Volumetric Convolution Forward Pass

In deep learning, volumetric convolution (also known as 3D convolution) extends the concept of traditional 2D convolution to three-dimensional data. This operation is fundamental for processing spatiotemporal data like video sequences, medical imaging scans (CT, MRI), and point cloud representations.

A volumetric convolution works by sliding a 3D kernel (a small cubic or rectangular prism of learnable weights) across an input 3D volume along all three spatial dimensions: depth, height, and width. At each position, the kernel computes a weighted sum (dot product) of the overlapping elements, producing a single output value.

Mathematical Formulation: For an input volume V with dimensions (D × H × W) and a kernel K with dimensions (Kd × Kh × Kw), the output at position (d, h, w) is computed as:

$$O[d, h, w] = \sum_{i=0}^{K_d-1} \sum_{j=0}^{K_h-1} \sum_{k=0}^{K_w-1} V[d \cdot s_d + i, h \cdot s_h + j, w \cdot s_w + k] \cdot K[i, j, k]$$

where (sₐ, sₕ, sᵥ) represents the stride in each dimension.

Output Dimensions: Given padding (pₐ, pₕ, pᵥ) and stride (sₐ, sₕ, sᵥ), the output dimensions are calculated as:

• Output Depth: ⌊(D + 2·pₐ - Kd) / sₐ⌋ + 1
• Output Height: ⌊(H + 2·pₕ - Kh) / sₕ⌋ + 1
• Output Width: ⌊(W + 2·pᵥ - Kw) / sᵥ⌋ + 1

Your Task: Implement a function that computes the forward pass of a volumetric convolution. Given an input 3D volume, a 3D kernel, stride values for each dimension, and padding values for each dimension, compute and return the resulting output feature map.

Key Concepts:

• Stride: Controls how far the kernel moves between positions. A stride of 2 means the kernel skips every other position, effectively downsampling the output.
• Padding: Zero values added around the input volume's borders to control the output size and preserve spatial dimensions at the boundaries.
• Feature Extraction: The kernel learns to detect specific 3D patterns like motion trajectories, volumetric textures, or anatomical structures.