cognitive/knowledge_base/technology/augmented_reality.md

type	id	created	modified	tags	aliases	complexity	processing_priority	semantic_relations
concept	augmented_reality_001	2024-03-15	2024-03-15	augmented-reality active-inference spatial-computing mixed-reality	ar mixed-reality spatial-augmentation	advanced	1	type links foundation active_inference free_energy_principle spatial_web type links implements spatial_computing computer_vision human_computer_interaction type links relates virtual_reality spatial_intelligence information_geometry

Augmented Reality

Overview

Augmented Reality (AR) represents the seamless integration of digital information with the physical world, increasingly understood through the framework of active inference. This approach reveals how AR systems minimize uncertainty in spatial registration, user interaction, and environmental understanding.

Mathematical Framework

1. Spatial Registration

Basic equations of AR registration:

\begin{aligned}
& \text{Registration Error:} \\
& E = ||T_{physical} - T_{virtual}|| \\
& \text{Free Energy:} \\
& F = \mathbb{E}_q[\ln q(s) - \ln p(o,s)] \\
& \text{Pose Estimation:} \\
& \dot{\mu} = -\nabla_\mu F
\end{aligned}

2. Visual Processing

Computer vision and tracking:

\begin{aligned}
& \text{Feature Detection:} \\
& I(x,y) * \nabla^2G(x,y,\sigma) \\
& \text{Optical Flow:} \\
& \frac{\partial I}{\partial t} + \nabla I \cdot \mathbf{v} = 0 \\
& \text{SLAM Update:} \\
& p(x_t|z_{1:t}) \propto p(z_t|x_t)p(x_t|z_{1:t-1})
\end{aligned}

3. Interaction Dynamics

User interaction modeling:

\begin{aligned}
& \text{Interaction Field:} \\
& \phi(x,t) = \sum_i w_i K(x-x_i) \\
& \text{Attention Model:} \\
& A(x) = \frac{\exp(-\beta V(x))}{\int \exp(-\beta V(y))dy} \\
& \text{Response Dynamics:} \\
& \tau\dot{r} = -r + f(I) + \eta(t)
\end{aligned}

Implementation Framework

1. AR Engine

class AugmentedReality:
    """Manages AR system using active inference"""
    def __init__(self,
                 vision_params: Dict[str, float],
                 tracking_params: Dict[str, float],
                 inference_params: Dict[str, float]):
        self.vision = vision_params
        self.tracking = tracking_params
        self.inference = inference_params
        self.initialize_system()
        
    def process_frame(self,
                     camera_input: np.ndarray,
                     sensors: Dict,
                     context: Dict) -> Dict:
        """Process AR frame"""
        # Initialize state
        state = self.initialize_state(camera_input)
        
        # Compute free energy
        F = self.compute_free_energy(state)
        
        # Update pose estimation
        pose = self.update_pose(state, F)
        
        # Process visual features
        features = self.process_features(camera_input)
        
        # Update tracking
        tracking = self.update_tracking(features, sensors)
        
        # Generate augmentations
        augmentations = self.generate_augmentations(
            pose, tracking, context)
            
        return {
            'pose': pose,
            'tracking': tracking,
            'augmentations': augmentations
        }
        
    def compute_free_energy(self,
                           state: Dict) -> float:
        """Compute AR free energy"""
        # Visual error
        E_visual = self.compute_visual_error(state)
        
        # Tracking error
        E_tracking = self.compute_tracking_error(state)
        
        # Prior term
        P = self.compute_prior_term(state)
        
        # Free energy
        F = E_visual + E_tracking + P
        
        return F

2. Visual Processor

class ARVision:
    """Processes visual information for AR"""
    def __init__(self):
        self.feature_detector = FeatureDetector()
        self.tracker = VisualTracker()
        self.slam = SLAMSystem()
        
    def process_vision(self,
                      frame: np.ndarray,
                      state: Dict) -> Dict:
        """Process visual information"""
        # Detect features
        features = self.feature_detector.detect(frame)
        
        # Track features
        tracking = self.tracker.update(features)
        
        # Update SLAM
        mapping = self.slam.update(tracking)
        
        return {
            'features': features,
            'tracking': tracking,
            'mapping': mapping
        }

3. Interaction Handler

class ARInteraction:
    """Manages AR interactions"""
    def __init__(self):
        self.gesture = GestureRecognition()
        self.physics = InteractionPhysics()
        self.feedback = HapticFeedback()
        
    def process_interaction(self,
                          user_input: Dict,
                          ar_state: Dict) -> Dict:
        """Process user interactions"""
        # Recognize gestures
        gestures = self.gesture.recognize(user_input)
        
        # Compute physics
        physics = self.physics.simulate(
            gestures, ar_state)
            
        # Generate feedback
        feedback = self.feedback.generate(
            physics)
            
        return {
            'gestures': gestures,
            'physics': physics,
            'feedback': feedback
        }

Advanced Concepts

1. Spatial Understanding

\begin{aligned}
& \text{Scene Graph:} \\
& G = (V,E,\phi) \\
& \text{Spatial Relations:} \\
& R(x,y) = f(d(x,y), \theta(x,y)) \\
& \text{Semantic Mapping:} \\
& p(s|x) = \frac{p(x|s)p(s)}{p(x)}
\end{aligned}

2. User Modeling

\begin{aligned}
& \text{Attention Model:} \\
& p(a|x) = \sigma(-\beta F(a,x)) \\
& \text{Learning Rate:} \\
& \eta(t) = \eta_0(1 + \alpha t)^{-\beta} \\
& \text{Performance:} \\
& P(t) = P_\infty(1 - e^{-t/\tau})
\end{aligned}

3. Display Optimization

\begin{aligned}
& \text{Rendering Quality:} \\
& Q = f(d, v, l) \\
& \text{Latency Compensation:} \\
& x_{pred} = x + v\Delta t + \frac{1}{2}a\Delta t^2 \\
& \text{Focus Depth:} \\
& \frac{1}{F} = \frac{1}{u} + \frac{1}{v}
\end{aligned}

Applications

1. Industrial AR

• Maintenance
• Assembly
• Quality control

2. Medical AR

• Surgical navigation
• Medical training
• Patient monitoring

3. Consumer AR

• Navigation
• Education
• Entertainment

Advanced Mathematical Extensions

1. Computer Vision

\begin{aligned}
& \text{Feature Detection:} \\
& \text{det}(H) = \lambda_1\lambda_2 \\
& \text{Pose Estimation:} \\
& \min_R \sum_i ||x_i - RX_i||^2 \\
& \text{Bundle Adjustment:} \\
& \min_{c,p} \sum_{i,j} ||x_{ij} - \pi(c_i, p_j)||^2
\end{aligned}

2. Information Theory

\begin{aligned}
& \text{Visual Information:} \\
& I(X;Y) = H(X) - H(X|Y) \\
& \text{Channel Capacity:} \\
& C = \max_{p(x)} I(X;Y) \\
& \text{Rate Distortion:} \\
& R(D) = \min_{p(y|x): \mathbb{E}[d(X,Y)]\leq D} I(X;Y)
\end{aligned}

3. Control Theory

\begin{aligned}
& \text{Tracking Control:} \\
& \dot{e} + Ke = 0 \\
& \text{Optimal Control:} \\
& J = \int_0^T (x^TQx + u^TRu)dt \\
& \text{Adaptive Control:} \\
& \dot{\hat{\theta}} = -\gamma e\phi
\end{aligned}

Implementation Considerations

1. Hardware Integration

• Displays
• Sensors
• Processing units

2. Software Architecture

• Real-time processing
• Rendering pipeline
• Sensor fusion

3. User Experience

• Interface design
• Interaction models
• Comfort and safety

References

• azuma_1997 - "A Survey of Augmented Reality"
• friston_2019 - "A Free Energy Principle for a Particular Physics"
• billinghurst_2015 - "Spatial Interfaces"
• hartley_2004 - "Multiple View Geometry"

See Also

• active_inference
• spatial_web
• virtual_reality
• computer_vision
• human_computer_interaction