Padding: Same vs. Valid

The problem, from first principles

01-single-filter-conv2d's output is always smaller than its input — a kernel can't hang off the edge, so every convolution shrinks the image a little. Stack enough convolutional layers and an image eventually shrinks to nothing, which is rarely what you want for a deep network's middle layers.

The fix is to give the kernel somewhere to go at the edges: surround the image with a border of zeros first, wide enough that the kernel can center itself on every original pixel, including the ones right at the edge. Do that and the output comes out the same size as the input went in.

From theory to code

Theory gives two modes: "valid" is exactly what 01-single-filter-conv2d already does — no border, output shrinks. "same" adds a zero border of the right width first, then runs the identical convolution.

Implement conv2d_with_padding(image, kernel, padding="valid") against that reasoning, reusing 01-single-filter-conv2d's conv2d_single_filter rather than reimplementing the sliding-window logic.

Constraints

• image: shape (H, W). kernel: shape (kH, kW), both dimensions odd.
• padding="valid": no padding, output shape (H - kH + 1, W - kW + 1).
• padding="same": zero-pad by (kH-1)//2 and (kW-1)//2 on each side per axis, output shape exactly (H, W).
• Any other padding value raises ValueError.
• image and kernel are never modified in place.

Hints

Open one at a time. Each gives away a little more than the last.