Effective convolution kernel size ($k_{\mathrm{eff}}=1+d(k-1)$)

The effective convolution kernel size is the bounding extent of a dilated convolution kernel along a spatial axis. For $k$ sampled coefficients with convolution dilation $d$, their offsets are $0,d,2d,\dots ,(k-1)d$. The first-to-last displacement is $d(k-1)$, so including the first position givesThis extent determines the kernel footprint used in spatial shape calculations; it is not generally the number of coefficients.^[ref]

For a two-dimensional example, choose a sampled kernel shape $(3,2)$ with dilation $(2,3)$. Its row offsets are $\{0,2,4\}$ and column offsets are $\{0,3\}$. The bounding shape is thereforeThere are six sampled spatial positions inside a twenty-position bounding box. Fourteen intermediate positions are absent from the sampling pattern.

With three input neural network channels and four output channels, ordinary dense channel connectivity usesweights and optionally four biases, giving $76$ parameters. A general dense five-by-four kernel with the same channel counts would have $240$ weights. Representing the dilated operation that way would require $168$ of them to be fixed zeros, rather than independently learned parameters.^[ref]

For an unpadded $9$-by-$10$ input and convolution stride one, the output grid isMore generally, for one axis with symmetric zero padding $p$ and stride $s$,when the padded input accommodates the footprint. The floor function counts admissible filter origins.^[ref]

The effective size is a geometric quantity for one layer. The complete neural network receptive field through several layers additionally depends on their kernels, strides and connections. Likewise, changing stride moves the spacing of origins; changing dilation changes the footprint at each origin. Keeping bounding extent, sampled-position count and learned-parameter count separate avoids confusing three different sizes.