The theoretical foundation of the Hough Transform lies in a mathematical approach that reinterprets pattern detection as an inverse problem:

The theoretical foundation of the Hough Transform lies in a mathematical approach that reinterprets pattern detection as an inverse problem: mapping a set of points into a shape's parameter space. In image processing, geometric shapes such as lines and circles are defined by collections of edge points, but detecting these shapes directly in image space is often hindered by noise and missing data. The Hough Transform addresses this by converting each point into a set of possible shapes—represented as curves in a parameter space (e.g., distance and angle for lines). Points that lie on the same shape in image space will correspond to curves that intersect at a single point in parameter space. This method exploits the duality between points and geometric primitives (lines or circles), transforming the original shape detection into the problem of identifying peak intersections in parameter space. These intersections indicate the most likely shapes. Conceptually, the Hough Transform can be viewed as a discrete version of the Radon Transform and shares statistical similarities with maximum likelihood estimation. Its voting-based mechanism allows for robust detection of shapes, even in noisy or incomplete data. It can be extended to circles, ellipses, and other forms, making it widely applicable in fields like medical imaging and robotics.