In mathematics, Legendre moments are a type of image moment and are achieved by using the Legendre polynomial. Legendre moments are used in areas of image processing including: pattern and object recognition, image indexing, line fitting, feature extraction, edge detection, and texture analysis. Legendre moments have been studied as a means to reduce image moment calculation complexity by limiting the amount of information redundancy through approximation. == Legendre moments == Source: With order of m + n, and object intensity function f(x,y): L m n = ( 2 m + 1 ) ( 2 n + 1 ) 4 ∫ − 1 1 ∫ − 1 1 P m ( x ) P n ( y ) f ( x , y ) d x d y {\displaystyle L_{mn}={\frac {(2m+1)(2n+1)}{4}}\int \limits _{-1}^{1}\int \limits _{-1}^{1}P_{m}(x)P_{n}(y)f(x,y)\,dx\,dy} where m,n = 1, 2, 3, ...∞ with the nth-order Legendre polynomials being: P n ( x ) = ∑ k = 0 n a k , n x k = ( − 1 ) n 2 n n ! ( d d x ) [ ( 1 − x 2 ) n ] {\displaystyle P_{n}(x)=\sum _{k=0}^{n}a_{k,n}x^{k}={\frac {(-1)^{n}}{2^{n}n!}}\left({\frac {d}{dx}}\right)[(1-x^{2})^{n}]} which can also be written: P n ( x ) = ∑ k = 0 D ( n ) ( − 1 ) k ( 2 n − 2 k ) ! 2 n k ! ( n − k ) ! ( n − 2 k ) ! x n − 2 k = ( 2 n ) ! 2 n ( n ! ) 2 x n − ( 2 n − 2 ) ! 2 n 1 ! ( n − 1 ) ! ( n − 2 ) ! x n − 2 + ⋯ {\displaystyle {\begin{aligned}P_{n}(x)&=\sum _{k=0}^{D(n)}(-1)^{k}{\frac {(2n-2k)!}{2^{n}k!(n-k)!(n-2k)!}}x^{n-2k}\\[5pt]&={\frac {(2n)!}{2^{n}(n!)^{2}}}x^{n}-{\frac {(2n-2)!}{2^{n}1!(n-1)!(n-2)!}}x^{n-2}+\cdots \end{aligned}}} where D(n) = floor(n/2). The set of Legendre polynomials {Pn(x)} form an orthogonal set on the interval [−1,1]: ∫ − 1 1 P n ( x ) P m ( x ) d x = 2 2 n + 1 δ n m {\displaystyle \int _{-1}^{1}P_{n}(x)P_{m}(x)\,dx={\frac {2}{2n+1}}\delta _{nm}} A recurrence relation can be used to compute the Legendre polynomial: ( n + 1 ) P n + 1 ( x ) − ( 2 n + 1 ) x P n ( x ) + n P n − 1 ( x ) = 0 {\displaystyle (n+1)P_{n+1}(x)-(2n+1)xP_{n}(x)+nP_{n-1}(x)=0} f(x,y) can be written as an infinite series expansion in terms of Legendre polynomials [−1 ≤ x,y ≤ 1.]: f ( x , y ) = ∑ m = 0 ∞ ∑ n = 0 ∞ λ m n P m ( x ) P n ( y ) {\displaystyle f(x,y)=\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }\lambda _{mn}P_{m}(x)P_{n}(y)}
Generative design
Generative design is an iterative design process that uses software to generate outputs that fulfill a set of constraints iteratively adjusted by a designer. Whether a human, test program, or artificial intelligence, the designer algorithmically or manually refines the feasible region of the program's inputs and outputs with each iteration to fulfill evolving design requirements. By employing computing power to evaluate more design permutations than a human alone is capable of, the process is capable of producing an optimal design that mimics nature's evolutionary approach to design through genetic variation and selection. The output can be images, sounds, architectural models, animation, and much more. It is, therefore, a fast method of exploring design possibilities that is used in various design fields such as art, architecture, communication design, and product design. Generative design has become more important, largely due to new programming environments or scripting capabilities that have made it relatively easy, even for designers with little programming experience, to implement their ideas. Additionally, this process can create solutions to substantially complex problems that would otherwise be resource-exhaustive with an alternative approach, making it a more attractive option for problems with a large or unknown solution set. It is also facilitated with tools in commercially available CAD packages. Not only are implementation tools more accessible, but also tools leveraging generative design as a foundation. Recent advancements have led to the development of Deep Generative Design, a framework that integrates topology optimization with deep learning models, such as Generative Adversarial Networks (GANs). Unlike traditional evolutionary methods that primarily focus on engineering performance, this approach uses deep generative models to enhance aesthetic diversity and novelty while simultaneously satisfying engineering constraints. For instance, research by Oh et al. (2019) proposed a framework using Boundary Equilibrium GANs (BEGAN) to generate diverse design options which are then refined through density-based topology optimization, allowing for the exploration of complex design spaces that balance structural integrity with visual variation. In practice, generative design does not solely aim to produce a single optimal solution, but involves iteratively refining the design problem by modifying parameters, constraints, and evaluation criteria within a computational model, resulting in multiple design alternatives from which the designer selects. == Use in architecture == Generative design in architecture is an iterative design process that enables architects to explore a wider solution space with more possibility and creativity. Architectural design has long been regarded as a wicked problem. Compared with traditional top-down design approach, generative design can address design problems efficiently, by using a bottom-up paradigm that uses parametric-defined rules to generate complex solutions. The solution itself then evolves to a good, if not optimal, solution. The advantage of using generative design as a design tool is that it does not construct fixed geometries, but take a set of design rules that can generate an infinite set of possible design solutions. The generated design solutions can be more sensitive, responsive, and adaptive to the problem. Generative design involves rule definition and result analysis that are integrated with the design process. By defining parameters and rules, the generative approach is able to provide optimized solution for both structural stability and aesthetics. Possible design algorithms include cellular automata, shape grammar, genetic algorithm, space syntax, and most recently, artificial neural network. Due to the high complexity of the solution generated, rule-based computational tools, such as finite element method and topology optimisation, are preferred to evaluate and optimise the generated solution. The iterative process provided by computer software enables the trial-and-error approach in design, and involves architects interfering with the optimisation process. Historically precedent work includes Antoni Gaudí's Sagrada Família, which used rule based geometrical forms for structures, and Buckminster Fuller's Montreal Biosphere where the rules were designed to generate individual components, rather than the final product. More recent generative-design cases include Foster and Partners' Queen Elizabeth II Great Court, where the tessellated glass roof was designed using a geometric schema to define hierarchical relationships, and then the generated solution was optimized based on geometrical and structural requirements. == Use in sustainable design == Generative design in sustainable design is an effective approach addressing energy efficiency and climate change at the early design stage, recognizing buildings contribute to approximately one-third of global greenhouse gas emissions and 30%-40% of total building energy use. It integrates environmental principles with algorithms, enabling exploration of countless design alternatives to enhance energy performance, reduce carbon footprints, and minimize waste. A key feature of generative design in sustainable design is its ability to incorporate Building Performance Simulations (BPS) into the design process. Simulation programs such as EnergyPlus, Ladybug Tools,, and so on, combined with generative algorithms, can optimize design solutions for cost-effective energy use and zero-carbon building designs. For example, the GENE_ARCH system used a Pareto algorithm with building energy simulation for the whole building design optimization. Generative design has improved sustainable facade design, as illustrated by the algorithm of cellular automata and daylight simulations in adaptive facade design. In addition, genetic algorithms were used with radiation simulations for energy-efficient photo-voltaic (PV) modules on high-rise building facades. Generative design is also applied to life cycle analysis (LCA), as demonstrated by a framework using grid search algorithms to optimize exterior wall design for minimum environmental impact. Multi-objective optimization embraces multiple diverse sustainability goals, such as interactive kinetic louvers using biomimicry and daylight simulations to enhance daylight, visual comfort, and energy efficiency. The study of PV and shading systems can maximize on-site electricity, improve visual quality, and daylight performance. Artificial intelligence (AI) and machine learning (ML) further improve computation efficiency in complex climate-responsive sustainable design. One study employed reinforcement learning to identify the relationship between design parameters and energy use for a sustainable campus, while other studies tried hybrid algorithms, such as using the genetic algorithm and GANs to balance daylight illumination and thermal comfort under different roof conditions. Other popular AI tools were also integrated, including deep reinforcement learning (DRL) and computer vision (CV), to generate an urban block according to direct sunlight hours and solar heat gains. These AI-driven generative design methods enable faster simulations and design decision making, resulting in designs that are environmentally responsible. == Use in additive manufacturing == Additive manufacturing (AM) is a process that creates physical models directly from three-dimensional (3D) data by joining materials layer by layer. It is used in industries to produce a variety of end-use parts, which are final components designed for direct application in products or systems. AM provides design flexibility and enables material reduction in lightweight applications, such as aerospace, automotive, medical, and portable electronic devices, where minimizing weight is critical for performance. Generative design, one of the four key methods for lightweight design in AM, is commonly applied to optimize structures for specific performance requirements. Generative design can help create optimized solutions that balance multiple objectives, such as enhancing performance while minimizing cost. In design for additive manufacturing (DfAM), multi-objective topology optimization is used to generate a set of candidate solutions. Designers then assess these options using their expertise and key performance indicators (KPIs) to select the best option for implementation. However, integrating AM constraints (e.g., speed of build, materials, build envelope, and accuracy) into generative design remains challenging, as ensuring all solutions are valid is complex. Balancing multiple design objectives while limiting computational costs adds further challenges for designers. To overcome these difficulties, researchers proposed a generative design method with manufacturing validation to improve decision-making efficiency. This method starts with a cons
Collateral freedom
Collateral freedom is an anti-censorship strategy that attempts to make it economically prohibitive for censors to block content on the Internet. This is achieved by hosting content on cloud services that are considered by censors to be "too important to block", and then using encryption to prevent censors from identifying requests for censored information that is hosted among other content, forcing censors to either allow access to the censored information or take down entire services.
Intrapixel and Interpixel processing
Intrapixel and Interpixel processing is used in the processing of computers graphics, as well as sensors and images in equipment such as cameras. For computer graphics, CMOS sensor processing is done in pixel level. This process includes two general categories: intrapixel processing, where the processing is performed on the individual pixel signals, and interpixel processing, where the processing is performed locally or globally on signals from several pixels. The purpose of interpixel processing is to perform early vision processing, not merely to capture images. Intrapixel and Interpixel processing is an integral part of spatial processing within the earth Mixed Spatial Attraction Model. This also includes use within hyperspectral image processing.
List of security hacking incidents
This list of security hacking incidents covers important or noteworthy events in the history of security hacking and cracking. == 1900 == === 1903 === Magician and inventor Nevil Maskelyne disrupts John Ambrose Fleming's public demonstration of Guglielmo Marconi's purportedly secure wireless telegraphy technology, sending insulting Morse code messages through the auditorium's projector. == 1930s == === 1932 === Polish cryptologists Marian Rejewski, Henryk Zygalski and Jerzy Różycki broke the Enigma machine code. === 1939 === Alan Turing, Gordon Welchman and Harold Keen worked together to develop the codebreaking device Bombe (based off of Rejewski's work on Bomba). The Enigma machine's use of a reliably small key space makes it vulnerable to brute force attacks. == 1940s == === 1943 === René Carmille, comptroller general of the Vichy French Army, hacked the punch card system used by the Nazis to locate Jews. === 1949 === The theory that underlies computer viruses was first made public in 1949, when computer pioneer John von Neumann presented a paper titled "Theory and Organization of Complicated Automata". In the paper, von Neumann speculated that computer programs could reproduce themselves. == 1950s == === 1955 === At MIT, "hack" first came to mean playing with machines. An April 1955 meeting of the Tech Model Railroad Club has one say that "Mr. Eccles requests that anyone working or hacking on the electrical system turn the power off to avoid fuse blowing." === 1957 === Joe "Joybubbles" Engressia, a blind seven-year-old boy with perfect pitch, discovered that whistling the fourth E above middle C (a frequency of 2600 Hz) would interfere with AT&T's automated telephone systems, thereby inadvertently opening the door for phreaking. == 1960s == Various phreaking boxes are used to interact with automated telephone systems. === 1963 === The first ever reference to malicious hacking is 'phreaking' in MIT's student newspaper, The Tech, containing hackers tying up the lines with Harvard, configuring the PDP-1 to make free calls, war dialing and accumulating large phone bills. === 1965 === William D. Mathews from MIT finds a vulnerability in a CTSS running on an IBM 7094. The standard text editor on the system was designed to be used by one user at a time, working in one directory, and so it created a temporary file with a constant name for all instances of the editor. The flaw was discovered when two system programmers were editing at the same time and the temporary files for the message of the day and the password file became swapped, causing the contents of the system CTSS password file to display to any user logging into the system. === 1967 === The first known incidence of network penetration hacking took place when members of a computer club at a suburban Chicago high school were provided access to IBM's APL network. In the fall of 1967, IBM (through Science Research Associates) approached Evanston Township High School with the offer of four 2741 Selectric teletypewriter-based terminals with dial-up modem connectivity to an experimental computer system which implemented an early version of the APL programming language. The APL network system was structured into workspaces which were assigned to various clients using the system. Working independently, the students quickly learned the language and the system. They were free to explore the system, often using existing code available in public workspaces as models for their own creations. Eventually, curiosity drove the students to explore the system's wider context. This first informal network penetration effort was later acknowledged as helping harden the security of one of the first publicly accessible networks:Science Research Associates undertook to write a full APL system for the IBM 1500. They modeled their system after APL/360, which had by that time been developed and seen substantial use inside of IBM, using code borrowed from MAT/1500 where possible. In their documentation, they acknowledge their gratitude to "a number of high school students for their compulsion to bomb the system". This was an early example of a kind of sportive, but very effective, debugging that was often repeated in the evolution of APL systems. == 1970s == === 1971 === John T. Draper (later nicknamed Captain Crunch), his friend Joe Engressia (also known as Joybubbles), and blue box phone phreaking hit the news with an Esquire magazine feature story. === 1979 === Kevin Mitnick breaks into his first major computer system, the Ark, which was the computer system Digital Equipment Corporation (DEC) used for developing their RSTS/E operating system software. == 1980s == === 1980 === The FBI investigates a breach of security at National CSS (NCSS). The New York Times, reporting on the incident in 1981, describes hackers as: Technical experts, skilled, often young, computer programmers who almost whimsically probe the defenses of a computer system, searching out the limits and the possibilities of the machine. Despite their seemingly subversive role, hackers are a recognized asset in the computer industry, often highly prized. The newspaper describes white hat activities as part of a "mischievous but perversely positive 'hacker' tradition". When a National CSS employee revealed the existence of his password cracker, which he had used on customer accounts, the company chastised him not for writing the software but for not disclosing it sooner. The letter of reprimand stated that "The Company realizes the benefit to NCSS and in fact encourages the efforts of employees to identify security weaknesses to the VP, the directory, and other sensitive software in files". === 1981 === Chaos Computer Club forms in Germany. Ian Murphy, aka Captain Zap, was the first cracker to be tried and convicted as a felon. Murphy broke into AT&T's computers in 1981 and changed the internal clocks that metered billing rates. People were getting late-night discount rates when they called at midday. Of course, the bargain-seekers who waited until midnight to call long distance were hit with high bills. === 1983 === The 414s break into 60 computer systems at institutions ranging from the Los Alamos National Laboratory to Manhattan's Memorial Sloan-Kettering Cancer Center. The incident appeared as the cover story of Newsweek with the title "Beware: Hackers at play". As a result, the U.S. House of Representatives held hearings on computer security and passed several laws. The group KILOBAUD is formed in February, kicking off a series of other hacker groups that formed soon after. The movie WarGames introduces the wider public to the phenomenon of hacking and creates a degree of mass paranoia about hackers and their supposed abilities to bring the world to a screeching halt by launching nuclear ICBMs. The U.S. House of Representatives begins hearings on computer security hacking. In his Turing Award lecture, Ken Thompson mentions "hacking" and describes a security exploit that he calls a "Trojan horse". === 1984 === Someone calling himself Lex Luthor founds the Legion of Doom. Named after a Saturday morning cartoon, the LOD had the reputation of attracting "the best of the best"—until one of the most talented members called Phiber Optik feuded with Legion of Doomer Erik Bloodaxe and got 'tossed out of the clubhouse'. Phiber's friends formed a rival group, the Masters of Deception. The Comprehensive Crime Control Act gives the Secret Service jurisdiction over computer fraud. The Cult of the Dead Cow forms in Lubbock, Texas, and begins publishing its underground ezine. The hacker magazine 2600 begins regular publication, right when TAP was putting out its final issue. The editor of 2600, "Emmanuel Goldstein" (whose real name is Eric Corley), takes his handle from the leader of the resistance in George Orwell's Nineteen Eighty-Four. The publication provides tips for would-be hackers and phone phreaks, as well as commentary on the hacker issues of the day. Today, copies of 2600 are sold at most large retail bookstores. The Chaos Communication Congress, the annual European hacker conference organized by the Chaos Computer Club, is held in Hamburg, Germany. William Gibson's groundbreaking science fiction novel Neuromancer, about "Case", a futuristic computer hacker, is published. Considered the first major cyberpunk novel, it brought into hacker jargon such terms as "cyberspace", "the matrix", "simstim", and "ICE". === 1985 === KILOBAUD is re-organized into P.H.I.R.M. and begins sysopping hundreds of bulletin board systems (BBSs) throughout the United States, Canada, and Europe. The online 'zine Phrack is established. The Hacker's Handbook is published in the UK. The FBI, Secret Service, Middlesex County NJ Prosecutor's Office and various local law enforcement agencies execute seven search warrants concurrently across New Jersey on July 12, 1985, seizing equipment from BBS operators and users alike for "complicity in computer theft", under a n
Ibotta
Ibotta, Inc. is an American mobile technology company headquartered in Denver, Colorado. Founded in 2011, the company offers cash back rewards on various purchases through its Ibotta Performance Network and direct to consumer app. Ibotta partners with CPG (consumer packaged goods) brands and network publishers to provide these rewards. As of 2024, the company operates solely in the United States. The company's rewards-as-a-service offering, the Ibotta Performance Network, went live in 2022. In August 2019, Ibotta received a $1 billion valuation after its Series D funding, and in 2023, the company surpassed $1.5 billion cash rewards paid to over 50 million consumers since the company's founding. Ibotta became a publicly traded company in April 2024 with a listing on the New York Stock Exchange. As of September 2025, Ibotta is trading at approximately $27.13 per share, marking a 69% decline from its initial public offering price of $88 per share on April 18, 2024. == History == === Founding through early 2019 === Ibotta was founded by current CEO Bryan Leach. The company was incorporated in 2011 and the app launched to both the App Store and Google Play stores in 2012. Early investors included entrepreneur and computer scientist Jim Clark and Tom “TJ” Jermoluk, Chairman of @Home Network. In 2015, Ibotta expanded beyond item level grocery, adding the ability to get cash back on in-store retail purchases. In 2016, in-app mobile commerce began, allowing users to navigate from the Ibotta app to its partners' apps to earn cash back on purchases. In 2016 with a Series C investment, Ibotta had raised over $73 million in funding. In March of that year, Ibotta partnered with Anheuser-Busch to offer cash back for adults who purchased its products. In May, the company partnered with LiveRamp so that companies could use their CRM data to create segmented, personalized campaigns. At the time, the company had around 200 full- and part-time employees and moved from offices in Lower Downtown Denver (LoDo) to a 40,000-square-foot office in the central Denver business district. A year later, the company had to expand to a second floor as it added almost another 100 employees. In 2017, Ibotta added cash back for Uber to its app as well as cash back rewards for online and mobile purchases. In 2018, Ibotta was listed on the Inc. 5,000 list as one of the fastest growing private companies in the U.S. A year later, in January 2019, the Ibotta app had been downloaded more than 30 million times with users receiving a reported $500 million in cash back rewards. That year, Ibotta was the largest mobile company in Colorado with six million monthly active users. === August 2019 to present === In August 2019, Ibotta was valued at $1 billion, following a Series D round of funding. The round was led by Koch Disruptive Technologies, a subsidiary of Koch Industries. 2019 was also the year the company introduced Pay with Ibotta, which allowed users to complete purchases at key retailers on the Ibotta app and earn instant cash back in the process. With that new service, users were able to enter their purchase total and use a QR code to checkout and receive immediate cash back. In 2020, the company partnered with Trees for the Future to plant up to 1 million trees as part of an Earth Month campaign to raise awareness about the waste of unused paper coupons. In response to the COVID-19 pandemic, Ibotta partnered with CPG brands in their “Here to Help” campaign and together committed over $10 million in cash back to American consumers. The company added the ability to earn cash back from online grocery pick-up and delivery orders. Later that year, Ibotta started its free Thanksgiving program, providing users with 100% cash back on select groceries needed for a Thanksgiving meal. By 2022, the company had provided approximately 10 million Thanksgiving meals. In 2021, Ibotta acquired the company OctoShop (originally InStok), a shopping browser extension company. The OctoShop app enables users to compare prices across stores and set restock and price-drop alerts. In April 2022, the Ibotta Performance Network (IPN) was launched. The IPN allows brands to deliver digital offers to consumers through third party publishers. Retailers including Walmart, Dollar General and Family Dollar, food delivery services including Instacart, and convenience stores including Shell are all part of the Ibotta Performance Network. This pay-per-sales or success-based performance network reaches over 200 million consumers. On April 18, 2024, Ibotta had its initial public offering (IPO), trading on the New York Stock Exchange (NYSE) under the ticker symbol IBTA. It was the largest technology IPO in Colorado history. In October 2025, Ibotta announced a partnership with technology and analytics company Circana, integrating Circana's Household Lift measurement into Ibotta campaigns to give CPG brands an increased understanding of the impact of their promotional campaigns. On November 3, 2025, Ibotta launched LiveLift, a tool for companies to measure the return on investment of digital promotions, in order to optimize performance marketing goals. === Athletic partnerships === Ibotta became the official jersey patch partner of the New Orleans Pelicans, a professional men's basketball team in the National Basketball Association (NBA), for the 2020–2021 and 2023–2024 seasons. Ibotta became the official jersey patch partner of the 2023 NBA champion Denver Nuggets baskeetball team beginning in the 2023–2024 season. In March 2023, F1 driver Logan Sargeant, the first U.S. racer to compete in F1 since 2015, partnered with Ibotta. The Ibotta logo was displayed on Sargeant's racing helmet throughout his F1 career. In June 2023, UConn Huskies women's basketball player Paige Bueckers entered into a "name, image, and likeness" (NIL) promotional agreement with Ibotta. According to a press release by Ibotta, the company has agreements with The Brandr Group, which finds NIL opportunities for women college athletes, and the Pearpop social media marketing platform to promote Ibotta. == Legal issues == In April 2025, shareholders filed a class action lawsuit—Fortune v. Ibotta, Inc., in the U.S. District Court for the District of Colorado (Case No. 25-cv-01213)—alleging that the registration statement in connection with Ibotta’s April 2024 initial public offering omitted material information. The complaint claims that, although Ibotta disclosed detailed terms for its contract with Walmart Inc., it failed to warn investors that its agreement with The Kroger Co., its second-largest client, was terminable at will and thus could be canceled without warning, creating a misleading impression of stability.
Tokken
Tokken is a payment system and mobile app most known for being a legal and secure option for businesses transactions within the cannabis industry, because of its compliance with bank requirements. The startup company was created by Lamine Zarrad, a former regulator at the Office of the Comptroller of the Currency. == Operability == In order for a person to start using the app, they need to provide evidence, in the form of bioidentification data and mobile carrier records, that they can legally purchase weed. After they have been verified, customers can pay directly through the app at any dispensary that is using Tokken. Tokken turns credit card transactions into a digital token, which can be exchanged back for money that can later be deposited into a bank account. All transactions are logged publicly through a blockchain leger, making the process both anonymous and verified. === Banking services === Tokken has a "pay taxes" function which enables dispensaries to pay their taxes directly to the department.