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Mathematical Model

The FaceLog recognition pipeline relies on mathematical transformations that convert images into numerical representations.

These representations allow the system to compare faces using distance metrics.

Image Representation​

An image is represented as a three-dimensional matrix:

X∈Rm×n×3X \in \mathbb{R}^{m \times n \times 3}

Where:

  • (m) = image height
  • (n) = image width
  • (3) = RGB color channels

Face Detection Using Convolution​

Face detection relies on convolution operations applied across the image.

S(i,j)=∑m=0k∑n=0kX(i+m,j+n)⋅W(m,n)S(i, j) = \sum_{m=0}^{k} \sum_{n=0}^{k} X(i + m, j + n) \cdot W(m, n)

Where:

  • (X) represents the image matrix
  • (W) represents the convolution kernel
  • (S(i,j)) represents the resulting feature map

This operation highlights visual patterns that indicate the presence of a face.

Embedding Generation​

After detecting a face, the system generates an embedding vector using a neural network.

e=fθ(X)e = f_\theta(X)

Where:

  • (X) is the input face image
  • (f_\theta) is the neural network
  • (e) is the resulting embedding vector

Embeddings encode facial characteristics into a numerical representation.

Face Comparison​

To determine whether two embeddings represent the same person, the system computes the Euclidean distance:

d(x,y)=∑i=1n(xi−yi)2d(x,y) = \sqrt{\sum_{i=1}^{n}(x_i - y_i)^2}

If the distance is sufficiently small, the embeddings are considered to represent the same individual.

note

Image Processing and Facial Recognition: - DeepFace v0.0.85: Library for facial analysis and recognition.