
from typing import SupportsIndex

import numpy as np
from scipy.ndimage import convolve





def 		SCREAMING_SNAKE_CASE__      ( __lowerCAmelCase      ,						__lowerCAmelCase      ,						__lowerCAmelCase							):
				snake_case__  =			np.stack((horizontal_flow, vertical_flow)      ,						2							)

				# Create a grid of all pixel coordinates and subtract the flow to get the
				# target pixels coordinates
				snake_case__  =			np.stack(
				    np.meshgrid(np.arange(0      ,						image.shape[1]							)      ,						np.arange(0      ,						image.shape[0]							)							)      ,						2							)
				snake_case__  =			np.round(grid - flow							).astype(np.intaa							)

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

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

				# Set pixels at invalid locations to 0
				snake_case__  =			0

				return warped





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

				# Initialize flow
				snake_case__  =			np.zeros_like(__UpperCamelCase							)
				snake_case__  =			np.zeros_like(__UpperCamelCase							)

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

								snake_case__  =			convolve(__UpperCamelCase      ,						__UpperCamelCase							)
								snake_case__  =			convolve(__UpperCamelCase      ,						__UpperCamelCase							)

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

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

				return horizontal_flow, vertical_flow


if __name__ == "__main__":
					import doctest

					doctest.testmod()





from __future__ import annotations

from abc import abstractmethod
from collections.abc import Iterable
from typing import Generic, Protocol, TypeVar




class    _SCREAMING_SNAKE_CASE					( __UpperCamelCase	):
      @abstractmethod
      def __lt__(     self						,		lowerCamelCase	):
          pass
      @abstractmethod
      def __gt__(     self						,		lowerCamelCase	):
          pass



      @abstractmethod
      def __eq__(     self						,		lowerCamelCase	):
          pass





__magic_name__           =					TypeVar('''T''', bound=Comparable)




class    _SCREAMING_SNAKE_CASE					( Generic[T]	):
      def __init__(     self	):
          snake_case__  =			[]
          snake_case__  =			0
      def __repr__(     self	):
          return str(self.h	)
      def A_     (     self						,		lowerCamelCase	):
          if child_idx > 0:
              return (child_idx - 1) // 2
          return None
      def A_     (     self						,		lowerCamelCase	):
          snake_case__  =			2 * parent_idx + 1
          if left_child_index < self.heap_size:
              return left_child_index
          return None
      def A_     (     self						,		lowerCamelCase	):
          snake_case__  =			2 * parent_idx + 2
          if right_child_index < self.heap_size:
              return right_child_index
          return None
      def A_     (     self						,		lowerCamelCase	):
          if index < self.heap_size:
              snake_case__  =			index
              snake_case__  =			self.left_child_idx(lowerCamelCase	)
              snake_case__  =			self.right_child_idx(lowerCamelCase	)
              # check which child is larger than its parent
              if left_child is not None and self.h[left_child] > self.h[violation]:
                  snake_case__  =			left_child
              if right_child is not None and self.h[right_child] > self.h[violation]:
                  snake_case__  =			right_child
              # if violation indeed exists
              if violation != index:
                  # swap to fix the violation
                  snake_case__  ,   snake_case__  =			self.h[index], self.h[violation]
                  # fix the subsequent violation recursively if any
                  self.max_heapify(lowerCamelCase	)
      def A_     (     self						,		lowerCamelCase	):
          snake_case__  =			list(lowerCamelCase	)
          snake_case__  =			len(self.h	)
          if self.heap_size > 1:
              # max_heapify from right to left but exclude leaves (last level)
              for i in range(self.heap_size // 2 - 1						,		-1						,		-1	):
                  self.max_heapify(lowerCamelCase	)
      def A_     (     self	):
          if self.heap_size >= 2:
              snake_case__  =			self.h[0]
              snake_case__  =			self.h.pop(-1	)
              self.heap_size -= 1
              self.max_heapify(0	)
              return me
          elif self.heap_size == 1:
              self.heap_size -= 1
              return self.h.pop(-1	)
          else:
              raise Exception("Empty heap"	)
      def A_     (     self						,		lowerCamelCase	):
          self.h.append(lowerCamelCase	)
          snake_case__  =			(self.heap_size - 1) // 2
          self.heap_size += 1
          while idx >= 0:
              self.max_heapify(lowerCamelCase	)
              snake_case__  =			(idx - 1) // 2



      def A_     (     self	):
          snake_case__  =			self.heap_size
          for j in range(size - 1						,		0						,		-1	):
              snake_case__  ,   snake_case__  =			self.h[j], self.h[0]
              self.heap_size -= 1
              self.max_heapify(0	)
          snake_case__  =			size


if __name__ == "__main__":
     import doctest

     # run doc test
     doctest.testmod()

     # demo
     for unsorted in [
         [0],
         [2],
         [3, 5],
         [5, 3],
         [5, 5],
         [0, 0, 0, 0],
         [1, 1, 1, 1],
         [2, 2, 3, 5],
         [0, 2, 2, 3, 5],
         [2, 5, 3, 0, 2, 3, 0, 3],
         [6, 1, 2, 7, 9, 3, 4, 5, 10, 8],
         [103, 9, 1, 7, 11, 15, 25, 201, 209, 107, 5],
         [-45, -2, -5],
     ]:
          print(F'''unsorted array: {unsorted}''')

          __magic_name__           =					Heap()
          heap.build_max_heap(unsorted)
          print(F'''after build heap: {heap}''')

          print(F'''max value: {heap.extract_max()}''')
          print(F'''after max value removed: {heap}''')

          heap.insert(100)
          print(F'''after new value 100 inserted: {heap}''')

          heap.heap_sort()
          print(F'''heap-sorted array: {heap}\n''')



