#### Horn Schunck

T
```"""
The Horn-Schunck method estimates the optical flow for every single pixel of
a sequence of images.
It works by assuming brightness constancy between two consecutive frames
and smoothness in the optical flow.

Useful resources:
Wikipedia: https://en.wikipedia.org/wiki/Horn%E2%80%93Schunck_method
Paper: http://image.diku.dk/imagecanon/material/HornSchunckOptical_Flow.pdf
"""

from typing import SupportsIndex

import numpy as np
from scipy.ndimage import convolve

def warp(
image: np.ndarray, horizontal_flow: np.ndarray, vertical_flow: np.ndarray
) -> np.ndarray:
"""
Warps the pixels of an image into a new image using the horizontal and vertical
flows.
Pixels that are warped from an invalid location are set to 0.

Parameters:
image: Grayscale image
horizontal_flow: Horizontal flow
vertical_flow: Vertical flow

Returns: Warped image

>>> warp(np.array([[0, 1, 2], [0, 3, 0], [2, 2, 2]]), \
np.array([[0, 1, -1], [-1, 0, 0], [1, 1, 1]]), \
np.array([[0, 0, 0], [0, 1, 0], [0, 0, 1]]))
array([[0, 0, 0],
[3, 1, 0],
[0, 2, 3]])
"""
flow = np.stack((horizontal_flow, vertical_flow), 2)

# Create a grid of all pixel coordinates and subtract the flow to get the
# target pixels coordinates
grid = np.stack(
np.meshgrid(np.arange(0, image.shape[1]), np.arange(0, image.shape[0])), 2
)
grid = np.round(grid - flow).astype(np.int32)

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

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

# Set pixels at invalid locations to 0
warped[invalid[:, :, 0] | invalid[:, :, 1]] = 0

return warped

def horn_schunck(
image0: np.ndarray,
image1: np.ndarray,
num_iter: SupportsIndex,
alpha: float | None = None,
) -> tuple[np.ndarray, np.ndarray]:
"""
This function performs the Horn-Schunck algorithm and returns the estimated
optical flow. It is assumed that the input images are grayscale and
normalized to be in [0, 1].

Parameters:
image0: First image of the sequence
image1: Second image of the sequence
alpha: Regularization constant
num_iter: Number of iterations performed

Returns: estimated horizontal & vertical flow

>>> np.round(horn_schunck(np.array([[0, 0, 2], [0, 0, 2]]), \
np.array([[0, 2, 0], [0, 2, 0]]), alpha=0.1, num_iter=110)).\
astype(np.int32)
array([[[ 0, -1, -1],
[ 0, -1, -1]],
<BLANKLINE>
[[ 0,  0,  0],
[ 0,  0,  0]]], dtype=int32)
"""
if alpha is None:
alpha = 0.1

# Initialize flow
horizontal_flow = np.zeros_like(image0)
vertical_flow = np.zeros_like(image0)

# Prepare kernels for the calculation of the derivatives and the average velocity
kernel_x = np.array([[-1, 1], [-1, 1]]) * 0.25
kernel_y = np.array([[-1, -1], [1, 1]]) * 0.25
kernel_t = np.array([[1, 1], [1, 1]]) * 0.25
kernel_laplacian = 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(num_iter):
warped_image = warp(image0, horizontal_flow, vertical_flow)
derivative_x = convolve(warped_image, kernel_x) + convolve(image1, kernel_x)
derivative_y = convolve(warped_image, kernel_y) + convolve(image1, kernel_y)
derivative_t = convolve(warped_image, kernel_t) + convolve(image1, -kernel_t)

avg_horizontal_velocity = convolve(horizontal_flow, kernel_laplacian)
avg_vertical_velocity = convolve(vertical_flow, kernel_laplacian)

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

horizontal_flow = avg_horizontal_velocity - derivative_x * update
vertical_flow = avg_vertical_velocity - derivative_y * update

return horizontal_flow, vertical_flow

if __name__ == "__main__":
import doctest

doctest.testmod()
```