Enhanced CEP-RLE: Multi-Orientation Scanning and Wavelet-Based Geometric Analysis

Mathematical Framework for Advanced CEP-RLE Extensions

1. Multi-Orientation Ensemble Scanning

1.1 Orientation Space Definition

Let Θ = {θ₁, θ₂, …, θₙ} be a set of scanning orientations where θᵢ ∈ [0, 2π).

For each orientation θᵢ, define the scanning line transformation:

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L_θᵢ(t, s) = p₀ + t·û_θᵢ + s·v̂_θᵢ

where:

1.2 Multi-Orientation Run Extraction

For each orientation θᵢ, extract continuous runs:

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R^(θᵢ) = {r^(θᵢ)_j,k | j = 1,...,m_θᵢ, k = 1,...,n^(θᵢ)_j}

where:

1.3 Ensemble Geometric Reconstruction

For a geometric feature F, its ensemble representation is:

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F_ensemble = {F^(θ₁), F^(θ₂), ..., F^(θₙ)}

Ensemble Boundary Estimation: For a point p on the boundary, collect all intersection measurements:

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I(p) = {(θᵢ, t^(θᵢ)_intersect) | p ∈ L_θᵢ(t^(θᵢ)_intersect, s_j)}

Robust Boundary Reconstruction: Use weighted least squares to estimate boundary position:

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p̂ = argmin_p Σᵢ wᵢ ||p - (p₀ + t^(θᵢ)_intersect·û_θᵢ + s_j·v̂_θᵢ)||²

where wᵢ are confidence weights based on measurement quality.

2. Island Run Topology and Evolution

2.1 Island Run Definition

An island run I is a connected component of runs across adjacent scanlines:

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I = {r_j,k | j ∈ J_I, overlap(r_j,k, r_{j+1,*}) > τ}

where:

2.2 Island Geometric Descriptors

For each island I, define:

Offset Evolution Function:

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δ_I(y) = x_center(I, y) - x_center(I, y₀)

where x_center(I, y) is the centroid x-coordinate of island I at scanline y.

Span Evolution Function:

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w_I(y) = x_end(I, y) - x_start(I, y)

where x_start(I, y) and x_end(I, y) are the leftmost and rightmost boundaries.

Island Lifespan:

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L_I = [y_birth, y_death] = [min(J_I), max(J_I)]

2.3 Island Topology Events

Birth Event: Island I appears at scanline y:

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Birth(I, y) ⟺ |{r_{y-1,k} : overlap(r_{y,*}, r_{y-1,k}) > τ}| = 0

Death Event: Island I disappears after scanline y:

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Death(I, y) ⟺ |{r_{y+1,k} : overlap(r_{y,*}, r_{y+1,k}) > τ}| = 0

Split Event: Island I splits into islands I₁, I₂ at scanline y:

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Split(I → I₁, I₂, y) ⟺ Connected(I, y-1) ∧ ¬Connected(I₁ ∪ I₂, y)

Merge Event: Islands I₁, I₂ merge into island I at scanline y:

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Merge(I₁, I₂ → I, y) ⟺ ¬Connected(I₁ ∪ I₂, y-1) ∧ Connected(I, y)

3. Wavelet Analysis of Geometric Profiles

3.1 Wavelet Transform of Island Profiles

For an island I with lifespan L_I = [y₁, y₂], define the profile functions:

Discrete Profile Sampling:

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δ̃_I = {δ_I(y₁), δ_I(y₁+1), ..., δ_I(y₂)} ∈ ℝ^{|L_I|}
w̃_I = {w_I(y₁), w_I(y₁+1), ..., w_I(y₂)} ∈ ℝ^{|L_I|}

Continuous Wavelet Transform (CWT):

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W_δ(a, b) = (1/√a) ∫ δ_I(y) ψ*((y-b)/a) dy
W_w(a, b) = (1/√a) ∫ w_I(y) ψ*((y-b)/a) dy

where:

3.2 Multi-Scale Geometric Descriptors

Scale-Space Feature Vector:

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F_I = [F_I^{(1)}, F_I^{(2)}, ..., F_I^{(J)}]

where F_I^{(j)} represents features at scale level j.

Scale-Specific Features:

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F_I^{(j)} = [||W_δ^{(j)}||₂, ||W_w^{(j)}||₂, E_δ^{(j)}, E_w^{(j)}, H_δ^{(j)}, H_w^{(j)}]

where:

3.3 Wavelet-Based Shape Classification

Shape Signature Matrix:

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S_I = [W_δ(a₁, b₁) ... W_δ(a₁, b_B)]
      [    ⋮      ⋱      ⋮    ]
      [W_δ(a_A, b₁) ... W_δ(a_A, b_B)]

Invariant Geometric Descriptors:

Translation Invariance: Use relative wavelet coefficients:

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W̃_δ(a, b) = W_δ(a, b) - W_δ(a, b₀)

Scale Invariance: Normalize by island span:

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Ŵ_δ(a, b) = W_δ(a, b) / max_y w_I(y)

Rotation Invariance: For multi-orientation ensemble:

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W_ensemble(a, b) = (1/N) Σᵢ W^{(θᵢ)}_δ(a, b)

4. Enhanced Expectation-Prior Mechanism

4.1 Multi-Orientation Statistical Models

For each spatial bin B_k and orientation θᵢ, maintain statistical models:

Profile Expectation Models:

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μ_δ^{(θᵢ)}(B_k) = E[δ_I(y) | I ∈ B_k, θᵢ]
Σ_δ^{(θᵢ)}(B_k) = Cov[δ_I(y) | I ∈ B_k, θᵢ]

Wavelet Coefficient Models:

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μ_W^{(θᵢ)}(a_j, B_k) = E[W_δ(a_j, ·) | I ∈ B_k, θᵢ]
Σ_W^{(θᵢ)}(a_j, B_k) = Cov[W_δ(a_j, ·) | I ∈ B_k, θᵢ]

4.2 Ensemble Prediction

Orientation-Weighted Prediction:

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P(δ_I(y+1) | H_y) = Σᵢ w_θᵢ P^{(θᵢ)}(δ_I(y+1) | H_y^{(θᵢ)})

where w_θᵢ are orientation confidence weights and H_y^{(θᵢ)} is the history for orientation θᵢ.

Wavelet-Informed Prediction:

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P(I_{y+1} | I_y, W_I) ∝ exp(-||W_I - μ_W(B_k)||²_{Σ_W(B_k)})

5. Computational Complexity Analysis

5.1 Multi-Orientation Scanning Complexity

Time Complexity:

Space Complexity:

5.2 Wavelet Analysis Complexity

Per Island Analysis:

5.3 Overall Algorithm Complexity

Enhanced CEP-RLE Total Complexity:

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T_total = O(NHW + I·J·L̄ log L̄ + N²R)

Memory Requirements:

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M_total = O(NR + I·J·L̄ + N²F)

6. Implementation Considerations

6.1 Orientation Selection Strategies

Uniform Sampling:

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Θ_uniform = {2πk/N | k = 0, 1, ..., N-1}

Adaptive Sampling:

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Θ_adaptive = argmin_Θ Σᵢ H(F^{(θᵢ)}) subject to |Θ| ≤ N_max

where H(F^{(θᵢ)}) is the entropy of features extracted at orientation θᵢ.

Hierarchical Sampling:

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Θ₁ = {0, π/2}  (coarse)
Θ₂ = Θ₁ ∪ {π/4, 3π/4}  (medium)
Θ₃ = Θ₂ ∪ {π/8, 3π/8, 5π/8, 7π/8}  (fine)

6.2 Wavelet Basis Selection

Shape-Specific Wavelets:

Adaptive Basis Selection:

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ψ_optimal = argmin_ψ ∫ |f(y) - f_reconstructed^{(ψ)}(y)|² dy

This mathematical framework provides the foundation for implementing multi-orientation scanning with wavelet-based geometric analysis, creating a comprehensive system for shape-aware spatial data compression and analysis.