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Intermediate•Deep Learning
CNNs
Convolutional Neural Networks for image recognition.
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Research-Level Deep Dive & Equations
In mathematical analysis, a **convolution** between a continuous input signal and a filter kernel is defined as the integral of the product of the two functions after one is reflected and shifted:
For discrete 2D applications like computer vision, a 2D convolution of an image with a kernel is defined as:
where . The spatial reflection of the kernel (represented by the minus sign in index subtraction) is necessary for convolution to satisfy the mathematical property of **associativity**: .
### Cross-Correlation in Deep Learning:
In practical deep learning libraries (e.g., PyTorch, TensorFlow), the operation implemented under the name "convolution" is actually **cross-correlation**, which omits the spatial reflection step:
Because the kernel parameters are learned from scratch during backpropagation, spatial reflection is mathematically redundant. If reflection were applied, the network would simply learn a flipped version of the kernel. Omitting the flip saves computational overhead.
### Output Size Formula with Dilation:
The spatial dimensions of a convolutional layer's output feature map depend on input width , kernel size , padding , stride , and **dilation rate** (which inserts spaces between kernel elements):
Dilation allows the filter to cover a much larger receptive field without increasing the number of trainable parameters or FLOPS, which is highly useful in semantic segmentation (e.g., Dilated Convolutions in DeepLab).
Key Equations
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