



'''simple docstring'''






from typing import SupportsIndex

import numpy as np
from scipy.ndimage import convolve




def 					snake_case_      (  lowerCAmelCase_ ,     lowerCAmelCase_ ,     lowerCAmelCase_	)->	np.ndarray:





       '''simple docstring'''


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

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

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

       # Set pixels at invalid locations to 0
       _UpperCAmelCase						:		int      =				0

       return warped




def 					snake_case_      (  lowerCAmelCase_ ,     lowerCAmelCase_ ,     lowerCAmelCase_ ,     lowerCAmelCase_ = None ,     )->	tuple[np.ndarray, np.ndarray]:





       '''simple docstring'''


       if alpha is None:
              _UpperCAmelCase						:		List[Any]      =				0.1

       # Initialize flow
       _UpperCAmelCase						:		Any      =				np.zeros_like(lowerCAmelCase_	)
       _UpperCAmelCase						:		str      =				np.zeros_like(lowerCAmelCase_	)

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

              _UpperCAmelCase						:		int      =				convolve(lowerCAmelCase_ ,     lowerCAmelCase_	)
              _UpperCAmelCase						:		Union[str, Any]      =				convolve(lowerCAmelCase_ ,     lowerCAmelCase_	)

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

              _UpperCAmelCase						:		Tuple      =				avg_horizontal_velocity - derivative_x * update
              _UpperCAmelCase						:		List[Any]      =				avg_vertical_velocity - derivative_y * update

       return horizontal_flow, vertical_flow


if __name__ == "__main__":
   import doctest

   doctest.testmod()








'''simple docstring'''






def 					snake_case_      (  lowerCAmelCase_	)->	tuple[int, int]:





       '''simple docstring'''


       if len(lowerCAmelCase_	) == 1:
              return a[0], a[0]
       else:
              # split A into half.
              _UpperCAmelCase						:		Union[str, Any]      =				a[: len(lowerCAmelCase_	) // 2]
              _UpperCAmelCase						:		str      =				a[len(lowerCAmelCase_	) // 2 :]

              # 2 sub problems, 1/2 of original size.
              _UpperCAmelCase  ,_UpperCAmelCase						:		str      =				max_difference(lowerCAmelCase_	)
              _UpperCAmelCase  ,_UpperCAmelCase						:		List[str]      =				max_difference(lowerCAmelCase_	)

              # get min of first and max of second
              # linear time
              _UpperCAmelCase						:		Dict      =				min(lowerCAmelCase_	)
              _UpperCAmelCase						:		List[Any]      =				max(lowerCAmelCase_	)

              # 3 cases, either (small1, big1),
              # (min_first, max_second), (small2, big2)
              # constant comparisons
              if biga - smalla > max_second - min_first and biga - smalla > biga - smalla:
                     return smalla, biga
              elif biga - smalla > max_second - min_first:
                     return smalla, biga
              else:
                     return min_first, max_second


if __name__ == "__main__":
   import doctest

   doctest.testmod()



