CrownGeneration

AI Pipeline for Crown Generation Module

📌 Our Architecture

📌 Abstract Representation of our Architecture

Our Architecture is a transformer-based encoder-decoder model for point cloud construction.

1. Input:
   • point cloud → X ∈ R^(B × N × 3)
   • Where B = batch size, N = number of points, and each point has 3D coordinates (x, y, z).

1️⃣ Feature Extraction (DGCNN)

• Extracts local features using a Dynamic Graph CNN (DGCNN).
• Converts each point into a feature vector.

📌 Mathematical Operation:
F = DGCNN(X)

📌 Shape Transformation:
X ∈ R^(B × N × 3) → F ∈ R^(B × N × d)

where d is the feature dimension.

2️⃣ Positional Embedding

• Encodes spatial information into the feature space.

📌 Mathematical Operation:
F' = F + PositionalEmbedding(X)

📌 Shape:
F ∈ R^(B × N × d) → F' ∈ R^(B × N × d)

(Positional encoding does not change dimensions.)

3️⃣ Geometry-Aware Transformer Encoder

• Self-Attention applied to point proxies.
• Models local and global geometric relations.

📌 Mathematical Operation:
V = Encoder(F')

📌 Shape:
F' ∈ R^(B × N × d) → V ∈ R^(B × N × d)

4️⃣ Query Generator (Dynamic Queries)

• Generates queries based on encoder output.
• Queries represent missing point proxies.

📌 Mathematical Operation:
Q = QueryGenerator(V)

📌 Shape:
V ∈ R^(B × N × d) → Q ∈ R^(B × M × d)

where M is the number of missing point proxies.

5️⃣ Geometry-Aware Transformer Decoder

• Cross-attention between encoder outputs and queries.
• Generates refined point proxy features.

📌 Mathematical Operation:
H = Decoder(Q, V)

📌 Shape:
Q ∈ R^(B × M × d), V ∈ R^(B × N × d) → H ∈ R^(B × M × d)

6️⃣ FoldingNet (Final Point Cloud Generation)

• Generates fine-grained point clouds from point proxies.
• Uses an MLP-based upsampling strategy.

📌 Mathematical Operation:
P = FoldingNet(H)

📌 Shape:
H ∈ R^(B × M × d) → P ∈ R^(B × M' × 3)

where M' is the number of final generated points.

📝 Summary Table (Shapes Per Module)

Module	Input Shape	Output Shape
1. Feature Extractor (DGCNN)	B × N × 3	B × N × d_lf
2. Positional Embedding	B × N × 3	B × N × d_pe
3. Transformer Encoder	B × N × d	B × N × d
4. Query Generator	B × N × d	B × M × d, B × M × 3
5. Transformer Decoder (Cross-Attn)	B × M × d_k, B × N × d_v	B × M × d
6. FoldingNet (Final Output)	B × M × 3	B × M × 3

🚀 Final Output

✅ Constructed 3D Point Cloud:
P ∈ R^(B × M × 3)

where M is the number of reconstructed points.

This provides the abstract mathematical operations and shape transformations for each module in Our Architecture.🚀