In mathematical morphology and digital image processing, 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.
Let be a grayscale image, mapping points from a Euclidean space or discrete grid E (such as R2 or Z2) into the real line. Let be a structuring element of grayscale.
Then, the white top-hat transform of f is given by:
where denotes the opening operation.
The black top-hat transform of f (sometimes called the bottom-hat transform [1] ) is given by:
where is the closing operation.
The white top-hat transform returns an image, containing those "objects" or "elements" of an input image that:
The black top-hat returns an image, containing the "objects" or "elements" that:
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 . The bigger the latter, the larger the elements extracted.
Both top-hat transforms are images that contain only non-negative values at all pixels.
One of its most important uses in image segmentation is to adjust nonuniform lighting conditions on an image and provide a better threshold value for separating objects.
Assume we are only interested in small blobs on the image and we want to remove the larger bright objects. In this case, the white top-hat transform can remove larger bright objects and retain small blobs by selecting the size of the structuring element that is between removed objects and objects of interest. The radius of six largest bright objects are approximately 50 to 100 pixels whereas the radius of objects of interest are around 2 to 4 pixels. In addition, the objects of interest are circular shapes so we choose a disk shaped structuring element with radius 5. However, selecting different shapes and sizes for the structuring element result in different images depending on whether objects fit in the structuring element or not.
The other example is that you have an image under nonuniform illumination and you want to extract objects separately from background. The common method for image segmentation is to threshold the input image based on intensity value. However, if the image is under nonuniform lighting, it is possible that segmentation errors might present themselves since some objects in darker area have close intensity values as background intensity values and would not be extracted by only utilizing threshold method. In this case, before Otsu's method is applied to input image, white top-hat transform should be implemented to correct nonuniform lighting condition and make obvious contrast between background and objects. Therefore, the objects can be extracted entirely from background without segmentation errors. The threshold values are 0.5216 and 0.2 and normalized to for original image and applied white top-hat transform respectively.
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Mathematical morphology (MM) is a theory and technique for the analysis and processing of geometrical structures, based on set theory, lattice theory, topology, and random functions. MM is most commonly applied to digital images, but it can be employed as well on graphs, surface meshes, solids, and many other spatial structures.
Edge detection includes a variety of mathematical methods that aim at identifying edges, curves in a digital image at which the image brightness changes sharply or, more formally, has discontinuities. The same problem of finding discontinuities in one-dimensional signals is known as step detection and the problem of finding signal discontinuities over time is known as change detection. Edge detection is a fundamental tool in image processing, machine vision and computer vision, particularly in the areas of feature detection and feature extraction.
The Hough transform is a feature extraction technique used in image analysis, computer vision, and digital image processing. The purpose of the technique is to find imperfect instances of objects within a certain class of shapes by a voting procedure. This voting procedure is carried out in a parameter space, from which object candidates are obtained as local maxima in a so-called accumulator space that is explicitly constructed by the algorithm for computing the Hough transform.
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Erosion is one of two fundamental operations in morphological image processing from which all other morphological operations are based. It was originally defined for binary images, later being extended to grayscale images, and subsequently to complete lattices. The erosion operation usually uses a structuring element for probing and reducing the shapes contained in the input image.
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In mathematical morphology, opening is the dilation of the erosion of a set A by a structuring element B:
In mathematical morphology, the closing of a set A by a structuring element B is the erosion of the dilation of that set,
In mathematical morphology, a structuring element is a shape, used to probe or interact with a given image, with the purpose of drawing conclusions on how this shape fits or misses the shapes in the image. It is typically used in morphological operations, such as dilation, erosion, opening, and closing, as well as the hit-or-miss transform.
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In mathematical morphology, hit-or-miss transform is an operation that detects a given configuration 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.
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