Dilated Convolution
The problem, from first principles
Growing a network's receptive field (how much of the input each output pixel "sees") the naive way costs something: bigger kernels mean more parameters, and stacking more layers means more depth and more downsampling (losing spatial resolution, which segmentation and dense-prediction tasks specifically can't afford to lose). Dilated ("atrous," from the French for "with holes") convolution grows the receptive field for free — same kernel, same parameter count — by spacing out where the kernel looks, skipping pixels between each tap instead of looking at a contiguous patch.
From theory to code
Theory says: instead of a kernel scanning a contiguous kH x kW patch, spread its taps dilation pixels apart, so the same kH x kW set of weights now spans a (kH-1)*dilation + 1 by (kW-1)*dilation + 1 region of the input — the kernel's effective footprint grows even though its parameter count (kH * kW) doesn't.
Implement dilated_conv2d(image, kernel, dilation=1) against that reasoning.
Constraints
image: shape(H, W).kernel: shape(kH, kW).dilation: positive integer;dilation=1is an ordinary, non-dilated convolution.- Output shape:
(H - eff_kH + 1, W - eff_kW + 1), whereeff_kH = (kH - 1) * dilation + 1(same formula foreff_kW). imageandkernelare never modified in place.
Hints
Open one at a time. Each gives away a little more than the last.
Hint 1
Compute eff_kH = (kH - 1) * dilation + 1 and eff_kW the same way, once, before the main loop — these are the true spatial span the kernel now reaches, even though the kernel array itself is still only kH x kW.
Hint 2
For output position (i, j), the slice image[i : i+eff_kH : dilation, j : j+eff_kW : dilation] — note the dilation step — picks out exactly the kH x kW pixels the spread-out kernel taps land on, ready to multiply elementwise against kernel and sum.
Keywords
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