Foundations — Feature maps and receptive fields

0. The grid: what an image actually is

Before any symbol, we need the object everything sits on: a grid of numbers.

Look at the figure: the pixel at row 1, column 2 is just one number. We are going to slide a small stencil over exactly this kind of grid. Every symbol below is a count of boxes on this grid.

1. $H$ and $W$ — the size of the grid

• Picture: the two edges of the rectangle in the figure above — one arrow pointing down (length $H$), one pointing right (length $W$).
• Why the topic needs it: the output-size formula starts from these two numbers. You cannot count where a stencil fits until you know how big the board is.

2. The filter and $k$ — the sliding stencil

• Picture: the small yellow square in the figure, sitting on top of a $k\times k$ block of the input. Under each yellow cell is one input box; we multiply pair-by-pair and add.
• Why the topic needs it: $k$ tells us how many boxes the stencil "eats" at each position. A bigger $k$ sees more at once but leaves fewer landing spots — this is exactly the trade-off inside the output formula.

3. Stride $s$ — how far the stencil jumps

• Picture: in the figure, the stencil's first position is at column 0, the next at column $s$. The gap between the two yellow outlines is the stride.
• Why the topic needs it: stride controls two things the parent uses. First, how many outputs fit (bigger jumps → fewer outputs). Second, it enters the receptive-field product $\prod s_i$ — each jump upstream stretches how far downstream neurons can "reach". More in 3.4.03-Padding-and-stride.

4. Padding $p$ — a border of fake boxes

• Picture: the pale outer frame in the figure of §2 — those cells were not in the original image; we added them so the stencil can sit centred on the very edge pixels.
• Why the topic needs it: padding adds $2p$ to each dimension (one ring on each side), which is why the parent's formula begins with $H+2p$. Without it, edge pixels get visited fewer times and the grid shrinks every layer. See 3.4.03-Padding-and-stride.

5. The floor bracket $\lfloor \rfloor$ — round down

• Picture: you can only land the stencil at whole box positions — half a jump is not a real landing. If the arithmetic gives $6.5$ landings, only $6$ are real, so we floor.
• Why the topic needs it: it appears around every output-size division. A stride that does not divide the board evenly leaves a partial step that simply does not count.

6. Channels $C_{\text{out}}$ — stacking many opinions

• Picture: imagine several transparent sheets, each the output of one stencil, layered into a loaf. Each sheet is one "opinion" (edges here, blobs there).
• Why the topic needs it: the parent describes a feature map as "one filter's opinion" — $C_{\text{out}}$ is the count of opinions, and it turns the 2D feature map into the 3D tensor CNNs actually pass along. See 3.4.05-CNN-architectures.

7. The layer index $l$, and $R_l$ — the receptive field number

• Picture: the fan-out cone in the figure. One box high up (deep layer) traces down through the sheets, widening at each level, until it covers a big square of the original image — that square's side is $R_l$.
• Why the topic needs it: this is the whole second half of the parent. Deeper neurons "see" wider context because their influence cone widens each layer.

8. The product $\prod$ and sum $\sum$ — accumulate over layers

• Picture: each upstream jump magnifies the step size in original-image space. Two jumps of stride 2 downstream means one downstream step slides $2\times 2=4$ original boxes — that multiplication is the $\prod$.
• Why the topic needs it: the recursive receptive-field rule scales each layer's extra reach $(k_l-1)$ by this cumulative jump product: $R_l=R_{l-1}+(k_l-1)\cdot {\prod }_{i=1}^{l-1}{s}_i$ Every symbol here was defined above: $R$ (§7), $k$ (§2), $s$ and the product (§3, §8).

The prerequisite map

Read top to bottom: the grid and filter come first; stride and padding modify how the filter lands; floor and the output formula turn that into a feature map; channels stack many; strides and the product feed the receptive field; both meet at the topic.

Equipment checklist

Cover the right side and test yourself — you are ready for the parent note only if every reveal matches.