2D Convolution Operation

The 2D convolution operation is one of the most fundamental building blocks in computer vision and deep learning. It enables neural networks to detect spatial patterns, edges, textures, and complex features within images and other grid-structured data.

In a 2D convolution, a small filter (called a kernel) slides across the input matrix, computing element-wise products and summing them at each position. This operation transforms the input into a new representation called a feature map, which highlights specific patterns based on the kernel's values.

The Convolution Process:

Given an input matrix I of dimensions H × W, a kernel K of dimensions kH × kW, padding p, and stride s, the output feature map O is computed as follows:

1. Padding: First, surround the input matrix with p layers of zeros on all sides, creating a padded input of dimensions (H + 2p) × (W + 2p)

2. Sliding Window: Starting from the top-left corner, place the kernel over the padded input. At each position (i, j), compute:

$$O_{i,j} = \sum_{m=0}^{kH-1} \sum_{n=0}^{kW-1} K_{m,n} \cdot I_{padded}[i \cdot s + m, j \cdot s + n]$$

3. Stride: After computing one output value, move the kernel by s positions horizontally. When reaching the end of a row, move down by s positions and start from the left again.

Output Dimensions:

The output feature map has dimensions:

• Output Height = ⌊(H + 2p - kH) / s⌋ + 1
• Output Width = ⌊(W + 2p - kW) / s⌋ + 1

Your Task: Implement a Python function that performs a 2D convolution operation. The function should properly apply zero-padding, slide the kernel with the specified stride, and compute the element-wise products and sums to produce the output feature map.