Multi-Camera Island Correspondence for 3D Object Localization

Mathematical Framework for Cross-Camera Island Matching

1. Camera System Setup and Calibration

1.1 Camera Model Definition

For each camera C_i (i = 1, 2, 3), define the perspective projection:

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π_i: ℝ³ → ℝ²
π_i(X) = K_i [R_i | t_i] [X; 1]

where:

1.2 View-Aligned Scanline Orientation Strategy

Camera View Vector:

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v_i = R_i^T [0, 0, 1]ᵀ  (optical axis in world coordinates)

Co-Perpendicular Plane Definition: For cameras C_i and C_j, the co-perpendicular plane normal is:

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n_ij = v_i × v_j / ||v_i × v_j||

Optimal Scanline Orientation: The scanline direction that maximizes geometric correspondence fidelity:

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θ_optimal^{(ij)} = atan2(n_ij[1], n_ij[0])

Multi-Camera Consensus Orientation: For three cameras, find the orientation that minimizes total geometric distortion:

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θ_consensus = argmin_θ Σ_{i<j} w_ij · Angular_Distance(θ, θ_optimal^{(ij)})

where w_ij are inter-camera weight factors based on baseline length and viewing angle.

1.3 Geometric Scanline Transformation

World-to-Scanline Coordinate System: For scanline orientation θ, define the transformation:

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T_scanline = [cos θ    sin θ   0]
             [-sin θ   cos θ   0]
             [0        0       1]

View-Aligned Scanning Lines:

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L_θ(t, s) = T_scanline^{-1} [t; s; 0] + origin_offset

This ensures that scanlines in different cameras sample geometrically corresponding regions of 3D objects.

1.4 Epipolar-Aligned Scanline Refinement

Epipolar Line Direction: For camera pair (C_i, C_j), the epipolar line direction at point p_i is:

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e_ij(p_i) = F_ij p_i / ||F_ij p_i||₂

Scanline-Epipolar Alignment Score:

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A_ij(θ) = (1/|Image|) ∫∫ |cos(angle(scanline_direction(θ), e_ij(p)))|² dp

Optimized Scanline Orientation:

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θ_final = argmax_θ Σ_{i<j} w_ij · A_ij(θ) · Geometric_Consistency_ij(θ)

2. View-Aligned Island Representation

2.1 Geometrically Consistent Island Extraction

For a 3D object O with boundary ∂O, its projection into camera C_i using view-aligned scanlines creates island I_i:

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I_i^{(θ_consensus)} = {π_i(X) | X ∈ ∂O ∩ Visible_i, sampled along L_θ_consensus}

Geometric Correspondence Preservation: The view-aligned scanning ensures that island boundaries in different cameras correspond to the same 3D geometric features:

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Correspondence_3D(I_i, I_j) = high when θ_i ≈ θ_j ≈ θ_consensus

2.2 Enhanced Island Descriptors with View Alignment

Scanline-Consistent Boundary Parameterization:

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b_i(s) = boundary_point_i(s) expressed in view-aligned coordinates

Cross-Camera Geometric Invariants: Using view-aligned scanlines, compute invariant descriptors:

Normalized Boundary Curvature:

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κ_normalized^{(i)}(s) = κ_i(s) · depth_compensation_i(s)

View-Corrected Aspect Ratios:

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aspect_corrected^{(i)} = aspect_raw^{(i)} / projection_distortion_i(θ_consensus)

Scanline-Aligned Wavelet Signatures:

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W_aligned^{(i)}(a,b) = CWT(profile_i(θ_consensus), a, b)

2.3 Geometric Consistency Metrics

Scanline Alignment Quality:

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Q_alignment = (1/3) Σ_i cos²(θ_consensus - θ_optimal^{(camera_i)})

Cross-View Sampling Coherence:

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C_sampling = Corr(sampling_density_i(θ_consensus), sampling_density_j(θ_consensus))

where sampling density accounts for perspective foreshortening.

3. Enhanced Epipolar-Constrained Island Correspondence

3.1 View-Aligned Epipolar Constraints

With view-aligned scanlines, the epipolar constraint becomes more geometrically meaningful:

Aligned Epipolar Distance:

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d_epipolar_aligned(I_i, I_j) = (1/|I_i|) Σ_{p_i∈I_i} min_{p_j∈I_j} |p_j^T F_ij p_i| · alignment_factor_ij

where:

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alignment_factor_ij = |cos(angle(scanline_i, epipolar_line_ij))|

Geometric Correspondence Probability:

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P_geometric(I_i ↔ I_j | θ_consensus) = exp(-λ · d_epipolar_aligned(I_i, I_j))

3.2 View-Aligned Multi-View Consistency

Triangulation Consistency with Scan Alignment:

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E_reproj_aligned(I_1, I_2, I_3) = Σ_{i=1}^3 ||p_i - π_i(X_consensus)||² · view_quality_i(θ_consensus)

where view_quality_i(θ_consensus) penalizes scanning orientations that create geometric distortion for camera i.

Three-Camera Geometric Consensus:

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Consensus_3D(I_1, I_2, I_3) = ∏_{i<j} Geometric_Compatibility_ij(θ_consensus)

3. Epipolar-Constrained Island Correspondence

3.1 Epipolar Constraint for Island Matching

For candidate island correspondence I_i ↔ I_j, the epipolar constraint requires:

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d_epipolar(I_i, I_j) = (1/|I_i|) ∑_{p_i∈I_i} min_{p_j∈I_j} |p_j^T F_ij p_i|

Constraint Satisfaction:

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Valid(I_i ↔ I_j) ⟺ d_epipolar(I_i, I_j) < τ_epipolar

3.2 Multi-View Geometric Consistency

For three-camera correspondence I_1 ↔ I_2 ↔ I_3, enforce triangulation consistency:

Triangulated 3D Point: For corresponding points p_1, p_2, p_3, solve:

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X* = argmin_X ∑_{i=1}^3 ||p_i - π_i(X)||²

Reprojection Error:

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E_reproj(I_1, I_2, I_3) = (1/3) ∑_{i=1}^3 ||p_i - π_i(X*)||²

Consistency Check:

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Consistent(I_1, I_2, I_3) ⟺ E_reproj(I_1, I_2, I_3) < τ_reproj

4. Advanced View-Aligned Matching Algorithm

4.1 Scanline-Coherent Hierarchical Matching

Level 0 (Coarse) - View-Aligned Geometric Matching:

Scanline-Corrected Centroid Distance:

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d_centroid_aligned(I_i, I_j) = ||transform_to_consensus(c_i) - transform_to_consensus(c_j)||

where transform_to_consensus projects centroids into the consensus scanline coordinate system.

Perspective-Corrected Size Matching:

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d_size_aligned(I_i, I_j) = |log(Area_normalized_i) - log(Area_normalized_j)|

where:

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Area_normalized_i = Area(I_i) / perspective_scaling_factor_i(θ_consensus)

View-Aligned Orientation Consistency:

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d_orientation_aligned(I_i, I_j) = |θ_principal_i(θ_consensus) - θ_principal_j(θ_consensus)|

4.2 Geometric Feature Correspondence Enhancement

Scanline-Aligned Boundary Matching:

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boundary_correspondence_ij = DTW(boundary_i(θ_consensus), boundary_j(θ_consensus))

using Dynamic Time Warping to handle sampling differences.

Cross-View Curvature Correlation:

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ρ_curvature_aligned = Corr(κ_i(s, θ_consensus), κ_j(s_aligned, θ_consensus))

Multi-Scale Wavelet Correspondence:

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W_correspondence = Σ_{scales} w_scale · Corr(W_i^{(scale)}, W_j^{(scale)})

where wavelets are computed on view-aligned profiles.

4.3 View-Aligned Matching Score

Enhanced Geometric Compatibility:

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S_geometric_aligned(I_i, I_j) = α₁ · S_centroid_aligned + α₂ · S_size_aligned + 
                                α₃ · S_orientation_aligned + α₄ · S_boundary_aligned +
                                α₅ · Q_alignment^{(ij)}

where Q_alignment^{(ij)} rewards good scanline alignment between cameras i and j.

5. Optimized Three-Way Correspondence with View Alignment

5.1 Scanline-Aware Triangle Scoring

Geometric Triangle Consistency:

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Score_triangle_aligned(I₁, I₂, I₃) = ∏_{i<j} S_geometric_aligned(I_i, I_j) · 
                                     Triangulation_Quality(I₁, I₂, I₃, θ_consensus)

Triangulation Quality Factor:

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Triangulation_Quality = 1 / (1 + β · E_reproj_aligned(I₁, I₂, I₃))

5.2 View-Alignment Optimization

Joint Optimization Problem:

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{θ_optimal, Correspondences} = argmax_{θ,M} Σ_{triplets} Score_triangle_aligned(triplet | θ)

subject to:

Iterative Refinement Algorithm:

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1. Initialize θ_consensus using co-perpendicular plane method
2. Extract view-aligned islands using θ_consensus
3. Compute initial correspondences
4. Refine θ_consensus based on correspondence quality
5. Re-extract islands and update correspondences
6. Repeat until convergence

6. Enhanced 3D Reconstruction with View-Aligned Data

6.1 Improved Triangulation Accuracy

View-Aligned Multi-View Triangulation:

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X_object = argmin_X Σ_{i=1}^3 w_i(θ_consensus) · ||p_i - π_i(X)||²

where w_i(θ_consensus) weights measurements based on scanline alignment quality for camera i.

Geometric Uncertainty Modeling:

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Σ_X = (Σ_{i=1}^3 w_i · J_i^T Σ_measurement_i^{-1} J_i)^{-1}

where J_i is the Jacobian of the projection function and Σ_measurement_i includes view-alignment uncertainty.

6.2 View-Aligned 3D Shape Reconstruction

Consensus 3D Boundary:

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∂O_3D = {X | ∃(p₁, p₂, p₃) ∈ (∂I₁ × ∂I₂ × ∂I₃), triangulate(p₁, p₂, p₃) = X}

3D Geometric Consistency Check:

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Valid_3D_Point(X) = Σ_{i=1}^3 Boundary_Distance_i(π_i(X), ∂I_i) < τ_3D

7. Computational Optimization for View-Aligned Processing

7.1 Efficient Scanline Orientation Selection

Precomputed Orientation Lookup:

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θ_LUT[camera_config] = precomputed optimal angles for standard camera arrangements

Fast Approximation:

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θ_fast = weighted_average({θ_optimal^{(12)}, θ_optimal^{(13)}, θ_optimal^{(23)}})

7.2 Parallel View-Aligned Processing

Concurrent Island Extraction:

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For each camera i in parallel:
    I_i = extract_islands(image_i, θ_consensus, precision_config_i)

Parallel Correspondence Computation:

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For each camera pair (i,j) in parallel:
    S_ij = compute_aligned_similarity(I_i, I_j, θ_consensus)

7.3 Memory-Efficient Implementation

Streaming Scanline Processing: Process one scanline at a time to reduce memory footprint while maintaining view alignment.

Hierarchical Island Storage: Store only necessary detail levels for each correspondence stage.

8. Validation and Quality Metrics

8.1 View-Alignment Quality Assessment

Scanline Coherence Metric:

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Coherence = (1/3) Σ_i Σ_j≠i |cos(angle(scanline_i, scanline_j))|

Geometric Distortion Measure:

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Distortion_i = ∫∫ |perspective_factor_i(u,v,θ_consensus) - 1| du dv

8.2 Correspondence Validation

Cross-View Reprojection Error:

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E_cross_view = (1/3) Σ_{i=1}^3 ||p_i - π_i(X_triangulated)||²

Temporal Consistency (for video):

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E_temporal = ||X_t - predict(X_{t-1}, motion_model)||²

This enhanced framework leverages view-aligned scanline orientations to dramatically improve geometric correspondence fidelity across multiple cameras, resulting in more accurate 3D object localization and robust multi-view analysis.

5. Three-Way Island Correspondence

5.1 Triangular Matching Graph

Define the correspondence graph G = (V, E) where:

5.2 Three-Way Consistency Enforcement

Transitivity Constraint: For triangle (I₁, I₂, I₃), require:

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S(I₁, I₂) · S(I₂, I₃) · S(I₃, I₁) > τ_triangle

Geometric Consistency:

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Consistent_3D(I₁, I₂, I₃) = exp(-βE_reproj(I₁, I₂, I₃))

5.3 Optimal Assignment Problem

Integer Programming Formulation:

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max ∑_{(I₁,I₂,I₃)} x_{123} · Score_total(I₁, I₂, I₃)

subject to:

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∑_{I₂,I₃} x_{123} ≤ 1  ∀I₁  (each island matched at most once)
x_{123} ∈ {0, 1}       (binary assignment variables)
Consistent_3D(I₁, I₂, I₃) > τ_3D  (geometric consistency)

where:

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Score_total(I₁, I₂, I₃) = w₁S(I₁,I₂) + w₂S(I₂,I₃) + w₃S(I₃,I₁) + w₄Consistent_3D(I₁,I₂,I₃)

6. 3D Object Localization and Reconstruction

6.1 Triangulation-Based 3D Reconstruction

For matched island triplet (I₁, I₂, I₃), reconstruct 3D object:

Multi-View Triangulation:

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X_object = argmin_X ∑_{i=1}^3 ∑_{p_i∈I_i} ||p_i - π_i(X)||²

Robust Estimation (RANSAC-based):

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X_robust = RANSAC(triangulate, {(p₁,p₂,p₃)}, τ_inlier)

6.2 3D Object Pose and Shape Estimation

Object Centroid:

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C_3D = (1/3) ∑_{i=1}^3 π_i^{-1}(c_i, Z_estimated)

Principal Axes from Multi-View:

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[U, Σ, V] = SVD([X₁ - C_3D, X₂ - C_3D, X₃ - C_3D])

3D Bounding Box:

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BBox_3D = {C_3D ± λ₁v₁, C_3D ± λ₂v₂, C_3D ± λ₃v₃}

7. Temporal Consistency for Video Sequences

7.1 Temporal Island Tracking

Cross-Frame Island Association:

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I_t^{(i)} ↔ I_{t+1}^{(j)} if S_temporal(I_t^{(i)}, I_{t+1}^{(j)}) > τ_temporal

Temporal Consistency Score:

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S_temporal(I_t, I_{t+1}) = exp(-γ||c_t - predicted_c_{t+1}||² - δ|Area_t - Area_{t+1}|)

7.2 3D Object Trajectory Estimation

Kalman Filter State:

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x_t = [X_t, Y_t, Z_t, Ẋ_t, Ẏ_t, Ż_t]ᵀ

State Transition Model:

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x_{t+1} = F x_t + w_t
z_t = H x_t + v_t

where z_t are the observed 3D positions from triangulation.

8. Algorithm Implementation Framework

8.1 Multi-Camera Island Correspondence Pipeline

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Input: {I₁, I₂, I₃} (island sets from three cameras)
       {K₁, R₁, t₁}, {K₂, R₂, t₂}, {K₃, R₃, t₃} (camera parameters)

1. Compute epipolar geometry: F₁₂, F₁₃, F₂₃
2. Extract multi-level descriptors for all islands
3. Hierarchical matching:
   a. Coarse matching using centroids and bounding boxes
   b. Medium matching using shape moments
   c. Fine matching using wavelet signatures
4. Three-way consistency enforcement
5. Optimal assignment solution
6. 3D triangulation and object reconstruction
7. Temporal tracking (for video sequences)

Output: Matched island triplets with 3D object locations

8.2 Computational Complexity

Matching Complexity:

Memory Requirements:

9. Robustness and Error Handling

9.1 Occlusion Handling

Partial Visibility Detection:

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Visibility_i(I) = ∑_{p∈I} depth_test(π_i^{-1}(p), scene_depth)

Two-Camera Fallback: When island missing in one camera:

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if |{I₁, I₂, I₃}| = 2:
    use stereo triangulation with higher uncertainty

9.2 Calibration Error Robustness

Adaptive Epipolar Thresholds:

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τ_epipolar_adaptive = τ_base + σ_calibration · confidence_factor

Iterative Bundle Adjustment:

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{R_i, t_i, X_objects} = argmin ∑_{i,j} ||p_i^{(j)} - π_i(X_j)||²

This framework provides a complete mathematical foundation for relating islands across multiple camera feeds to identify and locate the same objects in 3D space, with provisions for robustness, temporal consistency, and computational efficiency.