# Hit-or-miss transform

In mathematical morphology, hit-or-miss transform is an operation that detects a given configuration (or pattern) in a binary image, using the morphological erosion operator and a pair of disjoint structuring elements. The result of the hit-or-miss transform is the set of positions, where the first structuring element fits in the foreground of the input image, and the second structuring element misses it completely.

## Mathematical definition

In binary morphology, an image is viewed as a subset of an Euclidean space ${\displaystyle \mathbb {R} ^{d}}$ or the integer grid ${\displaystyle \mathbb {Z} ^{d}}$, for some dimension d. Let us denote this space or grid by E.

A structuring element is a simple, pre-defined shape, represented as a binary image, used to probe another binary image, in morphological operations such as erosion, dilation, opening, and closing.

Let ${\displaystyle C}$ and ${\displaystyle D}$ be two structuring elements satisfying ${\displaystyle C\cap D=\emptyset }$. The pair (C,D) is sometimes called a composite structuring element. The hit-or-miss transform of a given image A by B=(C,D) is given by:

${\displaystyle A\odot B=(A\ominus C)\cap (A^{c}\ominus D)}$,

where ${\displaystyle A^{c}}$ is the set complement of A.

That is, a point x in E belongs to the hit-or-miss transform output if C translated to x fits in A, and D translated to x misses A (fits the background of A).

## Some applications

### Thinning

Let ${\displaystyle E=\mathbb {Z} ^{2}}$, and consider the eight composite structuring elements, composed of:

${\displaystyle C_{1}=\{(0,0),(-1,-1),(0,-1),(1,-1)\}}$ and ${\displaystyle D_{1}=\{(-1,1),(0,1),(1,1)\}}$,
${\displaystyle C_{2}=\{(-1,0),(0,0),(-1,-1),(0,-1)\}}$ and ${\displaystyle D_{2}=\{(0,1),(1,1),(1,0)\}}$

and the three rotations of each by 90°, 180°, and 270°. The corresponding composite structuring elements are denoted ${\displaystyle B_{1},\ldots ,B_{8}}$.

For any i between 1 and 8, and any binary image X, define

${\displaystyle X\otimes B_{i}=X\setminus (X\odot B_{i}),}$

where ${\displaystyle \setminus }$ denotes the set-theoretical difference.

The thinning of an image A is obtained by cyclically iterating until convergence:

${\displaystyle A\otimes B_{1}\otimes B_{2}\otimes \ldots \otimes B_{8}\otimes B_{1}\otimes B_{2}\otimes \ldots }$

### Other applications

• Pattern detection. By definition, the hit-or-miss transform indicates the positions where a certain pattern (characterized by the composite structuring element B) occurs in the input image.
• Pruning. The hit-or-miss transform can be used to identify the end-points of a line to allow this line to be shrunk from each end to remove unwanted branches.
• Computing the Euler number.

## Bibliography

• An Introduction to Morphological Image Processing by Edward R. Dougherty, ISBN 0-8194-0845-X (1992)