAMING_SNAKE_CASE     ,						SCREAMING_SNAKE_CASE							) + convolve(SCREAMING_SNAKE_CASE     ,						SCREAMING_SNAKE_CASE							)
						UpperCAmelCase__		:str 		=    convolve(SCREAMING_SNAKE_CASE     ,						SCREAMING_SNAKE_CASE							) + convolve(SCREAMING_SNAKE_CASE     ,						SCREAMING_SNAKE_CASE							)
						UpperCAmelCase__		:Dict 		=    convolve(SCREAMING_SNAKE_CASE     ,						SCREAMING_SNAKE_CASE							) + convolve(SCREAMING_SNAKE_CASE     ,						-kernel_t							)

						UpperCAmelCase__		:Any 		=    convolve(SCREAMING_SNAKE_CASE     ,						SCREAMING_SNAKE_CASE							)
						UpperCAmelCase__		:Tuple 		=    convolve(SCREAMING_SNAKE_CASE     ,						SCREAMING_SNAKE_CASE							)

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

						UpperCAmelCase__		:Optional[int] 		=    avg_horizontal_velocity - derivative_x * update
						UpperCAmelCase__		:str 		=    avg_vertical_velocity - derivative_y * update

			return horizontal_flow, vertical_flow


if __name__ == "__main__":
						import doctest

						doctest.testmod()










from __future__ import annotations

from collections import defaultdict
from dataclasses import dataclass



@dataclass
class       _lowercase				:





	"""simple docstring"""
	__A		      =				42
	__A		      =				None
	__A		      =				None






def        a( )						->			TreeNode:
					"""simple docstring"""





					return TreeNode(3					,	TreeNode(9			)					,	TreeNode(20					,	TreeNode(15			)					,	TreeNode(7			)			)			)






def        a( A			:       TreeNode			)						->			list[int]:
					"""simple docstring"""





					def depth_first_search(
					    A			:       TreeNode | None					,	A			:       int					,	A			:       list[int]			) -> None:
										if not root:
															return

										if depth == len(A			):
															right_view.append(root.val			)

										depth_first_search(root.right					,	depth + 1					,	A			)
										depth_first_search(root.left					,	depth + 1					,	A			)

					a   	=  []
					if not root:
										return right_view

					depth_first_search(A					,	0					,	A			)
					return right_view






def        a( A			:       TreeNode			)						->			list[int]:
					"""simple docstring"""





					def depth_first_search(
					    A			:       TreeNode | None					,	A			:       int					,	A			:       list[int]			) -> None:
										if not root:
															return

										if depth == len(A			):
															left_view.append(root.val			)

										depth_first_search(root.left					,	depth + 1					,	A			)
										depth_first_search(root.right					,	depth + 1					,	A			)

					a   	=  []
					if not root:
										return left_view

					depth_first_search(A					,	0					,	A			)
					return left_view






def        a( A			:       TreeNode			)						->			list[int]:
					"""simple docstring"""





					def breadth_first_search(A			:       TreeNode					,	A			:       list[int]			) -> None:
										a   	=  [(root, 0)]
										a   	=  defaultdict(A			)

										while queue:
															a   	=  queue.pop(0			)
															a	,							a   	=  first

															lookup[hd].append(node.val			)

															if node.left:
																				queue.append((node.left, hd - 1)			)
															if node.right:
																				queue.append((node.right, hd + 1)			)

										for pair in sorted(lookup.items()					,	key=lambda A			: each[0]			):
															top_view.append(pair[1][0]			)

					a   	=  []
					if not root:
										return top_view

					breadth_first_search(A					,	A			)
					return top_view






def        a( A			:       TreeNode			)						->			list[int]:
					"""simple docstring"""





					from collections import defaultdict

					def breadth_first_search(A			:       TreeNode					,	A			:       list[int]			) -> None:
										a   	=  [(root, 0)]
										a   	=  defaultdict(A			)

										while queue:
															a   	=  queue.pop(0			)
															a	,							a   	=  first
															lookup[hd].append(node.val			)

															if node.left:
																				queue.append((node.left, hd - 1)			)
															if node.right:
																				queue.append((node.right, hd + 1)			)

										for pair in sorted(lookup.items()					,	key=lambda A			: each[0]			):
															bottom_view.append(pair[1][-1]			)

					a   	=  []
					if not root:
										return bottom_view

					breadth_first_search(A					,	A			)
					return bottom_view


if __name__ == "__main__":
				import doctest

				doctest.testmod()
