






import re
def       a__						(  A					):
   def docstring_decorator(A					):
      lowercase__						:	Any         = fn.__doc__
      lowercase__						:	int         = func_doc.split("""\n"""					)
      lowercase__						:	Any         = 0
      while i < len(A					) and re.search(R"""^\s*Examples?:\s*$"""		,				lines[i]					) is None:
         i += 1
      if i < len(A					):
         lowercase__						:	Tuple         = example_docstring
         lowercase__						:	Optional[Any]         = """\n""".join(A					)
      else:
         raise ValueError(
             F'''The function {fn} should have an empty \'Examples:\' in its docstring as placeholder, '''
             F'''current docstring is:\n{func_doc}'''					)
      lowercase__						:	Any         = func_doc
      return fn

   return docstring_decorator










from typing import SupportsIndex

import numpy as np
from scipy.ndimage import convolve
def       a__						(  A		,				A		,				A					):
   lowercase__						:	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
   lowercase__						:	Tuple         = np.stack(
       np.meshgrid(np.arange(0		,				image.shape[1]					)		,				np.arange(0		,				image.shape[0]					)					)		,				2					)
   lowercase__						:	Union[str, Any]         = np.round(grid - flow					).astype(np.intaa					)

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

   lowercase__						:	Tuple         = image[grid[:, :, 1], grid[:, :, 0]]

   # Set pixels at invalid locations to 0
   lowercase__						:	str         = 0

   return warped
def       a__						(  A		,				A		,				A		,				A = None		,				):
   if alpha is None:
      lowercase__						:	Tuple         = 0.1

   # Initialize flow
   lowercase__						:	Optional[int]         = np.zeros_like(A					)
   lowercase__						:	Tuple         = np.zeros_like(A					)

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

      lowercase__						:	str         = convolve(A		,				A					)
      lowercase__						:	Tuple         = convolve(A		,				A					)

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

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

   return horizontal_flow, vertical_flow


if __name__ == "__main__":
 import doctest

 doctest.testmod()


