In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset.
Digital geometry deals with discrete sets considered to be digitized models or images of objects of the 2D or 3D Euclidean space.
A distance transform, also known as distance map or distance field, is a derived representation of a digital image. The choice of the term depends on the point of view on the object in question: whether the initial image is transformed into another representation, or it is simply endowed with an additional map or field.
In geometry, a simple polygon is a polygon that does not intersect itself and has no holes. That is, it is a flat shape consisting of straight, non-intersecting line segments or "sides" that are joined pairwise to form a single closed path. If the sides intersect then the polygon is not simple. The qualifier "simple" is frequently omitted, with the above definition then being understood to define a polygon in general.
Level-set methods (LSM) are a conceptual framework for using level sets as a tool for numerical analysis of surfaces and shapes. The advantage of the level-set model is that one can perform numerical computations involving curves and surfaces on a fixed Cartesian grid without having to parameterize these objects. Also, the level-set method makes it very easy to follow shapes that change topology, for example, when a shape splits in two, develops holes, or the reverse of these operations. All these make the level-set method a great tool for modeling time-varying objects, like inflation of an airbag, or a drop of oil floating in water.
A point is said to be equidistant from a set of objects if the distances between that point and each object in the set are equal.
Pickover stalks are certain kinds of details to be found empirically in the Mandelbrot set, in the study of fractal geometry. They are so named after the researcher Clifford Pickover, whose "epsilon cross" method was instrumental in their discovery. An "epsilon cross" is a cross-shaped orbit trap.
The pruning algorithm is a technique used in digital image processing based on mathematical morphology. It is used as a complement to the skeleton and thinning algorithms to remove unwanted parasitic components (spurs). In this case 'parasitic' components refer to branches of a line which are not key to the overall shape of the line and should be removed. These components can often be created by edge detection algorithms or digitization. Common uses for pruning include automatic recognition of hand-printed characters. Often inconsistency in letter writing creates unwanted spurs that need to be eliminated for better characterization.
In computer vision and image processing, a feature is a piece of information about the content of an image; typically about whether a certain region of the image has certain properties. Features may be specific structures in the image such as points, edges or objects. Features may also be the result of a general neighborhood operation or feature detection applied to the image. Other examples of features are related to motion in image sequences, or to shapes defined in terms of curves or boundaries between different image regions.
The medial axis of an object is the set of all points having more than one closest point on the object's boundary. Originally referred to as the topological skeleton, it was introduced in 1967 by Harry Blum as a tool for biological shape recognition. In mathematics the closure of the medial axis is known as the cut locus.
NeuronStudio was a non-commercial program created at Icahn School of Medicine at Mount Sinai by the Computational Neurobiology and Imaging Center. This program performed automatic tracing and reconstruction of neuron structures from confocal image stacks. The resulting models were then be exported to file using standard formats for further processing, modeling, or for statistical analyses. NeuronStudio handled morphologic details on scales spanning local Dendritic spine geometry through complex tree topology to the gross spatial arrangement of multi-neuron networks. Its capability for automated digitization avoided the subjective errors inherent in manual tracing. The program ceased to be supported in 2012 and the project pages were eventually removed from the ISMMS Website. Its documentation and the Windows source code however are still available via the Internet Archive.
Digital topology deals with properties and features of two-dimensional (2D) or three-dimensional (3D) digital images that correspond to topological properties or topological features of objects.
In image processing, ridge detection is the attempt, via software, to locate ridges in an image, defined as curves whose points are local maxima of the function, akin to geographical ridges.
In geometry, a straight skeleton is a method of representing a polygon by a topological skeleton. It is similar in some ways to the medial axis but differs in that the skeleton is composed of straight line segments, while the medial axis of a polygon may involve parabolic curves. However, both are homotopy-equivalent to the underlying polygon.
In computer vision and computer graphics, 3D reconstruction is the process of capturing the shape and appearance of real objects. This process can be accomplished either by active or passive methods. If the model is allowed to change its shape in time, this is referred to as non-rigid or spatio-temporal reconstruction.
In digital image processing, morphological skeleton is a skeleton representation of a shape or binary image, computed by means of morphological operators.
In computational geometry and geometric graph theory, a β-skeleton or beta skeleton is an undirected graph defined from a set of points in the Euclidean plane. Two points p and q are connected by an edge whenever all the angles prq are sharper than a threshold determined from the numerical parameter β.
In image processing, the grassfire transform is the computation of the distance from a pixel to the border of a region. It can be described as "setting fire" to the borders of an image region to yield descriptors such as the region's skeleton or medial axis. Harry Blum introduced the concept in 1967.
Franz-Erich Wolter is a German computer scientist, chaired professor at Leibniz University Hannover, with research contributions especially in computational (differential) geometry and haptic/tactile Virtual reality.
Discrete Skeleton Evolution (DSE) describes an iterative approach to reducing a morphological or topological skeleton. It is a form of pruning in that it removes noisy or redundant branches (spurs) generated by the skeletonization process, while preserving information-rich "trunk" segments. The value assigned to individual branches varies from algorithm to algorithm, with the general goal being to convey the features of interest of the original contour with a few carefully chosen lines. Usually, clarity for human vision is valued as well. DSE algorithms are distinguished by complex, recursive decision-making processes with high computational requirements. Pruning methods such as by structuring element (SE) convolution and the Hough transform are general purpose algorithms which quickly pass through an image and eliminate all branches shorter than a given threshold. DSE methods are most applicable when detail retention and contour reconstruction are valued.
