PDE applications in image/video ingestion, preprocessing, detection, and tracking

 PDEs appear in vision pipelines mainly through anisotropic diffusion, optical flow, and level‑set evolution. Some examples follow:

1. PDE for Drone Video Preprocessing: Anisotropic Diffusion (Perona–Malik)

Used for denoising drone footage while preserving edges before detection/tracking.

python

import cv2
import numpy as np

def anisotropic_diffusion(img, n_iter=15, k=20, lambda_=0.25):
    img = img.astype(np.float32)
    for _ in range(n_iter):
        # Compute gradients
        nablaN = np.roll(img, -1, axis=0) - img
        nablaS = np.roll(img, 1, axis=0) - img
        nablaE = np.roll(img, -1, axis=1) - img
        nablaW = np.roll(img, 1, axis=1) - img

        # Perona–Malik conduction coefficients
        cN = np.exp(-(nablaN/k)**2)
        cS = np.exp(-(nablaS/k)**2)
        cE = np.exp(-(nablaE/k)**2)
        cW = np.exp(-(nablaW/k)**2)

        # Update PDE
        img += lambda_ * (
            cN * nablaN + cS * nablaS +
            cE * nablaE + cW * nablaW
        )
    return img

# Example: preprocess a drone frame
frame = cv2.imread("drone_frame.png", 0)
smooth = anisotropic_diffusion(frame)
cv2.imwrite("drone_frame_smooth.png", smooth)

When Drone footage is noisy (wind vibration, compression artifacts), Anisotropic diffusion PDE removes noise while keeping edges sharp: ideal before object detection or optical flow.

2. PDE for Motion Estimation: Optical Flow (Horn–Schunck)

This PDE estimates pixel‑wise motion—critical for drone tracking, stabilization, and moving‑object detection.

The Horn–Schunck optical flow PDE is:

I_xu + I_yv + I_t = 0,     α²∇²u = I_x(I_xu + I_yv + I_t),     α²∇²v = I_y(I_xu + I_yv + I_t)

Here is a minimal Python implementation:

python

def horn_schunck(im1, im2, alpha=10, n_iter=100):
    im1 = im1.astype(np.float32)
    im2 = im2.astype(np.float32)

    # Compute derivatives
    Ix = cv2.Sobel(im1, cv2.CV_32F, 1, 0, ksize=3)
    Iy = cv2.Sobel(im1, cv2.CV_32F, 0, 1, ksize=3)
    It = im2 - im1

    u = np.zeros_like(im1)
    v = np.zeros_like(im1)

    for _ in range(n_iter):
        # Laplacian smoothing (PDE regularization)
        u_avg = cv2.blur(u, (3,3))
        v_avg = cv2.blur(v, (3,3))

        # Update flow fields
        der = Ix*u_avg + Iy*v_avg + It
        u = u_avg - Ix * der / (alpha**2 + Ix**2 + Iy**2)
        v = v_avg - Iy * der / (alpha**2 + Ix**2 + Iy**2)

    return u, v

# Example: compute optical flow between two drone frames
f1 = cv2.imread("drone_frame_001.png", 0)
f2 = cv2.imread("drone_frame_002.png", 0)
u, v = horn_schunck(f1, f2)

Optical flow PDEs detect motion of vehicles, people, or other drones. They also stabilize drone footage and estimate ego‑motion when GPS is unreliable.

3. PDE for Object Detection/Tracking: Level‑Set Contour Evolution

Used for tracking moving objects in drone videos by evolving a contour according to a PDE:

∂ϕ/∂t = μ∇²ϕ − λF|∇ϕ|

Below is a minimal level‑set evolution loop:

python

def level_set_step(phi, img, mu=0.2, lambda_=5.0, dt=0.1):
    # Image-based speed term (edges)
    grad = cv2.Sobel(img, cv2.CV_32F, 1, 0) + cv2.Sobel(img, cv2.CV_32F, 0, 1)
    F = np.exp(-(grad**2) / 1000.0)

    # PDE terms
    lap = cv2.Laplacian(phi, cv2.CV_32F)
    grad_phi = np.sqrt(
        cv2.Sobel(phi, cv2.CV_32F, 1, 0)**2 +
        cv2.Sobel(phi, cv2.CV_32F, 0, 1)**2
    )

    # Level-set update
    dphi_dt = mu * lap - lambda_ * F * grad_phi
    return phi + dt * dphi_dt

# Example: track an object in drone video
phi = np.random.randn(480, 640).astype(np.float32)  # initial contour
frame = cv2.imread("drone_frame.png", 0)

for _ in range(200):
    phi = level_set_step(phi, frame)

Level‑set PDEs track moving cars, boats, or people from above, even under occlusion or changing lighting.

Summary of PDE Usage in Drone Vision Pipelines

PDE Method	Drone Use‑Case	Why It Matters
Anisotropic diffusion	Preprocessing, denoising	Removes noise while preserving edges for detection
Optical flow PDEs	Motion estimation, tracking, stabilization	Detects moving objects and drone ego‑motion
Level‑set PDEs	Object detection, contour tracking	Robust tracking under occlusion and noise

These are the exact PDE families used in classical UAV vision research before deep learning took over—and they still matter for preprocessing, robustness, and physics‑based tracking.

References: previous article: https://1drv.ms/w/c/d609fb70e39b65c8/IQBMGcHb0t_GRYmwWqOjDAu9Afyds1gCdYGDMsIaBXhN3fo?e=SiBqci