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  List of .NET libraries and frameworks

  This article contains a list of libraries that can be used in .NET languages. These languages require .NET Framework, Mono, or .NET, which provide a basis for software development, platform independence, language interoperability and extensive framework libraries. Standard Libraries (including the Base Class Library) are not included in this article. == Introduction == Apps created with .NET Framework or .NET run in a software environment known as the Common Language Runtime (CLR), an application virtual machine that provides services such as security, memory management, and exception handling. The framework includes a large class library called Framework Class Library (FCL). Thanks to the hosting virtual machine, different languages that are compliant with the .NET Common Language Infrastructure (CLI) can operate on the same kind of data structures. These languages can therefore use the FCL and other .NET libraries that are also written in one of the CLI compliant languages. When the source code of such languages are compiled, the compiler generates platform-independent code in the Common Intermediate Language (CIL, also referred to as bytecode), which is stored in CLI assemblies. When a .NET app runs, the just-in-time compiler (JIT) turns the CIL code into platform-specific machine code. To improve performance, .NET Framework also comes with the Native Image Generator (NGEN), which performs ahead-of-time compilation to machine code. This architecture provides language interoperability. Each language can use code written in other languages. Calls from one language to another are exactly the same as would be within a single programming language. If a library is written in one CLI language, it can be used in other CLI languages. Moreover, apps that consist only of pure .NET assemblies, can be transferred to any platform that contains an implementation of CLI and run on that platform. For example, apps written using .NET can run on Windows, macOS, and various versions of Linux. .NET apps or their libraries, however, may depend on native platform features, e.g. COM. As such, platform independence of .NET apps depends on the ability to transfer necessary native libraries to target platforms. In 2019, the Windows Forms and Windows Presentation Foundation portions of .NET Framework were made open source. === .NET implementations === There are four primary .NET implementations that are actively developed and maintained: .NET Framework: The original .NET implementation that has existed since 2002. While not yet discontinued, Microsoft does not plan on releasing its next major version, 5.0. Mono: A cross-platform implementation of .NET Framework by Ximian, introduced in 2004. It is free and open-source. It is now developed by Xamarin, a subsidiary of Microsoft. Universal Windows Platform (UWP): An implementation of .NET used for building UWP apps. It's designed to unify development for different targeted types of devices, including PCs, tablets, phablets, phones, and the Xbox. .NET: A cross-platform re-implementation of .NET Framework, introduced in 2016 and initially called .NET Core. It is free and open-source. .NET superseded .NET Framework with the release of .NET 5. Each implementation of .NET includes the following components: One or more runtime environments, e.g. Common Language Runtime (CLR) for .NET Framework and CoreCLR for .NET A class library The .NET Standard is a set of common APIs that are implemented in the Base Class Library of any .NET implementation. The class library of each implementation must implement the .NET Standard, but may also implement additional APIs. Traditionally, .NET apps targeted a certain version of a .NET implementation, e.g. .NET Framework 4.6. Starting with the .NET Standard, an app can target a version of the .NET Standard and then it could be used (without recompiling) by any implementation that supports that level of the standard. This enables portability across different .NET implementations. The following table lists the .NET implementations that adhere to the .NET Standard and the version number at which each implementation became compliant with a given version of .NET Standard. For example, according to this table, .NET Core 3.0 was the first version of .NET Core that adhered to .NET Standard 2.1. This means that any version of .NET Core bigger than 3.0 (e.g. .NET Core 3.1) also adheres to .NET Standard 2.1. == Web frameworks == === ASP.NET === First released in 2002, ASP.NET is an open-source server-side web application framework designed for web development to produce dynamic web pages. It is the successor to Microsoft's Active Server Pages (ASP) technology, built on the Common Language Runtime (CLR). === ASP.NET Core === ASP.NET was completely rewritten in 2016 as a modular web framework, together with other frameworks like Entity Framework. The re-written framework uses the new open-source .NET Compiler Platform (also known by its codename "Roslyn") and is cross platform. The programming models ASP.NET MVC, ASP.NET Web API, and ASP.NET Web Pages (a model using only Razor pages) were merged into a unified MVC 6. === Blazor === Blazor is a free and open-source web framework that enables developers to create Single-page Web apps using C# and HTML in ASP.NET Razor pages ("components"). Blazor is part of the ASP.NET Core framework. Blazor Server apps are hosted on a web server, while Blazor WebAssembly apps are downloaded to the client's web browser before running. In addition, a Blazor Hybrid framework is available with server-based and client-based application components. == Numerical libraries == === Open-source numerical libraries === ==== AForge.NET ==== This is a computer vision and artificial intelligence library. It implements a number of genetic, fuzzy logic and machine learning algorithms with several architectures of artificial neural networks with corresponding training algorithms. ==== ALGLIB ==== This is a cross-platform open source numerical analysis and data processing library. It consists of algorithm collections written in different programming languages (C++, C#, FreePascal, Delphi, VBA) and has dual licensing – commercial and GPL. ==== Math.NET Numerics ==== This library aims to provide methods and algorithms for numerical computations in science, engineering and everyday use. Covered topics include special functions, linear algebra, probability models, random numbers, interpolation, integral transforms and more. MIT/X11 license. ==== Meta.Numerics ==== This is a library for advanced scientific computation in the .NET Framework. ==== ML.NET ==== This is a free software machine learning library. The preview release of ML.NET included transforms for feature engineering like n-gram creation, and learners to handle binary classification, multi-class classification, and regression tasks. Additional ML tasks like anomaly detection and recommendation systems have since been added, and other approaches like deep learning will be included in future versions. === Proprietary numerical libraries === ==== ILNumerics.Net ==== This is a high performance, typesafe numerical array set of classes and functions for general math, FFT and linear algebra. The library, developed for .NET/Mono, aims to provide 32- and 64-bit script-like syntax in C#, 2D & 3D plot controls, and efficient memory management. It is released under GPLv3 or commercial license. ==== Measurement Studio ==== This is an integrated suite of UI controls and class libraries for use in developing test and measurement applications. The analysis class libraries provide various digital signal processing, signal filtering, signal generation, peak detection, and other general mathematical functionality. ==== NMath ==== This is a numerical component library for the .NET platform developed by CenterSpace Software. It includes signal processing (FFT) classes, a linear algebra (LAPACK & BLAS) framework, and a statistics package. == 3D graphics == === Open-source 3D graphics === ==== Open Toolkit (OpenTK) ==== This is a low-level C# binding for OpenGL, OpenGL ES and OpenAL. It runs on Windows, Linux, Mac OS X, BSD, Android and iOS. It can be used standalone or integrated into a GUI. ==== Windows Presentation Foundation (WPF) ==== This is a graphical subsystem for rendering user interfaces, developed by Microsoft. It also contains a 3D rendering engine. In addition, interactive 2D content can be overlaid on 3D surfaces natively. It only runs on Windows operating systems. === Proprietary 3D graphics === ==== Unity ==== This is a cross-platform game engine developed by Unity Technologies and used to develop video games for PC, consoles, mobile devices and websites. == Image processing == === AForge.NET === This is a computer vision and artificial intelligence library. It implements a number of image processing algorithms and filters. It is released under the LGPLv3 and partly GPLv3 license. Majority of the library is written in C# and th

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  PatchMatch

  PatchMatch is an algorithm used to quickly find correspondences (or matches) between small square regions (or patches) of an image. It has various applications in image editing, such as reshuffling or removing objects from images or altering their aspect ratios without cropping or noticeably stretching them. PatchMatch was first presented in a 2011 paper by researchers at Princeton University. == Algorithm == The goal of the algorithm is to find the patch correspondence by defining a nearest-neighbor field (NNF) as a function f : R 2 → R 2 {\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} ^{2}} of offsets, which is over all possible matches of patch (location of patch centers) in image A, for some distance function of two patches D {\displaystyle D} . So, for a given patch coordinate a {\displaystyle a} in image A {\displaystyle A} and its corresponding nearest neighbor b {\displaystyle b} in image B {\displaystyle B} , f ( a ) {\displaystyle f(a)} is simply b − a {\displaystyle b-a} . However, if we search for every point in image B {\displaystyle B} , the work will be too hard to complete. So the following algorithm is done in a randomized approach in order to accelerate the calculation speed. The algorithm has three main components. Initially, the nearest-neighbor field is filled with either random offsets or some prior information. Next, an iterative update process is applied to the NNF, in which good patch offsets are propagated to adjacent pixels, followed by random search in the neighborhood of the best offset found so far. Independent of these three components, the algorithm also uses a coarse-to-fine approach by building an image pyramid to obtain the better result. === Initialization === When initializing with random offsets, we use independent uniform samples across the full range of image B {\displaystyle B} . This algorithm avoids using an initial guess from the previous level of the pyramid because in this way the algorithm can avoid being trapped in local minima. === Iteration === After initialization, the algorithm attempted to perform iterative process of improving the N N F {\displaystyle NNF} . The iterations examine the offsets in scan order (from left to right, top to bottom), and each undergoes propagation followed by random search. === Propagation === We attempt to improve f ( x , y ) {\displaystyle f(x,y)} using the known offsets of f ( x − 1 , y ) {\displaystyle f(x-1,y)} and f ( x , y − 1 ) {\displaystyle f(x,y-1)} , assuming that the patch offsets are likely to be the same. That is, the algorithm will take new value for f ( x , y ) {\displaystyle f(x,y)} to be arg ⁡ min ( x , y ) D ( f ( x , y ) ) , D ( f ( x − 1 , y ) ) , D ( f ( x , y − 1 ) ) {\displaystyle \arg \min \limits _{(x,y)}{D(f(x,y)),D(f(x-1,y)),D(f(x,y-1))}} . So if f ( x , y ) {\displaystyle f(x,y)} has a correct mapping and is in a coherent region R {\displaystyle R} , then all of R {\displaystyle R} below and to the right of f ( x , y ) {\displaystyle f(x,y)} will be filled with the correct mapping. Alternatively, on even iterations, the algorithm search for different direction, fill the new value to be arg ⁡ min ( x , y ) { D ( f ( x , y ) ) , D ( f ( x + 1 , y ) ) , D ( f ( x , y + 1 ) ) } {\displaystyle \arg \min \limits _{(x,y)}\{D(f(x,y)),D(f(x+1,y)),D(f(x,y+1))\}} . === Random search === Let v 0 = f ( x , y ) {\displaystyle v_{0}=f(x,y)} , we attempt to improve f ( x , y ) {\displaystyle f(x,y)} by testing a sequence of candidate offsets at an exponentially decreasing distance from v 0 {\displaystyle v_{0}} u i = v 0 + w α i R i {\displaystyle u_{i}=v_{0}+w\alpha ^{i}R_{i}} where R i {\displaystyle R_{i}} is a uniform random in [ − 1 , 1 ] × [ − 1 , 1 ] {\displaystyle [-1,1]\times [-1,1]} , w {\displaystyle w} is a large window search radius which will be set to maximum picture size, and α {\displaystyle \alpha } is a fixed ratio often assigned as 1/2. This part of the algorithm allows the f ( x , y ) {\displaystyle f(x,y)} to jump out of local minimum through random process. === Halting criterion === The often used halting criterion is set the iteration times to be about 4~5. Even with low iteration, the algorithm works well.

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  Phase congruency

  Phase congruency is a measure of feature significance in computer images, a method of edge detection that is particularly robust against changes in illumination and contrast. == Foundations == Phase congruency reflects the behaviour of the image in the frequency domain. It has been noted that edgelike features have many of their frequency components in the same phase. The concept is similar to coherence, except that it applies to functions of different wavelength. For example, the Fourier decomposition of a square wave consists of sine functions, whose frequencies are odd multiples of the fundamental frequency. At the rising edges of the square wave, each sinusoidal component has a rising phase; the phases have maximal congruency at the edges. This corresponds to the human-perceived edges in an image where there are sharp changes between light and dark. == Definition == Phase congruency compares the weighted alignment of the Fourier components of a signal A n {\displaystyle A_{\rm {n}}} with the sum of the Fourier components. P C ( t ) = max ϕ ¯ ∑ n A n cos ⁡ ( ϕ n ( t ) − ϕ ¯ ) ∑ n A n {\displaystyle PC(t)=\max _{\bar {\phi }}{\frac {\sum _{\rm {n}}A_{\rm {n}}\cos(\phi _{\rm {n}}(t)-{\bar {\phi }})}{\sum _{\rm {n}}A_{n}}}} where ϕ n {\displaystyle \phi _{\rm {n}}} is the local or instantaneous phase as can be calculated using the Hilbert transform and A n {\displaystyle A_{\rm {n}}} are the local amplitude, or energy, of the signal. When all the phases are aligned, this is equal to 1. Several ways of implementing phase congruency have been developed, of which two versions are available in open source, one written for MATLAB and the other written in Java as a plugin for the ImageJ software. Given the different notations used for its formulation, a unified version has been recently presented, where a methodology for the parameter tuning is also presented. == Advantages == The square-wave example is naive in that most edge detection methods deal with it equally well. For example, the first derivative has a maximal magnitude at the edges. However, there are cases where the perceived edge does not have a sharp step or a large derivative. The method of phase congruency applies to many cases where other methods fail. A notable example is an image feature consisting of a single line, such as the letter "l". Many edge-detection algorithms will pick up two adjacent edges: the transitions from white to black, and black to white. On the other hand, the phase congruency map has a single line. A simple Fourier analogy of this case is a triangle wave. In each of its crests there is a congruency of crests from different sinusoidal functions. == Disadvantages == Calculating the phase congruency map of an image is very computationally intensive, and sensitive to image noise. Techniques of noise reduction are usually applied prior to the calculation.

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  Uniform convergence in probability

  Uniform convergence in probability is a form of convergence in probability in statistical asymptotic theory and probability theory. It means that, under certain conditions, the empirical frequencies of all events in a certain event-family uniformly converge to their theoretical probabilities. Uniform convergence in probability has applications to statistics as well as machine learning as part of statistical learning theory. Specifically, the Glivenko-Cantelli theorem and the homonymous classes of functions are fundamentally related to uniform convergence. The law of large numbers says that, for each single event A {\displaystyle A} , its empirical frequency in a sequence of independent trials converges (with high probability) to its theoretical probability. In many application however, the need arises to judge simultaneously the probabilities of events of an entire class S {\displaystyle S} from one and the same sample. Moreover, it, is required that the relative frequency of the events converge to the probability uniformly over the entire class of events S {\displaystyle S} . The Uniform Convergence Theorem gives a sufficient condition for this convergence to hold. Roughly, if the event-family is sufficiently simple (its VC dimension is sufficiently small) then uniform convergence holds. == Definitions == For a class of predicates H {\displaystyle H} defined on a set X {\displaystyle X} and a set of samples x = ( x 1 , x 2 , … , x m ) {\displaystyle x=(x_{1},x_{2},\dots ,x_{m})} , where x i ∈ X {\displaystyle x_{i}\in X} , the empirical frequency of h ∈ H {\displaystyle h\in H} on x {\displaystyle x} is Q ^ x ( h ) = 1 m | { i : 1 ≤ i ≤ m , h ( x i ) = 1 } | . {\displaystyle {\widehat {Q}}_{x}(h)={\frac {1}{m}}|\{i:1\leq i\leq m,h(x_{i})=1\}|.} The theoretical probability of h ∈ H {\displaystyle h\in H} is defined as Q P ( h ) = P { y ∈ X : h ( y ) = 1 } . {\displaystyle Q_{P}(h)=P\{y\in X:h(y)=1\}.} The Uniform Convergence Theorem states, roughly, that if H {\displaystyle H} is "simple" and we draw samples independently (with replacement) from X {\displaystyle X} according to any distribution P {\displaystyle P} , then with high probability, the empirical frequency will be close to its expected value, which is the theoretical probability. Here "simple" means that the Vapnik–Chervonenkis dimension of the class H {\displaystyle H} is small relative to the size of the sample. In other words, a sufficiently simple collection of functions behaves roughly the same on a small random sample as it does on the distribution as a whole. The Uniform Convergence Theorem was first proved by Vapnik and Chervonenkis using the concept of growth function. == Uniform Convergence Theorem == The statement of the Uniform Convergence Theorem is as follows: If H {\displaystyle H} is a set of { 0 , 1 } {\displaystyle \{0,1\}} -valued functions defined on a set X {\displaystyle X} and P {\displaystyle P} is a probability distribution on X {\displaystyle X} then for ε > 0 {\displaystyle \varepsilon >0} and m {\displaystyle m} a positive integer, we have: P m { | Q P ( h ) − Q x ^ ( h ) | ≥ ε for some h ∈ H } ≤ 4 Π H ( 2 m ) e − ε 2 m / 8 . {\displaystyle P^{m}\{|Q_{P}(h)-{\widehat {Q_{x}}}(h)|\geq \varepsilon {\text{ for some }}h\in H\}\leq 4\Pi _{H}(2m)e^{-\varepsilon ^{2}m/8}.} In the above, for any x ∈ X m , {\displaystyle x\in X^{m},} Q P ( h ) = P { ( y ∈ X : h ( y ) = 1 } , {\displaystyle Q_{P}(h)=P\{(y\in X:h(y)=1\},} Q ^ x ( h ) = 1 m | { i : 1 ≤ i ≤ m , h ( x i ) = 1 } | {\displaystyle {\widehat {Q}}_{x}(h)={\frac {1}{m}}|\{i:1\leq i\leq m,h(x_{i})=1\}|} and | x | = m . {\displaystyle |x|=m.} P m {\displaystyle P^{m}} indicates that the probability is taken over x {\displaystyle x} consisting of m {\displaystyle m} i.i.d. draws from the distribution P . {\displaystyle P.} Finally, the growth function Π H {\displaystyle \Pi _{H}} is defined in the following way, for any { 0 , 1 } {\displaystyle \{0,1\}} -valued functions H {\displaystyle H} over X {\displaystyle X} and for any natural number m {\displaystyle m} : Π H ( m ) = max | { h ∩ D : D ⊆ X , | D | = m , h ∈ H } | . {\displaystyle \Pi _{H}(m)=\max |\{h\cap D:D\subseteq X,|D|=m,h\in H\}|.} From the point of view of Learning Theory one can consider H {\displaystyle H} to be the Concept/Hypothesis class defined over the instance set X {\displaystyle X} . Crucially, the Sauer–Shelah lemma implies that Π H ( m ) ≤ m d {\displaystyle \Pi _{H}(m)\leq m^{d}} , where d {\displaystyle d} is the VC dimension of H {\displaystyle H} . == Proof of the Uniform Convergence Theorem == and are the sources of the proof below. Before we get into the details of the proof of the Uniform Convergence Theorem we will present a high level overview of the proof. Symmetrization: We transform the problem of analyzing | Q P ( h ) − Q ^ x ( h ) | ≥ ε {\displaystyle |Q_{P}(h)-{\widehat {Q}}_{x}(h)|\geq \varepsilon } into the problem of analyzing | Q ^ r ( h ) − Q ^ s ( h ) | ≥ ε / 2 {\displaystyle |{\widehat {Q}}_{r}(h)-{\widehat {Q}}_{s}(h)|\geq \varepsilon /2} , where r {\displaystyle r} and s {\displaystyle s} are i.i.d samples of size m {\displaystyle m} drawn according to the distribution P {\displaystyle P} . One can view r {\displaystyle r} as the original randomly drawn sample of length m {\displaystyle m} , while s {\displaystyle s} may be thought as the testing sample which is used to estimate Q P ( h ) {\displaystyle Q_{P}(h)} . Permutation: Since r {\displaystyle r} and s {\displaystyle s} are picked identically and independently, so swapping elements between them will not change the probability distribution on r {\displaystyle r} and s {\displaystyle s} . So, we will try to bound the probability of | Q ^ r ( h ) − Q ^ s ( h ) | ≥ ε / 2 {\displaystyle |{\widehat {Q}}_{r}(h)-{\widehat {Q}}_{s}(h)|\geq \varepsilon /2} for some h ∈ H {\displaystyle h\in H} by considering the effect of a specific collection of permutations of the joint sample x = r | | s {\displaystyle x=r||s} . Specifically, we consider permutations σ ( x ) {\displaystyle \sigma (x)} which swap x i {\displaystyle x_{i}} and x m + i {\displaystyle x_{m+i}} in some subset of 1 , 2 , . . . , m {\displaystyle {1,2,...,m}} . The symbol r | | s {\displaystyle r||s} means the concatenation of r {\displaystyle r} and s {\displaystyle s} . Reduction to a finite class: We can now restrict the function class H {\displaystyle H} to a fixed joint sample and hence, if H {\displaystyle H} has finite VC Dimension, it reduces to the problem to one involving a finite function class. We present the technical details of the proof. It should be stressed that this proof glosses over details like the measurability of the events V {\displaystyle V} and R {\displaystyle R} ; measurability is granted in the case of H {\displaystyle H} being finite or countable, but this is not normally the case in standard applications of the theorem (e.g. for statistical learning theory or to prove the Glivenko-Cantelli theorem). To get measurability, one needs to use a notion of separability of the underlying space, possibly related to H {\displaystyle H} . === Symmetrization === Lemma: Let V = { x ∈ X m : | Q P ( h ) − Q ^ x ( h ) | ≥ ε for some h ∈ H } {\displaystyle V=\{x\in X^{m}:|Q_{P}(h)-{\widehat {Q}}_{x}(h)|\geq \varepsilon {\text{ for some }}h\in H\}} and R = { ( r , s ) ∈ X m × X m : | Q r ^ ( h ) − Q ^ s ( h ) | ≥ ε / 2 for some h ∈ H } . {\displaystyle R=\{(r,s)\in X^{m}\times X^{m}:|{\widehat {Q_{r}}}(h)-{\widehat {Q}}_{s}(h)|\geq \varepsilon /2{\text{ for some }}h\in H\}.} Then for m ≥ 2 ε 2 {\displaystyle m\geq {\frac {2}{\varepsilon ^{2}}}} , P m ( V ) ≤ 2 P 2 m ( R ) {\displaystyle P^{m}(V)\leq 2P^{2m}(R)} . Proof: By the triangle inequality, if | Q P ( h ) − Q ^ r ( h ) | ≥ ε {\displaystyle |Q_{P}(h)-{\widehat {Q}}_{r}(h)|\geq \varepsilon } and | Q P ( h ) − Q ^ s ( h ) | ≤ ε / 2 {\displaystyle |Q_{P}(h)-{\widehat {Q}}_{s}(h)|\leq \varepsilon /2} then | Q ^ r ( h ) − Q ^ s ( h ) | ≥ ε / 2 {\displaystyle |{\widehat {Q}}_{r}(h)-{\widehat {Q}}_{s}(h)|\geq \varepsilon /2} . Therefore, P 2 m ( R ) ≥ P 2 m { ∃ h ∈ H , | Q P ( h ) − Q ^ r ( h ) | ≥ ε and | Q P ( h ) − Q ^ s ( h ) | ≤ ε / 2 } = ∫ V P m { s : ∃ h ∈ H , | Q P ( h ) − Q ^ r ( h ) | ≥ ε and | Q P ( h ) − Q ^ s ( h ) | ≤ ε / 2 } d P m ( r ) = A {\displaystyle {\begin{aligned}&P^{2m}(R)\\[5pt]\geq {}&P^{2m}\{\exists h\in H,|Q_{P}(h)-{\widehat {Q}}_{r}(h)|\geq \varepsilon {\text{ and }}|Q_{P}(h)-{\widehat {Q}}_{s}(h)|\leq \varepsilon /2\}\\[5pt]={}&\int _{V}P^{m}\{s:\exists h\in H,|Q_{P}(h)-{\widehat {Q}}_{r}(h)|\geq \varepsilon {\text{ and }}|Q_{P}(h)-{\widehat {Q}}_{s}(h)|\leq \varepsilon /2\}\,dP^{m}(r)\\[5pt]={}&A\end{aligned}}} since r {\displaystyle r} and s {\displaystyle s} are independent. Now for r ∈ V {\displaystyle r\in V} fix an h ∈ H {\displaystyle h\in H} such that | Q P ( h ) − Q ^ r ( h ) | ≥ ε {\displaystyle |Q_{P}(h)-{\widehat {Q}}_{r}(h)|\geq \varepsilon } . For this h {\displaystyle h} , we shall

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  Manifold regularization

  In machine learning, manifold regularization is a technique for using the shape of a dataset to constrain the functions that should be learned on that dataset. In many machine learning problems, the data to be learned do not cover the entire input space. For example, a facial recognition system may not need to classify any possible image, but only the subset of images that contain faces. The technique of manifold learning assumes that the relevant subset of data comes from a manifold, a mathematical structure with useful properties. The technique also assumes that the function to be learned is smooth: data with different labels are not likely to be close together, and so the labeling function should not change quickly in areas where there are likely to be many data points. Because of this assumption, a manifold regularization algorithm can use unlabeled data to inform where the learned function is allowed to change quickly and where it is not, using an extension of the technique of Tikhonov regularization. Manifold regularization algorithms can extend supervised learning algorithms in semi-supervised learning and transductive learning settings, where unlabeled data are available. The technique has been used for applications including medical imaging, geographical imaging, and object recognition. == Manifold regularizer == === Motivation === Manifold regularization is a type of regularization, a family of techniques that reduces overfitting and ensures that a problem is well-posed by penalizing complex solutions. In particular, manifold regularization extends the technique of Tikhonov regularization as applied to Reproducing kernel Hilbert spaces (RKHSs). Under standard Tikhonov regularization on RKHSs, a learning algorithm attempts to learn a function f {\displaystyle f} from among a hypothesis space of functions H {\displaystyle {\mathcal {H}}} . The hypothesis space is an RKHS, meaning that it is associated with a kernel K {\displaystyle K} , and so every candidate function f {\displaystyle f} has a norm ‖ f ‖ K {\displaystyle \left\|f\right\|_{K}} , which represents the complexity of the candidate function in the hypothesis space. When the algorithm considers a candidate function, it takes its norm into account in order to penalize complex functions. Formally, given a set of labeled training data ( x 1 , y 1 ) , … , ( x ℓ , y ℓ ) {\displaystyle (x_{1},y_{1}),\ldots ,(x_{\ell },y_{\ell })} with x i ∈ X , y i ∈ Y {\displaystyle x_{i}\in X,y_{i}\in Y} and a loss function V {\displaystyle V} , a learning algorithm using Tikhonov regularization will attempt to solve the expression arg min f ∈ H 1 ℓ ∑ i = 1 ℓ V ( f ( x i ) , y i ) + γ ‖ f ‖ K 2 {\displaystyle {\underset {f\in {\mathcal {H}}}{\arg \!\min }}{\frac {1}{\ell }}\sum _{i=1}^{\ell }V(f(x_{i}),y_{i})+\gamma \left\|f\right\|_{K}^{2}} where γ {\displaystyle \gamma } is a hyperparameter that controls how much the algorithm will prefer simpler functions over functions that fit the data better. Manifold regularization adds a second regularization term, the intrinsic regularizer, to the ambient regularizer used in standard Tikhonov regularization. Under the manifold assumption in machine learning, the data in question do not come from the entire input space X {\displaystyle X} , but instead from a nonlinear manifold M ⊂ X {\displaystyle M\subset X} . The geometry of this manifold, the intrinsic space, is used to determine the regularization norm. === Laplacian norm === There are many possible choices for the intrinsic regularizer ‖ f ‖ I {\displaystyle \left\|f\right\|_{I}} . Many natural choices involve the gradient on the manifold ∇ M {\displaystyle \nabla _{M}} , which can provide a measure of how smooth a target function is. A smooth function should change slowly where the input data are dense; that is, the gradient ∇ M f ( x ) {\displaystyle \nabla _{M}f(x)} should be small where the marginal probability density P X ( x ) {\displaystyle {\mathcal {P}}_{X}(x)} , the probability density of a randomly drawn data point appearing at x {\displaystyle x} , is large. This gives one appropriate choice for the intrinsic regularizer: ‖ f ‖ I 2 = ∫ x ∈ M ‖ ∇ M f ( x ) ‖ 2 d P X ( x ) {\displaystyle \left\|f\right\|_{I}^{2}=\int _{x\in M}\left\|\nabla _{M}f(x)\right\|^{2}\,d{\mathcal {P}}_{X}(x)} In practice, this norm cannot be computed directly because the marginal distribution P X {\displaystyle {\mathcal {P}}_{X}} is unknown, but it can be estimated from the provided data. === Graph-based approach of the Laplacian norm === When the distances between input points are interpreted as a graph, then the Laplacian matrix of the graph can help to estimate the marginal distribution. Suppose that the input data include ℓ {\displaystyle \ell } labeled examples (pairs of an input x {\displaystyle x} and a label y {\displaystyle y} ) and u {\displaystyle u} unlabeled examples (inputs without associated labels). Define W {\displaystyle W} to be a matrix of edge weights for a graph, where W i j {\displaystyle W_{ij}} is a similarity built from distance measure between the data points x i {\displaystyle x_{i}} and x j {\displaystyle x_{j}} (so that more close implies higher W i j {\displaystyle W_{ij}} ). Define D {\displaystyle D} to be a diagonal matrix with D i i = ∑ j = 1 ℓ + u W i j {\displaystyle D_{ii}=\sum _{j=1}^{\ell +u}W_{ij}} and L {\displaystyle L} to be the Laplacian matrix D − W {\displaystyle D-W} . Then, as the number of data points ℓ + u {\displaystyle \ell +u} increases, L {\displaystyle L} converges to the Laplace–Beltrami operator Δ M {\displaystyle \Delta _{M}} , which is the divergence of the gradient ∇ M {\displaystyle \nabla _{M}} . Then, if f {\displaystyle \mathbf {f} } is a vector of the values of f {\displaystyle f} at the data, f = [ f ( x 1 ) , … , f ( x l + u ) ] T {\displaystyle \mathbf {f} =[f(x_{1}),\ldots ,f(x_{l+u})]^{\mathrm {T} }} , the intrinsic norm can be estimated: ‖ f ‖ I 2 = 1 ( ℓ + u ) 2 f T L f {\displaystyle \left\|f\right\|_{I}^{2}={\frac {1}{(\ell +u)^{2}}}\mathbf {f} ^{\mathrm {T} }L\mathbf {f} } As the number of data points ℓ + u {\displaystyle \ell +u} increases, this empirical definition of ‖ f ‖ I 2 {\displaystyle \left\|f\right\|_{I}^{2}} converges to the definition when P X {\displaystyle {\mathcal {P}}_{X}} is known. === Solving the regularization problem with graph-based approach === Using the weights γ A {\displaystyle \gamma _{A}} and γ I {\displaystyle \gamma _{I}} for the ambient and intrinsic regularizers, the final expression to be solved becomes: arg min f ∈ H 1 ℓ ∑ i = 1 ℓ V ( f ( x i ) , y i ) + γ A ‖ f ‖ K 2 + γ I ( ℓ + u ) 2 f T L f {\displaystyle {\underset {f\in {\mathcal {H}}}{\arg \!\min }}{\frac {1}{\ell }}\sum _{i=1}^{\ell }V(f(x_{i}),y_{i})+\gamma _{A}\left\|f\right\|_{K}^{2}+{\frac {\gamma _{I}}{(\ell +u)^{2}}}\mathbf {f} ^{\mathrm {T} }L\mathbf {f} } As with other kernel methods, H {\displaystyle {\mathcal {H}}} may be an infinite-dimensional space, so if the regularization expression cannot be solved explicitly, it is impossible to search the entire space for a solution. Instead, a representer theorem shows that under certain conditions on the choice of the norm ‖ f ‖ I {\displaystyle \left\|f\right\|_{I}} , the optimal solution f ∗ {\displaystyle f^{}} must be a linear combination of the kernel centered at each of the input points: for some weights α i {\displaystyle \alpha _{i}} , f ∗ ( x ) = ∑ i = 1 ℓ + u α i K ( x i , x ) {\displaystyle f^{}(x)=\sum _{i=1}^{\ell +u}\alpha _{i}K(x_{i},x)} Using this result, it is possible to search for the optimal solution f ∗ {\displaystyle f^{}} by searching the finite-dimensional space defined by the possible choices of α i {\displaystyle \alpha _{i}} . === Functional approach of the Laplacian norm === The idea beyond the graph-Laplacian is to use neighbors to estimate the Laplacian. This method is akin to local averaging methods, that are known to scale poorly in high-dimensional problems. Indeed, the graph Laplacian is known to suffer from the curse of dimensionality. Luckily, it is possible to leverage expected smoothness of the function to estimate thanks to more advanced functional analysis. This method consists of estimating the Laplacian operator using derivatives of the kernel reading ∂ 1 , j K ( x i , x ) {\displaystyle \partial _{1,j}K(x_{i},x)} where ∂ 1 , j {\displaystyle \partial _{1,j}} denotes the partial derivatives according to the j-th coordinate of the first variable. This second approach to the Laplacian norm is to put in relation with meshfree methods, that contrast with the finite difference method in PDE. == Applications == Manifold regularization can extend a variety of algorithms that can be expressed using Tikhonov regularization, by choosing an appropriate loss function V {\displaystyle V} and hypothesis space H {\displaystyle {\mathcal {H}}} . Two commonly used examples are the families of support vector machines and regularized least squares algorithm

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  ByLock

  ByLock was a smartphone application that allowed users to communicate via a private, encrypted connection. It was launched in March 2014 on Google Play, Apple App Store The app was downloaded over 600,000 times from its launch in April 2014 until March 2016, when it was permanently shut down. The Turkish National Intelligence Organization (Turkish: Millî İstihbarat Teşkilatı, MİT) stated that the app was downloaded mainly in Turkey and the users were “Fetullahist Terror Organisation (FETÖ) which was formerly known as “Gülen movement” members. == Gülen Movement controversy == In Turkey, possession of the app is deemed evidence of membership in the Gülen Movement, which was allegedly connected to the failed Turkish coup d'état attempt in July 2016. Users of ByLock were deemed terrorists in Turkish courts. According to Deutsche Welle, of the 215,000 former ByLock users, an estimated 23,000 have been detained by Turkish authorities. Some believe that the MİT and other Turkish authorities manipulated the ByLock database in order to arrest suspected members of the Gülen Movement. Tuncay Beşikçi, a computer forensic expert in Turkey, emphasized that "the demands to investigate and analyze ByLock data from independent institutions are refused by the Turkish courts. But it is not normal". Tuncay Beşikçi believes that this application is precisely one of the channels for Gülen molecules to communicate and can also monitor the activities of other members of the organization. He also stated that the developers behind the Mor Beyin app, deliberately set a plan in motion that would put thousands of innocent people in prison as a cover for the Gülen movement. In December 2017, Turkish authorities revealed that almost half the people who had been prosecuted for having ByLock on their smartphones would have their legal cases reviewed, as they could have been redirected to the app without their knowledge. Following the failed coup attempt on 15 July 2016, the use of the ByLock messaging application by members of the Gülen Movement was the sole evidence in investigations and prosecutions to justify arrests and convictions for "membership in an armed terrorist organization." However, these decisions have been considered human rights violations by the European Court of Human Rights (ECHR), the United Nations Human Rights Committee, and the UN Working Group on Arbitrary Detention. Some of the relevant decisions include the following: === Decisions of the European Court of Human Rights === On 20 July 2021, in the case of Tekin Akgün v. Turkey, the European Court of Human Rights (ECHR) ruled that the use of the ByLock messaging application, unless supported by other evidence, does not create a reasonable suspicion of a crime. Based on this reasoning, the court found that the detention order violated Article 5 of the European Convention on Human Rights, which protects the right to liberty and security. In the Yüksel Yalçınkaya v. Turkey decision on 26 September 2023, the European Court of Human Rights (ECHR) examined an appeal against a conviction based on the use of ByLock. The Court ruled that the failure to provide an opportunity to challenge the authenticity of the ByLock data violated the right to a fair trial (Article 6 of the ECHR). The Court also stated that the mere use of ByLock could not be considered sufficient evidence for membership in an armed terrorist organization. It further noted that local courts had established an automatic presumption of guilt based solely on ByLock use, creating a broad and unpredictable interpretation of the law, making it nearly impossible for the accused to exonerate themselves. Therefore, the Court concluded that the conviction based on the use of ByLock violated the principle of no punishment without law (Article 7 of the ECHR). On 22 July 2025, in the Demirhan and 238 Others case, the European Court of Human Rights (ECHR) consolidated the applications of 239 individuals who had been convicted of "membership in an armed terrorist organization" based on their use of ByLock, as determined by 239 separate courts in Turkey. The Court ruled that the convictions violated the right to a fair trial under Article 6 and the principle of no punishment without law under Article 7 of the European Convention on Human Rights (ECHR). The ruling stated that the Turkish courts' "categorical approach" to the use of ByLock lacked legal foundation. In this context, it was emphasized that anyone who had used ByLock could not be convicted of membership in an armed terrorist organization based solely on this reasoning. The ruling also stated that, due to the large number of similar applications, the issue was "systemic in nature" and it called for a national solution to be developed by Turkey. While the Court did not order compensation for the 239 applicants, it emphasized that reopening the trial to ensure the enforcement of the violation ruling was the most appropriate remedy. This ruling, which confirms the violation finding in the Yüksel Yalçınkaya case of 26 September 2023, is considered a continuation of the ECHR's case law concerning trials based on ByLock evidence. === Decisions of the United Nations Human Rights Committee and Working Group === In the İsmet Özçelik and Turgay Karaman v. Turkey decision, dated 28 May 2019 (Application No. 2980/2017), the UN Human Rights Committee ruled that the use of ByLock and allegations of depositing money into Bank Asya could not justify the applicants' arrests. In the Mestan Yayman v. Turkey decision (Opinion No. 42/2018 – 21 August 2018) by the UN Human Rights Council Working Group on Arbitrary Detention, it was stated that using a public messaging application like ByLock cannot be considered criminal evidence, and that the use of such an application falls under the scope of freedom of thought and expression. The dozens of decisions later issued by the UN Human Rights Council Working Group are of the same nature.

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  Legal information retrieval

  Legal information retrieval is the science of information retrieval applied to legal text, including legislation, case law, and scholarly works. Accurate legal information retrieval is important to provide access to the law to laymen and legal professionals. Its importance has increased because of the vast and quickly increasing amount of legal documents available through electronic means. Legal information retrieval is a part of the growing field of legal informatics. In a legal setting, it is frequently important to retrieve all information related to a specific query. However, commonly used boolean search methods (exact matches of specified terms) on full text legal documents have been shown to have an average recall rate as low as 20 percent, meaning that only 1 in 5 relevant documents are actually retrieved. In that case, researchers believed that they had retrieved over 75% of relevant documents. This may result in failing to retrieve important or precedential cases. In some jurisdictions this may be especially problematic, as legal professionals are ethically obligated to be reasonably informed as to relevant legal documents. Legal Information Retrieval attempts to increase the effectiveness of legal searches by increasing the number of relevant documents (providing a high recall rate) and reducing the number of irrelevant documents (a high precision rate). This is a difficult task, as the legal field is prone to jargon, polysemes (words that have different meanings when used in a legal context), and constant change. Techniques used to achieve these goals generally fall into three categories: boolean retrieval, manual classification of legal text, and natural language processing of legal text. == Problems == Application of standard information retrieval techniques to legal text can be more difficult than application in other subjects. One key problem is that the law rarely has an inherent taxonomy. Instead, the law is generally filled with open-ended terms, which may change over time. This can be especially true in common law countries, where each decided case can subtly change the meaning of a certain word or phrase. Legal information systems must also be programmed to deal with law-specific words and phrases. Though this is less problematic in the context of words which exist solely in law, legal texts also frequently use polysemes, words may have different meanings when used in a legal or common-speech manner, potentially both within the same document. The legal meanings may be dependent on the area of law in which it is applied. For example, in the context of European Union legislation, the term "worker" has four different meanings: Any worker as defined in Article 3(a) of Directive 89/391/EEC who habitually uses display screen equipment as a significant part of his normal work. Any person employed by an employer, including trainees and apprentices but excluding domestic servants; Any person carrying out an occupation on board a vessel, including trainees and apprentices, but excluding port pilots and shore personnel carrying out work on board a vessel at the quayside; Any person who, in the Member State concerned, is protected as an employee under national employment law and in accordance with national practice; It also has the common meaning: A person who works at a specific occupation. Though the terms may be similar, correct information retrieval must differentiate between the intended use and irrelevant uses in order to return the correct results. Even if a system overcomes the language problems inherent in law, it must still determine the relevancy of each result. In the context of judicial decisions, this requires determining the precedential value of the case. Case decisions from senior or superior courts may be more relevant than those from lower courts, even where the lower court's decision contains more discussion of the relevant facts. The opposite may be true, however, if the senior court has only a minor discussion of the topic (for example, if it is a secondary consideration in the case). An information retrieval system must also be aware of the authority of the jurisdiction. A case from a binding authority is most likely of more value than one from a non-binding authority. Additionally, the intentions of the user may determine which cases they find valuable. For instance, where a legal professional is attempting to argue a specific interpretation of law, he might find a minor court's decision which supports his position more valuable than a senior courts position which does not. He may also value similar positions from different areas of law, different jurisdictions, or dissenting opinions. Overcoming these problems can be made more difficult because of the large number of cases available. The number of legal cases available via electronic means is constantly increasing (in 2003, US appellate courts handed down approximately 500 new cases per day), meaning that an accurate legal information retrieval system must incorporate methods of both sorting past data and managing new data. == Techniques == === Boolean searches === Boolean searches, where a user may specify terms such as use of specific words or judgments by a specific court, are the most common type of search available via legal information retrieval systems. They are widely implemented but overcome few of the problems discussed above. The recall and precision rates of these searches vary depending on the implementation and searches analyzed. One study found a basic boolean search's recall rate to be roughly 20%, and its precision rate to be roughly 79%. Another study implemented a generic search (that is, not designed for legal uses) and found a recall rate of 56% and a precision rate of 72% among legal professionals. Both numbers increased when searches were run by non-legal professionals, to a 68% recall rate and 77% precision rate. This is likely explained because of the use of complex legal terms by the legal professionals. === Manual classification === In order to overcome the limits of basic boolean searches, information systems have attempted to classify case laws and statutes into more computer friendly structures. Usually, this results in the creation of an ontology to classify the texts, based on the way a legal professional might think about them. These attempt to link texts on the basis of their type, their value, and/or their topic areas. Most major legal search providers now implement some sort of classification search, such as Westlaw's “Natural Language” or LexisNexis' Headnote searches. Additionally, both of these services allow browsing of their classifications, via Westlaw's West Key Numbers or Lexis' Headnotes. Though these two search algorithms are proprietary and secret, it is known that they employ manual classification of text (though this may be computer-assisted). These systems can help overcome the majority of problems inherent in legal information retrieval systems, in that manual classification has the greatest chances of identifying landmark cases and understanding the issues that arise in the text. In one study, ontological searching resulted in a precision rate of 82% and a recall rate of 97% among legal professionals. The legal texts included, however, were carefully controlled to just a few areas of law in a specific jurisdiction. The major drawback to this approach is the requirement of using highly skilled legal professionals and large amounts of time to classify texts. As the amount of text available continues to increase, some have stated their belief that manual classification is unsustainable. === Natural language processing === In order to reduce the reliance on legal professionals and the amount of time needed, efforts have been made to create a system to automatically classify legal text and queries. Adequate translation of both would allow accurate information retrieval without the high cost of human classification. These automatic systems generally employ Natural Language Processing (NLP) techniques that are adapted to the legal domain, and also require the creation of a legal ontology. Though multiple systems have been postulated, few have reported results. One system, “SMILE,” which attempted to automatically extract classifications from case texts, resulted in an f-measure (which is a calculation of both recall rate and precision) of under 0.3 (compared to perfect f-measure of 1.0). This is probably much lower than an acceptable rate for general usage. Despite the limited results, many theorists predict that the evolution of such systems will eventually replace manual classification systems. === Citation-Based ranking === In the mid-90s the Room 5 case law retrieval project used citation mining for summaries and ranked its search results based on citation type and count. This slightly pre-dated the PageRank algorithm at Stanford which was also a citation-based ranking. Ranking of results was based

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  CMU Pronouncing Dictionary

  The CMU Pronouncing Dictionary (also known as CMUdict) is an open-source pronouncing dictionary originally created by the Speech Group at Carnegie Mellon University (CMU) for use in speech recognition research. CMUdict provides a mapping orthographic/phonetic for English words in their North American pronunciations. It is commonly used to generate representations for speech recognition (ASR), e.g. the CMU Sphinx system, and speech synthesis (TTS), e.g. the Festival system. CMUdict can be used as a training corpus for building statistical grapheme-to-phoneme (g2p) models that will generate pronunciations for words not yet included in the dictionary. The most recent release is 0.7b; it contains over 134,000 entries. An interactive lookup version is available. == Database format == The database is distributed as a plain text file with one entry to a line in the format "WORD " with a two-space separator between the parts. If multiple pronunciations are available for a word, variants are identified using numbered versions (e.g. WORD(1)). The pronunciation is encoded using a modified form of the ARPABET system, with the addition of stress marks on vowels of levels 0, 1, and 2. A line-initial ;;; token indicates a comment. A derived format, directly suitable for speech recognition engines is also available as part of the distribution; this format collapses stress distinctions (typically not used in ASR). The following is a table of phonemes used by CMU Pronouncing Dictionary. == History == == Applications == The Unifon converter is based on the CMU Pronouncing Dictionary. The Natural Language Toolkit contains an interface to the CMU Pronouncing Dictionary. The Carnegie Mellon Logios tool incorporates the CMU Pronouncing Dictionary. PronunDict, a pronunciation dictionary of American English, uses the CMU Pronouncing Dictionary as its data source. Pronunciation is transcribed in IPA symbols. This dictionary also supports searching by pronunciation. Some singing voice synthesizer software like CeVIO Creative Studio and Synthesizer V uses modified version of CMU Pronouncing Dictionary for synthesizing English singing voices. Transcriber, a tool for the full text phonetic transcription, uses the CMU Pronouncing Dictionary 15.ai, a real-time text-to-speech tool using artificial intelligence, uses the CMU Pronouncing Dictionary

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  Language engineering

  Language engineering involves the creation of natural language processing systems, whose cost and outputs are measurable and predictable. It is a distinct field contrasted to natural language processing and computational linguistics. A recent trend of language engineering is the use of Semantic Web technologies for the creation, archiving, processing, and retrieval of machine processable language data. Meta-Language Engineering is a proposed extension of Language Engineering first recorded in 2025, associated with the work of Delyone de Paula Canedo Filho. The term is used to designate an approach that, in addition to natural language processing, encompasses the symbolic, cognitive, and epistemological structuring of language systems.

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  Connected-component labeling

  Connected-component labeling (CCL), connected-component analysis (CCA), blob extraction, region labeling, blob discovery, or region extraction is an algorithmic application of graph theory, where subsets of connected components are uniquely labeled based on a given heuristic. Connected-component labeling is not to be confused with segmentation. Connected-component labeling is used in computer vision to detect connected regions in binary digital images, although color images and data with higher dimensionality can also be processed. When integrated into an image recognition system or human-computer interaction interface, connected component labeling can operate on a variety of information. Blob extraction is generally performed on the resulting binary image from a thresholding step, but it can be applicable to gray-scale and color images as well. Blobs may be counted, filtered, and tracked. Blob extraction is related to but distinct from blob detection. == Overview == A graph, containing vertices and connecting edges, is constructed from relevant input data. The vertices contain information required by the comparison heuristic, while the edges indicate connected 'neighbors'. An algorithm traverses the graph, labeling the vertices based on the connectivity and relative values of their neighbors. Connectivity is determined by the medium; image graphs, for example, can be 4-connected neighborhood or 8-connected neighborhood. Following the labeling stage, the graph may be partitioned into subsets, after which the original information can be recovered and processed . == Definition == The usage of the term connected-component labeling (CCL) and its definition is quite consistent in the academic literature, whereas connected-component analysis (CCA) varies both in terminology and in its definition of the problem. Rosenfeld et al. define connected components labeling as the “[c]reation of a labeled image in which the positions associated with the same connected component of the binary input image have a unique label.” Shapiro et al. define CCL as an operator whose “input is a binary image and [...] output is a symbolic image in which the label assigned to each pixel is an integer uniquely identifying the connected component to which that pixel belongs.” There is no consensus on the definition of CCA in the academic literature. It is often used interchangeably with CCL. A more extensive definition is given by Shapiro et al.: “Connected component analysis consists of connected component labeling of the black pixels followed by property measurement of the component regions and decision making.” The definition for connected-component analysis presented here is more general, taking the thoughts expressed in into account. == Algorithms == The algorithms discussed can be generalised to arbitrary dimensions, albeit with increased time and space complexity. === One component at a time === This is a fast and very simple method to implement and understand. It is based on graph traversal methods in graph theory. In short, once the first pixel of a connected component is found, all the connected pixels of that connected component are labelled before going onto the next pixel in the image. This algorithm is part of Vincent and Soille's watershed segmentation algorithm, other implementations also exist. In order to do that a linked list is formed that will keep the indexes of the pixels that are connected to each other, steps (2) and (3) below. The method of defining the linked list specifies the use of a depth or a breadth first search. For this particular application, there is no difference which strategy to use. The simplest kind of a last in first out queue implemented as a singly linked list will result in a depth first search strategy. It is assumed that the input image is a binary image, with pixels being either background or foreground and that the connected components in the foreground pixels are desired. The algorithm steps can be written as: Start from the first pixel in the image. Set current label to 1. Go to (2). If this pixel is a foreground pixel and it is not already labelled, give it the current label and add it as the first element in a queue, then go to (3). If it is a background pixel or it was already labelled, then repeat (2) for the next pixel in the image. Pop out an element from the queue, and look at its neighbours (based on any type of connectivity). If a neighbour is a foreground pixel and is not already labelled, give it the current label and add it to the queue. Repeat (3) until there are no more elements in the queue. Go to (2) for the next pixel in the image and increment current label by 1. Note that the pixels are labelled before being put into the queue. The queue will only keep a pixel to check its neighbours and add them to the queue if necessary. This algorithm only needs to check the neighbours of each foreground pixel once and doesn't check the neighbours of background pixels. The pseudocode is: algorithm OneComponentAtATime(data) input : imageData[xDim][yDim] initialization : label = 0, labelArray[xDim][yDim] = 0, statusArray[xDim][yDim] = false, queue1, queue2; for i = 0 to xDim do for j = 0 to yDim do if imageData[i][j] has not been processed do if imageData[i][j] is a foreground pixel do check its four neighbors(north, south, east, west) : if neighbor is not processed do if neighbor is a foreground pixel do add it to queue1 else update its status to processed end if labelArray[i][j] = label (give label) statusArray[i][j] = true (update status) while queue1 is not empty do For each pixel in the queue do : check its four neighbors if neighbor is not processed do if neighbor is a foreground pixel do add it to queue2 else update its status to processed end if give it the current label update its status to processed remove the current element from queue1 copy queue2 into queue1 end While increase the label end if else update its status to processed end if end if end if end for end for === Two-pass === Relatively simple to implement and understand, the two-pass algorithm, (also known as the Hoshen–Kopelman algorithm) iterates through 2-dimensional binary data. The algorithm makes two passes over the image: the first pass to assign temporary labels and record equivalences, and the second pass to replace each temporary label by the smallest label of its equivalence class. The input data can be modified in situ (which carries the risk of data corruption), or labeling information can be maintained in an additional data structure. Connectivity checks are carried out by checking neighbor pixels' labels (neighbor elements whose labels are not assigned yet are ignored), or say, the north-east, the north, the north-west and the west of the current pixel (assuming 8-connectivity). 4-connectivity uses only north and west neighbors of the current pixel. The following conditions are checked to determine the value of the label to be assigned to the current pixel (4-connectivity is assumed) Conditions to check: Does the pixel to the left (west) have the same value as the current pixel? Yes – We are in the same region. Assign the same label to the current pixel No – Check next condition Do both pixels to the north and west of the current pixel have the same value as the current pixel but not the same label? Yes – We know that the north and west pixels belong to the same region and must be merged. Assign the current pixel the minimum of the north and west labels, and record their equivalence relationship No – Check next condition Does the pixel to the left (west) have a different value and the one to the north the same value as the current pixel? Yes – Assign the label of the north pixel to the current pixel No – Check next condition Do the pixel's north and west neighbors have different pixel values than current pixel? Yes – Create a new label id and assign it to the current pixel The algorithm continues this way, and creates new region labels whenever necessary. The key to a fast algorithm, however, is how this merging is done. This algorithm uses the union-find data structure which provides excellent performance for keeping track of equivalence relationships. Union-find essentially stores labels which correspond to the same blob in a disjoint-set data structure, making it easy to remember the equivalence of two labels by the use of an interface method E.g.: findSet(l). findSet(l) returns the minimum label value that is equivalent to the function argument 'l'. Once the initial labeling and equivalence recording is completed, the second pass merely replaces each pixel label with its equivalent disjoint-set representative element. A faster-scanning algorithm for connected-region extraction is presented below. On the first pass: Iterate through each element of the data by column, then by row (Raster Scanning) If the element is not the background Get the neighboring elements of the current element If there are no neighbors, uniquely

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  Second-order co-occurrence pointwise mutual information

  In computational linguistics, second-order co-occurrence pointwise mutual information (SOC-PMI) is a method used to measure semantic similarity, or how close in meaning two words are. The method does not require the two words to appear together in a text. Instead, it works by analyzing the "neighbor" words that typically appear alongside each of the two target words in a large body of text (corpus). If the two target words frequently share the same neighbors, they are considered semantically similar. For example, the words "cemetery" and "graveyard" may not appear in the same sentence often, but they both frequently appear near words like "buried," "dead," and "funeral." SOC-PMI uses this shared context to determine that they have a similar meaning. The method is called "second-order" because it doesn't look at the direct co-occurrence of the target words (which would be first-order), but at the co-occurrence of their neighbors (a second level of association). The strength of these associations is quantified using pointwise mutual information (PMI). == History == The method builds on earlier work like the PMI-IR algorithm, which used the AltaVista search engine to calculate word association probabilities. The key advantage of a second-order approach like SOC-PMI is its ability to measure similarity between words that do not co-occur often, or at all. The British National Corpus (BNC) has been used as a source for word frequencies and contexts for this method. == Methodology == The SOC-PMI algorithm measures the similarity between two words, w 1 {\displaystyle w_{1}} and w 2 {\displaystyle w_{2}} , in several steps. === Step 1: Score neighboring words with PMI === First, for each target word ( w 1 {\displaystyle w_{1}} and w 2 {\displaystyle w_{2}} ), the algorithm identifies its "neighbor" words within a certain text window (e.g., within 5 words to the left or right) across a large corpus. The strength of the association between a target word t i {\displaystyle t_{i}} and its neighbor w {\displaystyle w} is calculated using pointwise mutual information (PMI). A higher PMI value means the two words appear together more often than would be expected by chance. The PMI between a target word t i {\displaystyle t_{i}} and a neighbor word w {\displaystyle w} is calculated as: f pmi ( t i , w ) = log 2 ⁡ f b ( t i , w ) × m f t ( t i ) f t ( w ) {\displaystyle f^{\text{pmi}}(t_{i},w)=\log _{2}{\frac {f^{b}(t_{i},w)\times m}{f^{t}(t_{i})f^{t}(w)}}} where: f b ( t i , w ) {\displaystyle f^{b}(t_{i},w)} is the number of times t i {\displaystyle t_{i}} and w {\displaystyle w} appear together in the context window. f t ( t i ) {\displaystyle f^{t}(t_{i})} is the total number of times t i {\displaystyle t_{i}} appears in the corpus. f t ( w ) {\displaystyle f^{t}(w)} is the total number of times w {\displaystyle w} appears in the corpus. m {\displaystyle m} is the total number of tokens (words) in the corpus. === Step 2: Create a semantic 'signature' for each word === For each target word ( w 1 {\displaystyle w_{1}} and w 2 {\displaystyle w_{2}} ), the algorithm creates a list of its most significant neighbors. This is done by taking the top β {\displaystyle \beta } neighbor words, sorted in descending order by their PMI score with the target word. This list of top neighbors, X w {\displaystyle X^{w}} , acts as a semantic "signature" for the word w {\displaystyle w} . X w = { X i w } {\displaystyle X^{w}=\{X_{i}^{w}\}} , for i = 1 , 2 , … , β {\displaystyle i=1,2,\ldots ,\beta } The size of this list, β {\displaystyle \beta } , is a parameter of the method. === Step 3: Compare the signatures === The algorithm then compares the signatures of w 1 {\displaystyle w_{1}} and w 2 {\displaystyle w_{2}} . It looks for words that are present in both signatures. The similarity of w 1 {\displaystyle w_{1}} to w 2 {\displaystyle w_{2}} is calculated by summing the PMI scores of w 2 {\displaystyle w_{2}} with every word in w 1 {\displaystyle w_{1}} 's signature list. The β {\displaystyle \beta } -PMI summation function defines this score. The score for w 1 {\displaystyle w_{1}} with respect to w 2 {\displaystyle w_{2}} is: f ( w 1 , w 2 , β ) = ∑ i = 1 β ( f pmi ( X i w 1 , w 2 ) ) γ {\displaystyle f(w_{1},w_{2},\beta )=\sum _{i=1}^{\beta }(f^{\text{pmi}}(X_{i}^{w_{1}},w_{2}))^{\gamma }} This sum only includes terms where the PMI value is positive. The exponent γ {\displaystyle \gamma } (with a value > 1) is used to give more weight to neighbors that are more strongly associated with w 2 {\displaystyle w_{2}} . This calculation is done in both directions: The similarity of w 1 {\displaystyle w_{1}} with respect to w 2 {\displaystyle w_{2}} : f ( w 1 , w 2 , β 1 ) = ∑ i = 1 β 1 ( f pmi ( X i w 1 , w 2 ) ) γ {\displaystyle f(w_{1},w_{2},\beta _{1})=\sum _{i=1}^{\beta _{1}}(f^{\text{pmi}}(X_{i}^{w_{1}},w_{2}))^{\gamma }} The similarity of w 2 {\displaystyle w_{2}} with respect to w 1 {\displaystyle w_{1}} : f ( w 2 , w 1 , β 2 ) = ∑ i = 1 β 2 ( f pmi ( X i w 2 , w 1 ) ) γ {\displaystyle f(w_{2},w_{1},\beta _{2})=\sum _{i=1}^{\beta _{2}}(f^{\text{pmi}}(X_{i}^{w_{2}},w_{1}))^{\gamma }} === Step 4: Calculate final similarity score === Finally, the total semantic similarity is the average of the two scores from the previous step. S i m ( w 1 , w 2 ) = f ( w 1 , w 2 , β 1 ) β 1 + f ( w 2 , w 1 , β 2 ) β 2 {\displaystyle \mathrm {Sim} (w_{1},w_{2})={\frac {f(w_{1},w_{2},\beta _{1})}{\beta _{1}}}+{\frac {f(w_{2},w_{1},\beta _{2})}{\beta _{2}}}} This score can be normalized to fall between 0 and 1. For example, using this method, the words cemetery and graveyard achieve a high similarity score of 0.986 (with specific parameter settings).

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  Gradient vector flow

  Gradient vector flow (GVF), a computer vision framework introduced by Chenyang Xu and Jerry L. Prince, is the vector field that is produced by a process that smooths and diffuses an input vector field. It is usually used to create a vector field from images that points to object edges from a distance. It is widely used in image analysis and computer vision applications for object tracking, shape recognition, segmentation, and edge detection. In particular, it is commonly used in conjunction with active contour model. == Background == Finding objects or homogeneous regions in images is a process known as image segmentation. In many applications, the locations of object edges can be estimated using local operators that yield a new image called an edge map. The edge map can then be used to guide a deformable model, sometimes called an active contour or a snake, so that it passes through the edge map in a smooth way, therefore defining the object itself. A common way to encourage a deformable model to move toward the edge map is to take the spatial gradient of the edge map, yielding a vector field. Since the edge map has its highest intensities directly on the edge and drops to zero away from the edge, these gradient vectors provide directions for the active contour to move. When the gradient vectors are zero, the active contour will not move, and this is the correct behavior when the contour rests on the peak of the edge map itself. However, because the edge itself is defined by local operators, these gradient vectors will also be zero far away from the edge and therefore the active contour will not move toward the edge when initialized far away from the edge. Gradient vector flow (GVF) is the process that spatially extends the edge map gradient vectors, yielding a new vector field that contains information about the location of object edges throughout the entire image domain. GVF is defined as a diffusion process operating on the components of the input vector field. It is designed to balance the fidelity of the original vector field, so it is not changed too much, with a regularization that is intended to produce a smooth field on its output. Although GVF was designed originally for the purpose of segmenting objects using active contours attracted to edges, it has been since adapted and used for many alternative purposes. Some newer purposes including defining a continuous medial axis representation, regularizing image anisotropic diffusion algorithms, finding the centers of ribbon-like objects, constructing graphs for optimal surface segmentations, creating a shape prior, and much more. == Theory == The theory of GVF was originally described by Xu and Prince. Let f ( x , y ) {\displaystyle \textstyle f(x,y)} be an edge map defined on the image domain. For uniformity of results, it is important to restrict the edge map intensities to lie between 0 and 1, and by convention f ( x , y ) {\displaystyle \textstyle f(x,y)} takes on larger values (close to 1) on the object edges. The gradient vector flow (GVF) field is given by the vector field v ( x , y ) = [ u ( x , y ) , v ( x , y ) ] {\displaystyle \textstyle \mathbf {v} (x,y)=[u(x,y),v(x,y)]} that minimizes the energy functional In this equation, subscripts denote partial derivatives and the gradient of the edge map is given by the vector field ∇ f = ( f x , f y ) {\displaystyle \textstyle \nabla f=(f_{x},f_{y})} . Figure 1 shows an edge map, the gradient of the (slightly blurred) edge map, and the GVF field generated by minimizing E {\displaystyle \textstyle {\mathcal {E}}} . Equation 1 is a variational formulation that has both a data term and a regularization term. The first term in the integrand is the data term. It encourages the solution v {\displaystyle \textstyle \mathbf {v} } to closely agree with the gradients of the edge map since that will make v − ∇ f {\displaystyle \textstyle \mathbf {v} -\nabla f} small. However, this only needs to happen when the edge map gradients are large since v − ∇ f {\displaystyle \textstyle \mathbf {v} -\nabla f} is multiplied by the square of the length of these gradients. The second term in the integrand is a regularization term. It encourages the spatial variations in the components of the solution to be small by penalizing the sum of all the partial derivatives of v {\displaystyle \textstyle \mathbf {v} } . As is customary in these types of variational formulations, there is a regularization parameter μ > 0 {\displaystyle \textstyle \mu >0} that must be specified by the user in order to trade off the influence of each of the two terms. If μ {\displaystyle \textstyle \mu } is large, for example, then the resulting field will be very smooth and may not agree as well with the underlying edge gradients. Theoretical Solution. Finding v ( x , y ) {\displaystyle \textstyle \mathbf {v} (x,y)} to minimize Equation 1 requires the use of calculus of variations since v ( x , y ) {\displaystyle \textstyle \mathbf {v} (x,y)} is a function, not a variable. Accordingly, the Euler equations, which provide the necessary conditions for v {\displaystyle \textstyle \mathbf {v} } to be a solution can be found by calculus of variations, yielding where ∇ 2 {\displaystyle \textstyle \nabla ^{2}} is the Laplacian operator. It is instructive to examine the form of the equations in (2). Each is a partial differential equation that the components u {\displaystyle u} and v {\displaystyle v} of v {\displaystyle \mathbf {v} } must satisfy. If the magnitude of the edge gradient is small, then the solution of each equation is guided entirely by Laplace's equation, for example ∇ 2 u = 0 {\displaystyle \textstyle \nabla ^{2}u=0} , which will produce a smooth scalar field entirely dependent on its boundary conditions. The boundary conditions are effectively provided by the locations in the image where the magnitude of the edge gradient is large, where the solution is driven to agree more with the edge gradients. Computational Solutions. There are two fundamental ways to compute GVF. First, the energy function E {\displaystyle {\mathcal {E}}} itself (1) can be directly discretized and minimized, for example, by gradient descent. Second, the partial differential equations in (2) can be discretized and solved iteratively. The original GVF paper used an iterative approach, while later papers introduced considerably faster implementations such as an octree-based method, a multi-grid method, and an augmented Lagrangian method. In addition, very fast GPU implementations have been developed in Extensions and Advances. GVF is easily extended to higher dimensions. The energy function is readily written in a vector form as which can be solved by gradient descent or by finding and solving its Euler equation. Figure 2 shows an illustration of a three-dimensional GVF field on the edge map of a simple object (see ). The data and regularization terms in the integrand of the GVF functional can also be modified. A modification described in , called generalized gradient vector flow (GGVF) defines two scalar functions and reformulates the energy as While the choices g ( ∇ f | ) = μ {\displaystyle \textstyle g(\nabla f|)=\mu } and h ( | ∇ f | ) = | ∇ f | 2 {\displaystyle \textstyle h(|\nabla f|)=|\nabla f|^{2}} reduce GGVF to GVF, the alternative choices g ( | ∇ f | ) = exp ⁡ { − | ∇ f | / K } {\displaystyle \textstyle g(|\nabla f|)=\exp\{-|\nabla f|/K\}} and h ( ∇ f | ) = 1 − g ( | ∇ f | ) {\displaystyle \textstyle h(\nabla f|)=1-g(|\nabla f|)} , for K {\displaystyle K} a user-selected constant, can improve the tradeoff between the data term and its regularization in some applications. The GVF formulation has been further extended to vector-valued images in where a weighted structure tensor of a vector-valued image is used. A learning based probabilistic weighted GVF extension was proposed in to further improve the segmentation for images with severely cluttered textures or high levels of noise. The variational formulation of GVF has also been modified in motion GVF (MGVF) to incorporate object motion in an image sequence. Whereas the diffusion of GVF vectors from a conventional edge map acts in an isotropic manner, the formulation of MGVF incorporates the expected object motion between image frames. An alternative to GVF called vector field convolution (VFC) provides many of the advantages of GVF, has superior noise robustness, and can be computed very fast. The VFC field v V F C {\displaystyle \textstyle \mathbf {v} _{\mathrm {VFC} }} is defined as the convolution of the edge map f {\displaystyle f} with a vector field kernel k {\displaystyle \mathbf {k} } where The vector field kernel k {\displaystyle \textstyle \mathbf {k} } has vectors that always point toward the origin but their magnitudes, determined in detail by the function m {\displaystyle m} , decrease to zero with increasing distance from the origin. The beauty of VFC is that it can be computed very rapidly using a fast Fourier tra

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  List of software palettes

  This is a list of software palettes used by computers. Systems that use a 4-bit or 8-bit pixel depth can display up to 16 or 256 colors simultaneously. Many personal computers in the early 1990s displayed at most 256 different colors, freely selected by software (either by the user or by a program) from their wider hardware's RGB color palette. Usual selections of colors in limited subsets (generally 16 or 256) of the full palette includes some RGB level arrangements commonly used with the 8-bit palettes as master palettes or universal palettes (i.e., palettes for multipurpose uses). These are some representative software palettes, but any selection can be made in such of systems. For specific hardware color palettes, see the list of monochrome and RGB palettes, list of 8-bit computer hardware graphics, the list of 16-bit computer hardware graphics and the list of video game console palettes articles. Each palette is represented by an array of color patches. A one-pixel size version appears below each palette, to make it easy to compare palette sizes. For each unique palette, an image color test chart and sample image (truecolor original follows) rendered with that palette (without dithering) are given. The test chart shows the full 8-bit, 256 levels of the red, green, and blue (RGB) primary colors and cyan, magenta, and yellow complementary colors, along with a full 8-bit, 256 levels grayscale. Gradients of RGB intermediate colors (orange, lime green, sea green, sky blue, violet and fuchsia), and a full hue spectrum are also present. Color charts are not gamma corrected. These elements illustrate the color depth and distribution of the colors of any given palette, and the sample image indicates how the color selection of such palettes could represent real-life images. == System specifics == These are selections of colors officially employed as system palettes in some popular operating systems for personal computers that support 8-bit displays. === Microsoft Windows and IBM OS/2 default 16-color palette === Used by these platforms as a roughly backward compatible palette for the CGA, EGA and VGA text modes, but with colors arranged in a different order. Also, is the default palette for 16 color icons. The corresponding indices into this palette are: === Microsoft Windows default 20-color palette === In 256-color mode, there are four additional standard Windows colors, twenty system reserved colors in total; thus the system leaves 236 palette indexes free for applications to use. The system color entries inside a 256-color palette table are the first ten plus the last ten. In any case, the additional system colors do not seem to add a sharp color richness: they are only some intermediate shades of grayish colors. Since Windows 95, these additional colors can be changed by the system when a color scheme needs custom colors, reducing their utility as static, unchanging palette entries. The complete 20-color Windows system palette is: === Apple Macintosh default 16-color palette === When Apple Computer introduced the Macintosh II in 1987, this 16-color palette was included in System 4.1. === RISC OS default palette === Acorn RISC OS 2.x and 3.x provided this 16-color palette: === Solaris default 16-color palette === Solaris OS used this color palette: == RGB arrangements == These are selections of colors based in evenly ordered RGB levels which provide complete RGB combinations, mainly used as master palettes to display any kind of image within the limitations of the 8-bit pixel depth. === 6 level RGB === Having six levels for every primary, with 6³ = 216 combinations. The index can be addressed by (36×R)+(6×G)+B, with all R, G and B values in a range from 0 to 5. Intended as homogeneous RGB cube, it gives six true grays. Also, there is room for another sorts of 40 colors, so operating systems or programs can add extra colors. Systems that use this software palette are: Web-safe colors Apple Macintosh 256 color default palette. It also contains four gradients of ten shades each for gray, red, green and blue. === 6-7-6 levels RGB === This palette is constructed with six levels for red and blue primaries and seven levels for the green primary, giving 6×7×6 = 252 combinations. The index can be addressed by (42×R)+(6×G)+B, with R and B values in a range from 0 to 5 and G in a range from 0 to 6. The same case as the former, but with an added level of green due to the greater sensibility of the normal human eye to this frequency. It does not provide true grays, but remaining indexes can be filled with four intermediate grays. In any case, there is little room for any other color. === 6-8-5 levels RGB === This palette is constructed with six levels for red, eight levels for green and five levels for the blue primaries, giving 6×8×5 = 240 combinations. The index can be addressed by (40×R)+(5×G)+B, with R ranging from 0 to 5, G from 0 to 7 and B from 0 to 4. Levels are chosen in function of sensibility of the normal human eye to every primary color. Also, it does not provide true grays. Remaining indexes can be filled with sixteen intermediate grays or other fixed colors. In fact, this is the best balanced RGB master software palette, in a compromise between the RGB arrangement based in the human eye's sensibility and a sufficient remaining palette entries for another purposes. === 8-8-4 levels RGB === The 8-8-4 level RGB use eight levels for each of the red and green color components (3+3 high order bits), and four levels (2 low order bits) for the blue component, due to the lesser sensitivity of the normal human eye to this primary color. This results in an 8×8×4 = 256-color palette as follows: This RGB software palette occupies the full 8-bit range of possible palette entries, so there is no room for other fixed colors. Software using this palette must draw their user interface elements with the same colors used to show pictures. Also again, it does not provide true grays. == Other common uses of software palettes == === Grayscale palettes === Simple palette made doing every triplet RGB primaries having equal values as a continuous gradient from black to white through the full available palette entries. Here is the 8-bit, 256 levels palette: Used to display pure grayscale TIFF or JPEG images, for example. === Color gradient palettes === Palettes made of a continuous color gradient from darkest to lightest arbitrary hues. The pixel data is treated as if it were grayscale, but the color table plays with RGB color combinations, not only gray. The relationship between the original luminance and the mapped one can vary, but the lighting scale is preserved along all the palette entries. One very common case of such palettes is the sepia tone palette, which gives an image an old fashioned and aged look (left). Another gradient example, based on blue hues, is presented here (right), but any hue or mixing of hues can be used. Many cell phones with built-in cameras have options to take colorized photos using this technique. === Adaptive palettes === Those whose whole number of available indexes are filled with RGB combinations selected from the statistical order of appearance (usually balanced) of a concrete full true color original image. There exist many algorithms to pick the colors through color quantization; one well known is the Heckbert's median-cut algorithm. Here is the 8-bit, 256 color palette used with the color test chart and the image sample above: Adaptive palettes only work well with a unique image. Trying to display different images with adaptive palettes over an 8-bit display usually results in only one image with correct colors, because the images have different palettes and only one can be displayed at a time. Here is an example of what happens when an indexed color image is displayed with any color palette that is not its own adaptive palette: === False color palettes === Arbitrary gradient color scales, usually 256 shades, with no relationship with real colors of a given image. They are employed to artificially colorize a grayscale image to reveal details and/or to map the pixel level values to amounts of some physical magnitude (potential, temperature, altitude, etc.) Note, in the example above, that new details can be seen as blue over magenta in the background's dark areas of the original photograph. Here is the 8-bit, 256 color gradient palette used with the color test chart and the image sample above: There exist many false color palettes, some of them standardized, used mainly in scientific applications: astronomy and radioastronomy, satellite land imaging, thermography, study of materials, tomography and magnetic resonance imaging in medicine, etc.

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  AltStore

  AltStore is an alternative app store for the iOS and iPadOS[1] mobile operating systems, which allows users to download applications that are not available on the App Store, most commonly tweaked apps, jailbreak apps, and apps including paid apps on the app store. It was publicly announced on September 25, 2019, and launched on September 28. == History == Riley Testut is an American developer who began to work on AltStore after Apple declined to allow his Nintendo emulator Delta on the App Store. Since Xcode allowed him to temporarily install his Delta app to his iOS device for 7 days of testing, he created AltStore in 2019 to replicate this functionality, which could be extended to other .ipa files. As of 2022, AltStore had been downloaded 1.5 million times. In the following years, AltStore expanded beyond its initial sideloading functionality. The platform was founded by Testut, with Shane Gill later joining as co-founder. AltStore was initially supported through Patreon contributions from its user community, and later saw increased adoption following regulatory developments in the European Union that enabled broader third-party app distribution. The project has also been involved in notable industry collaborations, including a partnership with Epic Games. == Features == AltStore exploits a loophole in the Xcode developer platform, which allows developers to sideload their own apps which they are working on without needing to jailbreak. Sideloaded apps are signed like a developer project for testing and will expire after 7 days with a free account or one year with a paid developer account, by which they will need to be refreshed or reinstalled.

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  Ghana Post GPS

  GhanaPostGPS is a web and smartphone application, sponsored by the government of Ghana and developed by Vokacom, to provide a digital addresses and postal codes for every 5 squared meter location in Ghana. The digital address is a composite of the postcode (region, district & area code) plus a unique address. GhanaPostGPS is the first digital addressing system created by the government of Ghana. GhanaPost GPS is a mandatory requirement for obtaining the National Identification Card and other services.

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