





'''simple docstring'''

from typing import SupportsIndex

import numpy as np
from scipy.ndimage import convolve







def       a__       (       lowercase      :  np.ndarray,      lowercase      :  np.ndarray,      lowercase      :  np.ndarray   )	->	np.ndarray:





     """simple docstring"""

     _UpperCamelCase       					= 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       					= np.stack(
         np.meshgrid(np.arange(0,      image.shape[1]   ),      np.arange(0,      image.shape[0]   )   ),      2   )
     _UpperCamelCase       					= np.round(grid - flow   ).astype(np.intaa   )

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

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

     # Set pixels at invalid locations to 0
     _UpperCamelCase       					= 0

     return warped







def       a__       (       lowercase      :  np.ndarray,      lowercase      :  np.ndarray,      lowercase      :  SupportsIndex,      lowercase      :  float | None = None,      )	->	tuple[np.ndarray, np.ndarray]:





     """simple docstring"""

     if alpha is None:
          _UpperCamelCase       					= 0.1

     # Initialize flow
     _UpperCamelCase       					= np.zeros_like(lowercase   )
     _UpperCamelCase       					= np.zeros_like(lowercase   )

     # Prepare kernels for the calculation of the derivatives and the average velocity
     _UpperCamelCase       					= np.array([[-1, 1], [-1, 1]]   ) * 0.2_5
     _UpperCamelCase       					= np.array([[-1, -1], [1, 1]]   ) * 0.2_5
     _UpperCamelCase       					= np.array([[1, 1], [1, 1]]   ) * 0.2_5
     _UpperCamelCase       					= 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(lowercase   ):
          _UpperCamelCase       					= warp(lowercase,      lowercase,      lowercase   )
          _UpperCamelCase       					= convolve(lowercase,      lowercase   ) + convolve(lowercase,      lowercase   )
          _UpperCamelCase       					= convolve(lowercase,      lowercase   ) + convolve(lowercase,      lowercase   )
          _UpperCamelCase       					= convolve(lowercase,      lowercase   ) + convolve(lowercase,      -kernel_t   )

          _UpperCamelCase       					= convolve(lowercase,      lowercase   )
          _UpperCamelCase       					= convolve(lowercase,      lowercase   )

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

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

     return horizontal_flow, vertical_flow


if __name__ == "__main__":
   import doctest

   doctest.testmod()










'''simple docstring'''

from __future__ import annotations







def       a__       (       lowercase      :  list[int | float]   )	->	int | float:





     """simple docstring"""

     if len(lowercase   ) == 0:
          raise ValueError('''find_min() arg is an empty sequence'''   )
     _UpperCamelCase       					= nums[0]
     for num in nums:
          _UpperCamelCase       					= min(lowercase,      lowercase   )
     return min_num


if __name__ == "__main__":
   import doctest

   doctest.testmod(verbose=True)



