Top-hat transform

In mathematical morphology and digital image processing, a top-hat transform is an operation that extracts small elements and details from given images. There exist two types of top-hat transform: the white top-hat transform is defined as the difference between the input image and its opening by some structuring element, while the black top-hat transform is defined dually as the difference between the closing and the input image. Top-hat transforms are used for various image processing tasks, such as feature extraction, background equalization, image enhancement, and others.

Mathematical definitions

Let

f

:

E

↦

R

{\displaystyle f:E\mapsto \mathbb {R} }

be a grayscale image, mapping points from a Euclidean space or discrete grid E (such as

R

2

{\displaystyle \mathbb {R} ^{2}}

or

Z

2

{\displaystyle \mathbb {Z} ^{2}}

) into the real line. Let

b

(

x

)

{\displaystyle b(x)}

be a structuring element of grayscale.

Then, the white top-hat transform of f is given by:

T

w

(

f

)

=

f

−

f

∘

b

{\displaystyle T_{w}(f)=f-f\circ b}

,

where

∘

{\displaystyle \circ }

denotes the opening operation.

The black top-hat transform of f (sometimes called the bottom-hat transform

) is given by:

T

b

(

f

)

=

f

∙

b

−

f

{\displaystyle T_{b}(f)=f\bullet b-f}

,

where

∙

{\displaystyle \bullet }

is the closing operation.

Properties

The white top-hat transform returns an image, containing those "objects" or "elements" of an input image that:

Are "smaller" than the structuring element (i.e., places where the structuring element does not fit in), and

are brighter than their surroundings.

The black top-hat returns an image, containing the "objects" or "elements" that:

Are "smaller" than the structuring element, and

are darker than their surroundings.

The size, or width, of the elements that are extracted by the top-hat transforms can be controlled by the choice of the structuring element

b

{\displaystyle b}

.