



'''simple docstring'''

from typing import SupportsIndex

import numpy as np
from scipy.ndimage import convolve







def     __UpperCamelCase					(				_lowercase,					_lowercase,					_lowercase       )			->      Optional[int]:
	_lowercase       :							str  	=		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       :							Optional[Any]  	=		np.stack(
	    np.meshgrid(np.arange(0,					image.shape[1]       ),					np.arange(0,					image.shape[0]       )       ),					2       )
	_lowercase       :							List[str]  	=		np.round(grid - flow       ).astype(np.intaa       )

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

	_lowercase       :							Optional[int]  	=		image[grid[:, :, 1], grid[:, :, 0]]

	# Set pixels at invalid locations to 0
	_lowercase       :							Optional[int]  	=		0

	return warped







def     __UpperCamelCase					(				_lowercase,					_lowercase,					_lowercase,					_lowercase = None,					)			->      Optional[Any]:
	if alpha is None:
		_lowercase       :							Optional[Any]  	=		0.1

	# Initialize flow
	_lowercase       :							Union[str, Any]  	=		np.zeros_like(__a       )
	_lowercase       :							Any  	=		np.zeros_like(__a       )

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

		_lowercase       :							Optional[int]  	=		convolve(__a,					__a       )
		_lowercase       :							Any  	=		convolve(__a,					__a       )

		# This updates the flow as proposed in the paper (Step 12)
		_lowercase       :							Dict  	=		(
		    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       :							Optional[Any]  	=		avg_vertical_velocity - derivative_y * update

	return horizontal_flow, vertical_flow


if __name__ == "__main__":
		import doctest

		doctest.testmod()





from typing import Any

import numpy as np

class 			__snake_case   :







							'''simple docstring'''






							def __init__(				self	,	a_	,	a_		):
											# Check if RHS is negative
											if np.any(tableau[:, -1]	,	where=tableau[:, -1] < 0		):
															raise ValueError("""RHS must be > 0"""		)

											a__					   =		tableau
											a__   ,      a__					   =		tableau.shape

											# Number of decision variables x1, x2, x3...
											a__					   =		n_vars

											# Number of artificial variables to be minimised
											a__					   =		len(np.where(tableau[self.n_vars : -1] == -1		)[0]		)

											# 2 if there are >= or == constraints (nonstandard), 1 otherwise (std)
											a__					   =		(self.n_art_vars > 0) + 1

											# Number of slack variables added to make inequalities into equalities
											a__					   =		self.n_rows - self.n_stages

											# Objectives for each stage
											a__					   =		["""max"""]

											# In two stage simplex, first minimise then maximise
											if self.n_art_vars:
															self.objectives.append("""min"""		)

											a__					   =		[""""""]

											# Index of current pivot row and column
											a__					   =		None
											a__					   =		None

											# Does objective row only contain (non)-negative values?
											a__					   =		False


							@staticmethod
							def _a   (				*a_		):
											if len(a_		) != 3:
															raise ValueError("""Must provide n_vars, n_slack, and n_art_vars"""		)

											if not all(x >= 0 and isinstance(a_	,	a_		) for x in args		):
															raise ValueError("""All arguments must be non-negative integers"""		)

											# decision | slack | artificial
											a__					   =		["""x""", """s""", """a"""]
											a__					   =		[]
											for i in range(3		):
															for j in range(args[i]		):
																			titles.append(string_starts[i] + str(j + 1		)		)
											titles.append("""RHS"""		)
											return titles


							def _a   (				self	,	a_		):
											a__					   =		self.objectives[-1]

											# Find entries of highest magnitude in objective rows
											a__					   =		(objective == """min""") - (objective == """max""")
											a__					   =		np.argmax(sign * tableau[0, : self.n_vars]		)

											# Choice is only valid if below 0 for maximise, and above for minimise
											if sign * self.tableau[0, col_idx] <= 0:
															a__					   =		True
															return 0, 0

											# Pivot row is chosen as having the lowest quotient when elements of
											# the pivot column divide the right-hand side

											# Slice excluding the objective rows
											a__					   =		slice(self.n_stages	,	self.n_rows		)

											# RHS
											a__					   =		tableau[s, -1]

											# Elements of pivot column within slice
											a__					   =		tableau[s, col_idx]

											# Array filled with nans
											a__					   =		np.full(self.n_rows - self.n_stages	,	np.nan		)

											# If element in pivot column is greater than zeron_stages, return
											# quotient or nan otherwise
											a__					   =		np.divide(a_	,	a_	,	out=a_	,	where=divisor > 0		)

											# Arg of minimum quotient excluding the nan values. n_stages is added
											# to compensate for earlier exclusion of objective columns
											a__					   =		np.nanargmin(a_		) + self.n_stages
											return row_idx, col_idx


							def _a   (				self	,	a_	,	a_	,	a_		):
											a__					   =		tableau[row_idx].copy()

											a__					   =		piv_row[col_idx]

											# Entry becomes 1
											piv_row *= 1 / piv_val

											# Variable in pivot column becomes basic, ie the only non-zero entry
											for idx, coeff in enumerate(tableau[:, col_idx]		):
															tableau[idx] += -coeff * piv_row
											a__					   =		piv_row
											return tableau


							def _a   (				self	,	a_		):
											self.objectives.pop()

											if not self.objectives:
															return tableau

											# Slice containing ids for artificial columns
											a__					   =		slice(-self.n_art_vars - 1	,	-1		)

											# Delete the artificial variable columns
											a__					   =		np.delete(a_	,	a_	,	axis=1		)

											# Delete the objective row of the first stage
											a__					   =		np.delete(a_	,	0	,	axis=0		)

											a__					   =		1
											self.n_rows -= 1
											a__					   =		0
											a__					   =		False
											return tableau


							def _a   (				self		):
											# Stop simplex algorithm from cycling.
											for _ in range(100		):
															# Completion of each stage removes an objective. If both stages
															# are complete, then no objectives are left
															if not self.objectives:
																			a__					   =		self.generate_col_titles(
																			    self.n_vars	,	self.n_slack	,	self.n_art_vars		)

																			# Find the values of each variable at optimal solution
																			return self.interpret_tableau(self.tableau	,	self.col_titles		)

															a__   ,      a__					   =		self.find_pivot(self.tableau		)

															# If there are no more negative values in objective row
															if self.stop_iter:
																			# Delete artificial variable columns and rows. Update attributes
																			a__					   =		self.change_stage(self.tableau		)
															else:
																			a__					   =		self.pivot(self.tableau	,	a_	,	a_		)
											return {}







							def _a   (				self	,	a_	,	a_		):
											a__					   =		{"""P""": abs(tableau[0, -1]		)}

											for i in range(self.n_vars		):
															# Gives ids of nonzero entries in the ith column
															a__					   =		np.nonzero(tableau[:, i]		)
															a__					   =		len(nonzero[0]		)

															# First entry in the nonzero ids
															a__					   =		nonzero[0][0]
															a__					   =		tableau[nonzero_rowidx, i]

															# If there is only one nonzero value in column, which is one
															if n_nonzero == nonzero_val == 1:
																			a__					   =		tableau[nonzero_rowidx, -1]
																			a__					   =		rhs_val

        # Check for basic variables
											for title in col_titles:
															# Don't add RHS or slack variables to output dict
															if title[0] not in "R-s-a":
																			output_dict.setdefault(a_	,	0		)
											return output_dict


if __name__ == "__main__":
	import doctest

	doctest.testmod()

