Convolution Operator and Layer Explained in Deep Learning

What is a Convolution Layer in Deep Learning?

A convolution layer is a building block of Convolutional Neural Networks (CNNs). It's mostly used to process image data.

Instead of connecting every pixel of the input to every neuron (as in a fully connected layer), a convolution layer slides a small filter (kernel) across the image and extracts features like edges, textures, or patterns.

Key Terms

Input: The image or feature map (e.g., 6x6 pixels).
Kernel(Filter): A small matrix (e.g., 3x3 or 5x5) that moves across the image.
Stride: How many steps the filter moves at a time.
Padding: Adding extra pixels around the image to control the output size.
Feature Map: The result of the convolution operation.

How Convolution Works

Let’s walk through an example with no padding and stride = 1.

1. Input: 6x6 Matrix

Input:
[ [9, 4, 1, 6, 5],
  [1, 1, 1, 0, 2],
  [1, 2, 1, 1, 3],
  [2, 1, 0, 3, 0],
  [1, 4, 2, 5, 6] ]

2. Kernel: 3x3

Kernel:
[ [1, 2, 0],
  [0, 1, 4],
  [1, 0, 1] ]

3. Convolution Operation (3x3)

At each position:

Multiply overlapping numbers
Sum the result

Example at top-left corner (first 3x3 area):

Input Patch:
[ [9, 1, 1],
  [1, 1, 1],
  [1, 2, 1] ]

Calculation:
9*1 + 1*2 + 1*0 +
1*0 + 1*1 + 1*4 +
1*1 + 2*0 + 1*1
= 9 + 2 + 0 + 0 + 1 + 4 + 1 + 0 + 1 = 24

**Slide this filter across the whole input to produce the output feature map.

Output Size Formula

Example: 5x5 Kernel

Same 6x6 input, now with 5x5 kernel and no padding:

Only 2x2 positions to apply the kernel.

Why Use Convolution Instead of Fully Connected Layers?

1. Parameter Efficiency

Let’s say you have a 32x32 RGB image (i.e., 32x32x3 = 3072 inputs).

Fully Connected Layer (FC):
Every pixel connects to every neuron.

With 100 neurons: Parameters=3072x100=307,200

Convolution Layer (3x3, 3 input channels, 32 filters):

Parameters per filter=3x3x3=27 Total=27x32=864

** Way fewer parameters (307,200 → 864)!

2. Spatial Hierarchy

Convolution keeps spatial info (e.g., nearby pixels are related), while fully connected layers flatten everything.

How to Calculate Parameters in Convolution Layer

For each filter:

$\text{Parameters}=K\times K\times C_{\text{in}}+1\quad(\text{+1 for bias})$

Then multiply by number of filters:

$\text{Total}=(K\times K\times C_{\text{in}}+1)\times C_{\text{out}}$

Example:

Input Channels = 3 (RGB image)
Kernel Size = 5x5
Number of Filters = 64

${Total Params}=(5\times 5\times 3+1)\times 64=(75+1)\times 64=76\times 64=4864$

How to Calculate MACs

Each kernel application =

$K\times K\times C_{\text{in}}\text{MACs per position}$

Multiply by:

Number of output positions (H x W)
Number of filters

Example:

Input: 32x32x3
Kernel: 3x3
Filters: 16
Output: 30x30 (no padding)

$K\times K\times C_{\text{in}}\text{MACs per position}$$\text{MACs}=3\times 3\times 3\times 30\times 30\times 16=388,800$

References

Fuhg, J.N., Karmarkar, A., Kadeethum, T. et al. Deep convolutional Ritz method: parametric PDE surrogates without labeled data. Appl. Math. Mech.-Engl. Ed. 44 , 1151–1174 (2023). https://doi.org/10.1007/s10483-023-2992-6.

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