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
Google Cloud Dataflow
Google Cloud Dataflow is a fully managed service for executing Apache Beam pipelines within the Google Cloud Platform ecosystem. Dataflow provides a fully managed service for executing Apache Beam pipelines, offering features like autoscaling, dynamic work rebalancing, and a managed execution environment. Dataflow is suitable for large-scale, continuous data processing jobs, and is one of the major components of Google's big data architecture on the Google Cloud Platform. At its core, Dataflow's architecture is designed to abstract away infrastructure management, allowing developers to focus purely on the logic of their data processing tasks. When a pipeline written using the Apache Beam SDK is submitted, Dataflow translates this high-level definition into an optimized job graph. The service then provisions and manages a fleet of Google Compute Engine workers to execute this graph in a highly parallelized and fault-tolerant manner. This serverless approach, combined with intelligent autoscaling of both the number of workers (horizontal) and the resources per worker (vertical), ensures that jobs have the precise amount of computational power needed at any given time, optimizing both performance and cost. The service's deep integration with the Google Cloud ecosystem makes it a powerful tool for a variety of use cases beyond simple data movement. For real-time analytics, Dataflow can ingest unbounded streams of data from Cloud Pub/Sub, perform complex transformations, and load results into BigQuery for immediate querying. In machine learning workflows, it is commonly used to preprocess and transform massive datasets stored in Cloud Storage, preparing them for training models in Vertex AI. This versatility makes it the central processing engine for modern ETL (Extract, Transform, Load) operations, streaming analytics, and large-scale data preparation within the cloud. == History == Google Cloud Dataflow was announced in June, 2014 and released to the general public as an open beta in April, 2015. In January, 2016 Google donated the underlying SDK, the implementation of a local runner, and a set of IOs (data connectors) to access Google Cloud Platform data services to the Apache Software Foundation. The donated code formed the original basis for Apache Beam. In August 2022, there was an incident where user timers were broken for certain Dataflow streaming pipelines in multiple regions, which was later resolved. Throughout 2023 and 2024, there have been various other updates and incidents affecting Google Cloud Dataflow, as documented in the release notes and service health history. The donation of the Dataflow SDK to the Apache Software Foundation was a pivotal moment, establishing Apache Beam as a unified, open-source programming model for defining both batch and streaming data pipelines. This strategic move decoupled the pipeline definition from the execution engine. As a result, developers could write portable data processing logic that was not locked into Google's ecosystem. A Beam pipeline can be executed on various runners, including Apache Flink, Apache Spark, and, of course, the highly optimized Google Cloud Dataflow service, providing flexibility and future-proofing data processing investments. == Features == Google Cloud Dataflow supports both batch and streaming data processing pipelines. It automatically handles resource provisioning, data sharding, and scaling according to workload, reducing manual configuration needed for large-scale data operations. == Use cases == Dataflow is used for ETL (Extract, Transform, Load) data pipelines, real-time analytics, and event stream processing for companies in industries such as finance, advertising, and IoT.
Pydio
Pydio Cells, previously known as just Pydio and formerly known as AjaXplorer, is an open-source file-sharing and synchronisation software that runs on the user's own server or in the cloud. == Presentation == The project was created by musician Charles Du Jeu (current CEO and CTO) in 2007 under the name AjaXplorer. The name was changed in 2013 and became Pydio (an acronym for Put Your Data in Orbit). In May 2018, Pydio switched from PHP to Go with the release of Pydio Cells. The PHP version reached end-of-life state on 31 December 2019. Pydio Cells runs on any server supporting a recent Go version. Windows/Linux/macOS on the Intel architecture are directly supported; a fully functional working ARM implementation is under active development. Pydio Cells has been developed from scratch using the Go programming language; release 4.0.0 introduced code refactoring to fully support the Go modular structure as well as grid computing. Nevertheless, the web-based interface of Cells is very similar to the one from Pydio 8 (in PHP), and it successfully replicates most of its features, while adding a few more. There is also a new synchronisation client (also written in Go). The PHP version has been phased out as the company's focus is moving to Pydio Cells, with community feedback on the new features. According to the company, the switch to the new environment was made "to overcome inherent PHP limitations and provide you with a future-proof and modern solution for collaborating on documents". From a technical point of view, Pydio differs from solutions such as Google Drive or Dropbox. Pydio is not based on a public cloud; instead, the software connects to the user's existing storage (such as SAN / Local FS, SAMBA / CIFS, (s)FTP, NFS, S3-compatible cloud storage, Azure Blob Storage, Google Cloud Storage) as well as to the existing user directories (LDAP / AD, OAuth2 / OIDC SSO, SAML / Azure ADFS SSO, RADIUS, Shibboleth...), which allows companies to keep their data inside their infrastructure, according to their data security policy and user rights management. The software is built in a modular perspective; up to Pydio 8, various plugins allowed administrators to implement extra features. On the server side, Pydio Cells is deployed as a collection of independent microservices communicating among themselves using gRPC and logging user actions via Activity Streams 2.0 (AS2). Pydio Cells microservices are built with the Go Micro framework (using an embedded NATS server). A standard installation will deploy all required services on the same physical server, but for the purposes of performance, reliability and high availability, these can now be spread across several different servers (even in geographically separate locations) according to the 12-factors architecture pattern. Pydio Cells is available either through a free and open-source community distribution (Pydio Cells Home), or a commercially-licensed enterprise distribution (in two variants, Pydio Cells Connect and Pydio Cells Enterprise), which add features not available in the community distribution as well as additional levels of support beyond the community forums. == Features == File sharing between different internal users and across other Pydio instances SSL/TLS Encryption WebDAV file server Creation of dedicated workspaces, for each line of business / project / client, with a dedicated user rights management for each workspace. File-sharing with external users (private links, public links, password protection, download limitation, etc.) Online viewing and editing of documents with Collabora Office (Pydio Cells Enterprise also offers OnlyOffice integration) Preview and editing of image files Integrated audio and video reader Activity stream ('timeline') for all actions taken by users Integrated chat platform Client applications are available for all major desktop and mobile platforms.
Cloud-based integration
Cloud-based integration is a form of systems integration business delivered as a cloud computing service that addresses data, process, service-oriented architecture (SOA) and application integration. == Description == Integration platform as a service (iPaaS) is a suite of cloud services enabling customers to develop, execute and govern integration flows between disparate applications. Under the cloud-based iPaaS integration model, customers drive the development and deployment of integrations without installing or managing any hardware or middleware. The iPaaS model allows businesses to achieve integration without big investment into skills or licensed middleware software. iPaaS used to be regarded primarily as an integration tool for cloud-based software applications, used mainly by small to mid-sized business. Over time, a hybrid type of iPaaS—hybrid-IT iPaaS—that connects cloud to on-premises, is becoming increasingly popular. Additionally, large enterprises are exploring new ways of integrating iPaaS into their existing IT infrastructures. Cloud integration was created to break down the data silos, improve connectivity and optimize the business process. Cloud integration has increased in popularity as the usage of Software as a Service solutions has grown. Prior to the emergence of cloud computing in the early 2000s, integration could be categorized as either internal or business to business (B2B). Internal integration requirements were serviced through an on-premises middleware platform and typically utilized a service bus to manage exchange of data between systems. B2B integration was serviced through EDI gateways or value-added network (VAN). The advent of SaaS applications created a new kind of demand which was met through cloud-based integration. Since their emergence, many such services have also developed the capability to integrate legacy or on-premises applications, as well as function as EDI gateways. The following essential features were proposed by one marketing company: Deployed on a multi-tenant, elastic cloud infrastructure Subscription model pricing (operating expense, not capital expenditure) No software development (required connectors should already be available) Users do not perform deployment or manage the platform itself Presence of integration management and monitoring features The emergence of this sector led to new cloud-based business process management tools that do not need to build integration layers - since those are now a separate service. Drivers of growth include the need to integrate mobile app capabilities with proliferating API publishing resources and the growth in demand for the Internet of things functionalities as more 'things' connect to the Internet.
Peanut App
Peanut, a product of Peanut App Ltd. is an online community for women who are planning to become pregnant, women who are pregnant, women who have had children, and women who are experiencing menopause. Profiles of potential friends are displayed to users who can swipe up to show intent to connect. Users can also connect via discussion threads, groups, and live audio conversations. The app allows users to select their stage of life (trying to conceive, pregnancy, motherhood, or menopause), so as to meet women at a similar life stage, and to discover relevant content. Peanut was founded by Michelle Kennedy shortly after she left Bumble, a female-first dating app. She has described Peanut as, "the app she wishes she had when she first became a mother". == History == Peanut was initially launched in 2017 for mothers and pregnant women. The app focuses on helping users find others with shared interests, such as spoken languages, occupations, and hobbies. It also displays a woman's life stage, such as the age of her children, or the stage of pregnancy. In 2018, it launched a community discussion feature that intended to give women an "alternative to other social platforms". In 2019, it started to serve women who are trying to conceive. In April 2021, it integrated live audio, in response to the COVID-19 pandemic, and the restrictions around in-person socializing. in September 2021, it started to include women who are navigating perimenopause, menopause, and postmenopausal. Although it had initially catered for younger women navigating into new families, a large number of users had undergone surgically or chemically induced menopause due to medical conditions. In July 2021, Peanut launched an investment micro fund, Peanut StartHER, focused on investing in women-owned businesses, as well as other historically excluded founders. == Operation == The Peanut app is a social network exclusively for women, focusing on topics of pregnancy, motherhood, fertility, and menopause. It is available on iOS and Android devices. Users must prove their identity, in keeping with the primary function of in-app safety, and then they can create a profile to interact with other users. For pregnant users, the “Bump Buddies” feature helps connect them with other Peanut users who have a similar due date, which aimed to help expecting mothers combat loneliness during the COVID-19 pandemic. Peanut users also have the option to join “Groups” ‒ sub-sections of users focused on specific topics, including (but not limited to) location, life stage, pregnancy due date, and interests or hobbies. The live voice chat feature “Pods”, enables Peanut users to socialize without the pressure of photos or video chat. It offers features such as a muted audience of listeners who need to virtually raise their hand to speak, emoji reactions, and hosts who can moderate the conversations and invite people to speak.
Box blur
A box blur (also known as a box linear filter) is a spatial domain linear filter in which each pixel in the resulting image has a value equal to the average value of its neighboring pixels in the input image. It is a form of low-pass ("blurring") filter. A 3 by 3 box blur ("radius 1") can be written as matrix 1 9 [ 1 1 1 1 1 1 1 1 1 ] . {\displaystyle {\frac {1}{9}}{\begin{bmatrix}1&1&1\\1&1&1\\1&1&1\end{bmatrix}}.} Due to its property of using equal weights, it can be implemented using a much simpler accumulation algorithm, which is significantly faster than using a sliding-window algorithm. Box blurs are frequently used to approximate a Gaussian blur. By the central limit theorem, repeated application of a box blur will approximate a Gaussian blur. In the frequency domain, a box blur has zeros and negative components. That is, a sine wave with a period equal to the size of the box will be blurred away entirely, and wavelengths shorter than the size of the box may be phase-reversed, as seen when two bokeh circles touch to form a bright spot where there would be a dark spot between two bright spots in the original image. == Extensions == Gwosdek, et al. has extended Box blur to take a fractional radius: the edges of the 1-D filter are expanded with a fraction. It makes slightly better gaussian approximation possible due to the elimination of integer-rounding error. Mario Klingemann has a "stack blur" that tries to better emulate gaussian's look in one pass by stacking weights: 1 9 [ 1 2 3 2 1 ] {\displaystyle {\frac {1}{9}}{\begin{bmatrix}1&2&3&2&1\end{bmatrix}}} The triangular impulse response it forms decomposes to two rounds of box blur. Stacked Integral Image by Bhatia et al. takes the weighted average of a few box blurs to fit the gaussian response curve. == Implementation == The following pseudocode implements a 3x3 box blur. The example does not handle the edges of the image, which would not fit inside the kernel, so that these areas remain unblurred. In practice, the issue is better handled by: Introducing an alpha channel to represent the absence of colors; Extending the boundary by filling in values, ranked by quality: Fill in a mirrored image at the border Fill in a constant color extending from the last pixel Pad in a fixed color A number of optimizations can be applied when implementing the box blur of a radius r and N pixels: The box blur is a separable filter, so that only two 1D passes of averaging 2 r + 1 pixels will be needed, one horizontal and one vertical, for each pixel. This lowers the complexity from O(Nr2) to O(Nr). In digital signal processing terminology, each pass is a moving-average filter. Accumulation. Instead of discarding the sum for each pixel, the algorithm re-uses the previous sum, and updates it by subtracting away the old pixel and adding the new pixel in the blurring range. A summed-area table can be used similarly. This lowers the complexity from O(Nr) to O(N). When being used in multiple passes to approximate a Gaussian blur, the cascaded integrator–comb filter construction allows for doing the equivalent operation in a single pass.
Fatpaint
Fatpaint is a free, online (web-based) graphic design and desktop publishing software product and image editor. It includes integrated tools for creating page layout, painting, coloring and editing pictures and photos, drawing vector images, using dingbat vector clipart, writing rich text, creating ray traced 3D text logos and displaying graphics on products from Zazzle that can be purchased or sold. Fatpaint integrates desktop publishing features with brush painting, vector drawing and custom printed products in a single Flash application. It supports the use of a pressure-sensitive pen tablet and allows the user to add images by searching Wikimedia, Picasa, Flickr, Google, Yahoo, Bing, and Fatpaint's own collection of public domain images. The completed project can be saved on Fatpaint's server or locally. Fatpaint is affiliated with Zazzle, and owned by Mersica (also the developer of MakeWebVideo). == History == Fatpaint was launched in May 2010, after five years of development by Danish-Brazilian software developer, Mario Gomes Cavalcanti. After his departure, he was involved in the development of two of Denmark's most visited websites and is responsible for developing and running Fatpaint. Partner Kenneth Christensen mastered assembler and graphics programming on the Amiga computer. He spent years with Mario on the Amiga demo scene. According to the CEO, Kenneth helped him with the Linux servers while he handled the development, administration, promotion, video production, testing and content. The founder of Fatpaint also created "Make Web Video" (or Video Maker), a web application for creating video presentations for business, families and individuals. Video Maker allows users to give out the videos for personal or business use in a simple and affordable way. == Tools == Fatpaint provides free online logo maker, graphic design, vector drawing, photo editor and paint design in English, Danish and Portuguese. === Photo Editor === Users can change photo colours by manipulating R, G, B and A channels, saturation, contrast, brightness, hue, gamma, sharpness, tint and RGBA matrix. Users can also remove unwanted background and other artifacts by using the paint tools with added effects or by cloning. Multiple photos can be combined into a single image. Users can pick different blend modes and multiple layers. Users can also extract or change parts of the photo by cropping, resizing, skewing, bending, distorting and rotating in 2D and 3D. Hence, users' graphics can be printed on custom products that can be bought and sold for personal and business purposes. === Vector Drawing === Users can choose from 5000 vector images or draw vector graphics and art from scratch, using Fatpaint's vector shape creation tools. It also provides advanced symmetric vector transformation in 2D and 3D, as well as support for colour gradients. Multiple drawings can be combined to form complex vector shapes. Different blend modes and effects are supported. Vector drawings can be cropped, resized, skewed, distorted and rotated in 2D and 3D. Similar to Fatpaint's photo editor, vector graphics can be displayed on custom printed products that can be purchased and sold by the users for personal or business uses. === Paint Design === Fatpaint has full support for Pen Tablets and users can pick pen, brush, airbrush, paint bucket, clone painting, eraser and smudging tools. Fatpaint offers 8 palettes for painting, plus 13 palettes when clone painting. Fatpaint allows users to import or create their own brushes and thousands of free clipart drawings and brush sets that have dynamic brushes, effects and blend modes. Paintings can be combined in different layers and objects. Similarly, paintings can be cropped, resized, skewed, bent, distorted and rotated in 2D and 3D. Moreover, the graphics can be displayed on custom printed products, which users can buy or sell for personal or business uses. == Top Features == 3D Text objects: Create photorealistic, ray-traced 3D text logos and images. Image objects: Paint on multiple layers, import or create your own brushes, clone painting, and painting with effects. Vector drawing objects: Create vector images using multiple paths. Rich text objects with 981 fonts. Effect objects: Blur, Drop Shadow, Glow, Gradient Glow, Bevel, Gradient Bevel, Color manipulations. Page layout: Create multiple pages with a size limit of 64 megapixels, and arrange graphical objects on created pages (each object can be up to 7.8 megapixels in size). Nest graphical objects and transform them into 2D and 3D. Skew, bend and distort images and text. Design, purchase and sell custom-printed products. Fatpaint can send the projects to a printing company. Supports pressure-sensitive pen tablets. Fonts, public domain images, cliparts, and brushes. == Compatibility == Fatpaint supports Firefox, Google Chrome, Opera, and Internet Explorer with cookies and JavaScript enabled. Other browsers may not work correctly due to their support of Java Applets. Fatpaint requires Adobe's Flash 10 or newer and Sun's Java 6 or newer. It is recommended to run on Windows 7 and on Apple and Linux if Java has been disabled. The editor only works on Firefox on Linux. Java and Flash integration do not work on Linux and Apple browsers. WikiMedia search is disabled on those browsers. Fatpaint works best with at least 2 GB RAM and 1 GB video memory, as well as a decent graphics card.