Definition

Various shapes (straight lines, circles, ellipses, etc.) in the image can be described by a general equation:

$$f(x,y,\alpha_1,\cdots,\alpha_n)=0$$

where $x$ and $y$ are two variables representing the position in the image, while $\{\alpha_1, \cdots, \alpha_n\}$ is a set of $n$ parameters specifying the shape. Or, equivalently, the same equation can represent some different shape in an n-dimensional space spanned by $\{\alpha_1, \cdots, \alpha_n\}$ as $n$ variables, with $x$ and $y$ treated as two parameters.

A specific 2D image shape can therefore be represented either locally in the image space by the pixels $(x,y)$ on the boundary of the shape (such as edge detection), or globally in the parameter space by the parameters $(\alpha_1, \cdots, \alpha_n)$. These two representations are linked by the Hough transform:

• The forward transform

  Every point $(x,y)$ in the 2D image space is transformed to a hyper-surface in the n-dimensional parameter space;

• The inverse transform

  Every point $(\alpha_1, \cdots, \alpha_n)$ in the nD parameter space describes a specific instance of the shape of interest in the 2D image space.