Plant Nanny is a water tracker mobile application which reminds users to drink water. It was developed by Taiwanese app maker Fourdesire. The app was first released in 2013 and is available on the Apple App Store for iPhones and the Google Play Store for Android devices. == Description == Play Nanny uses a game method that allows users to turn their virtual selves into plants, which grows and thrives as the user drinks more water. The app sends occasional push notifications to remind users to drink water throughout the day. Users can choose from a wide range of plants, including cacti and carnations, and track their water intake. The app uses two resources, How to calculate how much water you should drink by Jennifer Stone (2018) and Human energy requirements by the Food and Agriculture Organization (2004), to calculate the recommended daily water intake for its users. Upon downloading the app, users are prompted to input basic personal information which is then used to calculate the recommended daily water intake and prompts them to drink the appropriate amount. == Accolades ==
Kernel density estimation
In statistics, kernel density estimation (KDE) is the application of kernel smoothing for probability density estimation, i.e., a non-parametric method to estimate the probability density function of a random variable based on kernels as weights. KDE answers a fundamental data smoothing problem where inferences about the population are made based on a finite data sample. In some fields such as signal processing and econometrics it is also termed the Parzen–Rosenblatt window method, after Emanuel Parzen and Murray Rosenblatt, who are usually credited with independently creating it in its current form. One of the famous applications of kernel density estimation is in estimating the class-conditional marginal densities of data when using a naive Bayes classifier, which can improve its prediction accuracy. == Definition == Let x = ( x 1 , x 2 , x 3 , . . . ) {\displaystyle \mathbf {x} =\left(x_{1},x_{2},x_{3},...\right)} be independent and identically distributed samples drawn from some univariate distribution with an unknown density f at any given point x. We are interested in estimating the shape of this function f. Its kernel density estimator is f ^ h ( x ) = 1 n ∑ i = 1 n K h ( x − x i ) = 1 n h ∑ i = 1 n K ( x − x i h ) , {\displaystyle {\hat {f}}_{h}(x)={\frac {1}{n}}\sum _{i=1}^{n}K_{h}(x-x_{i})={\frac {1}{nh}}\sum _{i=1}^{n}K{\left({\frac {x-x_{i}}{h}}\right)},} where K is the kernel — a non-negative function — and h > 0 is a smoothing parameter called the bandwidth or simply width. A kernel with subscript h is called the scaled kernel and defined as Kh(x) = 1/h K(x/h). Intuitively one wants to choose h as small as the data will allow; however, there is always a trade-off between the bias of the estimator and its variance. The choice of bandwidth is discussed in more detail below. A range of kernel functions are commonly used: uniform, triangular, biweight, triweight, Epanechnikov (parabolic), normal, and others. The Epanechnikov kernel is optimal in a mean square error sense, though the loss of efficiency is small for the kernels listed previously. Due to its convenient mathematical properties, the normal kernel is often used, which means K(x) = ϕ(x), where ϕ is the standard normal density function. The kernel density estimator then becomes f ^ h ( x ) = 1 n ∑ i = 1 n 1 h 2 π exp ( − ( x − x i ) 2 2 h 2 ) , {\displaystyle {\hat {f}}_{h}(x)={\frac {1}{n}}\sum _{i=1}^{n}{\frac {1}{h{\sqrt {2\pi }}}}\exp \left({\frac {-(x-x_{i})^{2}}{2h^{2}}}\right),} where h {\displaystyle h} is the standard deviation of the sample x {\displaystyle \mathbf {x} } . The construction of a kernel density estimate finds interpretations in fields outside of density estimation. For example, in thermodynamics, this is equivalent to the amount of heat generated when heat kernels (the fundamental solution to the heat equation) are placed at each data point locations xi. Similar methods are used to construct discrete Laplace operators on point clouds for manifold learning (e.g. diffusion map). == Example == Kernel density estimates are closely related to histograms, but can be endowed with properties such as smoothness or continuity by using a suitable kernel. The diagram below based on these 6 data points illustrates this relationship: For the histogram, first, the horizontal axis is divided into sub-intervals or bins which cover the range of the data: In this case, six bins each of width 2. Whenever a data point falls inside this interval, a box of height 1/12 is placed there. If more than one data point falls inside the same bin, the boxes are stacked on top of each other. For the kernel density estimate, normal kernels with a standard deviation of 1.5 (indicated by the red dashed lines) are placed on each of the data points xi. The kernels are summed to make the kernel density estimate (solid blue curve). The smoothness of the kernel density estimate (compared to the discreteness of the histogram) illustrates how kernel density estimates converge faster to the true underlying density for continuous random variables. == Bandwidth selection == The bandwidth of the kernel is a free parameter which exhibits a strong influence on the resulting estimate. To illustrate its effect, we take a simulated random sample from the standard normal distribution (plotted at the blue spikes in the rug plot on the horizontal axis). The grey curve is the true density (a normal density with mean 0 and variance 1). In comparison, the red curve is undersmoothed since it contains too many spurious data artifacts arising from using a bandwidth h = 0.05, which is too small. The green curve is oversmoothed since using the bandwidth h = 2 obscures much of the underlying structure. The black curve with a bandwidth of h = 0.337 is considered to be optimally smoothed since its density estimate is close to the true density. An extreme situation is encountered in the limit h → 0 {\displaystyle h\to 0} (no smoothing), where the estimate is a sum of n delta functions centered at the coordinates of analyzed samples. In the other extreme limit h → ∞ {\displaystyle h\to \infty } the estimate retains the shape of the used kernel, centered on the mean of the samples (completely smooth). The most common optimality criterion used to select this parameter is the expected L2 risk function, also termed the mean integrated squared error: MISE ( h ) = E [ ∫ ( f ^ h ( x ) − f ( x ) ) 2 d x ] {\displaystyle \operatorname {MISE} (h)=\operatorname {E} \!\left[\int \!{\left({\hat {f}}\!_{h}(x)-f(x)\right)}^{2}dx\right]} Under weak assumptions on f and K, (f is the, generally unknown, real density function), MISE ( h ) = AMISE ( h ) + o ( ( n h ) − 1 + h 4 ) {\displaystyle \operatorname {MISE} (h)=\operatorname {AMISE} (h)+{\mathcal {o}}{\left((nh)^{-1}+h^{4}\right)}} where o is the little o notation, and n the sample size (as above). The AMISE is the asymptotic MISE, i. e. the two leading terms, AMISE ( h ) = R ( K ) n h + 1 4 m 2 ( K ) 2 h 4 R ( f ″ ) {\displaystyle \operatorname {AMISE} (h)={\frac {R(K)}{nh}}+{\frac {1}{4}}m_{2}(K)^{2}h^{4}R(f'')} where R ( g ) = ∫ g ( x ) 2 d x {\textstyle R(g)=\int g(x)^{2}\,dx} for a function g, m 2 ( K ) = ∫ x 2 K ( x ) d x {\textstyle m_{2}(K)=\int x^{2}K(x)\,dx} and f ″ {\displaystyle f''} is the second derivative of f {\displaystyle f} and K {\displaystyle K} is the kernel. The minimum of this AMISE is the solution to this differential equation ∂ ∂ h AMISE ( h ) = − R ( K ) n h 2 + m 2 ( K ) 2 h 3 R ( f ″ ) = 0 {\displaystyle {\frac {\partial }{\partial h}}\operatorname {AMISE} (h)=-{\frac {R(K)}{nh^{2}}}+m_{2}(K)^{2}h^{3}R(f'')=0} or h AMISE = R ( K ) 1 / 5 m 2 ( K ) 2 / 5 R ( f ″ ) 1 / 5 n − 1 / 5 = C n − 1 / 5 {\displaystyle h_{\operatorname {AMISE} }={\frac {R(K)^{1/5}}{m_{2}(K)^{2/5}R(f'')^{1/5}}}n^{-1/5}=Cn^{-1/5}} Neither the AMISE nor the hAMISE formulas can be used directly since they involve the unknown density function f {\displaystyle f} or its second derivative f ″ {\displaystyle f''} . To overcome that difficulty, a variety of automatic, data-based methods have been developed to select the bandwidth. Several review studies have been undertaken to compare their efficacies, with the general consensus that the plug-in selectors and cross validation selectors are the most useful over a wide range of data sets. Substituting any bandwidth h which has the same asymptotic order n−1/5 as hAMISE into the AMISE gives that AMISE(h) = O(n−4/5), where O is the big O notation. It can be shown that, under weak assumptions, there cannot exist a non-parametric estimator that converges at a faster rate than the kernel estimator. Note that the n−4/5 rate is slower than the typical n−1 convergence rate of parametric methods. If the bandwidth is not held fixed, but is varied depending upon the location of either the estimate (balloon estimator) or the samples (pointwise estimator), this produces a particularly powerful method termed adaptive or variable bandwidth kernel density estimation. Bandwidth selection for kernel density estimation of heavy-tailed distributions is relatively difficult. === A rule-of-thumb bandwidth estimator === If Gaussian basis functions are used to approximate univariate data, and the underlying density being estimated is Gaussian, the optimal choice for h (that is, the bandwidth that minimises the mean integrated squared error) is: h = ( 4 σ ^ 5 3 n ) 1 / 5 ≈ 1.06 σ ^ n − 1 / 5 , {\displaystyle h={\left({\frac {4{\hat {\sigma }}^{5}}{3n}}\right)}^{1/5}\approx 1.06\,{\hat {\sigma }}\,n^{-1/5},} An h {\displaystyle h} value is considered more robust when it improves the fit for long-tailed and skewed distributions or for bimodal mixture distributions. This is often done empirically by replacing the standard deviation σ ^ {\displaystyle {\hat {\sigma }}} by the parameter A {\displaystyle A} below: A = min ( σ ^ , I Q R 1.34 ) {\displaystyle A=\min \left({\hat {\sigma }},{\frac {\mathrm {IQR} }{1.34}}\right)} where IQR is the
Language Server Protocol
The Language Server Protocol (LSP) is an open, JSON-RPC-based protocol for use between source-code editors or integrated development environments (IDEs) and servers that provide "language intelligence tools": programming language-specific features like code completion, syntax highlighting and marking of warnings and errors, as well as refactoring routines. The goal of the protocol is to allow programming language support to be implemented and distributed independently of any given editor or IDE. In the early 2020s, LSP quickly became a "norm" for language intelligence tools providers. == History == LSP was originally developed for Microsoft Visual Studio Code and is now an open standard. On June 27, 2016, Microsoft announced a collaboration with Red Hat and Codenvy to standardize the protocol's specification. Its specification is hosted and developed on GitHub. == Background == Modern IDEs provide programmers with sophisticated features like code completion, refactoring, navigating to a symbol's definition, syntax highlighting, and error and warning markers. For example, in a text-based programming language, a programmer might want to rename a method read. The programmer could either manually edit the respective source code files and change the appropriate occurrences of the old method name into the new name, or instead use an IDE's refactoring capabilities to make all the necessary changes automatically. To be able to support this style of refactoring, an IDE needs a sophisticated understanding of the programming language that the program's source is written in. A programming tool without such an understanding—for example, one that performs a naive search-and-replace instead—could introduce errors. When renaming a read method, for example, the tool should not replace the partial match in a variable that might be called readyState, nor should it replace the portion of a code comment containing the word "already". Neither should renaming a local variable read, for example, end up altering identically-named variables in other scopes. Conventional compilers or interpreters for a specific programming language are typically unable to provide these language services, because they are written with the goal of either transforming the source code into object code or immediately executing the code. Additionally, language services must be able to handle source code that is not well-formed, e.g. because the programmer is in the middle of editing and has not yet finished typing a statement, procedure, or other construct. Additionally, small changes to a source code file which are done during typing usually change the semantics of the program. In order to provide instant feedback to the user, the editing tool must be able to very quickly evaluate the syntactical and semantical consequences of a specific modification. Compilers and interpreters therefore provide a poor candidate for producing the information needed for an editing tool to consume. Prior to the design and implementation of the Language Server Protocol for the development of Visual Studio Code, most language services were generally tied to a given IDE or other editor. In the absence of the Language Server Protocol, language services are typically implemented by using a tool-specific extension API. Providing the same language service to another editing tool requires effort to adapt the existing code so that the service may target the second editor's extension interfaces. The Language Server Protocol allows for decoupling language services from the editor so that the services may be contained within a general-purpose language server. Any editor can inherit sophisticated support for many different languages by making use of existing language servers. Similarly, a programmer involved with the development of a new programming language can make services for that language available to existing editing tools. Making use of language servers via the Language Server Protocol thus also reduces the burden on vendors of editing tools, because vendors do not need to develop language services of their own for the languages the vendor intends to support, as long as the language servers have already been implemented. The Language Server Protocol also enables the distribution and development of servers contributed by an interested third party, such as end users, without additional involvement by either the vendor of the compiler for the programming language in use or the vendor of the editor to which the language support is being added. LSP is not restricted to programming languages. It can be used for any kind of text-based language, like specifications or domain-specific languages (DSL). == Technical overview == When a user edits one or more source code files using a language server protocol-enabled tool, the tool acts as a client that consumes the language services provided by a language server. The tool may be a text editor or IDE and the language services could be refactoring, code completion, etc. The client informs the server about what the user is doing, e.g., opening a file or inserting a character at a specific text position. The client can also request the server to perform a language service, e.g. to format a specified range in the text document. The server answers a client's request with an appropriate response. For example, the formatting request is answered either by a response that transfers the formatted text to the client or by an error response containing details about the error. The Language Server Protocol defines the messages to be exchanged between client and language server. They are JSON-RPC preceded by headers similar to HTTP. Messages may originate from the server or client. The protocol does not make any provisions about how requests, responses and notifications are transferred between client and server. For example, client and server could be components within the same process exchanging JSON strings via method calls. They could also be different processes on the same or on different machines communicating via network sockets. == Registry == There are lists of LSP-compatible implementations, maintained by the community-driven Langserver.org or Microsoft.
SlideRocket
SlideRocket was an online presentation platform that let users create, manage, share and measure presentations. SlideRocket was provided via a SaaS model. The company was acquired by VMware in April 2011, who sold it to ClearSlide, a similar SaaS application, in March 2013. It is no longer offering independent signups, as the platform is being integrated into ClearSlide. == History == SlideRocket was founded in Jan 2006, and launched as a private beta in March 2008 at the Under The Radar Spring event. A public beta was announced in September 2008 followed shortly by public release on October 28, 2008. SlideRocket is most commonly credited with inventing the PResuMÉ or Presentation Résumé in early 2009. On April 26, 2011, SlideRocket was acquired by VMware. On March 5, 2013, VMware sold SlideRocket to ClearSlide. SlideRocket is based in San Francisco.
Plug computer
A plug computer is a small-form-factor computer whose chassis contains the AC power plug, and thus plugs directly into the wall. Alternatively, the computer may resemble an AC adapter or a similarly small device. Plug computers are often configured for use in the home or office as compact computer. == Description == Plug computers consist of a high-performance, low-power system-on-a-chip processor, with several I/O hardware ports (USB ports, Ethernet connectors, etc.). Most versions do not have provisions for connecting a display and are best suited to running media servers, back-up services, or file sharing and remote access functions; thus acting as a bridge between in-home protocols (such as Digital Living Network Alliance (DLNA) and Server Message Block (SMB)) and cloud-based services. There are, however, plug computer offerings that have analog VGA monitor and/or HDMI connectors, which, along with multiple USB ports, permit the use of a display, keyboard, and mouse, thus making them full-fledged, low-power alternatives to desktop and laptop computers. They typically run any of a number of Linux distributions. Plug computers typically consume little power and are inexpensive. == History == A number of other devices of this type began to appear at the 2009 Consumer Electronics Show. On January 6, 2009 CTERA Networks launched a device called CloudPlug that provides online backup at local disk speeds and overlays a file sharing service. The device also transforms any external USB hard drive into a network-attached storage device. On January 7, 2009, Cloud Engines unveiled the Pogoplug network access server. On January 8, 2009, Axentra announced availability of their HipServ platform. On February 23, 2009, Marvell Technology Group announced its plans to build a mini-industry around plug computers. On August 19, 2009, CodeLathe announced availability of their TonidoPlug network access server. On November 13, 2009 QuadAxis launched its plug computing device product line and development platform, featuring the QuadPlug and QuadPC and running QuadMix, a modified Linux. On January 5, 2010, Iomega announced their iConnect network access server. On January 7, 2010 Pbxnsip launched its plug computing device the sipJack running pbxnsip: an IP Communications platform.
BeReal
BeReal (stylized on the app logo as BeReal.) is a French social-networking app released in 2020, developed by Alexis Barreyat and Kévin Perreau. Currently, it is owned by Voodoo. Its main feature is a daily notification that encourages users to share photos of themselves in their day-to-day life, on any randomly selected two-minute window every day. Critics noted its emphasis on authenticity, which some felt crossed the line into the mundane. The primary reference of its name relates to its focus on users uploading unpolished photos, with it being a pun of the term B-reel. According to the app's description on Apple's App Store, BeReal encourages its users to "show their friends who they really are, for once," by removing filters and opportunities to stage or edit photos. After a couple of years of relative obscurity, it rapidly gained popularity in early and mid-2022 growing from 21.6 million to 73.5 million users between July and August, before experiencing a decrease in use in 2023 and continuing to decline to 23 million users at the beginning of 2024. == History == The app was developed by Alexis Barreyat, a former employee at GoPro, and Kévin Perreau, a graduate from 42 in Paris. Initially released in 2020, it first gained widespread popularity in early 2022. It first spread widely on college campuses, partially due to a paid ambassador program. In late August 2022, the application had over 10 million active daily users and 21.6 million active monthly users. As of February 2023, the app has grown to 13 million active daily users and 47.8 million active monthly users. In June 2021, BeReal received a $30 million funding round led by Andreessen Horowitz and Accel. In May 2022, BeReal secured $85 million in a funding round led by Yuri Milner's DST Global, increasing its valuation to about $600 million. On July 25, 2022, BeReal topped Apple's free app list in the iOS App Store, and remained until September 2022. BeReal also received Apple's iPhone App of the Year in 2022. By late spring 2023, the app's momentum was waning, as daily users dropped to about 6 million, from 15 million in October 2022. In August 2024, there was a resurgence after a campaign at the Paris Olympics 2024, with the app reportedly gaining 1000 users. In June 2024, BeReal was acquired by the French company Voodoo for a reported €500 million. Alexis Barreyat is set to step down after a transition period. == Features == Once per day, BeReal notifies all users that a two-minute window to post is open. It asks users to create a post (known eponymously as a "BeReal") which, using mandatory simultaneous photos and now short videos from both the front and back cameras, provides a visual depiction of what they are doing at that moment, with an option to caption their post. The given window varies from day to day, and is not known to users before the notification is received. Once the daily notification is sent, users lose the ability to see others' BeReals from the previous day. Furthermore, users cannot see any of the current day's BeReals until they upload their own. On-time BeReals show the time it was uploaded, meanwhile, late BeReals uploaded after the two-minute window shows how late the BeReal was taken, but the user has to long-press the BeReal to reveal the time it was uploaded. Other users can also see how many attempts the poster took to take the BeReal, as well as their location when the BeReal was taken. Users only get one chance to delete their BeReal and post another one, and they used to not be able to post more than one at any time. However, in 2023, a feature was added that allowed users to post up to two extra BeReals on days when they posted their first BeReal within the 2-minute window. In July 2024, the number of bonus BeReals was increased to 5. [1] BeReal also features a "Discovery" section, wherein users are given the option to share to a much wider, public audience. This feature, however, is limited, as users are not able to interact with the posts through commenting—unlike the "My Friends" feature. In August 2023, in an attempt to make BeReal more social, another feature was added so that users are now able to see their friends of friends' BeReal. The app reportedly uses HiveAI to automate its image moderation process. However, there is also a report function that allows users to report a photo or another user if they are posting inappropriate content. === Comparison to other platforms === Because of its daily cycle of engagement, it has been compared to Wordle, which gained popularity earlier in 2022. It also supports a platform similar to Snapchat with a theme of impermanence and brevity. BeReal has been described as designed to compete with Instagram while simultaneously de-emphasising social media addiction and overuse. The app does not allow any photo filters or other editing, and has no follower counts. Marketing material from the company said that the app "can be addictive" and that "BeReal won't make you famous." Jacob Arnott, managing director of social agency We the People, describes BeReal as "an anti-Instagram" due to its raw and unedited nature. The app's foundation on friends rather than followers resembles Facebook's platform of adding friends, which comprise the content of a user's feed. This also resembles Instagram's "close friends" story feature. Further, rather than "liking" posts, BeReal uses "RealMojis" which involves taking a photo to interact with other posts. With the popularity of BeReal, other providers have launched similar features. In July 2022, Instagram launched a "Dual Camera" feature similar to BeReal, and in August 2022 it began testing a feature called "IG Candid Challenges", where users are prompted to post once a day within two minutes. As of September 2022, TikTok has also launched a feature called TikTok Now, following the same concept. In December 2022, similar to Spotify's "Wrapped," BeReal launched a feature involving a video of a compilation of users' BeReal posts of 2022. == User characteristics == BeReal is considered to be targeted towards Generation Z users, and attempts to minimise "social media fatigue", a feeling of numbness and disconnection from reality caused by constant interaction with an idealised version of others. This is a "core generational value" that this demographic holds compared to Millennials. Further, BeReal's users have been particularly strong across universities and university-aged students, and the majority of users are in the United States, the United Kingdom, and Germany. In 2022, the majority of users were female, with 43.2% of users falling within the age range of 16 to 25 and 55.1% of users being 26 to 44 years old. BeReal, the platform encourages users to share their real time moments by sending a daily notification that gives a least two minutes to post a unedited photo using bot the front and back camera, although users can post later and retake photos from when the notification happens, this action are still visible to friends, reinforcing transparency and genuine in the moment sharing. == Reception == Jason Koebler, a writer for Vice, wrote that in contrast to Instagram, which presents an unattainable view of people's lives, BeReal instead "makes everyone look extremely boring". Niklas Myhr, a professor of social media at Chapman University, argued that depth of engagement may determine whether the app is a passing trend or has "staying power". Kelsey Weekman, a reporter for BuzzFeed News, noted that the app's unwillingness to "glamorise the banality of life" made it feel "humbling" in its emphasis on authenticity. Niloufar Haidari for The Guardian comments similarly that where the app succeeds in being "drab" in perhaps a positive way, it fails in potentially "un-inspiring" users. Likewise, Dr. Brad Ridout, a behavioral psychologist at the University of Sydney, emphasizes that the "boring" experience is what the creators are targeting for the app and, in response to Instagram's platform of flawlessness, that "perfection is the enemy of happiness". === Criticisms === Some people regularly post after the two-minute notification expires, leading to some criticism of the app, as the ability to post late undermines its aims of authenticity. In addition, BeReal's daily two-minute window has been argued to contribute to social media fatigue and a need for self-exposure, as well as constant access to phones.
Digital image correlation and tracking
Digital image correlation and tracking is an optical method that employs tracking and image registration techniques for accurate 2D and 3D measurements of changes in 2D images or 3D volumes. This method is often used to measure full-field displacement and strains, and it is widely applied in many areas of science and engineering. Compared to strain gauges and extensometers, digital image correlation methods provide finer details about deformation, due to the ability to provide both local and average data. == Overview == Digital image correlation (DIC) techniques have been increasing in popularity, especially in micro- and nano-scale mechanical testing applications due to their relative ease of implementation and use. Advances in computer technology and digital cameras have been the enabling technologies for this method and while white-light optics has been the predominant approach, DIC can be and has been extended to almost any imaging technology. The concept of using cross-correlation to measure shifts in datasets has been known for a long time, and it has been applied to digital images since at least the early 1970s. The present-day applications are almost innumerable, including image analysis, image compression, velocimetry, and strain estimation. Much early work in DIC in the field of mechanics was led by researchers at the University of South Carolina in the early 1980s and has been optimized and improved in recent years. Commonly, DIC relies on finding the maximum of the correlation array between pixel intensity array subsets on two or more corresponding images, which gives the integer translational shift between them. It is also possible to estimate shifts to a finer resolution than the resolution of the original images, which is often called "sub-pixel" registration because the measured shift is smaller than an integer pixel unit. For sub-pixel interpolation of the shift, other methods do not simply maximize the correlation coefficient. An iterative approach can also be used to maximize the interpolated correlation coefficient by using non-linear optimization techniques. The non-linear optimization approach tends to be conceptually simpler and can handle large deformations more accurately, but as with most nonlinear optimization techniques, it is slower. The two-dimensional discrete cross correlation r i j {\displaystyle r_{ij}} can be defined in several ways, one possibility being: r i j = ∑ m ∑ n [ f ( m + i , n + j ) − f ¯ ] [ g ( m , n ) − g ¯ ] ∑ m ∑ n [ f ( m , n ) − f ¯ ] 2 ∑ m ∑ n [ g ( m , n ) − g ¯ ] 2 . {\displaystyle r_{ij}={\frac {\sum _{m}\sum _{n}[f(m+i,n+j)-{\bar {f}}][g(m,n)-{\bar {g}}]}{\sqrt {\sum _{m}\sum _{n}{[f(m,n)-{\bar {f}}]^{2}}\sum _{m}\sum _{n}{[g(m,n)-{\bar {g}}]^{2}}}}}.} Here f(m, n) is the pixel intensity or the gray-scale value at a point (m, n) in the original image, g(m, n) is the gray-scale value at a point (m, n) in the translated image, f ¯ {\displaystyle {\bar {f}}} and g ¯ {\displaystyle {\bar {g}}} are mean values of the intensity matrices f and g respectively. However, in practical applications, the correlation array is usually computed using Fourier-transform methods, since the fast Fourier transform is a much faster method than directly computing the correlation. F = F { f } , G = F { g } . {\displaystyle \mathbf {F} ={\mathcal {F}}\{f\},\quad \mathbf {G} ={\mathcal {F}}\{g\}.} Then taking the complex conjugate of the second result and multiplying the Fourier transforms together elementwise, we obtain the Fourier transform of the correlogram, R {\displaystyle \ R} : R = F ∘ G ∗ , {\displaystyle R=\mathbf {F} \circ \mathbf {G} ^{},} where ∘ {\displaystyle \circ } is the Hadamard product (entry-wise product). It is also fairly common to normalize the magnitudes to unity at this point, which results in a variation called phase correlation. Then the cross-correlation is obtained by applying the inverse Fourier transform: r = F − 1 { R } . {\displaystyle \ r={\mathcal {F}}^{-1}\{R\}.} At this point, the coordinates of the maximum of r i j {\displaystyle r_{ij}} give the integer shift: ( Δ x , Δ y ) = arg max ( i , j ) { r } . {\displaystyle (\Delta x,\Delta y)=\arg \max _{(i,j)}\{r\}.} == Deformation mapping == For deformation mapping, the mapping function that relates the images can be derived from comparing a set of subwindow pairs over the whole images. (Figure 1). The coordinates or grid points (xi, yj) and (xi, yj) are related by the translations that occur between the two images. If the deformation is small and perpendicular to the optical axis of the camera, then the relation between (xi, yj) and (xi, yj) can be approximated by a 2D affine transformation such as: x ∗ = x + u + ∂ u ∂ x Δ x + ∂ u ∂ y Δ y , {\displaystyle x^{}=x+u+{\frac {\partial u}{\partial x}}\Delta x+{\frac {\partial u}{\partial y}}\Delta y,} y ∗ = y + v + ∂ v ∂ x Δ x + ∂ v ∂ y Δ y . {\displaystyle y^{}=y+v+{\frac {\partial v}{\partial x}}\Delta x+{\frac {\partial v}{\partial y}}\Delta y.} Here u and v are translations of the center of the sub-image in the X and Y directions respectively. The distances from the center of the sub-image to the point (x, y) are denoted by Δ x {\displaystyle \Delta x} and Δ y {\displaystyle \Delta y} . Thus, the correlation coefficient rij is a function of displacement components (u, v) and displacement gradients ∂ u ∂ x , ∂ u ∂ y , ∂ v ∂ x , ∂ v ∂ y . {\displaystyle {\frac {\partial u}{\partial x}},{\frac {\partial u}{\partial y}},{\frac {\partial v}{\partial x}},{\frac {\partial v}{\partial y}}.} DIC has proven to be very effective at mapping deformation in macroscopic mechanical testing, where the application of specular markers (e.g. paint, toner powder) or surface finishes from machining and polishing provide the needed contrast to correlate images well. However, these methods for applying surface contrast do not extend to the application of free-standing thin films for several reasons. First, vapor deposition at normal temperatures on semiconductor grade substrates results in mirror-finish quality films with RMS roughnesses that are typically on the order of several nanometers. No subsequent polishing or finishing steps are required, and unless electron imaging techniques are employed that can resolve microstructural features, the films do not possess enough useful surface contrast to adequately correlate images. Typically this challenge can be circumvented by applying paint that results in a random speckle pattern on the surface, although the large and turbulent forces resulting from either spraying or applying paint to the surface of a free-standing thin film are too high and would break the specimens. In addition, the sizes of individual paint particles are on the order of μms, while the film thickness is only several hundred nanometers, which would be analogous to supporting a large boulder on a thin sheet of paper. == Digital volume correlation == Digital Volume Correlation (DVC, and sometimes called Volumetric-DIC) extends the 2D-DIC algorithms into three dimensions to calculate the full-field 3D deformation from a pair of 3D images. This technique is distinct from 3D-DIC, which only calculates the 3D deformation of an exterior surface using conventional optical images. The DVC algorithm is able to track full-field displacement information in the form of voxels instead of pixels. The theory is similar to above except that another dimension is added: the z-dimension. The displacement is calculated from the correlation of 3D subsets of the reference and deformed volumetric images, which is analogous to the correlation of 2D subsets described above. DVC can be performed using volumetric image datasets. These images can be obtained using confocal microscopy, X-ray computed tomography, Magnetic Resonance Imaging or other techniques. Similar to the other DIC techniques, the images must exhibit a distinct, high-contrast 3D "speckle pattern" to ensure accurate displacement measurement. DVC was first developed in 1999 to study the deformation of trabecular bone using X-ray computed tomography images. Since then, applications of DVC have grown to include granular materials, metals, foams, composites and biological materials. To date it has been used with images acquired by MRI imaging, Computer Tomography (CT), micro-CT, confocal microscopy, and lightsheet microscopy. DVC is currently considered to be ideal in the research world for 3D quantification of local displacements, strains, and stress in biological specimens. It is preferred because of the non-invasiveness of the method over traditional experimental methods. Two of the key challenges are improving the speed and reliability of the DVC measurement. The 3D imaging techniques produce noisier images than conventional 2D optical images, which reduces the quality of the displacement measurement. Computational speed is restricted by the file sizes of 3D images, which are significantly larger than 2D images. For example, an