Max Welling (born 1968) is a Dutch computer scientist in machine learning at the University of Amsterdam. In August 2017, the university spin-off Scyfer BV, co-founded by Welling, was acquired by Qualcomm. He has since then served as a Vice President of Technology at Qualcomm Netherlands. He is also a Distinguished Scientist at Microsoft Research AI4Science, based in Amsterdam. Welling received his PhD in physics with a thesis on quantum gravity under the supervision of Nobel laureate Gerard 't Hooft (1998) at the Utrecht University. He has published over 250 peer-reviewed articles in machine learning, computer vision, statistics and physics, and has most notably invented variational autoencoders (VAEs), together with Diederik P Kingma. In 2025 Welling was elected member of the Royal Netherlands Academy of Arts and Sciences.
Data commingling
Data commingling, in computer science, occurs when different items or kinds of data are stored in such a way that they become commonly accessible when they are supposed to remain separated. In cloud computing, this can occur where different customer data sits on the same server. Data that is commingled can present a security vulnerability. Data commingling can also occur due to high speed data transmission mixing. In this situation, data of one security level can inadvertently or purposely be mixed with data of a lower or higher security level on the same transmission portal. Portal vehicles can be wire, fiber optics, microwave or various radio frequency transmission portals. This commingling can cause breaches of security and become a source of legal issues to any entity, corporation or individual. Data commingling can also occur when personal computers and personal software programs are used for business, security, government, etc. uses. In the early formulation stages of entities, non-profit or profit corporations, LLC's, LLP's, etc., the creation and use of stand-alone computers and stand-alone networks, "absolutely unconnected" to involved individuals, is the easiest, and safest way to prevent Data Commingling.
Digital supply chain security
Digital supply chain security refers to efforts to enhance cyber security within the supply chain. It is a subset of supply chain security and is focused on the management of cyber security requirements for information technology systems, software and networks, which are driven by threats such as cyber-terrorism, malware, data theft and the advanced persistent threat (APT). Typical supply chain cyber security activities for minimizing risks include buying only from trusted vendors, disconnecting critical machines from outside networks, and educating users on the threats and protective measures they can take. The acting deputy undersecretary for the National Protection and Programs Directorate for the United States Department of Homeland Security, Greg Schaffer, stated at a hearing that he is aware that there are instances where malware has been found on imported electronic and computer devices sold within the United States. == Examples of supply chain cyber security threats == Network or computer hardware that is delivered with malware installed on it already. Malware that is inserted into software or hardware (by various means) Vulnerabilities in software applications and networks within the supply chain that are discovered by malicious hackers Counterfeit computer hardware == Related U.S. government efforts == Comprehensive National Cyber Initiative Defense Procurement Regulations: Noted in section 806 of the National Defense Authorization Act International Strategy for Cyberspace: White House lays out for the first time the U.S.’s vision for a secure and open Internet. The strategy outlines three main themes: diplomacy, development and defense. Diplomacy: The strategy sets out to “promote an open, interoperable, secure and reliable information and communication infrastructure” by establishing norms of acceptable state behavior built through consensus among nations. Development: Through this strategy the government seeks to “facilitate cybersecurity capacity-building abroad, bilaterally and through multilateral organizations.” The objective is to protect the global IT infrastructure and to build closer international partnerships to sustain open and secure networks. Defense: The strategy calls out that the government “will ensure that the risks associated with attacking or exploiting our networks vastly outweigh the potential benefits” and calls for all nations to investigate, apprehend and prosecute criminals and non-state actors who intrude and disrupt network systems. == Related government efforts around the world == Common Criteria offers with Evaluation Assurance Level(EAL) 4 an opportunity to evaluate all relevant aspects of the digital supply chain security like the product, the development environment, IT systems security, the processes in human resource, physical security and with the module ALC_FLR.3 (Systematic Flaw Remediation) also security update processes and methods even by physical site visits. EAL 4 is mutually recognized in countries that signed the SOGIS-MRA and up to ELA 2 in countries the signed the CCRA but including ALC_FRL.3. Russia: Russia has had non-disclosed functionality certification requirements for several years and has recently initiated the National Software Platform effort based on open-source software. This reflects the apparent desire for national autonomy, reducing dependence on foreign suppliers. India: Recognition of supply chain risk in its draft National Cybersecurity Strategy. Rather than targeting specific products for exclusion, it is considering Indigenous Innovation policies, giving preferences to domestic ITC suppliers in order to create a robust, globally competitive national presence in the sector. China: Deriving from goals in the 11th Five Year Plan (2006–2010), China introduced and pursued a mix of security-focused and aggressive Indigenous Innovation policies. China is requiring an indigenous innovation product catalog be used for its government procurement and implementing a Multi-level Protection Scheme (MLPS) which requires (among other things) product developers and manufacturers to be Chinese citizens or legal persons, and product core technology and key components must have independent Chinese or indigenous intellectual property rights. == Private sector efforts == SLSA (Supply-chain Levels for Software Artifacts) is an end-to-end framework for ensuring the integrity of software artifacts throughout the software supply chain. The requirements are inspired by Google’s internal "Binary Authorization for Borg" that has been in use for the past 8+ years and that is mandatory for all of Google's production workloads. The goal of SLSA is to improve the state of the industry, particularly open source, to defend against the most pressing integrity threats. With SLSA, consumers can make informed choices about the security posture of the software they consume. == Other references == Financial Sector Information Sharing and Analysis Center International Strategy for Cyberspace (from the White House) NSTIC SafeCode Whitepaper Archived 2013-10-21 at the Wayback Machine Trusted Technology Forum and the Open Trusted Technology Provider Standard (O-TTPS) Archived 2012-01-03 at the Wayback Machine Cyber Supply Chain Security Solution Malware Implants in Firmware Supply Chain in the Software Era INFORMATION AND COMMUNICATIONS TECHNOLOGY SUPPLY CHAIN RISK MANAGEMENT TASK FORCE: INTERIM REPORT
WhatsApp Messenger, commonly known simply as WhatsApp, is an American social media, instant messaging (IM), and Voice over IP (VoIP) service accessible via desktop and mobile app. Owned by Meta Platforms, the service allows users to send text messages, voice messages, and video messages, make voice and video calls, and share images, documents, user locations, and other content. The service requires a cellular mobile telephone number to register. WhatsApp was launched in May 2009. In January 2018, WhatsApp released a standalone business app called WhatsApp Business which can communicate with the standard WhatsApp client. As of May 2025, the service had 3 billion monthly active users, making it the most used messenger app. The name of the app is meant to sound like "what's up". The service was created by WhatsApp Inc. of Mountain View, California, which was acquired by Facebook in February 2014 for approximately US$19.3 billion. It became the world's most popular messaging application in 2015, with 900 million users, and had more than 2 billion active users worldwide in February 2020. WhatsApp Business had approximately 200 million monthly users in 2023. By 2016, it had become the primary means of Internet communication in regions including the Americas, the Indian subcontinent, and large parts of Europe and Africa. == History == === 2009–2014 === WhatsApp was founded by Brian Acton and Jan Koum, former employees of Yahoo. Koum incorporated WhatsApp Inc. in California on February 24, 2009. A month earlier, Koum had purchased an iPhone, and he and Acton decided to create an app for the App Store. The idea started off as an app that would display statuses in a phone's Contacts menu, showing if a person was at work or on a call. Their discussions often took place at the home of Koum's Russian friend Alex Fishman in West San Jose. They realized that to take the idea further, they would need an iPhone developer. Fishman visited RentACoder.com, found Russian developer Igor Solomennikov, and introduced him to Koum. Koum named the app WhatsApp to sound like "what's up" and it was published on the Apple App Store and BlackBerry App World in May and June 2009 respectively. However, when early versions of WhatsApp kept crashing, Koum considered giving up and looking for a new job. Acton encouraged him to wait for a "few more months". In June 2009, when the app had been downloaded by only a handful of Fishman's Russian-speaking friends, Apple launched push technology, allowing users to be pinged even when not using the app. Koum updated WhatsApp so that everyone in the user's network would be notified when a user's status changed. This new facility, to Koum's surprise, was used by users to ping "each other with jokey custom statuses like, 'I woke up late' or 'I'm on my way.'" Fishman said, "At some point it sort of became instant messaging". WhatsApp 2.0, released for iPhone in August 2009, featured a purpose-designed messaging component; the number of active users suddenly increased to 250,000. Although Acton was working on another startup idea, he decided to join the company. In October 2009, Acton persuaded five former friends at Yahoo! to invest $250,000 in seed funding, and Acton became a co-founder and was given a stake. He officially joined WhatsApp on November 1. Koum then hired a friend in Los Angeles, Chris Peiffer, to develop a BlackBerry version, which arrived two months later. Subsequently, WhatsApp for Symbian OS was added in May 2010, and for Android OS in August 2010. In 2010 Google made multiple acquisition offers for WhatsApp, which were all declined. To cover the cost of sending verification texts to users, WhatsApp was changed from a free service to a paid one. In December 2009, the ability to send photos was added to the iOS version. By early 2011, WhatsApp was one of the top 20 apps in the U.S. Apple App Store. In April 2011, Sequoia Capital invested about $8 million for more than 15% of the company, after months of negotiation by Sequoia partner Jim Goetz. By February 2013, WhatsApp had about 200 million active users and 50 staff members. Sequoia invested another $50 million at a $1.5 billion valuation. Some time in 2013 WhatsApp acquired Santa Clara–based startup SkyMobius, the developers of Vtok, a video and voice calling app. As of December 2013, the service had 400 million monthly active users. That year, the company had $148 million in expenses and a net loss of $138 million. === 2014–2015 === On February 19, 2014, one year after the venture capital financing round at a $1.5 billion valuation, Facebook, Inc. (now Meta Platforms) agreed to acquire the company for US$19 billion, its largest acquisition to date. At the time, it was the largest acquisition of a venture-capital-backed company in history. Sequoia Capital received an approximate 5,000% return on its initial investment. Facebook paid $4 billion in cash, $12 billion in Facebook shares, and an additional $3 billion in restricted stock units granted to WhatsApp's founders Koum and Acton. Employee stock was scheduled to vest over four years subsequent to closing. Days after the announcement, WhatsApp users experienced a loss of service, leading to anger across social media. The acquisition was influenced by the data provided by Onavo, Facebook's research app for monitoring competitors and trending usage of social activities on mobile phones, as well as startups that were performing "unusually well". The acquisition caused many users to try, or move to, other message services. Telegram claimed that it acquired 8 million new users, and Line, 2 million. At a keynote presentation at the Mobile World Congress in Barcelona in February 2014, Facebook CEO Mark Zuckerberg said that Facebook's acquisition of WhatsApp was closely related to the Internet.org vision. A TechCrunch article said about Zuckerberg's vision:The idea, he said, is to develop a group of basic internet services that would be free of charge to use – "a 911 for the internet". These could be a social networking service like Facebook, a messaging service, maybe search and other things like weather. Providing a bundle of these free of charge to users will work like a gateway drug of sorts – users who may be able to afford data services and phones these days just don't see the point of why they would pay for those data services. This would give them some context for why they are important, and that will lead them to pay for more services like this – or so the hope goes. Three days after announcing the Facebook purchase, Koum said they were working to introduce voice calls. He also said that new mobile phones would be sold in Germany with the WhatsApp brand, and that their ultimate goal was to be on all smartphones. In August 2014, WhatsApp was the most popular messaging app in the world, with more than 600 million users. By early January 2015, WhatsApp had 700 million monthly users and over 30 billion messages every day. In April 2015, Forbes predicted that between 2012 and 2018, the telecommunications industry would lose $386 billion because of "over-the-top" services like WhatsApp and Skype. That month, WhatsApp had over 800 million users. By September 2015, it had grown to 900 million; and by February 2016, one billion. On November 30, 2015, the Android WhatsApp client made links to Telegram unclickable and not copyable. Multiple sources confirmed that it was intentional, not a bug, and that it had been implemented when the Android source code that recognized Telegram URLs had been identified. (The word "telegram" appeared in WhatsApp's code.) Some considered it an anti-competitive measure; WhatsApp offered no explanation. === 2016–2019 === On January 18, 2016, WhatsApp's co-founder Jan Koum announced that it would no longer charge users a $1 annual subscription fee, in an effort to remove a barrier faced by users without payment cards. He also said that the app would not display any third-party ads, and that it would have new features such as the ability to communicate with businesses. On May 18, 2017, the European Commission announced that it was fining Facebook €110 million for "providing misleading information about WhatsApp takeover" in 2014. The Commission said that in 2014 when Facebook acquired the messaging app, it "falsely claimed it was technically impossible to automatically combine user information from Facebook and WhatsApp." However, in the summer of 2016, WhatsApp had begun sharing user information with its parent company, allowing information such as phone numbers to be used for targeted Facebook advertisements. Facebook acknowledged the breach, but said the errors in their 2014 filings were "not intentional". In September 2017, WhatsApp's co-founder Brian Acton left the company to start a nonprofit group, later revealed as the Signal Foundation, which developed the WhatsApp competitor Signal. He explained his reasons for leaving in an interview with Forbes a year later. WhatsApp also
T-vertices
T-vertices is a term used in computer graphics to describe a problem that can occur during mesh refinement or mesh simplification. The most common case occurs in naive implementations of continuous level of detail, where a finer-level mesh is "sewn" together with a coarser-level mesh by simply aligning the finer vertices on the edges of the coarse polygons. The result is a continuous mesh, however due to the nature of the z-buffer and certain lighting algorithms such as Gouraud shading, visual artifacts can often be detected. Some modeling algorithms such as subdivision surfaces will fail when a model contains T-vertices.
Richardson–Lucy deconvolution
The Richardson–Lucy algorithm, also known as Lucy–Richardson deconvolution, is an iterative procedure for recovering an underlying image that has been blurred by a known point spread function. It was named after William Richardson and Leon B. Lucy, who described it independently. == Description == When an image is produced using an optical system and detected using photographic film, a charge-coupled device or a CMOS sensor, for example, it is inevitably blurred, with an ideal point source not appearing as a point but being spread out into what is known as the point spread function. Extended sources can be decomposed into the sum of many individual point sources, thus the observed image can be represented in terms of a transition matrix p operating on an underlying image: d i = ∑ j p i , j u j , {\displaystyle d_{i}=\sum _{j}p_{i,j}u_{j},} where u j {\displaystyle u_{j}} is the intensity of the underlying image at pixel j {\displaystyle j} , and d i {\displaystyle d_{i}} is the detected intensity at pixel i {\displaystyle i} . In general, a matrix whose elements are p i , j {\displaystyle p_{i,j}} describes the portion of light from source pixel j that is detected in pixel i. In most good optical systems (or in general, linear systems that are described as shift-invariant) the transfer function p can be expressed simply in terms of the spatial offset between the source pixel j and the observation pixel i: p i , j = P ( i − j ) , {\displaystyle p_{i,j}=P(i-j),} where P ( Δ i ) {\displaystyle P(\Delta i)} is called a point spread function. In that case the above equation becomes a convolution. This has been written for one spatial dimension, but most imaging systems are two-dimensional, with the source, detected image, and point spread function all having two indices. So a two-dimensional detected image is a convolution of the underlying image with a two-dimensional point spread function P ( Δ x , Δ y ) {\displaystyle P(\Delta x,\Delta y)} plus added detection noise. In order to estimate u j {\displaystyle u_{j}} given the observed d i {\displaystyle d_{i}} and a known P ( Δ i x , Δ j y ) {\displaystyle P(\Delta i_{x},\Delta j_{y})} , the following iterative procedure is employed in which the estimate of u j {\displaystyle u_{j}} (called u ^ j ( t ) {\displaystyle {\hat {u}}_{j}^{(t)}} ) for iteration number t is updated as follows: u ^ j ( t + 1 ) = u ^ j ( t ) ∑ i d i c i p i j , {\displaystyle {\hat {u}}_{j}^{(t+1)}={\hat {u}}_{j}^{(t)}\sum _{i}{\frac {d_{i}}{c_{i}}}p_{ij},} where c i = ∑ j p i j u ^ j ( t ) , {\displaystyle c_{i}=\sum _{j}p_{ij}{\hat {u}}_{j}^{(t)},} and ∑ j p i j = 1 {\displaystyle \sum _{j}p_{ij}=1} is assumed. It has been shown empirically that if this iteration converges, it converges to the maximum likelihood solution for u j {\displaystyle u_{j}} . Writing this more generally for two (or more) dimensions in terms of convolution with a point spread function P: u ^ ( t + 1 ) = u ^ ( t ) ⋅ ( d u ^ ( t ) ⊗ P ⊗ P ∗ ) , {\displaystyle {\hat {u}}^{(t+1)}={\hat {u}}^{(t)}\cdot \left({\frac {d}{{\hat {u}}^{(t)}\otimes P}}\otimes P^{}\right),} where the division and multiplication are element-wise, ⊗ {\displaystyle \otimes } indicates a 2D convolution, and P ∗ {\displaystyle P^{}} is the mirrored point spread function, or the inverse Fourier transform of the Hermitian transpose of the optical transfer function. In problems where the point spread function p i j {\displaystyle p_{ij}} is not known a priori, a modification of the Richardson–Lucy algorithm has been proposed, in order to accomplish blind deconvolution. == Derivation == In the context of fluorescence microscopy, the probability of measuring a set of number of photons (or digitalization counts proportional to detected light) m = [ m 0 , … , m K ] {\displaystyle \mathbf {m} =[m_{0},\dots ,m_{K}]} for expected values E = [ E 0 , … , E K ] {\displaystyle \mathbf {E} =[E_{0},\dots ,E_{K}]} for a detector with K + 1 {\displaystyle K+1} pixels is given by P ( m ∣ E ) = ∏ i K Poisson ( E i ) = ∏ i K E i m i e − E i m i ! . {\displaystyle P(\mathbf {m} \mid \mathbf {E} )=\prod _{i}^{K}\operatorname {Poisson} (E_{i})=\prod _{i}^{K}{\frac {E_{i}^{m_{i}}e^{-E_{i}}}{m_{i}!}}.} Since in the context of maximum-likelihood estimation the aim is to locate the maximum of the likelihood function without concern for its absolute value, it is convenient to work with ln ( P ) {\displaystyle \ln(P)} : ln P ( m ∣ E ) = ∑ i K [ ( m i ln E i − E i ) − ln ( m i ! ) ] . {\displaystyle \ln P(\mathbf {m} \mid \mathbf {E} )=\sum _{i}^{K}[(m_{i}\ln E_{i}-E_{i})-\ln(m_{i}!)].} Moreover, since ln ( m i ! ) {\displaystyle \ln(m_{i}!)} is a constant, it does not give any additional information regarding the position of the maximum, so consider α ( m ∣ E ) = ∑ i K [ m i ln E i − E i ] , {\displaystyle \alpha (\mathbf {m} \mid \mathbf {E} )=\sum _{i}^{K}[m_{i}\ln E_{i}-E_{i}],} where α {\displaystyle \alpha } is something that shares the same maximum position as P ( m ∣ E ) {\displaystyle P(\mathbf {m} \mid \mathbf {E} )} . Now consider that E {\displaystyle \mathbf {E} } comes from a ground truth x {\displaystyle \mathbf {x} } and a measurement H {\displaystyle \mathbf {H} } which is assumed to be linear. Then E = H x , {\displaystyle \mathbf {E} =\mathbf {H} \mathbf {x} ,} where a matrix multiplication is implied. This can also be written in the form E m = ∑ n K H m n x n , {\displaystyle E_{m}=\sum _{n}^{K}H_{mn}x_{n},} where it can be seen how H {\displaystyle H} mixes or blurs the ground truth. It can also be shown that the derivative of an element of E {\displaystyle \mathbf {E} } , ( E i ) {\displaystyle (E_{i})} with respect to some other element of x j {\displaystyle x_{j}} can be written as It is easy to see this by writing a matrix H {\displaystyle \mathbf {H} } of, say, 5 × 5 and two arrays E {\displaystyle \mathbf {E} } and x {\displaystyle \mathbf {x} } of 5 elements and check it. This last equation can be interpreted as how much one element of x {\displaystyle \mathbf {x} } , say element i {\displaystyle i} , influences the other elements j ≠ i {\displaystyle j\neq i} (and of course the case i = j {\displaystyle i=j} is also taken into account). For example, in a typical case an element of the ground truth x {\displaystyle \mathbf {x} } will influence nearby elements in E {\displaystyle \mathbf {E} } but not the very distant ones (a value of 0 {\displaystyle 0} is expected on those matrix elements). Now, the key and arbitrary step: x {\displaystyle \mathbf {x} } is not known but may be estimated by x ^ {\displaystyle {\hat {\mathbf {x} }}} . Let's call x ^ old {\displaystyle {\hat {\mathbf {x} }}_{\text{old}}} and x ^ new {\displaystyle {\hat {\mathbf {x} }}_{\text{new}}} the estimated ground truths while using the RL algorithm, where the hat symbol is used to distinguish ground truth from estimator of the ground truth where ∂ ∂ x {\displaystyle {\frac {\partial }{\partial \mathbf {x} }}} stands for a K {\displaystyle K} -dimensional gradient. Performing the partial derivative of α ( m ∣ E ( x ) ) {\displaystyle \alpha (\mathbf {m} \mid \mathbf {E} (\mathbf {x} ))} yields the following expression: ∂ α ( m ∣ E ( x ) ) ∂ x j = ∂ ∂ x j ∑ i K [ m i ln E i − E i ] = ∑ i K [ m i E i ∂ ∂ x j E i − ∂ ∂ x j E i ] = ∑ i K ∂ E i ∂ x j [ m i E i − 1 ] . {\displaystyle {\frac {\partial \alpha (\mathbf {m} \mid \mathbf {E} (\mathbf {x} ))}{\partial x_{j}}}={\frac {\partial }{\partial x_{j}}}\sum _{i}^{K}[m_{i}\ln E_{i}-E_{i}]=\sum _{i}^{K}\left[{\frac {m_{i}}{E_{i}}}{\frac {\partial }{\partial x_{j}}}E_{i}-{\frac {\partial }{\partial x_{j}}}E_{i}\right]=\sum _{i}^{K}{\frac {\partial E_{i}}{\partial x_{j}}}\left[{\frac {m_{i}}{E_{i}}}-1\right].} By substituting (1), it follows that ∂ α ( m ∣ E ( x ) ) ∂ x j = ∑ i K H i j [ m i E i − 1 ] . {\displaystyle {\frac {\partial \alpha (\mathbf {m} \mid \mathbf {E} (\mathbf {x} ))}{\partial x_{j}}}=\sum _{i}^{K}H_{ij}\left[{\frac {m_{i}}{E_{i}}}-1\right].} Note that H j i T = H i j {\displaystyle H_{ji}^{T}=H_{ij}} by the definition of a matrix transpose. And hence Since this equation is true for all j {\displaystyle j} spanning all the elements from 1 {\displaystyle 1} to K {\displaystyle K} , these K {\displaystyle K} equations may be compactly rewritten as a single vectorial equation ∂ α ( m ∣ E ( x ) ) ∂ x = H T [ m E − 1 ] , {\displaystyle {\frac {\partial \alpha (\mathbf {m} \mid \mathbf {E} (\mathbf {x} ))}{\partial \mathbf {x} }}=\mathbf {H} ^{T}\left[{\frac {\mathbf {m} }{\mathbf {E} }}-\mathbf {1} \right],} where H T {\displaystyle \mathbf {H} ^{T}} is a matrix, and m {\displaystyle \mathbf {m} } , E {\displaystyle \mathbf {E} } and 1 {\displaystyle \mathbf {1} } are vectors. Now, as a seemingly arbitrary but key step, let where 1 {\displaystyle \mathbf {1} } is a vector of ones of size K {\displaystyle K} (same as m {\displaystyle \mathbf {m} } , E {\displaystyle \mathbf {E} } and x {\displaystyle \mathbf {x} } ), and the d
Space partitioning
In geometry, space partitioning is the process of dividing an entire space (usually a Euclidean space) into two or more disjoint subsets (see also partition of a set). In other words, space partitioning divides a space into non-overlapping regions. Any point in the space can then be identified to lie in exactly one of the regions. == Overview == Space-partitioning systems are often hierarchical, meaning that a space (or a region of space) is divided into several regions, and then the same space-partitioning system is recursively applied to each of the regions thus created. The regions can be organized into a tree, called a space-partitioning tree. Most space-partitioning systems use planes (or, in higher dimensions, hyperplanes) to divide space: points on one side of the plane form one region, and points on the other side form another. Points exactly on the plane are usually arbitrarily assigned to one or the other side. Recursively partitioning space using planes in this way produces a BSP tree, one of the most common forms of space partitioning. == Uses == === In computer graphics === Space partitioning is particularly important in computer graphics, especially heavily used in ray tracing, where it is frequently used to organize the objects in a virtual scene. A typical scene may contain millions of polygons. Performing a ray/polygon intersection test with each would be a very computationally expensive task. Storing objects in a space-partitioning data structure (k-d tree or BSP tree for example) makes it easy and fast to perform certain kinds of geometry queries—for example in determining whether a ray intersects an object, space partitioning can reduce the number of intersection test to just a few per primary ray, yielding a logarithmic time complexity with respect to the number of polygons. Space partitioning is also often used in scanline algorithms to eliminate the polygons out of the camera's viewing frustum, limiting the number of polygons processed by the pipeline. There is also a usage in collision detection: determining whether two objects are close to each other can be much faster using space partitioning. === In integrated circuit design === In integrated circuit design, an important step is design rule check. This step ensures that the completed design is manufacturable. The check involves rules that specify widths and spacings and other geometry patterns. A modern design can have billions of polygons that represent wires and transistors. Efficient checking relies heavily on geometry query. For example, a rule may specify that any polygon must be at least n nanometers from any other polygon. This is converted into a geometry query by enlarging a polygon by n/2 at all sides and query to find all intersecting polygons. === In probability and statistical learning theory === The number of components in a space partition plays a central role in some results in probability theory. See Growth function for more details. === In geography and GIS === There are many studies and applications where Geographical Spatial Reality is partitioned by hydrological criteria, administrative criteria, mathematical criteria or many others. In the context of cartography and GIS - Geographic Information System, is common to identify cells of the partition by standard codes. For example the for HUC code identifying hydrographical basins and sub-basins, ISO 3166-2 codes identifying countries and its subdivisions, or arbitrary DGGs - discrete global grids identifying quadrants or locations. == Data structures == Common space-partitioning systems include: BSP trees Quadtrees Octrees k-d trees Bins == Number of components == Suppose the n-dimensional Euclidean space is partitioned by r {\displaystyle r} hyperplanes that are ( n − 1 ) {\displaystyle (n-1)} -dimensional. What is the number of components in the partition? The largest number of components is attained when the hyperplanes are in general position, i.e, no two are parallel and no three have the same intersection. Denote this maximum number of components by C o m p ( n , r ) {\displaystyle Comp(n,r)} . Then, the following recurrence relation holds: C o m p ( n , r ) = C o m p ( n , r − 1 ) + C o m p ( n − 1 , r − 1 ) {\displaystyle Comp(n,r)=Comp(n,r-1)+Comp(n-1,r-1)} C o m p ( 0 , r ) = 1 {\displaystyle Comp(0,r)=1} - when there are no dimensions, there is a single point. C o m p ( n , 0 ) = 1 {\displaystyle Comp(n,0)=1} - when there are no hyperplanes, all the space is a single component. And its solution is: C o m p ( n , r ) = ∑ k = 0 n ( r k ) {\displaystyle Comp(n,r)=\sum _{k=0}^{n}{r \choose k}} if r ≥ n {\displaystyle r\geq n} C o m p ( n , r ) = 2 r {\displaystyle Comp(n,r)=2^{r}} if r ≤ n {\displaystyle r\leq n} (consider e.g. r {\displaystyle r} perpendicular hyperplanes; each additional hyperplane divides each existing component to 2). which is upper-bounded as: C o m p ( n , r ) ≤ r n + 1 {\displaystyle Comp(n,r)\leq r^{n}+1}