In computer programming, genetic representation is a way of presenting solutions/individuals in evolutionary computation methods. The term encompasses both the concrete data structures and data types used to realize the genetic material of the candidate solutions in the form of a genome, and the relationships between search space and problem space. In the simplest case, the search space corresponds to the problem space (direct representation). The choice of problem representation is tied to the choice of genetic operators, both of which have a decisive effect on the efficiency of the optimization. Genetic representation can encode appearance, behavior, physical qualities of individuals. Difference in genetic representations is one of the major criteria drawing a line between known classes of evolutionary computation. Terminology is often analogous with natural genetics. The block of computer memory that represents one candidate solution is called an individual. The data in that block is called a chromosome. Each chromosome consists of genes. The possible values of a particular gene are called alleles. A programmer may represent all the individuals of a population using binary encoding, permutational encoding, encoding by tree, or any one of several other representations. == Representations in some popular evolutionary algorithms == Genetic algorithms (GAs) are typically linear representations; these are often, but not always, binary. Holland's original description of GA used arrays of bits. Arrays of other types and structures can be used in essentially the same way. The main property that makes these genetic representations convenient is that their parts are easily aligned due to their fixed size. This facilitates simple crossover operation. Depending on the application, variable-length representations have also been successfully used and tested in evolutionary algorithms (EA) in general and genetic algorithms in particular, although the implementation of crossover is more complex in this case. Evolution strategy uses linear real-valued representations, e.g., an array of real values. It uses mostly gaussian mutation and blending/averaging crossover. Genetic programming (GP) pioneered tree-like representations and developed genetic operators suitable for such representations. Tree-like representations are used in GP to represent and evolve functional programs with desired properties. Human-based genetic algorithm (HBGA) offers a way to avoid solving hard representation problems by outsourcing all genetic operators to outside agents, in this case, humans. The algorithm has no need for knowledge of a particular fixed genetic representation as long as there are enough external agents capable of handling those representations, allowing for free-form and evolving genetic representations. === Common genetic representations === binary array integer or real-valued array binary tree natural language parse tree directed graph == Distinction between search space and problem space == Analogous to biology, EAs distinguish between problem space (corresponds to phenotype) and search space (corresponds to genotype). The problem space contains concrete solutions to the problem being addressed, while the search space contains the encoded solutions. The mapping from search space to problem space is called genotype-phenotype mapping. The genetic operators are applied to elements of the search space, and for evaluation, elements of the search space are mapped to elements of the problem space via genotype-phenotype mapping. == Relationships between search space and problem space == The importance of an appropriate choice of search space for the success of an EA application was recognized early on. The following requirements can be placed on a suitable search space and thus on a suitable genotype-phenotype mapping: === Completeness === All possible admissible solutions must be contained in the search space. === Redundancy === When more possible genotypes exist than phenotypes, the genetic representation of the EA is called redundant. In nature, this is termed a degenerate genetic code. In the case of a redundant representation, neutral mutations are possible. These are mutations that change the genotype but do not affect the phenotype. Thus, depending on the use of the genetic operators, there may be phenotypically unchanged offspring, which can lead to unnecessary fitness determinations, among other things. Since the evaluation in real-world applications usually accounts for the lion's share of the computation time, it can slow down the optimization process. In addition, this can cause the population to have higher genotypic diversity than phenotypic diversity, which can also hinder evolutionary progress. In biology, the Neutral Theory of Molecular Evolution states that this effect plays a dominant role in natural evolution. This has motivated researchers in the EA community to examine whether neutral mutations can improve EA functioning by giving populations that have converged to a local optimum a way to escape that local optimum through genetic drift. This is discussed controversially and there are no conclusive results on neutrality in EAs. On the other hand, there are other proven measures to handle premature convergence. === Locality === The locality of a genetic representation corresponds to the degree to which distances in the search space are preserved in the problem space after genotype-phenotype mapping. That is, a representation has a high locality exactly when neighbors in the search space are also neighbors in the problem space. In order for successful schemata not to be destroyed by genotype-phenotype mapping after a minor mutation, the locality of a representation must be high. === Scaling === In genotype-phenotype mapping, the elements of the genotype can be scaled (weighted) differently. The simplest case is uniform scaling: all elements of the genotype are equally weighted in the phenotype. A common scaling is exponential. If integers are binary coded, the individual digits of the resulting binary number have exponentially different weights in representing the phenotype. Example: The number 90 is written in binary (i.e., in base two) as 1011010. If now one of the front digits is changed in the binary notation, this has a significantly greater effect on the coded number than any changes at the rear digits (the selection pressure has an exponentially greater effect on the front digits). For this reason, exponential scaling has the effect of randomly fixing the "posterior" locations in the genotype before the population gets close enough to the optimum to adjust for these subtleties. == Hybridization and repair in genotype-phenotype mapping == When mapping the genotype to the phenotype being evaluated, domain-specific knowledge can be used to improve the phenotype and/or ensure that constraints are met. This is a commonly used method to improve EA performance in terms of runtime and solution quality. It is illustrated below by two of the three examples. == Examples == === Example of a direct representation === An obvious and commonly used encoding for the traveling salesman problem and related tasks is to number the cities to be visited consecutively and store them as integers in the chromosome. The genetic operators must be suitably adapted so that they only change the order of the cities (genes) and do not cause deletions or duplications. Thus, the gene order corresponds to the city order and there is a simple one-to-one mapping. === Example of a complex genotype-phenotype mapping === In a scheduling task with heterogeneous and partially alternative resources to be assigned to a set of subtasks, the genome must contain all necessary information for the individual scheduling operations or it must be possible to derive them from it. In addition to the order of the subtasks to be executed, this includes information about the resource selection. A phenotype then consists of a list of subtasks with their start times and assigned resources. In order to be able to create this, as many allocation matrices must be created as resources can be allocated to one subtask at most. In the simplest case this is one resource, e.g., one machine, which can perform the subtask. An allocation matrix is a two-dimensional matrix, with one dimension being the available time units and the other being the resources to be allocated. Empty matrix cells indicate availability, while an entry indicates the number of the assigned subtask. The creation of allocation matrices ensures firstly that there are no inadmissible multiple allocations. Secondly, the start times of the subtasks can be read from it as well as the assigned resources. A common constraint when scheduling resources to subtasks is that a resource can only be allocated once per time unit and that the reservation must be for a contiguous period of time. To achieve this in a timely manner, which is a c
Direct voice input
Direct voice input (DVI), sometimes called voice input control (VIC), is a style of human–machine interaction "HMI" in which the user makes voice commands to issue instructions to the machine through speech recognition. In the field of military aviation, DVI has been introduced into the cockpits of several modern military aircraft, such as the Eurofighter Typhoon, the Lockheed Martin F-35 Lightning II, the Dassault Rafale, the KF-21 Boramae and the Saab JAS 39 Gripen. Such systems have also been used for various other purposes, including industry control systems and speech recognition assistance for impaired individuals. == Overview == DVI systems can be divided into two major categories of functionality: "user-dependent" or "user-independent". A user-dependent system requires that a personal voice template to be generated for a specific person; the template for this individual has to be loaded onto their assigned machine prior to use of the DVI system for it to function properly. In contrast, a user-independent system does not require any personal voice template, being intended to respond correctly to the voice of any user. They can also be categorised between "discrete recognition" and "continuous recognition". Users of a discrete recognition system must pause between each word so that the DVI system can identify the separations between each word, while a continuous speech recognition system is capable of understanding a normal rate of speech. During the mid-2000s, researchers at the National Aerospace Laboratory in the Netherlands examined the use of DVI in the "GRACE" simulator; a total of twelve pilots participated in the ensuing experiment. The tests performed reportedly revealed that, while the hardware itself functioned well, several improvements were desirable prior to real-world deployment on aircraft since DVI operations actually consumed more time in comparison to traditional existing methods. Recommendations for improvements included the adoption of simpler syntax, the achievement of a greater recognition rate, and a decrease in response times; all of the issues encountered were determined to be of a technological nature, and were deemed feasible to resolve. The researchers concluded that in cockpits, especially during emergencies where pilots have to operate entirely on their own, a DVI system could be highly relevant, but that it was not of crucial importance during most other conceivable scenarios. Around the same time, evaluations of DVI systems for civil aviation purposes were conducted within the framework of Project SafeSound, coordinated by the European Union. It involved the observation of pilot workloads in real-world cockpits and contrasting them against pilot activity in flight simulators using both conventional systems and DVI assistance. The project aimed to enhance aviation safety and to decrease the workload in both ground and flight operations via the application of enhanced audio functions. == Applications == === Aviation === Prior to its widespread deployment, a handful of conventional military aircraft were converted to trial DVI systems; examples include the Harrier AV-8B and F-16 VISTA. In another case, a General Dynamics F-16 Fighting Falcon simulator was modified with DVI for a voice control study that was undertaken by the Royal Netherlands Air Force. DVI trials have also been conducted on helicopters, including the Boeing AH-64 Apache, showing the potential to improve flight safety and mission effectiveness. Numerous modern fighter aircraft have been outfitted with DVI systems, often in combination with various other man-machine interface schemes, such as HOTAS-compliant controls and other advanced control technologies. The combination of Voice and HOTAS control schemes has sometimes been referred to as the "V-TAS" concept. A prominent fighter aircraft to be furnished with a V-TAS cockpit is the Eurofighter Typhoon. The Lockheed Martin F-35 Lightning II also features a DVI system, which was developed by Adacel. Other examples includes the Dassault Rafale and the Saab JAS 39 Gripen. Numerous aircraft have been planned to use DVI. At one stage, the United States Air Force had sought to integrate DVI upon the Lockheed Martin F-22 Raptor; however, the technology was eventually judged to pose too many technical risks at that point in time, and thus such efforts were abandoned. === Personal === By 1990, working prototypes of speech recognition systems were being demonstrated; these were being promoted for the purpose of providing an effective man-machine interface for individuals with impaired speech. Techniques employed included time-encoded digital speech and automatic token set selection. Investigations of these early DVI systems reportedly included the use of automatic diagnostic routines and limited-scale trials using volunteers. During the 2010s, various companies were offering voice recognition systems to the general public in the form of personal digital assistants. One example is the Google Voice service, which allows users to pose questions via a DVI package installed on either a personal computer, tablet, or mobile phone. Numerous digital assistants have been developed, such as Amazon Echo, Siri, and Cortana, that use DVI to interact with users. === Commercial === DVI technology has enabled automated telephone systems to be widely deployed. Many companies commonly use centralised phone systems that route callers to the correct department via such methods. Various car manufacturers have also furnished their road vehicles with DVI systems; these typically allow drivers to control infotainment systems and interact with mobile phones with more convenience than legacy methods. During the late 1980s, investigations into the use of DVI systems for controlling CNC machines and other manufacturing apparatus were underway. During the 2010s, such systems were being used for logistics and warehouse management purposes.
Outline of automation
The following outline is provided as an overview of and topical guide to automation: Automation – use of control systems and information technologies to reduce the need for human work in the production of goods and services. In the scope of industrialization, automation is a step beyond mechanization. == Essence of automation == Control system – a device, or set of devices to manage, command, direct or regulate the behavior of other devices or systems. Industrial control system (ICS) – encompasses several types of control systems used in industrial production, including supervisory control and data acquisition (SCADA) systems, distributed control systems (DCS), and other smaller control system configurations such as skid-mounted programmable logic controllers (PLC) often found in industrial sectors and critical infrastructures. Industrialization – period of social and economic change that transforms a human group from an agrarian society into an industrial one. Numerical control (NC) – refers to the automation of machine tools that are operated by abstractly programmed commands encoded on a storage medium, as opposed to controlled manually via handwheels or levers, or mechanically automated via cams alone. Robotics – the branch of technology that deals with the design, construction, operation, structural disposition, manufacture and application of robots and computer systems for their control, sensory feedback, and information processing. == Branches of automation == === General purpose === Autonomous automation – autonomous software agents to adapt the controllers of computer controlled industrial machinery and processes Banking automation Broadcast automation Building automation – advanced functionality provided by the control system of a building. A building automation system (BAS) is an example of a distributed control system. Home automation – control system of a home. Office automation – the varied computer machinery and software used to digitally create, collect, store, manipulate, and relay office information needed for accomplishing basic tasks such as business process automation and robotic process automation. Console automation Database automation Integrated library system Laboratory automation === Specific purpose === Automated attendant Automated guided vehicle Autonomous mobile robot Automated highway system Automated pool cleaner Automated teller machine Automatic painting (robotic) Pop music automation Remotely operated vehicle Robotic lawn mower Telephone switchboard Vending machine == Fields contributing to automation == Cybernetics – the interdisciplinary study of the structure of regulatory systems. Cognitive science – interdisciplinary scientific study of the mind and its processes. It examines what cognition is, what it does and how it works. Robotics – the branch of technology that deals with the design, construction, operation, structural disposition, manufacture and application of robots and computer systems for their control, sensory feedback, and information processing. == History of automation == History of mass production – Prerequisites of mass production were interchangeable parts, machine tools and power, especially in the form of electricity. Mass production was popularized in the 1910s and 1920s by Henry Ford's Ford Motor Company, which introduced electric motors to the then-well-known technique of chain or sequential production. History of home automation == Automated machines == Machine to Machine OLE for process control (OPC) Process control – a statistics and engineering discipline that deals with architectures, mechanisms and algorithms for maintaining the output of a specific process within a desired range. Run Book Automation (RBA) Robot – a mechanical or virtual intelligent agent that can perform tasks automatically or with guidance, typically by remote control. == Automated machine components == Artificial intelligence – the intelligence of machines and the branch of computer science that aims to create it. Friendly artificial intelligence – an artificial intelligence that has a positive rather than negative effect on humanity, and the field of knowledge required to build such an artificial intelligence. === Automation tools === Artificial neural network (ANN) – mathematical model or computational model that is inspired by the structure or functional aspects of biological neural networks. Human machine interface (HMI) – operator level local control panel that monitors field devices Laboratory information management system (LIMS) – software package that offers a set of key features that support a modern laboratory's operations. Industrial control system – encompasses several types of control systems used in industrial production, including supervisory control and data acquisition (SCADA) systems, distributed control systems (DCS), and other smaller control system configurations such as skid-mounted programmable logic controllers (PLC) often found in the industrial sectors and critical infrastructures. Distributed control system (DCS) – control system usually of a manufacturing system, process or any kind of dynamic system, in which the controller elements are not central in location (like the brain) but are distributed throughout the system with each component sub-system controlled by one or more controllers. Manufacturing execution system (MES) – system that manages manufacturing operations in a factory, including management of resources, scheduling production processes, dispatching production orders, execution of production orders, etc. Programmable automation controller (PAC) – digital computer used for automation of electromechanical processes, such as control of machinery on factory assembly lines, amusement rides, or light fixtures. Programmable logic controller (PLC)A Programmable Logic Controller, PLC or Programmable Controller is a digital computer used for automation of electromechanical processes, such as control of machinery on factory assembly lines, amusement rides, or light fixtures. The abbreviation "PLC" and the term "Programmable Logic Controller" are registered trademarks of the Allen-Bradley Company (Rockwell Automation). PLCs are used in many industries and machines. Unlike general-purpose computers, the PLC is designed for multiple inputs and output arrangements, extended temperature ranges, immunity to electrical noise, and resistance to vibration and impact. Programs to control machine operation are typically stored in battery-backed-up or non-volatile memory. A PLC is an example of a hard real time system since output results must be produced in response to input conditions within a limited time, otherwise unintended operation will result. Supervisory control and data acquisition (SCADA) – generally refers to industrial control systems (ICS): computer systems that monitor and control industrial, infrastructure, or facility-based processes, as described below: Industrial processes include those of manufacturing, production, power generation, fabrication, and refining, and may run in continuous, batch, repetitive, or discrete modes. Simulation § Engineering Technology simulation or Process simulation == Social movements == Automation-related social movement – a movement that advocates semi- or fully automatic systems to provide for human needs globally. For example, automation of farming and food distribution throughout the world so that no one will go hungry. One goal is to automate all mundane labor, to free humans to engage in more creative activities (or less work). The Technocracy movement – social movement active from the Great Depression (1930s) to date that proposes replacing politicians and business people with scientists and engineers who have the technical expertise to manage the economy. The Zeitgeist Movement – movement advocating the replacement of the market economy with an economy in which all resources are equitably, commonly and sustainably shared. == Automation in the future == Android – a robot or synthetic organism designed to look and act like a human, and with a body having a flesh-like resemblance Technological singularity – the hypothetical future emergence of greater-than-human intelligence through technological means Semi-automation – using a centralized computer controller to orchestrate the activities of man and machine. == Automation-related publications == IEEE Spectrum – the flagship publication of the Institute of Electrical and Electronics Engineers (IEEE), explores the development, applications and implications of new technologies, and provides a forum for understanding, discussion and leadership in these areas. IEEE Transactions on Information Theory – peer-reviewed scientific journal published by the Institute of Electrical and Electronics Engineers (IEEE), focused on the study of information theory, the mathematics of communications, including computer communications, robotics communications, etc. IEEE Transactions on Control S
ViEWER
ViEWER, the Virtual Environment Workbench for Education and Research, is a proprietary, freeware computer program for Microsoft Windows written by researchers at the University of Idaho for the study of visual perception and complex immersive three-dimensional environments. It was created using C++ and OpenGL, and has been used by Dr. Brian Dyre, Dr. Steffen Werner, Dr. Ernesto Bustamante, Dr. Ben Barton, and their undergraduate and graduate researchers in visual perception, signal detection, and child-safety experiments.
Dating app
An online dating application, commonly known as a dating app, is an online dating service presented through a mobile phone application. These apps often take advantage of a smartphone's GPS location capabilities, always on-hand presence, and access to mobile wallets. These apps aim to speed up the online dating process of sifting through potential dating partners, chatting, flirting, and potentially meeting or becoming romantically involved. Online dating apps are now mainstream in the United States. As of 2017, online dating (which included both apps and other online dating services) was the principal method by which new couples in the U.S. met. The percentage of couples meeting online is predicted to increase to 70% by 2040. == Origins == The first computerized dating service was launched in 1964, the St. James Computer Dating Service, which became known as Com-Pat. The first U.S. dating service that used computerized match making was Operation Match. It required men and women to complete a questionnaire and was launched in 1965. Operation Match inspired the methodology of Dateline, which became popular in the 1970s and 1980s. Match.com was launched in 1995 and turned computerized match making into a profitable business. Grindr targeted gay and bisexual men at launch. Tinder, launched in 2012, led to a growth of online dating applications by both new providers and existing online dating services that expanded into the mobile app market. == Usage by demographic group == Online dating applications typically target a younger demographic group, though some apps, like Senior Match and Silver Singles are geared toward the 50 and up demographic. In 2016, almost 50% of people knew of someone who use the services or had met their loved one through the service. After the iPhone launch in 2007, online dating data has mushroomed as application usage increased. In 2005, only 10% of 18-24 year olds reported to have used online dating services; this number quickly grew to over 27%, making this target demographic the largest number of users for most applications. When Pew Research Center conducted a study in 2016, they found that 59% of U.S. adults agreed that online dating is a good way to meet people compared to 44% in 2005. This explosion in usage can be explained by the increased use of smartphones. By the end of 2022, it is expected there will be 413 million active users of online dating services worldwide. A 2023 Pew Research Center survey of 6,034 American adults found that 30% had ever used an online dating site or app, including 53% of those aged 18 to 29, 37% of those aged 30 to 49, and 17% of those aged 50 and over. Lesbian, gay and bisexual respondents reported using dating apps at nearly twice the rate of straight respondents (51% versus 28%), and 36% of divorced, separated or widowed adults had used one, compared with 16% of married adults. The increased use of smartphones by those 65 and older has also driven that population to the use dating apps. The Pew Research Center found that usage increase by 8 points since last surveyed in 2012. A study in 2021 found that more than one-third of seniors have dated in the past 5 years, and roughly one-third of those dating seniors have turned to dating apps. During the COVID-19 pandemic, Morning Consult found that more Americans were using online dating apps than ever before. In one survey in April 2020, the company discovered that 53% of U.S. adults who use online dating apps have been using them more during the pandemic. As of February 2021, that share increased to 71 percent. Research using Hofstede's cultural dimensions theory has indicated that norms about online dating applications tend to differ across cultures. A study published in the Journal of Creative Communications looked into the relationships between dating-app advertisements from over 51 countries and the cultural dimensions of these countries. The results revealed that dating-app advertisements appealed to multiple cultural needs, including the needs for relationships, friendship, entertainment, sex, status, design and identity. The use of these appeals was found to be 'congruent with ... the individualism/collectivism and uncertainty avoidance cultural dimensions.' == Popular applications == Following Tinder's success, other companies released dating applications. Raya was released in 2015 as a membership-based dating app, allowing entrance only through referrals, which was marketed as a dating app for celebrities. In early 2026, Hily surpassed Bumble to become the third most-used dating application in the United States and the fifth highest-grossing overall, driven largely by growing popularity among Generation Z users, while remaining behind Tinder and Hinge, both owned by Match Group. A number of dating apps have been created targeting adherents of particular religions seeking partners of the same religion, such as Muzmatch for Muslims, Christian Mingle, SALT, and Christian Connection for Christians, and JSwipe and JDate for Jews. === VR Dating === VR Dating is an application of Social VR where people can exist, collaborate, and perform various activities together. Virtual reality apps use virtual and augmented realities to make the dating experience more lifelike and more effective, as well as allow people to expand what is already possible in the world of online dating. There are several online platforms of VR Dating. The VR dating app Nevermet is the VR equivalent of Tinder, where people can search and find on dates. However, instead of actual real-life pictures, users will update pictures of virtual selves and will be interacting with avatars rather than real faces. Flirtual is a self-contained social VR app that serves to match users who then decide where and how to meet in VR. Flirtual hosts speed dating and social events in VR. == Effects on dating == The usage of online dating applications can have both advantages and disadvantages: === Advantages === Many of the applications provide personality tests for matching or use algorithms to match users. These factors enhance the possibility of users getting matched with a compatible candidate. Users are in control; they are provided with many options so there are enough matches that fit their particular type. Users can simply choose to not match the candidates that they know they are not interested in. Narrowing down options is easy. Once users think they are interested, they are able to chat and get to know the potential candidate. This form of communication can reduce the time, cost, and uncertainty often associated with traditional dating methods. Online dating offers convenience; people want dating to work around their schedules. Online dating can also increase self-confidence; even if users get rejected, they know there are hundreds of other candidates that will want to match with them so they can simply move on to the next option. In fact, 60% of U.S. adults agree that online dating is a good way to meet people and 66% say they have gone on a real date with someone they met through an application. Today, 5% of married Americans or Americans in serious relationships said they met their significant other online. The 39% of online dating users (representing 12% of all U.S. adults) say they have been in a committed relationship or married someone they met on a dating site or app. ==== Rejection sensitive individuals ==== Individuals high in rejection sensitivity are more likely to use online dating applications. As they tend to expect, perceive and overreact to rejection, rejection sensitive individuals struggle with traditional dating. Online dating applications allow for them to better reveal their true selves, potentially increasing their dating success. Online dating applications also obscure rejection cues, alleviating the rejection-related distress experienced by rejection sensitive individuals. ==== Anxiously attached individuals ==== Individuals high in attachment anxiety are also more likely to use online dating applications. While they harbour negative self-views, anxiously attached individuals are also more eager to enter and commit to relationships. Online dating applications allow for them to present "an authentic yet ideal version of themselves", mitigating their tendencies to view themselves as undesirable. This increases their romantic confidence, and potentially alleviates their anxiety when searching for a romantic partner. === Disadvantages === Sometimes having too many options can be overwhelming. With so many options available, users can get lost in their choices and end up spending too much time looking for the "perfect" candidate instead of using that time to start a real relationship. In addition, the algorithms and matching systems put in place may not always be as accurate as users think. There is no perfect system that can match two people's personalities perfectly every time. Communication online also lacks the physical chemistry aspec
Phase correlation
Phase correlation is an approach to estimate the relative translative offset between two similar images (digital image correlation) or other data sets. It is commonly used in image registration and relies on a frequency-domain representation of the data, usually calculated by fast Fourier transforms. The term is applied particularly to a subset of cross-correlation techniques that isolate the phase information from the Fourier-space representation of the cross-correlogram. == Example == The following image demonstrates the usage of phase correlation to determine relative translative movement between two images corrupted by independent Gaussian noise. The image was translated by (20,23) pixels. Accordingly, one can clearly see a peak in the phase-correlation representation at approximately (20,23). == Method == Given two input images g a {\displaystyle \ g_{a}} and g b {\displaystyle \ g_{b}} : Apply a window function (e.g., a Hamming window) on both images to reduce edge effects (this may be optional depending on the image characteristics). Then, calculate the discrete 2D Fourier transform of both images. G a = F { g a } , G b = F { g b } {\displaystyle \ \mathbf {G} _{a}={\mathcal {F}}\{g_{a}\},\;\mathbf {G} _{b}={\mathcal {F}}\{g_{b}\}} Calculate the cross-power spectrum by taking the complex conjugate of the second result, multiplying the Fourier transforms together elementwise, and normalizing this product elementwise. R = G a ∘ G b ∗ | G a ∘ G b ∗ | {\displaystyle \ R={\frac {\mathbf {G} _{a}\circ \mathbf {G} _{b}^{}}{|\mathbf {G} _{a}\circ \mathbf {G} _{b}^{}|}}} Where ∘ {\displaystyle \circ } is the Hadamard product (entry-wise product) and the absolute values are taken entry-wise as well. Written out entry-wise for element index ( j , k ) {\displaystyle (j,k)} : R j k = G a , j k ⋅ G b , j k ∗ | G a , j k ⋅ G b , j k ∗ | {\displaystyle \ R_{jk}={\frac {G_{a,jk}\cdot G_{b,jk}^{}}{|G_{a,jk}\cdot G_{b,jk}^{}|}}} Obtain the normalized cross-correlation by applying the inverse Fourier transform. r = F − 1 { R } {\displaystyle \ r={\mathcal {F}}^{-1}\{R\}} Determine the location of the peak in r {\displaystyle \ r} . ( Δ x , Δ y ) = arg max ( x , y ) { r } {\displaystyle \ (\Delta x,\Delta y)=\arg \max _{(x,y)}\{r\}} === Subpixel registration === Commonly, interpolation methods are used to estimate the peak location in the cross-correlogram to non-integer values, despite the fact that the data are discrete, and this procedure is often termed 'subpixel registration'. A large variety of subpixel interpolation methods are given in the technical literature. Common peak interpolation methods such as parabolic interpolation have been used, and the OpenCV computer vision package uses a centroid-based method, though these generally have inferior accuracy compared to more sophisticated methods. Because the Fourier representation of the data has already been computed, it is especially convenient to use the Fourier shift theorem with real-valued (sub-integer) shifts for this purpose, which essentially interpolates using the sinusoidal basis functions of the Fourier transform. An especially popular FT-based estimator is given by Foroosh et al. In this method, the subpixel peak location is approximated by a simple formula involving peak pixel value and the values of its nearest neighbors, where r ( 0 , 0 ) {\displaystyle r_{(0,0)}} is the peak value and r ( 1 , 0 ) {\displaystyle r_{(1,0)}} is the nearest neighbor in the x direction (assuming, as in most approaches, that the integer shift has already been found and the comparand images differ only by a subpixel shift). Δ x = r ( 1 , 0 ) r ( 1 , 0 ) ± r ( 0 , 0 ) {\displaystyle \ \Delta x={\frac {r_{(1,0)}}{r_{(1,0)}\pm r_{(0,0)}}}} The Foroosh et al. method is quite fast compared to most methods, though it is not always the most accurate. Some methods shift the peak in Fourier space and apply non-linear optimization to maximize the correlogram peak, but these tend to be very slow since they must apply an inverse Fourier transform or its equivalent in the objective function. It is also possible to infer the peak location from phase characteristics in Fourier space without the inverse transformation, as noted by Stone. These methods usually use a linear least squares (LLS) fit of the phase angles to a planar model. The long latency of the phase angle computation in these methods is a disadvantage, but the speed can sometimes be comparable to the Foroosh et al. method depending on the image size. They often compare favorably in speed to the multiple iterations of extremely slow objective functions in iterative non-linear methods. Since all subpixel shift computation methods are fundamentally interpolative, the performance of a particular method depends on how well the underlying data conform to the assumptions in the interpolator. This fact also may limit the usefulness of high numerical accuracy in an algorithm, since the uncertainty due to interpolation method choice may be larger than any numerical or approximation error in the particular method. Subpixel methods are also particularly sensitive to noise in the images, and the utility of a particular algorithm is distinguished not only by its speed and accuracy but its resilience to the particular types of noise in the application. == Rationale == The method is based on the Fourier shift theorem. Let the two images g a {\displaystyle \ g_{a}} and g b {\displaystyle \ g_{b}} be circularly-shifted versions of each other: g b ( x , y ) = d e f g a ( ( x − Δ x ) mod M , ( y − Δ y ) mod N ) {\displaystyle \ g_{b}(x,y)\ {\stackrel {\mathrm {def} }{=}}\ g_{a}((x-\Delta x){\bmod {M}},(y-\Delta y){\bmod {N}})} (where the images are M × N {\displaystyle \ M\times N} in size). Then, the discrete Fourier transforms of the images will be shifted relatively in phase: G b ( u , v ) = G a ( u , v ) e − 2 π i ( u Δ x M + v Δ y N ) {\displaystyle \mathbf {G} _{b}(u,v)=\mathbf {G} _{a}(u,v)e^{-2\pi i({\frac {u\Delta x}{M}}+{\frac {v\Delta y}{N}})}} One can then calculate the normalized cross-power spectrum to factor out the phase difference: R ( u , v ) = G a G b ∗ | G a G b ∗ | = G a G a ∗ e 2 π i ( u Δ x M + v Δ y N ) | G a G a ∗ e 2 π i ( u Δ x M + v Δ y N ) | = G a G a ∗ e 2 π i ( u Δ x M + v Δ y N ) | G a G a ∗ | = e 2 π i ( u Δ x M + v Δ y N ) {\displaystyle {\begin{aligned}R(u,v)&={\frac {\mathbf {G} _{a}\mathbf {G} _{b}^{}}{|\mathbf {G} _{a}\mathbf {G} _{b}^{}|}}\\&={\frac {\mathbf {G} _{a}\mathbf {G} _{a}^{}e^{2\pi i({\frac {u\Delta x}{M}}+{\frac {v\Delta y}{N}})}}{|\mathbf {G} _{a}\mathbf {G} _{a}^{}e^{2\pi i({\frac {u\Delta x}{M}}+{\frac {v\Delta y}{N}})}|}}\\&={\frac {\mathbf {G} _{a}\mathbf {G} _{a}^{}e^{2\pi i({\frac {u\Delta x}{M}}+{\frac {v\Delta y}{N}})}}{|\mathbf {G} _{a}\mathbf {G} _{a}^{}|}}\\&=e^{2\pi i({\frac {u\Delta x}{M}}+{\frac {v\Delta y}{N}})}\end{aligned}}} since the magnitude of an imaginary exponential always is one, and the phase of G a G a ∗ {\displaystyle \ \mathbf {G} _{a}\mathbf {G} _{a}^{}} always is zero. The inverse Fourier transform of a complex exponential is a Dirac delta function, i.e. a single peak: r ( x , y ) = δ ( x + Δ x , y + Δ y ) {\displaystyle \ r(x,y)=\delta (x+\Delta x,y+\Delta y)} This result could have been obtained by calculating the cross correlation directly. The advantage of this method is that the discrete Fourier transform and its inverse can be performed using the fast Fourier transform, which is much faster than correlation for large images. === Benefits === Unlike many spatial-domain algorithms, the phase correlation method is resilient to noise, occlusions, and other defects typical of medical or satellite images. The method can be extended to determine rotation and scaling differences between two images by first converting the images to log-polar coordinates. Due to properties of the Fourier transform, the rotation and scaling parameters can be determined in a manner invariant to translation. === Limitations === In practice, it is more likely that g b {\displaystyle \ g_{b}} will be a simple linear shift of g a {\displaystyle \ g_{a}} , rather than a circular shift as required by the explanation above. In such cases, r {\displaystyle \ r} will not be a simple delta function, which will reduce the performance of the method. In such cases, a window function (such as a Gaussian or Tukey window) should be employed during the Fourier transform to reduce edge effects, or the images should be zero padded so that the edge effects can be ignored. If the images consist of a flat background, with all detail situated away from the edges, then a linear shift will be equivalent to a circular shift, and the above derivation will hold exactly. The peak can be sharpened by using edge or vector correlation. For periodic images (such as a chessboard or picket fence), phase correlation may yield ambiguous results with several peaks in the resulting output. == Applications == Phase correlation is the preferred m
Odor source localization
Odor source localization (OSL) is the problem of locating the origin of an airborne or waterborne chemical plume using one or more mobile sensors, typically robots equipped with chemical sensors. The task sits at the intersection of robotics, fluid dynamics and machine olfaction. Chemical plumes in turbulent flows are intermittent and patchy, and most chemical sensors respond slowly and have limited selectivity, so the instantaneous reading available to a moving sensor is a poor proxy for the underlying time-averaged concentration field. Robotic OSL has been studied since the late 1980s and has applications including the detection of gas leaks, search and rescue after industrial accidents, and environmental monitoring of industrial emissions. == History == Robotic odor search emerged in the late 1980s and 1990s, drawing on earlier work in chemical ecology that had described how moths and other insects locate distant pheromone sources. R. A. Russell at Monash University was among the first to build mobile robots that followed chemical trails on the floor and tracked airborne odor plumes. Distributed and multi-robot odor search were investigated by Hayes, Martinoli and Goodman at the California Institute of Technology and EPFL, who studied cooperative plume-tracing on simulated and physical robot swarms. In 2007 Vergassola, Villermaux and Shraiman introduced infotaxis, an information-theoretic search strategy in which a sensor moves so as to maximize the expected information gain about source location, rather than following a chemical concentration gradient; the paper appeared in Nature and prompted substantial follow-up work in the robotics community. From the mid-2010s, multi-rotor unmanned aerial vehicles carrying lightweight chemical sensors became a common experimental platform for OSL research. == Problem formulation == OSL is generally decomposed into three sub-problems: plume detection (deciding whether a chemical signal is present), plume traversal (moving so as to remain in contact with the plume), and source declaration (deciding when the source has been reached). The mathematical difficulty depends strongly on the assumed dispersion model. In laminar or low-Reynolds number flows a Gaussian advection–diffusion model gives a smooth concentration field with a well-defined gradient. In turbulent flows, which dominate most realistic environments, the plume is filamentary: the sensor receives short, randomly spaced bursts of chemical separated by periods of zero signal, and the time-averaged field is not a useful guide on the time scales at which a robot must act. Source-term estimation, surveyed by Hutchinson and colleagues, additionally aims to recover both the position and the release rate of the source from the observed concentrations, often using probabilistic filters. == Biological inspiration == Many OSL strategies are explicitly modeled on the behavior of male moths flying upwind toward a pheromone source. As reviewed by Cardé and Willis, moths combine an upwind surge whenever they detect a filament of pheromone with a wider crosswind cast when contact is lost, producing a characteristic zig-zag trajectory that has been transposed onto mobile robots by several groups. Other biological models draw on the search behavior of dogs and of marine animals such as blue crabs and lobsters, which integrate chemical and bilateral hydrodynamic cues over much shorter ranges. == Algorithms and strategies == === Reactive strategies === Reactive strategies select the next motion as a direct function of the current sensor reading. Chemotaxis steers along the locally estimated concentration gradient, which is effective in laminar plumes but degrades severely in turbulence. Anemotaxis exploits a measured wind direction by surging upwind when chemical contact is made. The bio-inspired cast-and-surge family combines anemotaxis with a deterministic crosswind cast on contact loss, and is the dominant reactive approach for turbulent environments. === Probabilistic and information-theoretic strategies === Probabilistic methods maintain a posterior distribution over possible source locations and choose actions that improve that distribution. The infotaxis strategy of Vergassola, Villermaux and Shraiman selects the move that maximizes the expected reduction in entropy of the source-location posterior, and is effective in regimes where the spatial gradient is unusable. Bayesian source-term estimation extends this idea by inferring both source position and release rate, typically using particle filters or sequential Monte Carlo. === Map-based strategies === Map-based methods build a spatial model of the time-averaged gas distribution from sensor readings collected along the robot's trajectory and search for local maxima in that model. Lilienthal and colleagues describe a family of kernel-based gas distribution mapping techniques in which point measurements are convolved with a Gaussian kernel to produce a spatially extrapolated estimate. Such methods are most useful when the source can be assumed quasi-stationary and the robot is able to revisit locations. === Multi-robot and swarm strategies === Multiple robots searching cooperatively can shorten search times. Cooperative formations spread the sensors across the crosswind axis, making detection of an intermittent plume more likely. Swarm-based approaches, reviewed by Wang and colleagues, deploy larger numbers of simpler agents and rely on collective behavior rather than centralized planning; reported advantages include improved coverage of the search area and the possibility of locating multiple sources in parallel. == Sensors and platforms == Most OSL systems use metal-oxide semiconductor (MOX) sensors, photoionization detectors or electrochemical cells, which trade off sensitivity, selectivity, response time and power consumption. Ishida and colleagues describe how these sensors interact with airflow around the robot body, an effect that motivates careful aerodynamic design and active sampling. Mobile platforms include wheeled ground robots for indoor and structured outdoor environments, multi-rotor unmanned aerial vehicles for open spaces and elevated sources, and autonomous underwater vehicles for chemical plumes in the marine environment. == Notable systems == Among the early demonstrations, R. A. Russell's series of differential-drive robots at Monash University localized volatile sources in still and ventilated rooms during the 1990s. The Smelling Nano Aerial Vehicle reported by Burgués and colleagues used a Crazyflie nano-quadcopter (approximately 27 grams in mass and 10 cm across) carrying a custom MOX gas sensing board, and built three-dimensional gas distribution maps of indoor releases from sweeping flights of less than three minutes. The GADEN simulator, released by Monroy and colleagues, couples three-dimensional dispersion computed from an OpenFOAM CFD solver with models of MOX and photo-ionization gas sensors, and is widely used to test mobile-robot olfaction algorithms in simulation. == Applications == Reported applications include the localization of natural-gas and methane leaks in urban infrastructure, search for chemical contamination after industrial accidents, search and rescue, and environmental monitoring of industrial emissions. Drug- and explosives-detection robots are an adjacent application area, although these typically rely on close-range sniffing rather than long-range plume tracking. == Open challenges == Open challenges identified in recent reviews include the limited speed, selectivity and stability of available chemical sensors; the scarcity of standardized, large-scale benchmarks comparable to those available in computer vision; reliable handling of multi-source environments, where standard single-source assumptions fail; and the integration of OSL with other autonomous-vehicle subsystems such as obstacle avoidance and navigation in three-dimensional turbulent flow.