What is the math behind computer vision algorithms?

Computer vision algorithms rely heavily on linear algebra, calculus, and probability. At their core, these algorithms process images as numerical data—typically represented as matrices of pixel values—and apply mathematical operations to extract patterns or features. For example, an image might be stored as a 3D tensor (height × width × color channels), and operations like convolution or matrix multiplication are used to detect edges, textures, or shapes. Linear algebra underpins transformations such as scaling, rotation, and translation, which are essential for tasks like image alignment or object detection. Matrix operations also drive techniques like principal component analysis (PCA) for dimensionality reduction or singular value decomposition (SVD) for compressing image data.

Calculus and optimization are critical for training models like convolutional neural networks (CNNs). Gradients, computed via partial derivatives, enable backpropagation—the process of adjusting network weights to minimize prediction errors. For instance, in edge detection, a Sobel filter applies convolution kernels to approximate image gradients, highlighting areas of rapid intensity change. Optimization algorithms like stochastic gradient descent (SGD) adjust parameters iteratively to reduce loss functions, such as cross-entropy for classification tasks. Even non-neural-network methods, like optical flow (tracking pixel motion between frames), rely on solving systems of equations derived from partial derivatives of image intensity over time.

Probability and statistics handle uncertainty in tasks like object recognition or segmentation. Bayesian networks model relationships between variables, such as the likelihood of a pixel belonging to a specific object class. For example, Gaussian Mixture Models (GMMs) cluster pixels based on color distributions to separate foreground and background. Modern architectures like YOLO (You Only Look Once) use probabilistic bounding box predictions and confidence scores to localize objects. Additionally, techniques like non-maximum suppression—a statistical method to eliminate overlapping predictions—ensure clean outputs. Even basic operations like histogram equalization, which adjusts image contrast, rely on redistributing pixel intensity probabilities to enhance visibility. These mathematical foundations enable algorithms to reason about noisy, real-world visual data effectively.