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EyeOS

eyeOS was a web desktop for cloud computing, whose main purpose is to enable collaboration and communication among users. It is mainly written in PHP, XML, and JavaScript. It is a private-cloud application platform with a web-based desktop interface. eyeOS delivers a whole desktop from the cloud with file management, personal management information tools, and collaborative tools, with the integration of the client's applications. == History == The first publicly available eyeOS version was released on August 1, 2005, as eyeOS 0.6.0 in Olesa de Montserrat, Barcelona (Spain). A worldwide community of developers soon took part in the project and helped improve it by translating, testing, and developing it. After two years of development, the eyeOS Team published eyeOS 1.0 on June 4, 2007. Compared with previous versions, eyeOS 1.0 introduced a complete reorganization of the code and some new web technologies, like eyeSoft, a portage-based web software installation system. Moreover, eyeOS also included the eyeOS Toolkit, a set of libraries allowing easy and fast development of new web applications. With the release of eyeOS 1.1 on July 2, 2007, eyeOS changed its license and migrated from GNU GPL Version 2 to Version 3. Version 1.2 was released just a month after the 1.1 version and integrated full compatibility with Microsoft Word files. eyeOS 1.5 Gala was released on January 15, 2008. This version was the first to support both Microsoft Office and OpenOffice.org file formats for documents, presentations, and spreadsheets. With this version, eyeOS also gained the ability to import and export documents in both formats using server-side scripting. eyeOS 1.6 was released on April 25, 2008, and included many improvements such as synchronization with local computers, drag and drop, a mobile version, and more. eyeOS 1.8 Lars was released on January 7, 2009, and featured a completely rewritten file manager and a new sound API to develop media-rich applications. Later, on April 1, 2009, 1.8.5 was released with a new default theme and some rewritten apps, such as the Word Processor and the Address Book. On July 13, 2009, 1.8.6 was released with an interface for the iPhone and a new version of eyeMail with support for POP3 and IMAP. eyeOS 1.9 was released on December 29, 2009. It was followed up with the 1.9.0.1 release with minor fixes on February 18, 2010. These releases were the last of the "classic desktop" interfaces. A major re-work was completed in March 2010, now called eyeOS 2.x. However, a small group of eyeOS developers still maintain the code within the eyeOS forum, where support is provided, but the eyeOS group itself has stopped active 1.x development. It is now available as the On-eye project on GitHub. Active development was halted on 1.x as of February 3, 2010. eyeOS 2.0 release took place on March 3, 2010. This was a total restructure of the operating system. The 2.x stable is the new series of eyeOS, which is in active development and will replace 1.x as stable in a few months. It includes live collaboration and more social capabilities than eyeOS 1.x. eyeOS then released 2.2.0.0 on July 28, 2010. On December 14, 2010, a working group inside the eyeOS open-source development community began the structure development and further upgrade of eyeOS 1.9.x. The group's main goal is to continue the work eyeOS has stopped on 1.9.x. eyeOS released 2.5 on May 17, 2011. This was the last release under an open source license. It is available on SourceForge for download under another project called eyeOS 2.5 Open Source Version. On April 1, 2014, Telefónica announced their acquisition of eyeOS. eyeOS would maintain its headquarters in the Catalonia, Spain, where their staff would continue to work but now as part of Telefónica. After its integration into Telefónica, eyeOS would continue to function as an independent subsidiary under CEO Michel Kisfaludi. == Structure and API == For developers, EyeOS provides the eyeOS Toolkit, a set of libraries and functions to develop applications for eyeOS. Using the integrated Portage-based eyeSoft system, one can create their own repository for eyeOS and distribute applications through it. Each core part of the desktop is its own application, using JavaScript to send server commands as the user interacts. As actions are performed using AJAX (such as launching an application), it sends event information to the server. The server then sends back tasks for the client to do in XML format, such as drawing a widget. On the server, eyeOS uses XML files to store information. This makes it simple for a user to set up on the server, as it requires zero configuration other than the account information for the first user, making it simple to deploy. To avoid bottlenecks that flat files present, each user's information and settings are stored in different files, preventing resource starvation from occurring, though this in turn may create issues in high volume user environments due to host operating system open file descriptor limits. == Professional edition == A Professional Edition of eyeOS was launched on September 15, 2011, as an operating system for businesses. It uses a new version number and was released under version 1.0 instead of continuing with the next version number in the open source project. The Professional Edition retains the web desktop interface used by the open source version while targeting enterprise users. A host of new features designed for enterprises, like file sharing and synchronization (called eyeSync), Active Directory/LDAP connectivity, system-wide administration controls, and a local file execution tool called eyeRun were introduced. A new suite of Web Apps (a mail client, calendar, instant messaging, and collaboration tools) was also introduced, specific to the enterprise edition for the web desktop. With eyeOS Professional Edition 1.1, a to-do task manager tool, Citrix XenApp integration, and a Facebook like 'wall' for collaboration were introduced. == Awards == 2007 – Received the Softpedia's Pick award. 2007 – Finalist at SourceForge's 2007 Community Choice Awards at the "Best Project" category. The winner for that category was 7-Zip. 2007 – Won the Yahoo! Spain Web Revelation award in the Technology category. 2008 – Finalist for the Webware 100 awards by CNET, under the "Browsing" category. 2008 – Finalist at the SourceForge's 2008 Community Choice Awards at the "Most Likely to Change the World" category. The winner for that category was Linux. 2009 – Selected Project of the Month (August 2009) by SourceForge. 2009 – BMW Innovation Award. 2010 – Winner of Accelera (Ernst & Young). 2010 – Asturias & Girona Spanish Prince award “IMPULSA”. 2011 – Winner of MIT's TR35 award as Innovator of the Year in Spain. == Community == eyeOS community is formed with the eyeOS forums, which reached 10,000 members on April 4, 2008; the eyeOS wiki; and the eyeOS Application Communities, available at the eyeOS-Apps website, hosted and provided by openDesktop.org as well as Softpedia.
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Scripped

Scripped was an online screenplay services company offering three services: script writing, script registration, and script coverage. Scripped did not facilitate collaboration among screenwriters. It combined with Zhura in 2010. According to Techcrunch, Scripped had more than 60,000 writers as of March 2010. Scripped was administered by Sunil Rajaraman, Ryan Buckley and Zak Freer. Actor, writer, and director Edward Burns and screenwriter Steven E. de Souza joined Scripped's Board of Advisers in May 2008. In 2008, the company formed a partnership with Write Brothers, makers of Movie Magic Screenwriter software. On March 29, 2010, Scripped announced that it closed $250,000 in private investment and merged with competitor Zhura. Scripped's CEO, Sunil Rajaraman, remains the merged company's Chief Executive Officer. On April 1, 2015, citing a serious technical failure, Scripped shuttered its service. As part of the announcement, it was disclosed that their backup servers had failed as well, losing all of its users' stored scripts. The website URL currently redirects to WriterDuet's website, another online scriptwriting service; Scripped had advertised WriterDuet in Scripped's shutdown open letter. == Features == The Scripped Writer provided a built-in screenplay template which formatted the document to a standard for scripts as recommended by the AMPAS. The screenplay document was composed of seven elements: scene, action, character, dialog, parenthetical, transition and general. Each element had a specific style to which the Scripped Writer conformed as text was entered. Like other client-side screenplay software, Scripped offered Tab-Enter toggling between screenplay elements, making the writing process much faster. Text files could be imported into the Scripped Writer and automatically conformed to the screenplay template. Completed scripts could be exported as PDF files. In May 2011 the administrators of Scripped launched Scripted.com - a sister site focused on freelance writing jobs. Subsequent to the service's launch, the company was renamed to Scripted, Inc.
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Azure Data Lake

Azure Data Lake is a scalable data storage and analytics service. The service is hosted in Azure, Microsoft's public cloud. == History == Azure Data Lake service was released on November 16, 2016. It is based on COSMOS, which is used to store and process data for applications such as Azure, AdCenter, Bing, MSN, Skype and Windows Live. COSMOS features a SQL-like query engine called SCOPE upon which U-SQL was built. == Storage == Data Lake Storage is a cloud service to store structured, semi-structured or unstructured data produced from applications including social networks, relational data, sensors, videos, web apps, mobile or desktop devices. A single account can store trillions of files where a single file can be greater than a petabyte in size. == Analytics == Data Lake Analytics is a parallel on-demand job service. The parallel processing system is based on Microsoft Dryad. Dryad can represent arbitrary Directed Acyclic Graphs (DAGs) of computation. Data Lake Analytics provides a distributed infrastructure that can dynamically allocate resources so that customers pay for only the services they use. The system uses Apache YARN, the part of Apache Hadoop which governs resource management across clusters. Data Lake Store supports any application that uses the Hadoop Distributed File System (HDFS) interface. == U-SQL == U-SQL is a query language for Data Lake Analytics parallel data transformation and processing programs. It combines SQL and C#: it is and an evolution of the declarative SQL language with native extensibility through user code written in C#. U-SQL uses C# data types and the C# expression language. == Retirement == In 2021, Microsoft announced the 2024 retirement of the original Azure Data Lake Storage, now called "Gen1". The related Azure Data Lake Analytics / U-SQL technologies are also being retired. Azure Data Lake Storage Gen2, an extension of Azure Storage, will continue. The suggested replacement technologies are Azure Synapse Analytics and Apache Spark.
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Chirplet transform

In signal processing, the chirplet transform is an inner product of an input signal with a family of analysis primitives called chirplets. Similar to the wavelet transform, chirplets are usually generated from (or can be expressed as being from) a single mother chirplet (analogous to the so-called mother wavelet of wavelet theory). == Definitions == The term chirplet transform was coined by Steve Mann, as the title of the first published paper on chirplets. The term chirplet itself (apart from chirplet transform) was also used by Steve Mann, Domingo Mihovilovic, and Ronald Bracewell to describe a windowed portion of a chirp function. In Mann's words: A wavelet is a piece of a wave, and a chirplet, similarly, is a piece of a chirp. More precisely, a chirplet is a windowed portion of a chirp function, where the window provides some time localization property. In terms of time–frequency space, chirplets exist as rotated, sheared, or other structures that move from the traditional parallelism with the time and frequency axes that are typical for waves (Fourier and short-time Fourier transforms) or wavelets. The chirplet transform thus represents a rotated, sheared, or otherwise transformed tiling of the time–frequency plane. Although chirp signals have been known for many years in radar, pulse compression, and the like, the first published reference to the chirplet transform described specific signal representations based on families of functions related to one another by time–varying frequency modulation or frequency varying time modulation, in addition to time and frequency shifting, and scale changes. In that paper, the Gaussian chirplet transform was presented as one such example, together with a successful application to ice fragment detection in radar (improving target detection results over previous approaches). The term chirplet (but not the term chirplet transform) was also proposed for a similar transform, apparently independently, by Mihovilovic and Bracewell later that same year. == Applications == The first practical application of the chirplet transform was in water-human-computer interaction (WaterHCI) for marine safety, to assist vessels in navigating through ice-infested waters, using marine radar to detect growlers (small iceberg fragments too small to be visible on conventional radar, yet large enough to damage a vessel). Other applications of the chirplet transform in WaterHCI include the SWIM (Sequential Wave Imprinting Machine). More recently other practical applications have been developed, including image processing (e.g. where there is periodic structure imaged through projective geometry), as well as to excise chirp-like interference in spread spectrum communications, in EEG processing, and Chirplet Time Domain Reflectometry. == Extensions == The warblet transform is a particular example of the chirplet transform introduced by Mann and Haykin in 1992 and now widely used. It provides a signal representation based on cyclically varying frequency modulated signals (warbling signals).
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Jive (software)

Jive (formerly known as Clearspace, then Jive SBS, then Jive Engage) is a commercial Java EE-based Enterprise 2.0 collaboration and knowledge management tool produced by Jive Software. It was first released as "Clearspace" in 2006, then renamed SBS (for "Social Business Software") in March 2009, then renamed "Jive Engage" in 2011, and renamed simply to "Jive" in 2012. Jive integrates the functionality of online communities, microblogging, social networking, discussion forums, blogs, wikis, and IM under one unified user interface. Content placed into any of the systems (blog, wiki, documentation, etc.) can be found through a common search interface. Other features include RSS capability, email integration, a reputation and reward system for participation, personal user profiles, JAX-WS web service interoperability, and integration with the Spring Framework. The product is a pure-Java server-side web application and will run on any platform where Java (JDK 1.5 or higher) is installed. It does not require a dedicated server - users have reported successful deployment in both shared environments and multiple machine clusters. As of Jive 8, released March 30, 2015, there is a Jive-n version which is for internal use (hosted by the consumer or hosted by Jive as a service) and a Jive-x version which is an external version hosted as a service. Jive no longer supports wiki markup language. == Server requirements for Jive 8-n == The following are the server requirements for Jive 8-n Operating systems: RHEL version 6 or 7 for x86_64, CentOS version 6 or 7 for x86_64 or SuSE Enterprise Linux Server (SLES) 11 and 12 for x86_64 Application Servers: Jive ships with its own embedded Apache HTTPD and Tomcat servers as part of the install package. It is not possible to deploy the application onto other appservers. Databases: MySQL (5.1, 5.5, 5.6) Oracle (11gR2, 12c) Postgres (9.0, 9.1, 9.2, 9.3, 9.4 - 9.2 or higher recommended) Microsoft SQL Server (2008R2, 2012, 2014) Environment: Jive recommends a server with at least 4GB of RAM and a dual-core 2 GHz processor with x86_64 architecture The product integrates with an LDAP repository or Active Directory For optimal deployment with a large community Jive Software recommends: using dedicated cache and document-conversion servers hosting the application and database servers separately == Releases == Jive 8, released on March 30, 2015 Jive 7, released in October 2013 Jive 9.0.x, released in November 2016 Jive 9, released in November 2016, supported now
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Cloud computing

Cloud computing is defined by the International Organization for Standardization (ISO) as "a paradigm for enabling network access to a scalable and elastic pool of shareable physical or virtual resources with self-service provisioning and administration on demand". It is commonly referred to as "the cloud". == Characteristics == In 2011, the National Institute of Standards and Technology (NIST) identified five "essential characteristics" for cloud systems. Below are the exact definitions according to NIST: On-demand self-service: "A consumer can unilaterally provision computing capabilities, such as server time and network storage, as needed automatically without requiring human interaction with each service provider." Broad network access: "Capabilities are available over the network and accessed through standard mechanisms that promote use by heterogeneous thin or thick client platforms (e.g., mobile phones, tablets, laptops, and workstations)." Resource pooling: " The provider's computing resources are pooled to serve multiple consumers using a multi-tenant model, with different physical and virtual resources dynamically assigned and reassigned according to consumer demand." Rapid elasticity: "Capabilities can be elastically provisioned and released, in some cases automatically, to scale rapidly outward and inward commensurate with demand. To the consumer, the capabilities available for provisioning often appear unlimited and can be appropriated in any quantity at any time." Measured service: "Cloud systems automatically control and optimize resource use by leveraging a metering capability at some level of abstraction appropriate to the type of service (e.g., storage, processing, bandwidth, and active user accounts). Resource usage can be monitored, controlled, and reported, providing transparency for both the provider and consumer of the utilized service. By 2023, the International Organization for Standardization (ISO) had expanded and refined the list. == History == The history of cloud computing extends to the 1960s, with the initial concepts of time-sharing becoming popularized via remote job entry (RJE). The "data center" model, where users submitted jobs to operators to run on mainframes, was predominantly used during this era. This period saw broad experimentation with making large-scale computing power more accessible through time-sharing, while optimizing infrastructure, platforms, and applications to improve efficiency for end users. The "cloud" metaphor for virtualized services dates to 1994, when it was used by General Magic for the universe of "places" that mobile agents in the Telescript environment could "go". The metaphor is credited to David Hoffman, a General Magic communications specialist, based on its long-standing use in networking and telecom. The expression cloud computing became more widely known in 1996 when Compaq Computer Corporation drew up a business plan for future computing and the Internet. The company's ambition was to supercharge sales with "cloud computing-enabled applications". The business plan foresaw that online consumer file storage would likely be commercially successful. As a result, Compaq decided to sell server hardware to internet service providers. In the 2000s, the application of cloud computing began to take shape with the establishment of Amazon Web Services (AWS) in 2002, which allowed developers to build applications independently. In 2006 Amazon Simple Storage Service, known as Amazon S3, and the Amazon Elastic Compute Cloud (EC2) were released. In 2008 NASA's development of the first open-source software for deploying private and hybrid clouds. The following decade saw the launch of various cloud services. In 2010, Microsoft launched Microsoft Azure, and Rackspace Hosting and NASA initiated an open-source cloud-software project, OpenStack. IBM introduced the IBM SmartCloud framework in 2011, and Oracle announced the Oracle Cloud in 2012. In December 2019, Amazon launched AWS Outposts, a service that extends AWS infrastructure, services, APIs, and tools to customer data centers, co-location spaces, or on-premises facilities. == Value proposition == Cloud computing can shorten time to market by offering pre-configured tools, scalable resources, and managed services, allowing users to focus on core business value rather than maintaining infrastructure. Cloud platforms can enable organizations and individuals to reduce upfront capital expenditures on physical infrastructure by shifting to an operational expenditure model, where costs scale with usage. Cloud platforms also offer managed services and tools, such as artificial intelligence, data analytics, and machine learning, which might otherwise require significant in-house expertise and infrastructure investment. While cloud computing can offer cost advantages through effective resource optimization, organizations often face challenges such as unused resources, inefficient configurations, and hidden costs without proper oversight and governance. Many cloud platforms provide cost management tools, such as AWS Cost Explorer and Azure Cost Management, and frameworks like FinOps have emerged to standardize financial operations in the cloud. Cloud computing also facilitates collaboration, remote work, and global service delivery by enabling secure access to data and applications from any location with an internet connection. Cloud providers offer various redundancy options for core services, such as managed storage and managed databases, though redundancy configurations often vary by service tier. Advanced redundancy strategies, such as cross-region replication or failover systems, typically require explicit configuration and may incur additional costs or licensing fees. Cloud environments operate under a shared responsibility model, where providers are typically responsible for infrastructure security, physical hardware, and software updates, while customers are accountable for data encryption, identity and access management (IAM), and application-level security. These responsibilities vary depending on the cloud service model—Infrastructure as a Service (IaaS), Platform as a Service (PaaS), or Software as a Service (SaaS)—with customers typically having more control and responsibility in IaaS environments and progressively less in PaaS and SaaS models, often trading control for convenience and managed services. == Adoption and suitability == The decision to adopt cloud computing or maintain on-premises infrastructure depends on factors such as scalability, cost structure, latency requirements, regulatory constraints, and infrastructure customization. Organizations with variable or unpredictable workloads, limited capital for upfront investments, or a focus on rapid scalability benefit from cloud adoption. Startups, SaaS companies, and e-commerce platforms often prefer the pay-as-you-go operational expenditure (OpEx) model of cloud infrastructure. Additionally, companies prioritizing global accessibility, remote workforce enablement, disaster recovery, and leveraging advanced services such as AI/ML and analytics are well-suited for the cloud. In recent years, some cloud providers have started offering specialized services for high-performance computing and low-latency applications, addressing some use cases previously exclusive to on-premises setups. On the other hand, organizations with strict regulatory requirements, highly predictable workloads, or reliance on deeply integrated legacy systems may find cloud infrastructure less suitable. Businesses in industries like defense, government, or those handling highly sensitive data often favor on-premises setups for greater control and data sovereignty. Additionally, companies with ultra-low latency requirements, such as high-frequency trading (HFT) firms, rely on custom hardware (e.g., FPGAs) and physical proximity to exchanges, which most cloud providers cannot fully replicate despite recent advancements. Similarly, tech giants like Google, Meta, and Amazon build their own data centers due to economies of scale, predictable workloads, and the ability to customize hardware and network infrastructure for optimal efficiency. However, these companies also use cloud services selectively for certain workloads and applications where it aligns with their operational needs. In practice, many organizations are increasingly adopting hybrid cloud architectures, combining on-premises infrastructure with cloud services. This approach allows businesses to balance scalability, cost-effectiveness, and control, offering the benefits of both deployment models while mitigating their respective limitations. == Challenges and limitations == One of the primary challenges of cloud computing, compared with traditional on-premises systems, is maintaining data security and privacy. Cloud users entrust their sensitive data to third-party providers, who may not have adequate measures to protect it from unau
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Color moments

Color moments are measures that characterise color distribution in an image in the same way that central moments uniquely describe a probability distribution. Color moments are mainly used for color indexing purposes as features in image retrieval applications in order to compare how similar two images are based on color. Usually one image is compared to a database of digital images with pre-computed features in order to find and retrieve a similar Image. Each comparison between images results in a similarity score, and the lower this score is the more identical the two images are supposed to be. == Overview == Color moments are scaling and rotation invariant. It is usually the case that only the first three color moments are used as features in image retrieval applications as most of the color distribution information is contained in the low-order moments. Since color moments encode both shape and color information they are a good feature to use under changing lighting conditions, but they cannot handle occlusion very successfully. Color moments can be computed for any color model. Three color moments are computed per channel (e.g. 9 moments if the color model is RGB and 12 moments if the color model is CMYK). Computing color moments is done in the same way as computing moments of a probability distribution. === Mean === The first color moment can be interpreted as the average color in the image, and it can be calculated by using the following formula E i = ∑ j = 1 N 1 N p i j {\displaystyle E_{i}=\textstyle \sum _{j=1}^{N}{\frac {1}{N}}p_{ij}} where N is the number of pixels in the image and p i j {\displaystyle p_{ij}} is the value of the j-th pixel of the image at the i-th color channel. === Standard Deviation === The second color moment is the standard deviation, which is obtained by taking the square root of the variance of the color distribution. σ i = ( 1 N ∑ j = 1 N ( p i j − E i ) 2 ) {\displaystyle \sigma _{i}={\sqrt {({\frac {1}{N}}\textstyle \sum _{j=1}^{N}(p_{ij}-E_{i})^{2})}}} where E i {\displaystyle E_{i}} is the mean value, or first color moment, for the i-th color channel of the image. === Skewness === The third color moment is the skewness. It measures how asymmetric the color distribution is, and thus it gives information about the shape of the color distribution. Skewness can be computed with the following formula: s i = ( 1 N ∑ j = 1 N ( p i j − E i ) 3 ) 3 σ i {\displaystyle s_{i}={\frac {\sqrt[{3}]{\left({\frac {1}{N}}\textstyle \sum _{j=1}^{N}(p_{ij}-E_{i})^{3}\right)}}{\sigma _{i}}}} === Kurtosis === Kurtosis is the fourth color moment, and, similarly to skewness, it provides information about the shape of the color distribution. More specifically, kurtosis is a measure of how extreme the tails are in comparison to the normal distribution. === Higher-order color moments === Higher-order color moments are usually not part of the color moments feature set in image retrieval tasks as they require more data in order to obtain a good estimate of their value, and also the lower-order moments generally provide enough information. == Applications == Color moments have significant applications in image retrieval. They can be used in order to compare how similar two images are. This is a relatively new approach to color indexing. The greatest advantage of using color moments comes from the fact that there is no need to store the complete color distribution. This greatly speeds up image retrieval since there are less features to compare. In addition, the first three color moments have the same units, which allows for comparison between them. === Color indexing === Color indexing is the main application of color moments. Images can be indexed, and the index will contain the computed color moments. Then, if someone has a particular image and wants to find similar images in the database, the color moments of the image of interest will also be computed. After that the following function will be used in order to compute a similarity score between the image of interest and all the images in the database: d m o m ( H , I ) = ∑ i = 1 r w i 1 | E i 1 − E i 2 | + w i 2 | σ i 1 − σ i 2 | + w i 3 | s i 1 − s i 2 | {\displaystyle d_{mom}(H,I)=\textstyle \sum _{i=1}^{r}w_{i1}|E_{i}^{1}-E_{i}^{2}|+w_{i2}|\sigma _{i}^{1}-\sigma _{i}^{2}|+w_{i3}|s_{i}^{1}-s_{i}^{2}|} where: H and I are the color distributions of the two images that are being compared i is the channel index and r is the total number of channels E i 1 {\displaystyle E_{i}^{1}} and E i 2 {\displaystyle E_{i}^{2}} are the first order moments computed for the image distributions. σ i 1 {\displaystyle \sigma _{i}^{1}} and σ i 2 {\displaystyle \sigma _{i}^{2}} are the second order moments computed for the image distributions. s_i^1 and s_i^2 are the third order moments computed for the image distributions. w i 1 {\displaystyle w_{i1}} , w i 2 {\displaystyle w_{i2}} , and w i 3 {\displaystyle w_{i3}} are weights, specified by the user, for each of the three color moments used. Finally, the images in the database will be ranked according to the computed similarity score with the image of interest, and the database images with the lowest d m o m ( H , I ) {\displaystyle d_{mom}(H,I)} value should be retrieved. "A retrieval based on d m o m ( H , I ) {\displaystyle d_{mom}(H,I)} may produce false positives because the index contains no information about the correlation between the color channels". == Example == A simple and concise example of the use of color moments for image retrieval tasks is illustrated in. Consider having several test images in a database and a "New Image". The goal is to retrieve images from the database that are similar to the "New Image". The first three color moments are used as features. There are several steps in this computation. Image preprocessing (Optional) - The image preprocessing step of the computation process is optional. For example, in this step all images could be modified to be the same size (in terms of pixels). However, since color moments are invariant to scaling, it is not necessary to make all images the same width and height. Computing the features - Use the color moments formulae in order to compute the first three moments for each of the color channels in the image. For example, if the HSV color space is used, this means that for each of the images, 9 features in total will be computed (the first three order moments for the Hue, Saturation, and Value channels). Calculating the similarity score - After computing the color moments the weights for each of the moments in the d m o m ( H , I ) {\displaystyle d_{mom}(H,I)} function should be determined by the user. The weights have to be adjusted each time in accordance with the application or condition and quality of the images. Following that the d m o m ( H , I ) {\displaystyle d_{mom}(H,I)} function is used to calculate a similarity score for the "New Image" and each of the images in the database. Ranking and image retrieval - From the previous step the d m o m ( H , I ) {\displaystyle d_{mom}(H,I)} values were obtained. Now a comparison of these values can be made in order to decide which of the images in the database are more similar to the "New Image", and thus rank the database images accordingly. The smaller the d m o m ( H , I ) {\displaystyle d_{mom}(H,I)} value is the more similar the two color distributions are supposed to be. Finally, some of the top ranked images (the ones with the smallest d m o m ( H , I ) {\displaystyle d_{mom}(H,I)} value) from the database are retrieved.
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Linux Trace Toolkit

The Linux Trace Toolkit (LTT) is a set of tools that is designed to log program execution details from a patched Linux kernel and then perform various analyses on them, using console-based and graphical tools. LTT has been mostly superseded by its successor LTTng (Linux Trace Toolkit Next Generation). LTT allows the user to see in-depth information about the processes that were running during the trace period, including when context switches occurred, how long the processes were blocked for, and how much time the processes spent executing vs. how much time the processes were blocked. The data is logged to a text file and various console-based and graphical (GTK+) tools are provided for interpreting that data. In order to do data collection, LTT requires a patched Linux kernel. The authors of LTT claim that the performance hit for a patched kernel compared to a regular kernel is minimal; Their testing has reportedly shown that this is less than 2.5% on a "normal use" system (measured using batches of kernel makes) and less than 5% on a file I/O intensive system (measured using batches of tar). == Usage == === Collecting trace data === Data collection is Started by: trace 15 foo This command will cause the LTT tracedaemon to do a trace that lasts for 15 seconds, writing trace data to foo.trace and process information from the /proc filesystem to foo.proc. The trace command is actually a script which runs the program tracedaemon with some common options. It is possible to run tracedaemon directly and in that case, the user can use a number of command-line options to control the data which is collected. For the complete list of options supported by tracedaemon, see the online manual page for tracedaemon. === Viewing the results === Viewing the results of a trace can be accomplished with: traceview foo This command will launch a graphical (GTK+) traceview tool that will read from foo.trace and foo.proc. This tool can show information in various interesting ways, including Event Graph, Process Analysis, and Raw Trace. The Event Graph is perhaps the most interesting view, showing the exact timing of events like page faults, interrupts, and context switches, in a simple graphical way. The traceview command is a wrapper for a program called tracevisualizer. For the complete list of options supported by tracevisualizer, see the online manual page for tracevisualizer.
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Drop shadow

In graphic design and computer graphics, a drop shadow is a visual effect consisting of a drawing element which looks like the shadow of an object, giving the impression that the object is raised above the objects behind it. The drop shadow is often used for elements of a graphical user interface such as windows or menus, and for simple text. The text label for icons on desktops in many desktop environments has a drop shadow, as this effect effectively distinguishes the text from any colored background it may be in front of. A simple way of drawing a drop shadow of a rectangular object is to draw a gray or black area underneath and offset from the object. In general, a drop shadow is a copy in black or gray of the object, drawn in a slightly different position. Realism may be increased by: Darkening the colors of the pixels where the shadow casts instead of making them gray. This can be done with alpha blending the shadow with the area it is cast on. Softening the edges of the shadow. This can be done by adding Gaussian blur to the shadow's alpha channel before blending. Inset drop shadows are a type which draws the shadows inside the element. This allows the interface element to appear as if it is sunken into the interface. == Photo editing == In photo editing or photography post-production, a drop shadow may be added right beneath a model or product in the image. It is used to create contrast between the background and the subject. To add a drop shadow, retouchers use graphic editing tools like Adobe Photoshop. Drop shadows are often used as a visual effect in e-commerce. This is done to improve the presentation of product images and create depth in the image. == Use == Generally, window managers which are capable of compositing allow drop shadow effects, whereas incapable window managers do not. In some operating systems like macOS, drop shadow is used to differentiate between active and inactive windows. Websites are able to use drop shadow effects through the CSS properties box-shadow, text-shadow, and drop-shadow() filter function in filter. The first two are used for elements and text respectively, while the filter applies to the element's content, letting it support oddly shaped elements or transparent images.
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Software design

Software design is the process of conceptualizing how a software system will work before it is implemented or modified. Software design also refers to the direct result of the design process – the concepts of how the software will work which may be formally documented or may be maintained less formally, including via oral tradition. The design process enables a designer to model aspects of a software system before it exists with the intent of making the effort of writing the code more efficiently. Creativity, past experience, a sense of what makes "good" software, and a commitment to quality are success factors for a competent design. A software design can be compared to an architected plan for a house. High-level plans represent the totality of the house (e.g., a three-dimensional rendering of the house). Lower-level plans provide guidance for constructing each detail (e.g., the plumbing lay). Similarly, the software design model provides a variety of views of the proposed software solution. == Part of the overall process == In terms of the waterfall development process, software design is the activity that occurs after requirements analysis and before coding. Requirements analysis determines what the system needs to do without determining how it will do it, and thus, multiple designs can be imagined that satisfy the requirements. The design can be created while coding, without a plan or requirements analysis, but for more complex projects this is less feasible. Completing a design prior to coding allows for multidisciplinary designers and subject-matter experts to collaborate with programmers to produce software that is useful and technically sound. Sometimes, a simulation or prototype is created to model the system in an effort to determine a valid and good design. == Code as design == A common point of confusion with the term design in software is that the process applies at multiple levels of abstraction such as a high-level software architecture and lower-level components, functions and algorithms. A relatively formal process may occur at high levels of abstraction but at lower levels, the design process is almost always less formal where the only artifact of design may be the code itself. To the extent that this is true, software design refers to the design of the design. Edsger W. Dijkstra referred to this layering of semantic levels as the "radical novelty" of computer programming, and Donald Knuth used his experience writing TeX to describe the futility of attempting to design a program prior to implementing it: TEX would have been a complete failure if I had merely specified it and not participated fully in its initial implementation. The process of implementation constantly led me to unanticipated questions and to new insights about how the original specifications could be improved. == Artifacts == A design process may include the production of art Software design documentation such as flow chart, use case, Pseudocode, Unified Modeling Language model and other Fundamental modeling concepts. For user centered software, design may involve user experience design yielding a storyboard to help determine those specifications. Documentation may be reviewed to allow constraints, specifications and even requirements to be adjusted prior to coding. == Iterative design == Software systems inherently deal with uncertainties, and the size of software components can significantly influence a system's outcomes, both positively and negatively. Neal Ford and Mark Richards propose an iterative approach to address the challenge of identifying and right-sizing components. This method emphasizes continuous refinement as teams develop a more nuanced understanding of system behavior and requirements. The approach typically involves a cycle with several stages: A high-level partitioning strategy is established, often categorized as technical or domain-based. Guidelines for the smallest meaningful deployable unit, referred to as "quanta," are defined. While these foundational decisions are made early, they may be revisited later in the cycle if necessary. Initial components are identified based on the established strategy. Requirements are assigned to the identified components. The roles and responsibilities of each component are analyzed to ensure clarity and minimize overlap. Architectural characteristics, such as scalability, fault tolerance, and maintainability, are evaluated. Components may be restructured based on feedback from development teams. This cycle serves as a general framework and can be adapted to different domains. == Design principles == Design principles enable a software engineer to navigate the design process. Davis suggested principles which have been refined over time as: The design process should not suffer from "tunnel vision" A good designer should consider alternative approaches, judging each based on the requirements of the problem, the resources available to do the job. The design should be traceable to the analysis model Because a single element of the design model can often be traced back to multiple requirements, it is necessary to have a means for tracking how requirements have been satisfied by the design model. The design should not reinvent the wheel Systems are constructed using a set of design patterns, many of which have likely been encountered before. These patterns should always be chosen as an alternative to reinvention. Time is short and resources are limited; design time should be invested in representing (truly new) ideas by integrating patterns that already exist (when applicable). The design should "minimize the intellectual distance" between the software and the problem as it exists in the real world That is, the structure of the software design should, whenever possible, mimic the structure of the problem domain. The design should exhibit uniformity and integration A design is uniform if it appears fully coherent. In order to achieve this outcome, rules of style and format should be defined for a design team before design work begins. A design is integrated if care is taken in defining interfaces between design components. The design should be structured to accommodate change The design concepts discussed in the next section enable a design to achieve this principle. The design should be structured to degrade gently, even when aberrant data, events, or operating conditions are encountered Well-designed software should never "bomb"; it should be designed to accommodate unusual circumstances, and if it must terminate processing, it should do so in a graceful manner. Design is not coding, coding is not design Even when detailed procedural designs are created for program components, the level of abstraction of the design model is higher than the source code. The only design decisions made at the coding level should address the small implementation details that enable the procedural design to be coded. The design should be assessed for quality as it is being created, not after the fact A variety of design concepts and design measures are available to assist the designer in assessing quality throughout the development process. The design should be reviewed to minimize conceptual (semantic) errors There is sometimes a tendency to focus on minutiae when the design is reviewed, missing the forest for the trees. A design team should ensure that major conceptual elements of the design (omissions, ambiguity, inconsistency) have been addressed before worrying about the syntax of the design model. == Design concepts == Design concepts provide a designer with a foundation from which more sophisticated methods can be applied. Design concepts include: Abstraction Reducing the information content of a concept or an observable phenomenon, typically to retain only information that is relevant for a particular purpose. It is an act of Representing essential features without including the background details or explanations. Architecture The overall structure of the software and the ways in which that structure provides conceptual integrity for a system. Good software architecture will yield a good return on investment with respect to the desired outcome of the project, e.g. in terms of performance, quality, schedule and cost. Control hierarchy A program structure that represents the organization of a program component and implies a hierarchy of control. Data structure Representing the logical relationship between elements of data. Design pattern A designer may identify a design aspect of the system that has solved in the past. The reuse of such patterns can increase software development velocity. Information hiding Modules should be specified and designed so that information contained within a module is inaccessible to other modules that have no need for such information. Modularity Dividing the solution into parts (modules). Refinement The process of elaboration. A hierarchy is developed by decomposing a macrosco
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Actionstep

Actionstep is a cloud-based legal practice management software for law firms and compliance-focused businesses. Actionstep is built to be a comprehensive practice management software with features for workflow automation as well as automatic document generation == History == Actionstep was created by Ted Jordan, CEO of Actionstep, in 2004. It was first used commercially in 2005 by a New Zealand construction franchise as well as a law firm. Actionstep soon expanded into central government and a wider range of small business users (mainly in New Zealand and Australia). After a few years the expanse of their legal client base prompted the company to add key legal specific features to the product with the aim of further expanding their legal market. Through Actionstep's tenure as a practice management software they have gradually expanded from their headquarters in New Zealand and offices located in the United Kingdom and the United States of America. In October 2020, private equity firm Serent Capital Partners purchased 84.25% stake in Actionstep. In April 2022, the company announced unlimited annual leave to its staff == Product == The premise of Actionstep is that it saves companies from having to purchase software tailored to their work flow and instead allows companies to modify the program without additional coding.{{Citation needed}} The founder and CEO Ted Jordan used cloud technology to allow the software to be continuously updated without the need to purchase or redesign new software. This theoretically allows businesses to remain current all the time and cut external I.T. costs.{{Citation needed}} Actionstep also integrates with software from other companies, such as Xero accounting, Microsoft Office & Office 365, Gmail, Google Drive, Dropbox, NetDocuments, QuickBooks, LawPay, BundleDocs, Box, HotDocs, Infotrack, GlobalX, PEXA, JOSEF and Zapier. Actionstep contains workflow automation features aimed at increasing office efficiency. These automated processes include automatic task assignment, information collection, document generation & automation, cataloguing, and matter generation. == Awards == Actionstep was named First International Best of SaaS Showplace Award Winner in 2009. Actionstep has also been a finalist in the ComputerWorld Excellence Awards (2007), and the Vero Excellence in Business Support (2010).
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Actifsource

Actifsource is a domain-specific modeling workbench. It is realized as plug-in for the software development environment Eclipse. Actifsource supports the creation of multiple domain models which can be linked together. It comes with a UML-like graphical editor to create domain-specific languages and a general graphical editor to edit structures in the created languages. It supports code generation using user-defined generic code templates which are directly linked to the domain models. Code generation is integrated into Eclipse's incremental build process. == Interoperability == Actifsource can use models from other modelling tools by importing and exporting the ecore format which is defined by the Eclipse Modeling Framework. == Licensing policy == There are two versions of actifsource available: The free community edition which can be used freely for non-commercial projects and the enterprise edition which contains additional features. The enterprise edition comes with customer support and maintenance for a limited period of time. This package allows the customers to upgrade to new versions and maintenance releases during their support period.
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Mastodon (social network)

Mastodon is a free and open-source software platform for decentralized social networking with microblogging features similar to Twitter. It operates as a federated network of independently managed servers that communicate using the ActivityPub protocol, allowing users to connect across different instances within the Fediverse. Each Mastodon instance establishes its own moderation policies and content guidelines, distinguishing it from centrally controlled social media platforms. First released in 2016 by Eugen Rochko, Mastodon has positioned itself as an alternative to mainstream social media, particularly for users seeking decentralized, community-driven spaces. The platform has experienced multiple surges in adoption, most notably following the Twitter acquisition by Elon Musk in 2022, as users sought alternatives to Twitter. It is part of a broader shift toward decentralized social networks, including Bluesky and Lemmy. Mastodon emphasizes user privacy and moderation flexibility, offering features such as granular post visibility controls, content warning options, and local community-driven moderation. The software is written in Ruby on Rails and Node.js, with a web interface built using React and Redux. It is interoperable with other ActivityPub-based platforms, such as Threads, and supports various third-party applications on desktop and mobile devices. == Functionality == Users post short-form status messages, historically known as "toots", for others to see and interact with. On a standard Mastodon instance, these messages can include up to 500 text-based characters, greater than Twitter's 280-character limit. Some instances support even longer messages. Images, audio files, videos or polls can also be added to a message. Users join a specific Mastodon server, rather than a single centralized website or application. The servers are connected as nodes in a network, and each server can administer its own rules, account privileges, and whether to share messages to and from other servers. Users can communicate and follow each other across connected Mastodon servers with usernames similar in format to full email addresses. Since version 2.9.0, Mastodon's web user interface has offered a single-column mode for new users by default. In advanced mode, the interface approximates the microblogging interface of TweetDeck. === Privacy === Mastodon includes a number of specific privacy features. Each message has a variety of privacy options available, and users can choose whether the message is public or private. Messages can display public on a global feed, known as a timeline, or can be shared only to the user's followers. Messages can also be marked as unlisted from timelines or direct between users. Users can also mark their accounts as completely private. In the timeline, messages can display with an optional content warning feature, which requires readers to click on the hidden main body of the message to reveal it. Mastodon servers have used this feature to hide spoilers, trigger warnings, and not safe for work (NSFW) content, though some accounts use the feature to hide links and thoughts others might not want to read. Mastodon aggregates messages in local and federated timelines in real time. The local timeline shows messages from users on a singular server, while the federated timeline shows messages across all participating Mastodon servers. === Content moderation === In early 2017, journalists like Sarah Jeong distinguished Mastodon from Twitter for its approach to combating harassment. Mastodon uses community-based moderation, in which each server can limit or filter out undesirable types of content, while Twitter uses a single, global policy on content moderation. Servers can choose to limit or filter out messages with disparaging content. The founder of Mastodon, Eugen Rochko, believes that small, closely related communities deal with unwanted behavior more effectively than a large company's small safety team. In Move Slowly and Build Bridges, Robert W. Gehl argues that predominantly white participation has shaped Mastodon in ways that affect how reports of racism are received and limit its ability to replicate Black Twitter on Twitter. Users can also block and report others to administrators, much like on Twitter. Instance administrators can block other instances from interacting with their own, an action called defederation. By posting toots hashtagged with #fediblock, some instance administrators and users alert others of issues requiring moderation. === Searching === Mastodon by default allows searching for hashtags and mentioned accounts in the Fediverse. Server administrators can optionally enable Elasticsearch to search the full-text of public posts that have opted in to being indexed. == Versions == In September 2018, with the release of version 2.5 with redesigned public profile pages, Mastodon marked its 100th release. Mastodon 2.6 was released in October 2018, introducing the possibilities of verified profiles and live, in-stream link previews for images and videos. Version 2.7, in January 2019, made it possible to search for multiple hashtags at once, instead of searching for just a single hashtag, with more robust moderation capabilities for server administrators and moderators, while accessibility, such as contrast for users with sight issues, was improved. The ability for users to create and vote in polls, as well as a new invitation system to manage registrations was integrated in April 2019. Mastodon 2.8.1, released in May 2019, made images with content warnings blurred instead of completely hidden. In version 2.9 in June 2019, an optional single-column view was added. This view became the default displayed to new users, with a user "preferences" option to switch to a multiple-column-based view. In August 2020, Mastodon 3.2 was released. It included a redesigned audio player with custom thumbnails and the ability to add personal notes to one's profile. In July 2021, an official client for iOS devices was released. According to the project's then CEO, Eugen Rochko, the release was part of an effort to attract new users. Mastodon 4.0 was released in November 2022, including language support for translating posts, editing posts and following hashtags. Mastodon 4.5 was released in November 2025. Among other features it introduced quote posts, which were previously rejected from being implemented due to concerns about toxicity and harassment. To mitigate these issues Mastodon's quote post feature has been designed in a way that lets users decide if and by whom their posts can be quoted. == Software == Mastodon is published as free and open-source software under the Affero GPL license, allowing anyone to use the software or modify it as they wish. Servers can be run by any individual or organization, and users can join these servers as they wish. The server software itself is powered by Ruby on Rails and Node.js, with its web client being written in React.js and Redux. The only database software supported is PostgreSQL, with Redis being used for job processing and various actions that Mastodon needs to process. The service is interoperable with the fediverse, a collection of social networking services which use the ActivityPub protocol for communication between each other, with previous versions containing support for OStatus. Client apps for interacting with the Mastodon API are available for desktop computer operating systems, including Windows, macOS and the Linux family of operating systems, as well as mobile phones running iOS and Android. The API is open for anyone to utilize, allowing clients to be built for any operating system that can connect to the internet. === Integration with Fediverse === Mastodon uses the ActivityPub protocol for federation; this allows users to communicate between independent Mastodon instances and other ActivityPub compatible services. Thus, Mastodon is generally considered to be a part of the Fediverse. Services utilizing the ActivityPub protocol exist which allow for searching all posts on all instances as long as users opt-in. For similar reasons, only hashtags can appear in a Mastodon instance's trending topics, not arbitrary popular words. Trending topics vary between instances, since individual instances are aware of different subsets of posts from the whole fediverse. === Security concerns === While Mastodon's decentralized structure is one of its most distinctive features, it also poses additional security challenges. Since many Mastodon instances are run by volunteers, some security experts are concerned about data security and responsiveness to new threats and vulnerabilities across the network, considering the difficulty of configuring and maintaining an instance as well as uneven skill levels among administrators. Administrators of an instance also have access to the private information of any users that are either registered with that instance or have federated
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1DayLater

1DayLater was a free, web-based software that was focused on professional invoicing. The company was formed in 2009 and closed in October 2013. The main function of 1DayLater was to help users create invoices for clients. It could also be used to track time and other expenses, work to budgets, and to track projects. Multiple users could simultaneously work on the same projects together. PC Magazine (PCMag) voted 1DayLater as one of the 'Best Free Software of 2010'. == History == The software was developed by two brothers, Paul and David King; after they experienced similar frustrations while working freelance, the brothers wanted to create a product that would let them track time, expenses and business miles in a single online location. == Media coverage == 1DayLater had the following press coverage: BBC Webscape (July 2010) - Kate Russell gives her latest selection of the best sites on the World Wide Web PCMag (March 2010) - The best free software of 2010 Lifehacker (February 2010) - "A worthy addition to our 'Top Ten Tips and Tools for Freelancers'" Gigaom (February 2010) - Taking a closer look with 1DayLater The Journal (May 2009) - "Top Ten Brands of the North East" (UK) Techcrunch (January 2009) - "A 'feisty time tracking solution from the North East of England'"
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Focus recovery based on the linear canonical transform

For digital image processing, the Focus recovery from a defocused image is an ill-posed problem since it loses the component of high frequency. Most of the methods for focus recovery are based on depth estimation theory. The Linear canonical transform (LCT) gives a scalable kernel to fit many well-known optical effects. Using LCTs to approximate an optical system for imaging and inverting this system, theoretically permits recovery of a defocused image. == Depth of field and perceptual focus == In photography, depth of field (DOF) means an effective focal length. It is usually used for stressing an object and deemphasizing the background (and/or the foreground). The important measure related to DOF is the lens aperture. Decreasing the diameter of aperture increases focus and lowers resolution and vice versa. == The Huygens–Fresnel principle and DOF == The Huygens–Fresnel principle describes diffraction of wave propagation between two fields. It belongs to Fourier optics rather than geometric optics. The disturbance of diffraction depends on two circumstance parameters, the size of aperture and the interfiled distance. Consider a source field and a destination field, field 1 and field 0, respectively. P1(x1,y1) is the position in the source field, P0(x0,y0) is the position in the destination field. The Huygens–Fresnel principle gives the diffraction formula for two fields U(x0,y0), U(x1,y1) as following: U ( x 0 , y 0 ) = 1 j λ ∫ ∫ U ( x 1 , y 1 ) e j k r 01 r 01 cos ⁡ θ d x 1 d y 1 {\displaystyle \mathbf {U} (x_{0},y_{0})={\frac {1}{j\lambda }}\int \!\int \mathbf {U} (x_{1},y_{1}){\frac {e^{jkr_{01}}}{r_{01}}}\cos \theta dx_{1}dy_{1}} where θ denotes the angle between r 01 {\displaystyle r_{01}} and z {\displaystyle z} . Replace cos θ by r 01 z {\displaystyle {\frac {r_{01}}{z}}} and r 01 {\displaystyle r_{01}} by [ ( x 0 − x 1 ) 2 + ( y 0 − y 1 ) 2 + z 2 ] 1 / 2 {\displaystyle [(x_{0}-x_{1})^{2}+(y_{0}-y_{1})^{2}+z^{2}]^{1/2}} we get U ( x 0 , y 0 ) = 1 j λ z ∫ ∫ U ( x 1 , y 1 ) exp ⁡ ( j k z [ 1 + ( x 0 − x 1 z ) 2 + ( y 0 − y 1 z ) 2 ] 1 / 2 ) 1 + ( x 0 − x 1 z ) 2 + ( y 0 − y 1 z ) 2 d x 1 d y 1 {\displaystyle \mathbf {U} (x_{0},y_{0})={\frac {1}{j\lambda z}}\int \!\int \mathbf {U} (x_{1},y_{1}){\frac {\exp(jkz[1+({\frac {x_{0}-x_{1}}{z}})^{2}+({\frac {y_{0}-y_{1}}{z}})^{2}]^{1/2})}{1+({\frac {x_{0}-x_{1}}{z}})^{2}+({\frac {y_{0}-y_{1}}{z}})^{2}}}dx_{1}dy_{1}} The further distance z or the smaller aperture (x1,y1) causes a greater diffraction. A larger DOF can lead to a more effective focused wave distribution. This seems to be a conflict. Here are the notations: Diffraction In a real imaging environment, the depths of objects comparing to the aperture are usually not enough to lead to serious diffraction. However, a long enough depth of the object can truly blurs the image. Effective Focus Small aperture, small blurring radius, few wave information. Loses details in comparing to a large aperture. In conclusion, diffraction explains a micro behavior whereas DOF shows a macro behavior. Both of them are related to aperture size. == Linear canonical transform == As the meaning of "canonical", the linear canonical transform (LCT) is a scalable transform that connects to many important kernels such as the Fresnel transform, Fraunhofer transform and the fractional Fourier transform. It can be easily controlled by its four parameters, a, b, c, d (3 degrees of freedom). The definition: L M ( f ( u ) ) = ∫ L M ( u , u ′ ) f ( u ′ ) d u ′ {\displaystyle L_{M}(f(u))=\int L_{M}(u,u')f(u')du'} where L M ( u , u ′ ) = { 1 b e − j π / 4 e [ j π ( d b u 2 ) − 2 1 b u u ′ + a b u ′ 2 ] , if b ≠ 0 d e j 2 c d u 2 δ ( u ′ − d u ) , if b = 0 {\displaystyle L_{M}(u,u')={\begin{cases}{\sqrt {\frac {1}{b}}}e^{-j\pi /4}e^{[j\pi ({\frac {d}{b}}u^{2})-2{\frac {1}{b}}uu'+{\frac {a}{b}}u'^{2}]},&{\mbox{if }}b\neq 0\\{\sqrt {d}}e^{{\frac {j}{2}}cdu^{2}}\delta (u'-du),&{\mbox{if }}b=0\end{cases}}} Consider a general imaging system with object distance z0, focal length of the thin lens f and an imaging distance z1. The effect of the propagation in freespace acts as nearly a chirp convolution, that is, the formula of diffraction. Besides, the effect of the propagation in thin lens acts as a chirp multiplication. The parameters are all simplified as paraxial approximations while meeting the freespace propagation. It does not consider aperture size. From the properties of the LCT, it is possible to obtain those 4 parameters for this optical system as: [ 1 − z 1 f λ z 0 − λ z 0 z 1 f + λ z 1 − 1 λ f 1 − z 0 f ] {\displaystyle {\begin{bmatrix}1-{\frac {z_{1}}{f}}\quad &\lambda z_{0}-{\frac {\lambda z_{0}z_{1}}{f}}+\lambda z_{1}\\-{\frac {1}{\lambda f}}\quad &1-{\frac {z_{0}}{f}}\end{bmatrix}}} Once the values of z1, z0 and f are known, the LCT can simulate any optical system.
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