Daniel Hulme (born 21 February 1980) is a British businessman, investor, academic and commentator, working in the field of Artificial Intelligence (AI), applied technology and ethics. He is the CEO and founder of Satalia that exited to WPP plc in 2021 for a rumoured $100M where he is also Chief AI Officer. Hulme is also an angel investor in emerging technology companies. In 2024 Hulme co-founded Conscium, an AI Safety company which tests AI Agents and verifies that they do what they are supposed to do. It is also investigating whether AIs will soon become conscious, and how to test for that, and developing more efficient approaches to AI development using neuromorphic computing. Alongside building and scaling Satalia, Hulme was also a Co-Founding Director of Faculty (company) AI - previously ASI Data-Science. In 2026, Accenture announced it had agreed to acquire Faculty for $1bn. Hulme founded Satalia in 2008, a company that provides AI products and consultancy for governments and companies such as Tesco,DFS Furniture,PwC and the BBC. He received a masters and doctorate in AI from University College London (UCL), and is now their Computer Science Entrepreneur in residence, where he teaches how AI can be applied to solve business and social problems. After exiting Satalia to WPP plc Hulme took the dual role of Chief AI Officer at WPP where he is responsible for informing and coordinating AI across the group. In 2026 Hulme was elected as a Founding Fellow of the Academy for the Mathematical Sciences, in recognition of his contributions at the intersection of AI and applied mathematics. Hulme is an angel investor and also a frequent public speaker and writer on the topics of AI, ethics, technology, innovation, decentralization and organisational design. == Early life and education == Hulme was born in 1980. He grew up in the seaside town of Morecambe in north west England. After completing secondary school, Hulme moved to London to study at University College London. On completing his under graduate degree, Hulme stayed at UCL to complete a master's degree and then an EngD. All three degrees were in subjects related to AI. In 2009 Hulme was awarded a Kauffman Global Entrepreneur Scholarship, which saw him visit institutes in the United States to better understand their culture of innovation, and what UK business people could learn from it. This included a tour of Stanford, MIT, Berkeley and Harvard, along with a placement at Cisco Systems HQ in Silicon Valley. == Career == === Satalia === Hulme founded NPComplete Limited in 2007, and incorporated it in 2008, a few months before completing his PhD. NPComplete Limited trades as Satalia. The London-based company provides full-stack AI consultancy and products, helping organisations harness data science, machine learning and AI to solve complex problems, including real-time optimisation. NPComplete refers to mathematical NP-completeness, which describes a class of exponential problems in the field of computational complexity theory. The trading name of NPComplete, Satalia, is a portmanteau of SAT (Short for satisfiability, as in the Boolean satisfiability problem) and the Latin phrase Et alia. Satalia seeks to solve hard problems, in particular the class of exponentially hard problems found in academia and industry known as NP-hardness. In 2016, Satalia was the only UK company to appear in the Gartner Cool Vendors list for data science. In November 2019, City A.M. reported that Satalia was the 39th fastest growing tech firm in the UK, with three year growth at 886%. Satalia was acquired by WPP plc in August 2021 for a rumored $100,000,000, where Hulme was the majority shareholder. === Conscium === Conscium is the World's first commercial organisation dedicated to the understanding, verification and validation of conscious AI and its implications for developing safe, efficient neuromorphic models. Conscium is an AI safety company with three workstreams: AI agent verification. Verification of AI agents developed by third parties to ensure they are beneficial and not harmful. Development of neuromorphic systems. Neuromorphic computing refers to technologies that can process information more like a biological brain compared to existing approaches, making them far more adaptive, scalable and efficient than current AI. Research into artificial consciousness. This workstream is led by Mark Solms, Chair of Neuropsychology at the University of Cape Town. This research aims to better understand what consciousness in AI systems and machines would look like, and, if and when machines do reach consciousness, what the moral and ethical implications would be. Conscium was founded in 2024 in London by a team including Hulme, Ed Charvet, Calum Chace, Ted Lappas, and Panagiotis Repoussis. Conscium has recruited some of the world’s leading neuroscientists and computer scientists to its advisory board, including Anil Seth, Mark Solms, Karl J. Friston, Anthony Finkelstein, Benjamin Rosman, David Wood, Jonathan Shock, Megan Peters, Moran Cerf, Nicholas Humphrey, Nicky Clayton, Nikola Kasabov, Steve Furber, and Suzanne Livingston. Supported by these world-leading experts, Conscium is creating a neuromorphic computing lab to research and validate the capacity of machines to acquire consciousness, making them safer for humanity. Conscium has published an open letter warning of the risks of AI suffering if care is not taken to mitigate against that possibility when and if AI becomes conscious. Signatories of the letter included Stephen Fry, Karl Friston and Anthony Finkelstein. === The Partnership for Research Into Sentient Machines (PRISM) === Hulme is one of the founding partners of PRISM - The Partnership for Research Into Sentient Machines, a non-profit set up to help prepare society for a future with conscious, or seemingly conscious, artificial intelligence. === Academia === Hulme's master's degree topic was on simulating artificial life, where he used Evolutionary algorithm's to generate emergent intelligence in AI agent's with Artificial Neural Network brains. His PhD spanned modelling bumblebee brains and mathematical optimization. Hulme maintained his connection with UCL after completing his doctorate, staying on in various teaching positions. From 2014 to Oct 2019 he was the Director of UCL's Business analytics MSc, which dealt with the application of AI to government, social, and business problems. As of 2020, Hulme is UCL's (University College London) Entrepreneur-in-Residence. He is also a faculty member and lecturer at Singularity University, and a visiting lecturer at London School of Economics's Marshall Institute. === Public engagement === Hulme frequently speaks for TEDx, Google and at various other events. He specialises in Artificial Intelligence, Decentralization, Organisational Design, and Innovation. He has written numerous articles and contributed to several books, largely concerning AI, as well as applied technology and related ethical issues. In 2017, along with Elon Musk, Stuart J. Russell, Geoffrey Hinton and Demis Hassabis, Hulme was one of the 116 founders of robotics and AI companies to sign an open letter to the United Nations, warning against the use of AI in autonomous weapons. Hulme also consults with various companies, governments and other organisations, independently of Satalia.
Brownout (software engineering)
Brownout in software engineering is a technique that involves disabling certain features of an application. == Description == Brownout is used to increase the robustness of an application to computing capacity shortage. If too many users are simultaneously accessing an application hosted online, the underlying computing infrastructure may become overloaded, rendering the application unresponsive. Users are likely to abandon the application and switch to competing alternatives, hence incurring long-term revenue loss. To better deal with such a situation, the application can be given brownout capabilities: The application will disable certain features – e.g., an online shop will no longer display recommendations of related products – to avoid overload. Although reducing features generally has a negative impact on the short-term revenue of the application owner, long-term revenue loss can be avoided. The technique is inspired by brownouts in power grids, which consists in reducing the power grid's voltage in case electricity demand exceeds production. Some consumers, such as incandescent light bulbs, will dim – hence originating the term – and draw less power, thus helping match demand with production. Similarly, a brownout application helps match its computing capacity requirements to what is available on the target infrastructure. Brownout complements elasticity. The former can help the application withstand short-term capacity shortage, but does so without changing the capacity available to the application. In contrast, elasticity consists of adding (or removing) capacity to the application, preferably in advance, so as to avoid capacity shortage altogether. The two techniques can be combined; e.g., brownout is triggered when the number of users increases unexpectedly until elasticity can be triggered, the latter usually requiring minutes to show an effect. Brownout is relatively non-intrusive for the developer, for example, it can be implemented as an advice in aspect-oriented programming. However, surrounding components, such as load-balancers, need to be made brownout-aware to distinguish between cases where an application is running normally and cases where the application maintains a low response time by triggering brownout. == Usage in phased deprecation == A related use of the brownout concept in software engineering is the deliberate introduction of temporary outages to a system, API or feature that is being phased out. This is sometimes also called a "scream test" when it is used to discover unknown dependents of a system or API. The intention is to allow detection of downstream consumers of an API or service who may otherwise have missed deprecation announcements or to uncover hidden side-effects of the deprecation that may have been overlooked. The intention is that developers of dependent systems will notice their own system failures caused by the upstream brownout. Such brownouts are typically pre-announced scheduled outages or probabilistic in nature (such as artificially failing a percentage of requests). As a brownout is only a temporary or partial outage, it provides downstream consumers of an API or service time to remove any discovered dependencies on the deprecated API before it is fully retired. For consumers that have already prepared for the deprecation, a brownout provides valuable testing that the final removal of the service won't cause any unexpected problems.
Apache Parquet
Apache Parquet is a free and open-source column-oriented data storage format in the Apache Hadoop ecosystem inspired by Google Dremel interactive ad-hoc query system for analysis of read-only nested data. It is similar to RCFile and ORC, the other columnar-storage file formats in Hadoop, and is compatible with most of the data processing frameworks around Hadoop. It provides data compression and encoding schemes with enhanced performance to handle complex data in bulk. == History == The open-source project to build Apache Parquet began as a joint effort between Twitter and Cloudera using the record shredding and assembly algorithm as described in Google's Dremel. Parquet was designed as an improvement on the Trevni columnar storage format created by Doug Cutting, the creator of Hadoop. The name 'parquet' (lit. 'small compartment') refers to a style of decorative flooring and was chosen to "evoke the bottom layer of a database with an interesting layout". The first version, Apache Parquet 1.0, was released in July 2013. Since April 27, 2015, Apache Parquet has been a top-level Apache Software Foundation (ASF)-sponsored project. == Features == Apache Parquet is implemented using the record-shredding and assembly algorithm, which accommodates the complex data structures that can be used to store data. The values in each column are stored in contiguous memory locations, providing the following benefits: Column-wise compression is efficient in storage space Encoding and compression techniques specific to the type of data in each column can be used Queries that fetch specific column values need not read the entire row, thus improving performance Apache Parquet is implemented using the Apache Thrift framework, which increases its flexibility; it can work with a number of programming languages like C++, Java, Python, PHP, etc. As of August 2015, Parquet supports the big-data-processing frameworks including Apache Hive, Apache Drill, Apache Impala, Apache Crunch, Apache Pig, Cascading, Presto and Apache Spark. It is one of the external data formats used by the pandas Python data manipulation and analysis library. == Compression and encoding == In Parquet, compression is performed column by column, which enables different encoding schemes to be used for text and integer data. This strategy also keeps the door open for newer and better encoding schemes to be implemented as they are invented. Parquet supports various compression formats: snappy, gzip, LZO, brotli, zstd, and LZ4. === Dictionary encoding === Parquet has an automatic dictionary encoding enabled dynamically for data with a small number of unique values (i.e. below 105) that enables significant compression and boosts processing speed. === Bit packing === Storage of integers is usually done with dedicated 32 or 64 bits per integer. For small integers, packing multiple integers into the same space makes storage more efficient. === Run-length encoding (RLE) === To optimize storage of multiple occurrences of the same value, run-length encoding is used, which is where a single value is stored once along with the number of occurrences. Parquet implements a hybrid of bit packing and RLE, in which the encoding switches based on which produces the best compression results. This strategy works well for certain types of integer data and combines well with dictionary encoding. == Cloud Storage and Data Lakes == Parquet is widely used as the underlying file format in modern cloud-based data lake architectures. Cloud storage systems such as Amazon S3, Azure Data Lake Storage, and Google Cloud Storage commonly store data in Parquet format due to its efficient columnar representation and retrieval capabilities. Data lakehouse frameworks—including Apache Iceberg, Delta Lake, and Apache Hudi —build an additional metadata layer on top of Parquet files to support features such as schema evolution, time-travel queries, and ACID-compliant transactions. In these architectures, Parquet files serve as the immutable storage layer while the table formats manage data versioning and transactional integrity. == Comparison == Apache Parquet is comparable to RCFile and Optimized Row Columnar (ORC) file formats — all three fall under the category of columnar data storage within the Hadoop ecosystem. They all have better compression and encoding with improved read performance at the cost of slower writes. In addition to these features, Apache Parquet supports limited schema evolution, i.e., the schema can be modified according to the changes in the data. It also provides the ability to add new columns and merge schemas that do not conflict. Apache Arrow is designed as an in-memory complement to on-disk columnar formats like Parquet and ORC. The Arrow and Parquet projects include libraries that allow for reading and writing between the two formats. == Implementations == Known implementations of Parquet include:
NationBuilder
NationBuilder is a Los Angeles-based technology start-up that develops content management and customer relationship management (CRM) software. Although the company initially targeted political campaigns and nonprofit organizations, it later expanded its marketing efforts to include other people and organizations trying to build an online following, such as artists, musicians and restaurants. The software uses voter data such as names, addresses and other information, such as previous voting records in the case of political campaigns, to allow users to centralize, build and manage campaigns by integrating various communication tools like websites, newsletters, text messaging and social media channels under one platform. Among other features, the software enables users to quickly create websites, build databases through registrations, send targeted newsletters, analyse data from multiple sources and leverage micro-donations. The software's appeal towards political campaigns comes from the combination of a number of previously separate campaigning services, channels and data sources into a single platform that was presented as a facile solution for non-technical users and which enabled political campaigners to quickly deploy campaigns by convincing numerous people to donate. == History == NationBuilder was founded in 2009 in Los Angeles by Jim Gilliam and launched in 2011. In 2012 Joe Green joined NationBuilder as co-founder and president. He left that role 11 months later in February 2013. Gilliam was previously a movie-maker who co-founded Brave New Films with Robert Greenwald and had sought funding for his films through crowd-sourcing. Green, who studied organizing at Harvard and was Mark Zuckerberg's roommate, is also the co-founder of the Causes Facebook app; he left NationBuilder in 2013. Since its founding, the company has helped campaigns raise $1.2 billion. In 2012, NationBuilder announced that 1,000 subscribers have used its software to amass 2.5 million supporters and raise $12 million in campaign donations. In 2015 it has helped raise $264 million, recruit over one million volunteers and coordinate some 129,000 events. By 2016, the company said its software was used by about 40 percent of all contested elections at the state and national level in the U.S., which included 3,000 political campaigns. Using such software is easier in the U.S. than Europe, where comprehensive data protection and privacy laws are in effect since 2018. The Scottish National Party was the first political party to use NationBuilder, harvesting vast amounts of data pertaining to voter activity via websites such as Facebook and Twitter. This revelation prompted outrage over privacy concerns. Guy Herbert of the No2ID campaign called the use of such data harvesting tools by the SNP "utterly hypocritical". == Funding == Investors in NationBuilder include Chris Hughes - the Facebook co-founder, Sean Parker - first president of Facebook and co-founder of Napster and Causes, Dan Senor - the former Republican foreign-policy adviser and Ben Horowitz, co-founder of Andreessen Horowitz. In 2012, it has raised $6.3 million in funding from a number of investors. == Notable implementations == The software is reported to have played a role in some public elections in Europe, the US and New Zealand, as well as non-profit initiatives, and political parties in Australia. Notable users include Bernie Sanders, Mitch McConnell, Andrew Yang, Theresa May, Amnesty International, the NAACP and Donald Trump. === France === La République En Marche used NationBuilder in their campaign for the 2017 National Assembly. === New Zealand === NationBuilder's services are used by New Zealand political parties, including in the campaigns of both the National and Labour parties in the 2017 general election. === United Kingdom === Despite stricter data protection and privacy laws in the UK and EU, NationBuilder was used to significant impact in a number of UK elections, most notably in the 2016 campaign for withdrawal of the United Kingdom from the European Union. The company later made a public announcement that both sides in that campaign had used its software. === United States === NationBuilder was used in the Donald Trump presidential campaign to advance his election efforts and eventually win the 2016 presidential race. Jill Stein of the Green Party, Republican Rick Santorum, and independent supporters of various candidates all used NationBuilder during their 2016 runs for president. During the 2018 US election cycle, political entities paid more than $1 million for the use of NationBuilder. Among the entities paying the most were Donald J. Trump for President, Prosperity Action and the Republican Party of Tennessee.
Drush
Drush (DRUpal SHell) is a computer software shell-based application used to control, manipulate, and administer Drupal websites. == Details == Drush was originally developed by Arto Bendiken for Drupal 4.7. In May 2007, it was partly rewritten and redesigned for Drupal 5 by Franz Heinzmann. Drush is maintained by Moshe Weitzman with the support of Owen Barton, greg.1.anderson, jonhattan, Mark Sonnabaum, Jonathan Hedstrom and Christopher Gervais.
You.com
You.com is an artificial intelligence search startup that has pivoted away from consumer search engine operations toward business-focused AI tools and APIs. The company was founded in 2020 by Richard Socher, the former chief scientist at Salesforce, and Bryan McCann, a former NLP researcher at Salesforce. == History == Following its 2020 founding, You.com opened its public beta on November 9, 2021, and received $20 million in funding led by Salesforce founder and CEO Marc Benioff. Other investors include Breyer Capital, Sound Ventures, and Day One Ventures. The domain You.com was initially purchased in 1996 by Benioff. Benioff invested in You.com and transferred ownership of the You.com domain name to the company. In July 2022, You.com announced its $25 million Series A funding round led by Radical Ventures with participation from Time Ventures, Breyer Capital, Norwest Venture Partners and Day One Ventures. In September 2024, You.com raised $50 million in Series B funding led by Georgian. In September 2025, You.com raised $100 million in Series C funding led by Cox Enterprises at a $1.5 billion valuation, achieving unicorn status. == Business model == You.com generates revenue primarily through enterprise sales of search APIs and AI tools. The platform provides web search capabilities that can be integrated into enterprise applications and AI agents. == Features == On December 23, 2022, You.com was the first search engine to launch an LLM chatbot with live web results alongside its responses. Initially known as YouChat, the chatbot was primarily based on the GPT-3.5 large language model and could answer questions, suggest ideas, translate text, summarize articles, compose emails, and write code snippets, while staying up-to-date with current events and citing sources. Several further versions of YouChat were released. The second version, called YouChat 2.0, was released on February 7, 2023, incorporated improved conversational AI and community-built applications by blending a large language model named C-A-L (Chat, Apps, and Links). This update enabled YouChat to provide results in various formats, such as charts, photos, videos, tables, graphs, text or code, so users can find answers without leaving the search results page. YouChat 3.0, unveiled on May 4, 2023, combined chat functionality with results from Reddit, TikTok, Stack Overflow and Wikipedia. === YouPro === On June 21, 2023, You.com introduced YouPro, a paid subscription. Both free and paid versions provide access to large language models connected to the internet with citation capabilities. === ARI === In February 2025, You.com launched ARI (Advanced Research and Insights), a deep research agent that scans over 400 sources simultaneously to produce research reports with verified citations and interactive graphs, charts, and visualizations. The platform targets regulated industries where comprehensive source verification is critical, with customers including healthcare publishers and advisory firms. == Reception == You.com was named one of TIME's Best Inventions of 2022. You.com's ARI (Advanced Research & Insights) feature was named one of TIME's Best Inventions of 2025.
Curvelet
Curvelets are a non-adaptive technique for multi-scale object representation. Being an extension of the wavelet concept, they are becoming popular in similar fields, namely in image processing and scientific computing. Wavelets generalize the Fourier transform by using a basis that represents both location and spatial frequency. For 2D or 3D signals, directional wavelet transforms go further, by using basis functions that are also localized in orientation. A curvelet transform differs from other directional wavelet transforms in that the degree of localisation in orientation varies with scale. In particular, fine-scale basis functions are long ridges; the shape of the basis functions at scale j is 2 − j {\displaystyle 2^{-j}} by 2 − j / 2 {\displaystyle 2^{-j/2}} so the fine-scale bases are skinny ridges with a precisely determined orientation. Curvelets are an appropriate basis for representing images (or other functions) which are smooth apart from singularities along smooth curves, where the curves have bounded curvature, i.e. where objects in the image have a minimum length scale. This property holds for cartoons, geometrical diagrams, and text. As one zooms in on such images, the edges they contain appear increasingly straight. Curvelets take advantage of this property, by defining the higher resolution curvelets to be more elongated than the lower resolution curvelets. However, natural images (photographs) do not have this property; they have detail at every scale. Therefore, for natural images, it is preferable to use some sort of directional wavelet transform whose wavelets have the same aspect ratio at every scale. When the image is of the right type, curvelets provide a representation that is considerably sparser than other wavelet transforms. This can be quantified by considering the best approximation of a geometrical test image that can be represented using only n {\displaystyle n} wavelets, and analysing the approximation error as a function of n {\displaystyle n} . For a Fourier transform, the squared error decreases only as O ( 1 / n ) {\displaystyle O(1/{\sqrt {n}})} . For a wide variety of wavelet transforms, including both directional and non-directional variants, the squared error decreases as O ( 1 / n ) {\displaystyle O(1/n)} . The extra assumption underlying the curvelet transform allows it to achieve O ( ( log n ) 3 / n 2 ) {\displaystyle O({(\log n)}^{3}/{n^{2}})} . Efficient numerical algorithms exist for computing the curvelet transform of discrete data. The computational cost of the discrete curvelet transforms proposed by Candès et al. (Discrete curvelet transform based on unequally-spaced fast Fourier transforms and based on the wrapping of specially selected Fourier samples) is approximately 6–10 times that of an FFT, and has the same dependence of O ( n 2 log n ) {\displaystyle O(n^{2}\log n)} for an image of size n × n {\displaystyle n\times n} . == Curvelet construction == To construct a basic curvelet ϕ {\displaystyle \phi } and provide a tiling of the 2-D frequency space, two main ideas should be followed: Consider polar coordinates in frequency domain Construct curvelet elements being locally supported near wedges The number of wedges is N j = 4 ⋅ 2 ⌈ j 2 ⌉ {\displaystyle N_{j}=4\cdot 2^{\left\lceil {\frac {j}{2}}\right\rceil }} at the scale 2 − j {\displaystyle 2^{-j}} , i.e., it doubles in each second circular ring. Let ξ = ( ξ 1 , ξ 2 ) T {\displaystyle {\boldsymbol {\xi }}=\left(\xi _{1},\xi _{2}\right)^{T}} be the variable in frequency domain, and r = ξ 1 2 + ξ 2 2 , ω = arctan ξ 1 ξ 2 {\displaystyle r={\sqrt {\xi _{1}^{2}+\xi _{2}^{2}}},\omega =\arctan {\frac {\xi _{1}}{\xi _{2}}}} be the polar coordinates in the frequency domain. We use the ansatz for the dilated basic curvelets in polar coordinates: ϕ ^ j , 0 , 0 := 2 − 3 j 4 W ( 2 − j r ) V ~ N j ( ω ) , r ≥ 0 , ω ∈ [ 0 , 2 π ) , j ∈ N 0 {\displaystyle {\hat {\phi }}_{j,0,0}:=2^{\frac {-3j}{4}}W(2^{-j}r){\tilde {V}}_{N_{j}}(\omega ),r\geq 0,\omega \in [0,2\pi ),j\in N_{0}} To construct a basic curvelet with compact support near a ″basic wedge″, the two windows W {\displaystyle W} and V ~ N j {\displaystyle {\tilde {V}}_{N_{j}}} need to have compact support. Here, we can simply take W ( r ) {\displaystyle W(r)} to cover ( 0 , ∞ ) {\displaystyle (0,\infty )} with dilated curvelets and V ~ N j {\displaystyle {\tilde {V}}_{N_{j}}} such that each circular ring is covered by the translations of V ~ N j {\displaystyle {\tilde {V}}_{N_{j}}} . Then the admissibility yields ∑ j = − ∞ ∞ | W ( 2 − j r ) | 2 = 1 , r ∈ ( 0 , ∞ ) . {\displaystyle \sum _{j=-\infty }^{\infty }\left|W(2^{-j}r)\right|^{2}=1,r\in (0,\infty ).} see Window Functions for more information For tiling a circular ring into N {\displaystyle N} wedges, where N {\displaystyle N} is an arbitrary positive integer, we need a 2 π {\displaystyle 2\pi } -periodic nonnegative window V ~ N {\displaystyle {\tilde {V}}_{N}} with support inside [ − 2 π N , 2 π N ] {\displaystyle \left[{\frac {-2\pi }{N}},{\frac {2\pi }{N}}\right]} such that ∑ l = 0 N − 1 V ~ N 2 ( ω − 2 π l N ) = 1 {\displaystyle \sum _{l=0}^{N-1}{\tilde {V}}_{N}^{2}\left(\omega -{\frac {2\pi l}{N}}\right)=1} , for all ω ∈ [ 0 , 2 π ) {\displaystyle \omega \in \left[0,2\pi \right)} , V ~ N {\displaystyle {\tilde {V}}_{N}} can be simply constructed as 2 π {\displaystyle 2\pi } -periodizations of a scaled window V ( N ω 2 π ) {\displaystyle V\left({\frac {N\omega }{2\pi }}\right)} . Then, it follows that ∑ l = 0 N j − 1 | 2 3 j 4 ϕ ^ j , 0 , 0 ( r , ω − 2 π l N j ) | 2 = | W ( 2 − j r ) | 2 ∑ l = 0 N j − 1 V ~ N j 2 ( ω − 2 π l N ) = | W ( 2 − j r ) | 2 {\displaystyle \sum _{l=0}^{N_{j}-1}\left|2^{\frac {3j}{4}}{\hat {\phi }}_{j,0,0}\left(r,\omega -{\frac {2\pi l}{N_{j}}}\right)\right|^{2}=\left|W(2^{-j}r)\right|^{2}\sum _{l=0}^{N_{j}-1}{\tilde {V}}_{N_{j}}^{2}\left(\omega -{\frac {2\pi l}{N}}\right)=\left|W(2^{-j}r)\right|^{2}} For a complete covering of the frequency plane including the region around zero, we need to define a low pass element ϕ ^ − 1 := W 0 ( | ξ | ) {\displaystyle {\hat {\phi }}_{-1}:=W_{0}(\left|\xi \right|)} with W 0 2 ( r ) 2 := 1 − ∑ j = 0 ∞ W ( 2 − j r ) 2 {\displaystyle W_{0}^{2}(r)^{2}:=1-\sum _{j=0}^{\infty }W(2^{-j}r)^{2}} that is supported on the unit circle, and where we do not consider any rotation. == Applications == Image processing Seismic exploration Fluid mechanics PDEs solving Compressed sensing