XLeratorDB is a suite of database function libraries that enable Microsoft SQL Server to perform a wide range of additional (non-native) business intelligence and ad hoc analytics. The libraries, which are embedded and run centrally on the database, include more than 450 individual functions similar to those found in Microsoft Excel spreadsheets. The individual functions are grouped and sold as six separate libraries based on usage: finance, statistics, math, engineering, unit conversions and strings. WestClinTech, the company that developed XLeratorDB, claims it is "the first commercial function package add-in for Microsoft SQL Server." == Company history == WestClinTech (LLC), founded by software industry veterans Charles Flock and Joe Stampf in 2008, is located in Irvington, New York, United States. Flock was a co-founder of The Frustum Group, developer of the OPICS enterprise banking and trading platform, which was acquired by London-based Misys, PLC in 1996. Stampf joined Frustum in 1994 and with Flock remained active with the company after acquisition, helping to develop successive generations of OPICS now employed by over 150 leading financial institutions worldwide. Following a full year of research, development and testing, WestClinTech introduced and recorded its first commercial sale of XLeratorDB in April 2009. In September 2009, XLeratorDB became available to all Federal agencies through NASA's Strategic Enterprise-Wide Procurement (SEWP-IV) program, a government-wide acquisition contract. == Technology == XLeratorDB uses Microsoft SQL CLR(Common Language Runtime) technology. SQL CLR allows managed code to be hosted by, and run in, the Microsoft SQL Server environment. SQL CLR relies on the creation, deployment and registration of .NET Framework assemblies that are physically stored in managed code dynamic-link libraries (DLL). The assemblies may contain .NET namespaces, classes, functions, and properties. Because managed code compiles to native code prior to execution, functions using SQL CLR can achieve significant performance increases versus the equivalent functions written in T-SQL in some scenarios. XLeratorDB requires Microsoft SQL Server 2005 or SQL Server 2005 Express editions, or later (compatibility mode 90 or higher). The product installs with PERMISSION_SET=SAFE. SAFE mode, the most restrictive permission set, is accessible by all users. Code executed by an assembly with SAFE permissions cannot access external system resources such as files, the network, the internet, environment variables, or the registry. == Functions == In computer science, a function is a portion of code within a larger program which performs a specific task and is relatively independent of the remaining code. As used in database and spreadsheet applications these functions generally represent mathematical formulas widely used across a variety of fields. While this code may be user-generated, it is also embedded as a pre-written sub-routine in applications. These functions are typically identified by common nomenclature which corresponds to their underlying operations: e.g. IRR identifies the function which calculates Internal Rate of Return on a series of periodic cash flows. === Function uses === As subroutines, functions can be integrated and used in a variety of ways, and as part of larger, more complicated applications. Within large enterprise applications they may, for example, play an important role in defining business rules or risk management parameters, while remaining virtually invisible to end users. Within database management systems and spreadsheets, however, these kinds of functions also represent discrete sets of tools; they can be accessed directly and utilized on a stand-alone basis, or in more complex, user-defined configurations. In this context, functions can be used for business intelligence and ad hoc analysis of data in fields such as finance, statistics, engineering, math, etc. === Function types === XLeratorDB uses three kinds of functions to perform analytic operations: scalar, aggregate, and a hybrid form which WestClinTech calls Range Queries. Scalar functions take a single value, perform an operation and return a single value. An example of this type of function is LOG, which returns the logarithm of a number to a specified base. Aggregate functions operate on a series of values but return a single, summarizing value. An example of this type of function is AVG, which returns the average of values in a specified group. In XLeratorDB there are some functions which have characteristics of aggregate functions (operating on multiple series of values) but cannot be processed in SQL CLR using single column inputs, such as AVG does. For example, irregular internal rate of return (XIRR), a financial function, operates on a collection of cash flow values from one column, but must also apply variable period lengths from another column and an initial iterative assumption from a third, in order to return a single, summarizing value. WestClinTech documentation notes that Range Queries specify the data to be included in the result set of the function independently of the WHERE clause associated with the T-SQL statement, by incorporating a SELECT statement into the function as a string argument; the function then traps that SELECT statement, executes it internally and processes the result. Some XLeratorDB functions that employ Range Queries are: NPV, XNPV, IRR, XIRR, MIRR, MULTINOMIAL, and SERIESSUM. Within the application these functions are identified by a "_q" naming convention: e.g. NPV_q, IRR_q, etc. == Analytic functions == === SQL Server functions === Microsoft SQL Server is the #3 selling database management system (DBMS), behind Oracle and IBM. (While versions of SQL Server have been on the market since 1987, XLeratorDB is compatible with only the 2005 edition and later.) Like all major DBMS, SQL Server performs a variety of data mining operations by returning or arraying data in different views (also known as drill-down). In addition, SQL Server uses Transact-SQL (T-SQL) to execute four major classes of pre-defined functions in native mode. Functions operating on the DBMS offer several advantages over client layer applications like Excel: they utilize the most up-to-date data available; they can process far larger quantities of data; and, the data is not subject to exporting and transcription errors. SQL Server 2008 includes a total of 58 functions that perform relatively basic aggregation (12), math (23) and string manipulation (23) operations useful for analytics; it includes no native functions that perform more complex operations directly related to finance, statistics or engineering. === Excel functions === Microsoft Excel, a component of Microsoft Office suite, is one of the most widely used spreadsheet applications on the market today. In addition to its inherent utility as a stand-alone desktop application, Excel overlaps and complements the functionality of DBMS in several ways: storing and arraying data in rows and columns; performing certain basic tasks such as pivot table and aggregating values; and facilitating sharing, importing and exporting of database data. Excel's chief limitation relative to a true database is capacity; Excel 2003 is limited to some 65k rows and 256 columns; Excel 2007 extends this capacity to roughly 1million rows and 16k columns. By comparison, SQL Server is able to manage over 500k terabytes of memory. Excel offers, however, an extensive library of specialized pre-written functions which are useful for performing ad hoc analysis on database data. Excel 2007 includes over 300 of these pre-defined functions, although customized functions can also be created by users, or imported from third party developers as add-ons. Excel functions are grouped by type: === Excel business intelligence functions === Operating on the client computing layer Excel plays an important role as a business intelligence tool because it: performs a wide array of complex analytic functions not native to most DBMS software offers far greater ad hoc reporting and analytic flexibility than most enterprise software provides a medium for sharing and collaborating because of its ubiquity throughout the enterprise Microsoft reinforces this positioning with Business Intelligence documentation that positions Excel in a clearly pivotal role. === XLeratorDB vs. Excel functions === While operating within the database environment, XLeratorDB functions utilize the same naming conventions and input formats, and in most cases, return the same calculation results as Excel functions. XLeratorDB, coupled with SQL Server's native capabilities, compares to Excel's function sets as follows:
Secure environment
In computing, a secure environment is any system which implements the controlled storage and use of information. In the event of computing data loss, a secure environment is used to protect personal or confidential data. It may also be known as a trusted execution environment (TEE). Often, secure environments employ cryptography as a means to protect information. This is typically used for processing confidential or restricted information. Some secure environments employ cryptographic hashing, simply to verify that the information has not been altered since it was last modified.
MovieRide FX
MovieRide FX is a patented automated special visual effects video compositing engine used in the MovieRide FX mobile application for Android (requires Android 2.3 or later) and iOS (compatible with iPhone 4 and up, iPad, and iPod Touch (new generation), requires iOS 7 or later). MovieRide FX allows the user to personalize a "Hollywood-style" movie clip by inserting themself into the clip as the "actor". == Features == The MovieRide FX app uses the relevant mobile device's camera to record a video of the user and insert it into a pre-packaged "Hollywood style" movie clip. The "actor" is extracted from their recorded video clip through various known effects such as masking, keying, and motion tracking. The "actor" is then inserted into one of the pre-packaged movie clips created by the MovieRide FX visual effects artists. This is done through an automated process requiring little or no artistic or technical skill from the user. The custom movie clips pre-packaged with MovieRide FX offer the user a variety of movie scenarios. Additional clips based on popular television and movie themes are continually being developed and are available on a freemium basis. == Sharing == Once the user's footage has automatically been composited into a movie clip and rendered as an .mp4 file, it can be shared via social media, such as Facebook, YouTube, and Twitter, and by e-mail. == History == === 2012 === MovieRide FX was created by Grant Waterston and Johann Mynhardt, who started development in 2012. === 2013 === The beta version was released on Google Play in July 2013. In August 2013 MovieRide FX was a New Media Award winner in the "New Media" category of the Accolade International Awards in Los Angeles. In October 2013 MovieRide FX was awarded exhibitor space in the ‘start-up village’ at the Apps-World Expo in London. === 2014 === MovieRide FX reached the 100 000 – 500 000 downloads category on the Google Play Store in June 2014. The official Android version was launched in July 2014. iOS version released in August 2014. MovieRide FX was selected as one of the "Top 150" startups at the Pioneer Festival in Vienna in September 2014. In November 2014 MovieRide FX was shortlisted for the Appster Awards in the "Best Entertainment App" and "Most Innovative App" categories and was awarded exhibitor space at the ‘start-up village’ at the Apps-World Expo in London. Patent applications were filed in South Africa, the EU and USA in April 2014. === 2015 === In September 2015 MovieRide FX was shortlisted for "Best Software innovation" at The Technology Expo Awards in London. === 2016 === In April 2016 MovieRide FX was nominated for a National Science and Technology Forum (NSTF) award for 'Research leading to Innovation by a corporate organization' In August 2016 Movie Ride FX won two Gold Awards at the 2016 Mobile Marketing Awards (MMA Smarties SA). These two Gold awards were for the 'Innovation' and 'Best in Show’ categories. In December 2016 FlicJam Inc. was formed in the US to access the larger global market. EU patent application was published in March 2016. === 2017 === South African patent was granted in February 2017. === 2018 === US patent was granted in March 2018.
Supertoroid
In geometry and computer graphics, a supertoroid or supertorus is usually understood to be a family of doughnut-like surfaces (technically, a topological torus) whose shape is defined by mathematical formulas similar to those that define the superellipsoids. The plural of "supertorus" is either supertori or supertoruses. The family was described and named by Alan Barr in 1994. Barr's supertoroids have been fairly popular in computer graphics as a convenient model for many objects, such as smooth frames for rectangular things. One quarter of a supertoroid can provide a smooth and seamless 90-degree joint between two superquadric cylinders. However, they are not algebraic surfaces (except in special cases). == Formulas == Alan Barr's supertoroids are defined by parametric equations similar to the trigonometric equations of the torus, except that the sine and cosine terms are raised to arbitrary powers. Namely, the generic point P(u, v) of the surface is given by P ( u , v ) = ( X ( u , v ) Y ( u , v ) Z ( u , v ) ) = ( ( a + C u s ) C v t ( b + C u s ) S v t S u s ) {\displaystyle P(u,v)=\left({\begin{array}{c}X(u,v)\\Y(u,v)\\Z(u,v)\end{array}}\right)=\left({\begin{array}{c}(a+C_{u}^{s})C_{v}^{t}\\(b+C_{u}^{s})S_{v}^{t}\\S_{u}^{s}\end{array}}\right)} where C θ ε = sgn ( cos θ ) | cos θ | ε , S θ ε = sgn ( sin θ ) | sin θ | ε , {\displaystyle {\begin{aligned}C_{\theta }^{\varepsilon }&=\operatorname {sgn} (\cos \theta )\,\left|\,\cos \theta \,\right|^{\varepsilon },\\S_{\theta }^{\varepsilon }&=\operatorname {sgn} (\sin \theta )\ \left|\,\sin \theta \ \right|^{\varepsilon },\end{aligned}}} sgn is the sign function, and the parameters u, v range from 0 to 360 degrees (0 to 2π radians). In these formulas, the parameter s > 0 controls the "squareness" of the vertical sections, t > 0 controls the squareness of the horizontal sections, and a, b ≥ 1 are the major radii in the x and y directions. With s = t = 1 and a = b = R one obtains the ordinary torus with major radius R and minor radius 1, with the center at the origin and rotational symmetry about the z-axis. In general, the supertorus defined as above spans the intervals: − ( a + 1 ) ≤ x ≤ + ( a + 1 ) − ( b + 1 ) ≤ y ≤ + ( b + 1 ) − 1 ≤ z ≤ + 1 {\displaystyle {\begin{array}{rcccl}-(a+1)&\leq &x&\leq &+(a+1)\\[4pt]-(b+1)&\leq &y&\leq &+(b+1)\\[4pt]-1&\leq &z&\leq &+1\end{array}}} The whole shape is symmetric about the planes x = 0, y = 0, and z = 0. The hole runs in the z direction and spans the intervals − ( a − 1 ) ≤ x ≤ + ( a − 1 ) − ( b − 1 ) ≤ y ≤ + ( b − 1 ) − ∞ ≤ z ≤ + ∞ {\displaystyle {\begin{array}{rcccl}-(a-1)&\leq &x&\leq &+(a-1)\\[4pt]-(b-1)&\leq &y&\leq &+(b-1)\\[4pt]-\infty &\leq &z&\leq &+\infty \end{array}}} A curve of constant u on this surface is a horizontal Lamé curve with exponent 2 t , {\displaystyle {\tfrac {2}{t}},} scaled in x and y and displaced in z. A curve of constant v, projected on the plane x = 0 or y = 0, is a Lamé curve with exponent 2 s , {\displaystyle {\tfrac {2}{s}},} scaled and horizontally shifted. If v = 0, the curve is planar and spans the intervals: a − 1 ≤ x ≤ a + 1 − 1 ≤ z ≤ + 1 {\displaystyle {\begin{array}{rcccl}a-1&\leq &x&\leq &a+1\\[4pt]-1&\leq &z&\leq &+1\end{array}}} and similarly if v = 90°, 180°, 270°. The curve is also planar if a = b. In general, if a ≠ b and v is not a multiple of 90 degrees, the curve of constant v will not be planar; and, conversely, a vertical plane section of the supertorus will not be a Lamé curve. The basic supertoroid shape defined above is often modified by non-uniform scaling to yield supertoroids of specific width, length, and vertical thickness. == Plotting code == The following GNU Octave code generates plots of a supertorus:
Glow (app)
Glow is a fertility awareness and period-tracking app. It is part of a suite of mobile apps focused on women's reproductive health and childcare, which includes Eve by Glow (a dedicated period tracker), Glow Nurture (a pregnancy tracker), and Glow Baby (a baby development tracker). The Glow company also operates an online shop that sells several fertility-related products, including ovulation test strips, pregnancy tests, and wearable breast pumps. In 2024, Glow was reported to have approximately 25 million users across its various apps and community message boards. == History == Glow debuted in August 2013 as an iOS app. It was founded by Michael Huang and Max Levchin and launched with $6 million in Series A funding from venture capital firms Founders Fund and Andreesen Horowitz. In 2014, Glow raised an additional $17 million in Series B funding, with Formation 8 joining existing investors. In 2015, Glow launched Ruby, an app dedicated to sexual health. That year, Wired reported that the company had added features to their apps allowing men to monitor their fertility. Glow subsequently released an additional set of apps focused on pregnancy tracking and infant development. In 2016, Glow reported that it had a total of approximately 3 million users; by 2018, this had grown to 15 million. Vox described it as one of the “big two” period and fertility tracking apps and the one that had started the “boom” in the femtech space. == Application and features == Glow was initially described as a fertility application that applied data-driven methods to menstrual and ovulation tracking. Core features include cycle logging, ovulation prediction, and symptom tracking. The app also provides educational content related to reproductive health and childcare, as well as a set of online message boards that allow individuals to share experiences and seek peer support. == Privacy and legal issues == Glow has received significant media attention for its privacy and security practices. In 2016, Consumer Reports identified potential exploits in the Glow app that they claimed could have exposed private user data to hackers. Glow subsequently reported that it had fixed the vulnerabilities and told The Washington Post they had no evidence that user data had been compromised. In September 2020, the California Attorney General announced a settlement with Glow related to Consumer Reports’ findings, which included a $250,000 civil penalty. Following the US Supreme Court's 2022 Dobbs v. Jackson ruling, which legalized state-level bans on abortion, Glow (and other fertility trackers, such as Clue and Flo) came under additional scrutiny over concerns that user data on abortions could be reported to law enforcement. After this surge of media interest, a research team affiliated with the University of New South Wales conducted an investigation into the privacy practices of several popular fertility apps, including Glow. Their review of Glow was mixed, noting that they provided several privacy settings and de-identified sensitive data, but that user information could still be disclosed in the future if the app was sold. Glow rejected that claim, telling the Australian Associated Press that it "did not share" personal data. The company also cited several internal security measures it had implemented and its apps' offline data protection setting, which allows users to permanently delete their health-related data. == Reception == In 2014, Fast Company reported that 20,000 women had used Glow to conceive. Later that year, The Guardian included Glow Nurture on its list of the best iPhone apps of 2014. Media coverage often praised Glow's array of menstrual tracking options, although some reviews also noted that fertility apps are not birth control tools and cautioned against relying on them for that purpose. In 2019, Cosmopolitan singled Glow's community of users as one of its standout features.
Generative art
Generative art is post-conceptual art that has been created (in whole or in part) with the use of an autonomous system. An autonomous system in this context is generally one that is non-human and can independently determine features of an artwork that would otherwise require decisions made directly by the artist. In some cases the human creator may claim that the generative system represents their own artistic idea, and in others that the system takes on the role of the creator. "Generative art" often refers to algorithmic art (algorithmically determined computer generated artwork) and synthetic media (general term for any algorithmically generated media), but artists can also make generative art using systems of chemistry, biology, mechanics and robotics, smart materials, manual randomization, mathematics, data mapping, symmetry, and tiling. Generative algorithms, algorithms programmed to produce artistic works through predefined rules, stochastic methods, or procedural logic, often yielding dynamic, unique, and contextually adaptable outputs—are central to many of these practices. == History == The use of the word "generative" in the discussion of art has developed over time. The use of "Artificial DNA" defines a generative approach to art focused on the construction of a system able to generate unpredictable events, all with a recognizable common character. The use of autonomous systems, required by some contemporary definitions, focuses a generative approach where the controls are strongly reduced. This approach is also named "emergent". Margaret Boden and Ernest Edmonds have noted the use of the term "generative art" in the broad context of automated computer graphics in the 1960s, beginning with artwork exhibited by Georg Nees and Frieder Nake in 1965: A. Michael Noll did his initial computer art, combining randomness with order, in 1962, and exhibited it along with works by Bell Julesz in 1965. The terms "generative art" and "computer art" have been used in tandem, and more or less interchangeably, since the very earliest days. The first such exhibition showed the work of Nees in February 1965, which some claim was titled "Generative Computergrafik". While Nees does not himself remember, this was the title of his doctoral thesis published a few years later. The correct title of the first exhibition and catalog was "computer-grafik". "Generative art" and related terms was in common use by several other early computer artists around this time, including Manfred Mohr and Ken Knowlton. Vera Molnár (born 1924) is a French media artist of Hungarian origin. Molnar is widely considered to be a pioneer of generative art, and is also one of the first women to use computers in her art practice. The term "Generative Art" with the meaning of dynamic artwork-systems able to generate multiple artwork-events was clearly used the first time for the "Generative Art" conference in Milan in 1998. The term has also been used to describe geometric abstract art where simple elements are repeated, transformed, or varied to generate more complex forms. Thus defined, generative art was practiced by the Argentinian artists Eduardo Mac Entyre and Miguel Ángel Vidal in the late 1960s. In 1972 the Romanian-born Paul Neagu created the Generative Art Group in Britain. It was populated exclusively by Neagu using aliases such as "Hunsy Belmood" and "Edward Larsocchi". In 1972 Neagu gave a lecture titled 'Generative Art Forms' at the Queen's University, Belfast Festival. In 1970 the School of the Art Institute of Chicago created a department called Generative Systems. As described by Sonia Landy Sheridan the focus was on art practices using the then new technologies for the capture, inter-machine transfer, printing and transmission of images, as well as the exploration of the aspect of time in the transformation of image information. Also noteworthy is John Dunn, first a student and then a collaborator of Sheridan. In 1988 Clauser identified the aspect of systemic autonomy as a critical element in generative art: It should be evident from the above description of the evolution of generative art that process (or structuring) and change (or transformation) are among its most definitive features, and that these features and the very term 'generative' imply dynamic development and motion. (the result) is not a creation by the artist but rather the product of the generative process - a self-precipitating structure. In 1989 Celestino Soddu defined the Generative Design approach to Architecture and Town Design in his book Citta' Aleatorie. In 1989 Franke referred to "generative mathematics" as "the study of mathematical operations suitable for generating artistic images." From the mid-1990s Brian Eno popularized the terms generative music and generative systems, making a connection with earlier experimental music by Terry Riley, Steve Reich and Philip Glass. From the end of the 20th century, communities of generative artists, designers, musicians and theoreticians began to meet, forming cross-disciplinary perspectives. The first meeting about generative Art was in 1998, at the inaugural International Generative Art conference at Politecnico di Milano University, Italy. In Australia, the Iterate conference on generative systems in the electronic arts followed in 1999. On-line discussion has centered around the eu-gene mailing list, which began late 1999, and has hosted much of the debate which has defined the field. These activities have more recently been joined by the Generator.x conference in Berlin starting in 2005. In 2012 the new journal GASATHJ, Generative Art Science and Technology Hard Journal was founded by Celestino Soddu and Enrica Colabella jointing several generative artists and scientists in the editorial board. Some have argued that as a result of this engagement across disciplinary boundaries, the community has converged on a shared meaning of the term. As Boden and Edmonds put it in 2011: Today, the term "Generative Art" is still current within the relevant artistic community. Since 1998 a series of conferences have been held in Milan with that title (Generativeart.com), and Brian Eno has been influential in promoting and using generative art methods (Eno, 1996). Both in music and in visual art, the use of the term has now converged on work that has been produced by the activation of a set of rules and where the artist lets a computer system take over at least some of the decision-making (although, of course, the artist determines the rules). In the call of the Generative Art conferences in Milan (annually starting from 1998), the definition of Generative Art by Celestino Soddu: Generative Art is the idea realized as genetic code of artificial events, as construction of dynamic complex systems able to generate endless variations. Each Generative Project is a concept-software that works producing unique and non-repeatable events, like music or 3D Objects, as possible and manifold expressions of the generating idea strongly recognizable as a vision belonging to an artist / designer / musician / architect /mathematician. Discussion on the eu-gene mailing list was framed by the following definition by Adrian Ward from 1999: Generative art is a term given to work which stems from concentrating on the processes involved in producing an artwork, usually (although not strictly) automated by the use of a machine or computer, or by using mathematic or pragmatic instructions to define the rules by which such artworks are executed. A similar definition is provided by Philip Galanter: Generative art refers to any art practice where the artist creates a process, such as a set of natural language rules, a computer program, a machine, or other procedural invention, which is then set into motion with some degree of autonomy contributing to or resulting in a completed work of art. Around the 2020s, generative AI models learned to imitate the distinct style of particular authors. For example, a generative image model such as Stable Diffusion is able to model the stylistic characteristics of an artist like Pablo Picasso (including his particular brush strokes, use of colour, perspective, and so on), and a user can engineer a prompt such as "an astronaut riding a horse, by Picasso" to cause the model to generate a novel image applying the artist's style to an arbitrary subject. Generative image models have received significant backlash from artists who object to their style being imitated without their permission, arguing that this harms their ability to profit from their own work. The emergence of text-to-image generative AI systems has expanded debates over authorship, copyright, and artistic labor. The main issues in these debates include the eligibility of AI-generated outputs for copyright protection and the legal and ethical questions of using existing copyrighted works as training data for generative AI systems. == Types == === Music === Johann Kirnberger's Mu
Image texture
An image texture is the small-scale structure perceived on an image, based on the spatial arrangement of color or intensities. It can be quantified by a set of metrics calculated in image processing. Image texture metrics give us information about the whole image or selected regions. Image textures can be artificially created or found in natural scenes captured in an image. Image textures are one way that can be used to help in segmentation or classification of images. For more accurate segmentation the most useful features are spatial frequency and an average grey level. To analyze an image texture in computer graphics, there are two ways to approach the issue: structured approach and statistical approach. == Structured approach == A structured approach sees an image texture as a set of primitive texels in some regular or repeated pattern. This works well when analyzing artificial textures. To obtain a structured description a characterization of the spatial relationship of the texels is gathered by using Voronoi tessellation of the texels. == Statistical approach == A statistical approach sees an image texture as a quantitative measure of the arrangement of intensities in a region. In general this approach is easier to compute and is more widely used, since natural textures are made of patterns of irregular subelements. === Edge detection === The use of edge detection is to determine the number of edge pixels in a specified region, helps determine a characteristic of texture complexity. After edges have been found the direction of the edges can also be applied as a characteristic of texture and can be useful in determining patterns in the texture. These directions can be represented as an average or in a histogram. Consider a region with N pixels. the gradient-based edge detector is applied to this region by producing two outputs for each pixel p: the gradient magnitude Mag(p) and the gradient direction Dir(p). The edgeness per unit area can be defined by F e d g e n e s s = | { p | M a g ( p ) > T } | N {\displaystyle F_{edgeness}={\frac {|\{p|Mag(p)>T\}|}{N}}} for some threshold T. To include orientation with edgeness histograms for both gradient magnitude and gradient direction can be used. Hmag(R) denotes the normalized histogram of gradient magnitudes of region R, and Hdir(R) denotes the normalized histogram of gradient orientations of region R. Both are normalized according to the size NR Then F m a g , d i r = ( H m a g ( R ) , H d i r ( R ) ) {\displaystyle F_{mag,dir}=(H_{mag}(R),H_{dir}(R))} is a quantitative texture description of region R. === Co-occurrence matrices === The co-occurrence matrix captures numerical features of a texture using spatial relations of similar gray tones. Numerical features computed from the co-occurrence matrix can be used to represent, compare, and classify textures. The following are a subset of standard features derivable from a normalized co-occurrence matrix: A n g u l a r 2 n d M o m e n t = ∑ i ∑ j p [ i , j ] 2 C o n t r a s t = ∑ i = 1 N g ∑ j = 1 N g n 2 p [ i , j ] , where | i − j | = n C o r r e l a t i o n = ∑ i = 1 N g ∑ j = 1 N g ( i j ) p [ i , j ] − μ x μ y σ x σ y E n t r o p y = − ∑ i ∑ j p [ i , j ] l n ( p [ i , j ] ) {\displaystyle {\begin{aligned}Angular{\text{ }}2nd{\text{ }}Moment&=\sum _{i}\sum _{j}p[i,j]^{2}\\Contrast&=\sum _{i=1}^{Ng}\sum _{j=1}^{Ng}n^{2}p[i,j]{\text{, where }}|i-j|=n\\Correlation&={\frac {\sum _{i=1}^{Ng}\sum _{j=1}^{Ng}(ij)p[i,j]-\mu _{x}\mu _{y}}{\sigma _{x}\sigma _{y}}}\\Entropy&=-\sum _{i}\sum _{j}p[i,j]ln(p[i,j])\\\end{aligned}}} where p [ i , j ] {\displaystyle p[i,j]} is the [ i , j ] {\displaystyle [i,j]} th entry in a gray-tone spatial dependence matrix, and Ng is the number of distinct gray-levels in the quantized image. One negative aspect of the co-occurrence matrix is that the extracted features do not necessarily correspond to visual perception. It is used in dentistry for the objective evaluation of lesions [DOI: 10.1155/2020/8831161], treatment efficacy [DOI: 10.3390/ma13163614; DOI: 10.11607/jomi.5686; DOI: 10.3390/ma13173854; DOI: 10.3390/ma13132935] and bone reconstruction during healing [DOI: 10.5114/aoms.2013.33557; DOI: 10.1259/dmfr/22185098; EID: 2-s2.0-81455161223; DOI: 10.3390/ma13163649]. === Laws texture energy measures === Another approach is to use local masks to detect various types of texture features. Laws originally used four vectors representing texture features to create sixteen 2D masks from the outer products of the pairs of vectors. The four vectors and relevant features were as follows: L5 = [ +1 +4 6 +4 +1 ] (Level) E5 = [ -1 -2 0 +2 +1 ] (Edge) S5 = [ -1 0 2 0 -1 ] (Spot) R5 = [ +1 -4 6 -4 +1 ] (Ripple) To these 4, a fifth is sometimes added: W5 = [ -1 +2 0 -2 +1 ] (Wave) From Laws' 4 vectors, 16 5x5 "energy maps" are then filtered down to 9 in order to remove certain symmetric pairs. For instance, L5E5 measures vertical edge content and E5L5 measures horizontal edge content. The average of these two measures is the "edginess" of the content. The resulting 9 maps used by Laws are as follows: L5E5/E5L5 L5R5/R5L5 E5S5/S5E5 S5S5 R5R5 L5S5/S5L5 E5E5 E5R5/R5E5 S5R5/R5S5 Running each of these nine maps over an image to create a new image of the value of the origin ([2,2]) results in 9 "energy maps," or conceptually an image with each pixel associated with a vector of 9 texture attributes. === Autocorrelation and power spectrum === The autocorrelation function of an image can be used to detect repetitive patterns of textures. == Texture segmentation == The use of image texture can be used as a description for regions into segments. There are two main types of segmentation based on image texture, region based and boundary based. Though image texture is not a perfect measure for segmentation it is used along with other measures, such as color, that helps solve segmenting in image. === Region based === Attempts to group or cluster pixels based on texture properties. === Boundary based === Attempts to group or cluster pixels based on edges between pixels that come from different texture properties.