Computer vision relies on several core areas of mathematics to process, analyze, and interpret visual data. The foundational math includes linear algebra, calculus, probability, and geometry. These tools enable tasks like image manipulation, object detection, and 3D reconstruction. While not every developer needs deep expertise, understanding the basics helps in choosing algorithms, debugging models, and implementing custom solutions.
Linear algebra is essential for representing and transforming images. Images are stored as matrices (grids of pixel values), and operations like rotations or scaling use matrix multiplication. Techniques like Singular Value Decomposition (SVD) reduce noise or compress images by simplifying matrix data. Convolutional Neural Networks (CNNs) rely heavily on tensor operations (multi-dimensional arrays) to apply filters for edge detection or texture analysis. For example, a CNN layer might multiply a 3x3 filter matrix with image patches to extract features like edges. Geometry is equally important for tasks like camera calibration, where 3D world points are projected onto 2D images using transformation matrices. Homography matrices, for instance, align images taken from different viewpoints in panorama stitching.
Calculus and probability underpin many optimization and inference techniques. Training neural networks involves calculus to compute gradients (partial derivatives) for backpropagation, adjusting weights to minimize prediction errors. Edge detection algorithms like the Sobel operator use gradients to identify intensity changes in images. Probability models help handle uncertainty, such as classifying objects in noisy images. Bayesian networks estimate the likelihood of a pixel belonging to an object, while clustering algorithms like k-means group similar pixels for segmentation. Statistics also provides metrics like precision-recall curves to evaluate model performance, ensuring reliable object detection in real-world applications.
Finally, optimization methods tie these concepts together. Algorithms like gradient descent adjust parameters to minimize loss functions, balancing speed and accuracy. In traditional computer vision, RANSAC (Random Sample Consensus) optimizes model fitting by iteratively selecting the best subset of data points—for example, finding the best line to fit detected edges in a scene. Understanding these mathematical principles helps developers troubleshoot issues (e.g., why a model fails to converge) and adapt techniques to new problems, such as implementing real-time object tracking with Kalman filters. While frameworks abstract much of the math, knowing the fundamentals empowers developers to innovate beyond pre-built solutions.