Add the Horn-Schunck algorithm (#5333)

* Added implementation of the Horn-Schunck algorithm

* Cleaner variable names

* added doctests

* Fix doctest

* Update horn_schunck.py

* Update horn_schunck.py

* Update horn_schunck.py

* Update horn_schunck.py

Co-authored-by: John Law <johnlaw.po@gmail.com>

computer_vision/horn_schunck.py

@@ -0,0 +1,130 @@
+"""
+    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
+"""
+
+import numpy as np
+from scipy.ndimage.filters import convolve
+from typing_extensions import SupportsIndex
+
+
+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()