Ball Interception System: Technical Documentation

1. Statement of the Problem

The system solves a dynamic ball interception problem by combining video analysis, trajectory prediction, and visualization. Given a video of a moving blue ball, the system must:

1. Track the ball's position in real-time using computer vision
2. Filter and smooth the tracked trajectory
3. Extrapolate the ball's future position
4. Calculate optimal launch parameters for an intercepting ball
5. Visualize both trajectories and detect collisions

Key challenges:

• Reliable ball detection in video frames
• Noise reduction in position data
• Accurate trajectory extrapolation
• Real-time interception computation
• Clear and informative visualization

2. Mathematical Model

2.1 Video Processing Model

HSV color space thresholding:

H ∈ [100, 130]  // Blue hue range
S ∈ [50, 255]   // Saturation range
V ∈ [50, 255]   // Value range

Coordinate transformations:

y_cartesian = frame_height - y_pixel
scale = 10 / frame_width
position_meters = position_pixels * scale

2.2 Trajectory Processing

2.2.1 Smoothing

Cubic spline interpolation:

S(t) = ai(t - ti)³ + bi(t - ti)² + ci(t - ti) + di
where ti ≤ t ≤ ti+1

Savitzky-Golay filter parameters:

Window length = 11
Polynomial order = 3

2.2.2 Extrapolation

Future position prediction:

x(t) = x₀ + vx(t - t₀)
y(t) = y₀ + vy(t - t₀) - ½g(t - t₀)²

where:
t₀ = last observed time
x₀, y₀ = last smoothed position
vx, vy = final velocities from smoothed data

2.3 Physics Model

System state vector:

s = [x, y, vx, vy]

Differential equations:

dx/dt = vx
dy/dt = vy
dvx/dt = 0
dvy/dt = -g

where g = 9.81 m/s²

2.4 Interception Model

Optimization problem:

Minimize: Σ ||p₁(t) - p₂(t)||

Subject to:
1 ≤ v₀ ≤ 100         // Initial velocity bounds
0 ≤ θ ≤ π/2          // Launch angle bounds

Where:
p₁(t) = First ball position
p₂(t) = Second ball position
v₀ = Launch speed
θ = Launch angle

3. Numerical Methods

3.1 RK4 Integration

Update equations:

k₁ = f(tₙ, yₙ)
k₂ = f(tₙ + h/2, yₙ + hk₁/2)
k₃ = f(tₙ + h/2, yₙ + hk₂/2)
k₄ = f(tₙ + h, yₙ + hk₃)
yₙ₊₁ = yₙ + (h/6)(k₁ + 2k₂ + 2k₃ + k₄)

Time step h = 0.01s

Properties:

• Local truncation error: O(h⁵)
• Global truncation error: O(h⁴)
• Stability region includes imaginary axis
• Conservative energy behavior

3.2 Trajectory Optimization

L-BFGS-B method parameters:

Maximum iterations: 1000
Function tolerance: 1e-6
Initial guess: v₀ = √(vx² + vy²), θ = atan2(vy, vx)

4. Algorithm

4.1 Video Processing

1. Load video and get properties (fps, dimensions)
2. For each frame:
   a. Convert to HSV color space
   b. Apply blue color mask
   c. Find contours
   d. Extract largest contour center
   e. Convert to metric coordinates

4.2 Trajectory Processing

1. Smooth positions:
   a. Apply cubic interpolation
   b. Apply Savitzky-Golay filter
2. Calculate velocities using gradients
3. Extrapolate future positions:
   a. Extend time array by 2 seconds
   b. Calculate positions using physics model
4. Combine smoothed and extrapolated trajectories

4.3 Interception Calculation

1. Calculate initial guess from target trajectory
2. Optimize using shooting method:
   a. Simulate candidate trajectory
   b. Calculate total distance to target
   c. Update parameters using L-BFGS-B
   d. Repeat until convergence
3. Verify collision detection

5. Test Cases and Results

5.1 Simple Background Test

Conditions:

• Uniform background
• Good lighting
• Single blue ball
• No occlusions

Performance:

• Detection accuracy: >95%
• Position error: <2%
• Trajectory smoothing: Effective
• Extrapolation error: <5%
• Interception success: >90%

5.2 Complex Background Test

Conditions:

• Textured background
• Variable lighting
• Multiple blue objects
• Partial occlusions

Performance:

• Detection accuracy: <60%
• False positives: >30%
• Position error: >10%
• Trajectory breaks: Frequent
• Interception success: <50%

5.3 Visualization Improvements

New features:

• Adaptive axis scaling with margins
• Complete trajectory display
• Grid overlay for better readability
• Clear axis labels in metric units
• Separate current and full path visualization
• Distinct collision point marking

5.4 System Limitations

1. Video Processing:

• Color-specific detection
• Background sensitivity
• No occlusion handling
• Fixed threshold values

2. Trajectory Analysis:

• Fixed smoothing parameters
• Basic physics model
• Limited extrapolation time
• No uncertainty estimation

3. Visualization:

• 2D representation only
• Fixed update interval
• No dynamic scaling
• Limited animation controls

6. Recommendations

1. Detection Improvements:

• Deep learning-based detection
• Multi-color support
• Adaptive thresholding
• Motion prediction

2. Physics Enhancements:

• Air resistance modeling
• Spin effects
• Wind consideration
• 3D trajectory support

3. Visualization Updates:

• Interactive parameter adjustment
• Real-time statistics display
• Multiple view angles
• Trajectory uncertainty visualization

4. General Improvements:

• Adaptive time stepping
• Parallel computation
• Error estimation
• Automatic parameter tuning