






import math
def       a__						(  A					):
   if n <= 0:
      raise ValueError("""Only positive integers have prime factors"""					)
   lowercase__						:	Optional[Any]         = []
   while n % 2 == 0:
      pf.append(2					)
      lowercase__						:	Tuple         = int(n / 2					)
   for i in range(3		,				int(math.sqrt(A					)					) + 1		,				2					):
      while n % i == 0:
         pf.append(A					)
         lowercase__						:	Dict         = int(n / i					)
   if n > 2:
      pf.append(A					)
   return pf
def       a__						(  A					):
   if n <= 0:
      raise ValueError("""Only positive numbers are accepted"""					)
   lowercase__						:	Dict         = 1
   lowercase__						:	Optional[Any]         = 1
   while n % 2 == 0:
      temp += 1
      lowercase__						:	str         = int(n / 2					)
   div *= temp
   for i in range(3		,				int(math.sqrt(A					)					) + 1		,				2					):
      lowercase__						:	Tuple         = 1
      while n % i == 0:
         temp += 1
         lowercase__						:	Dict         = int(n / i					)
      div *= temp
   if n > 1:
      div *= 2
   return div
def       a__						(  A					):
   if n <= 0:
      raise ValueError("""Only positive numbers are accepted"""					)
   lowercase__						:	Tuple         = 1
   lowercase__						:	Optional[Any]         = 1
   while n % 2 == 0:
      temp += 1
      lowercase__						:	List[str]         = int(n / 2					)
   if temp > 1:
      s *= (2**temp - 1) / (2 - 1)
   for i in range(3		,				int(math.sqrt(A					)					) + 1		,				2					):
      lowercase__						:	List[Any]         = 1
      while n % i == 0:
         temp += 1
         lowercase__						:	Tuple         = int(n / i					)
      if temp > 1:
         s *= (i**temp - 1) / (i - 1)
   return int(A					)
def       a__						(  A					):
   lowercase__						:	Dict         = n
   for x in set(prime_factors(A					)					):
      s *= (x - 1) / x
   return int(A					)


if __name__ == "__main__":
 import doctest

 doctest.testmod()










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()


