




'''simple docstring'''






from typing import SupportsIndex

import numpy as np
from scipy.ndimage import convolve

def      __lowerCAmelCase   (       snake_case__   , snake_case__   , snake_case__		):
			__UpperCamelCase :     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 :     List[Any]										=  np.stack(
			    np.meshgrid(np.arange(0   , image.shape[1]		)   , np.arange(0   , image.shape[0]		)		)   , 2		)
			__UpperCamelCase :     Dict										=  np.round(grid - flow		).astype(np.intaa		)

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

			__UpperCamelCase :     List[Any]										=  image[grid[:, :, 1], grid[:, :, 0]]

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

			return warped

def      __lowerCAmelCase   (       snake_case__   , snake_case__   , snake_case__   , snake_case__ = None   , ):
			if alpha is None:
						__UpperCamelCase :     Tuple										=  0.1

			# Initialize flow
			__UpperCamelCase :     str										=  np.zeros_like(snake_case__		)
			__UpperCamelCase :     Dict										=  np.zeros_like(snake_case__		)

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

						__UpperCamelCase :     List[str]										=  convolve(snake_case__   , snake_case__		)
						__UpperCamelCase :     Dict										=  convolve(snake_case__   , snake_case__		)

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

						__UpperCamelCase :     Tuple										=  avg_horizontal_velocity - derivative_x * update
						__UpperCamelCase :     Optional[Any]										=  avg_vertical_velocity - derivative_y * update

			return horizontal_flow, vertical_flow


if __name__ == "__main__":
				import doctest

				doctest.testmod()









'''simple docstring'''






def      __lowerCAmelCase   (       snake_case__		):
			__UpperCamelCase :     int										=  word.upper()
			__UpperCamelCase :     List[str]										=  0
			__UpperCamelCase :     int										=  {}
			__UpperCamelCase :     Tuple										=  []

			for letter in word:
						if letter not in letter_nums:
									__UpperCamelCase :     Tuple										=  str(snake_case__		)
									next_num += 1
						word_pattern.append(letter_nums[letter]		)
			return ".".join(snake_case__		)


if __name__ == "__main__":
				import pprint
				import time

				_lowerCAmelCase       =   time.time()
				with open('''dictionary.txt''') as in_file:
								_lowerCAmelCase       =   in_file.read().splitlines()

				_lowerCAmelCase       =   {}
				for word in word_list:
								_lowerCAmelCase       =   get_word_pattern(word)
								if pattern in all_patterns:
												all_patterns[pattern].append(word)
								else:
												_lowerCAmelCase       =   [word]

				with open('''word_patterns.txt''', '''w''') as out_file:
								out_file.write(pprint.pformat(all_patterns))

				_lowerCAmelCase       =   round(time.time() - start_time, 2)
				print(f'Done!  {len(all_patterns):,} word patterns found in {total_time} seconds.')
				# Done!  9,581 word patterns found in 0.58 seconds.



