Top-hat transform

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In the morphology of mathematics and processing of digital images, top-hat transform is an operation that extracts small elements and details from a given image. There are two types of top-hat transformations: white top-hat transform is defined as the difference between the input image and its opening by some element of arrangement; Black hat-hat transform is defined periodically as the difference between closure and the input image. Top-hat transforms are used for various image processing tasks, such as feature extraction, background alignment, image enhancement, and more.

Video Top-hat transform

Definisi matematis

Biarkan ${\ displaystyle f: E \ mapsto R}  ÂÂ$ menjadi gambar grayscale, memetakan titik dari ruang Euclidean atau kotak diskrit E (seperti R < sup> 2 atau Z ² ) ke garis nyata. Biarkan ${\ displaystyle b (x)}  ÂÂ$ menjadi elemen penataan grayscale.

Kemudian, transformasi top-hat putih f diberikan oleh:

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

di mana ${\ displaystyle \ circ}  ÂÂ$ menunjukkan operasi pembukaan.

Transformasi top-hat hitam f (kadang-kadang disebut bottom-hat transform) diberikan oleh:

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

di mana ${\ displaystyle \ bullet}  ÂÂ$ adalah operasi penutupan.

Maps Top-hat transform

Properti

The white top hat transformation returns an image, containing "objects" or "elements" of the input image that:

• Is "smaller" than the element of setup (that is, the place where the element of the arrangement does not match), and
• more sunny from the surroundings.

Black top-hat returns an image, containing "object" or "element" that:

• Is "smaller" than structuring elements, and
• darker than it was around.

The size, or width, of an element extracted by a top-hat transformation can be controlled by the choice of the ${\ displaystyle b}$ . The larger the latter, the larger the extracted element.

The two top-hat transformations are images that contain only non-negative values ​​on all pixels.

One of the most important uses in image segmentation is to adjust the non-uniform lighting conditions in the image and provide a better threshold value for separating objects.

src: www.met.rdg.ac.uk

Example

Assume we are only interested in small clumps on the image and we want to remove larger bright objects. In this case, white top-hat transforms can remove larger bright objects and retain small clumps by selecting the size of the structuring element that is between deleted objects and objects of interest. The radius of the six largest bright objects is about 50 to 100 pixels while the radius of the object of interest is about 2 to 4 pixels. In addition, an interesting object is a circle shape so we choose the disk-shaped arrangement element with radius 5. However, selecting different shapes and sizes for the setup elements produces different images depending on whether the object fits within the structuring element or not.

Another example is you have an image under non-uniform lighting and you want to extract the object separately from the background. A common method for image segmentation is to threshold the input image based on the intensity value. However, if the image is under non-uniform lighting, it is possible that segmentation errors may exist because some objects in darker areas have close intensity values ​​as background intensity values ​​and will not be extracted by using only threshold methods. In this case, before the Otsu method is applied to the input image, a white top-hat transformation must be applied to improve the non-uniform lighting conditions and create a clear contrast between the background and the object. Therefore, the object can be extracted completely from the background without any segmentation error. The threshold values ​​are 0.5216 and 0.2 and normalized to ${\ displaystyle [0,1]}$ for the original image and the top-hat white changed respectively.

src: tophat.com

References

• Mathematical Image Analysis and Morphology by Jean Serra, ISBNÃ, 0-12-637240-3 (1982)
• Mathematical Image Analysis and Morphology, Volume 2: Theoretical Progress by Jean Serra, ISBNÃ, 0-12-637241-1 (1988)
• Introduction to the Morphological Imagery Processing by Edward R. Dougherty, ISBNÃ, 0-8194-0845-X (1992)
• Practical Morphological Image Processing by Edward R. Dougherty and R. Lotufo, ISBNÃ, 0-8194-4720-X (2003)

• Digital Image Processing ( Third Edition ) by Rafael C. Gonzalez and Richard E. Woods, ISBN 978-93-325-7032-0 (2008 )

Source of the article : Wikipedia