How do convolutional neural networks work?

CNNs are specifically designed to work with grid-like data structures, such as images, making them a powerful tool in the realm of computer vision.

Theoretical Background

• Convolutional Layer: The convolution operation involves a small matrix called a filter or kernel, which slides over the input image to produce a feature map. This operation helps in detecting features such as edges and textures. The mathematical operation can be expressed as: $(f * g)(t) = \int f(\tau)g(t - \tau) \, d\tau$ where (*) denotes the convolution operation.

• Pooling Layer: This layer is responsible for down-sampling the feature maps to reduce their dimensions and make the computation more feasible. Common pooling techniques include max pooling, which selects the maximum value in each region, and average pooling.

• Fully Connected Layer: After several convolutional and pooling operations, the feature maps are flattened and passed through a fully connected layer to perform classification or regression tasks.

Practical Applications

CNNs are extensively used in image classification, object detection, and semantic segmentation. For example, in medical imaging, CNNs can be used to detect tumors in MRI scans.

Code Example

A simple example using a CNN in Keras for image classification could look like this:

from keras.models import Sequential
from keras.layers import Conv2D, MaxPooling2D, Flatten, Dense

model = Sequential([
    Conv2D(32, (3, 3), activation='relu', input_shape=(64, 64, 3)),
    MaxPooling2D(pool_size=(2, 2)),
    Flatten(),
    Dense(128, activation='relu'),
    Dense(10, activation='softmax')
])

Diagrams

Below is a simple diagram illustrating the flow of data through a CNN:

graph TD;
    A[Input Image] --> B[Convolutional Layer];
    B --> C[Pooling Layer];
    C --> D[Flatten];
    D --> E[Fully Connected Layer];
    E --> F[Output Layer];

External References

• Stanford's CS231n Convolutional Neural Networks for Visual Recognition
• Dive into Deep Learning - CNNs

In summary, CNNs leverage the spatial structure of images, which makes them highly effective for tasks in computer vision, distinguishing them from traditional neural networks that lack such spatial awareness.