A neural radiance field (NeRF) is a neural field for reconstructing a three-dimensional representation of a scene from two-dimensional images. The NeRF model enables downstream applications of novel view synthesis, scene geometry reconstruction, and obtaining the reflectance properties of the scene. Additional scene properties such as camera poses may also be jointly learned. First introduced in 2020, it has since gained significant attention for its potential applications in computer graphics and content creation. == Algorithm == The NeRF algorithm represents a scene as a radiance field parametrized by a deep neural network (DNN). The network predicts a volume density and view-dependent emitted radiance given the spatial location ( x , y , z ) {\displaystyle (x,y,z)} and viewing direction in Euler angles ( θ , Φ ) {\displaystyle (\theta ,\Phi )} of the camera. By sampling many points along camera rays, traditional volume rendering techniques can produce an image. === Data collection === A NeRF needs to be retrained for each unique scene. The first step is to collect images of the scene from different angles and their respective camera pose. These images are standard 2D images and do not require a specialized camera or software. Any camera is able to generate datasets, provided the settings and capture method meet the requirements for SfM (Structure from Motion). This requires tracking of the camera position and orientation, often through some combination of SLAM, GPS, or inertial estimation. Researchers often use synthetic data to evaluate NeRF and related techniques. For such data, images (rendered through traditional non-learned methods) and respective camera poses are reproducible and error-free. === Training === For each sparse viewpoint (image and camera pose) provided, camera rays are marched through the scene, generating a set of 3D points with a given radiance direction (into the camera). For these points, volume density and emitted radiance are predicted using the multi-layer perceptron (MLP). An image is then generated through classical volume rendering. Because this process is fully differentiable, the error between the predicted image and the original image can be minimized with gradient descent over multiple viewpoints, encouraging the MLP to develop a coherent model of the scene. == Variations and improvements == Early versions of NeRF were slow to optimize and required that all input views were taken with the same camera in the same lighting conditions. These performed best when limited to orbiting around individual objects, such as a drum set, plants or small toys. Since the original paper in 2020, many improvements have been made to the NeRF algorithm, with variations for special use cases. === Fourier feature mapping === In 2020, shortly after the release of NeRF, the addition of Fourier Feature Mapping improved training speed and image accuracy. Deep neural networks struggle to learn high frequency functions in low dimensional domains; a phenomenon known as spectral bias. To overcome this shortcoming, points are mapped to a higher dimensional feature space before being fed into the MLP. γ ( v ) = [ a 1 cos ( 2 π B 1 T v ) a 1 sin ( 2 π B 1 T v ) ⋮ a m cos ( 2 π B m T v ) a m sin ( 2 π B m T v ) ] {\displaystyle \gamma (\mathrm {v} )={\begin{bmatrix}a_{1}\cos(2{\pi }{\mathrm {B} }_{1}^{T}\mathrm {v} )\\a_{1}\sin(2\pi {\mathrm {B} }_{1}^{T}\mathrm {v} )\\\vdots \\a_{m}\cos(2{\pi }{\mathrm {B} }_{m}^{T}\mathrm {v} )\\a_{m}\sin(2{\pi }{\mathrm {B} }_{m}^{T}\mathrm {v} )\end{bmatrix}}} Where v {\displaystyle \mathrm {v} } is the input point, B i {\displaystyle \mathrm {B} _{i}} are the frequency vectors, and a i {\displaystyle a_{i}} are coefficients. This allows for rapid convergence to high frequency functions, such as pixels in a detailed image. === Bundle-adjusting neural radiance fields === One limitation of NeRFs is the requirement of knowing accurate camera poses to train the model. Often times, pose estimation methods are not completely accurate, nor is the camera pose even possible to know. These imperfections result in artifacts and suboptimal convergence. So, a method was developed to optimize the camera pose along with the volumetric function itself. Called Bundle-Adjusting Neural Radiance Field (BARF), the technique uses a dynamic low-pass filter (DLPF) to go from coarse to fine adjustment, minimizing error by finding the geometric transformation to the desired image. This corrects imperfect camera poses and greatly improves the quality of NeRF renders. === Multiscale representation === Conventional NeRFs struggle to represent detail at all viewing distances, producing blurry images up close and overly aliased images from distant views. In 2021, researchers introduced a technique to improve the sharpness of details at different viewing scales known as mip-NeRF (comes from mipmap). Rather than sampling a single ray per pixel, the technique fits a gaussian to the conical frustum cast by the camera. This improvement effectively anti-aliases across all viewing scales. mip-NeRF also reduces overall image error and is faster to converge at about half the size of ray-based NeRF. === Learned initializations === In 2021, researchers applied meta-learning to assign initial weights to the MLP. This rapidly speeds up convergence by effectively giving the network a head start in gradient descent. Meta-learning also allowed the MLP to learn an underlying representation of certain scene types. For example, given a dataset of famous tourist landmarks, an initialized NeRF could partially reconstruct a scene given one image. === NeRF in the wild === Conventional NeRFs are vulnerable to slight variations in input images (objects, lighting) often resulting in ghosting and artifacts. As a result, NeRFs struggle to represent dynamic scenes, such as bustling city streets with changes in lighting and dynamic objects. In 2021, researchers at Google developed a new method for accounting for these variations, named NeRF in the Wild (NeRF-W). This method splits the neural network (MLP) into three separate models. The main MLP is retained to encode the static volumetric radiance. However, it operates in sequence with a separate MLP for appearance embedding (changes in lighting, camera properties) and an MLP for transient embedding (changes in scene objects). This allows the NeRF to be trained on diverse photo collections, such as those taken by mobile phones at different times of day. === Relighting === In 2021, researchers added more outputs to the MLP at the heart of NeRFs. The output now included: volume density, surface normal, material parameters, distance to the first surface intersection (in any direction), and visibility of the external environment in any direction. The inclusion of these new parameters lets the MLP learn material properties, rather than pure radiance values. This facilitates a more complex rendering pipeline, calculating direct and global illumination, specular highlights, and shadows. As a result, the NeRF can render the scene under any lighting conditions with no re-training. === Plenoctrees === Although NeRFs had reached high levels of fidelity, their costly compute time made them useless for many applications requiring real-time rendering, such as VR/AR and interactive content. Introduced in 2021, Plenoctrees (plenoptic octrees) enabled real-time rendering of pre-trained NeRFs through division of the volumetric radiance function into an octree. Rather than assigning a radiance direction into the camera, viewing direction is taken out of the network input and spherical radiance is predicted for each region. This makes rendering over 3000x faster than conventional NeRFs. === Sparse Neural Radiance Grid === Similar to Plenoctrees, this method enabled real-time rendering of pretrained NeRFs. To avoid querying the large MLP for each point, this method bakes NeRFs into Sparse Neural Radiance Grids (SNeRG). A SNeRG is a sparse voxel grid containing opacity and color, with learned feature vectors to encode view-dependent information. A lightweight, more efficient MLP is then used to produce view-dependent residuals to modify the color and opacity. To enable this compressive baking, small changes to the NeRF architecture were made, such as running the MLP once per pixel rather than for each point along the ray. These improvements make SNeRG extremely efficient, outperforming Plenoctrees. === Instant NeRFs === In 2022, researchers at Nvidia enabled real-time training of NeRFs through a technique known as Instant Neural Graphics Primitives. An innovative input encoding reduces computation, enabling real-time training of a NeRF, an improvement orders of magnitude above previous methods. The speedup stems from the use of spatial hash functions, which have O ( 1 ) {\displaystyle O(1)} access times, and parallelized architectures which run fast on modern GPUs. == Related techniques == === Plenoxels === Plen
DreamLab
DreamLab was a volunteer computing Android and iOS app launched in 2015 by Imperial College London and the Vodafone Foundation. It was discontinued on 2nd April 2025. == Description == The app helped to research cancer, COVID-19, new drugs and tropical cyclones. To do this, DreamLab accessed part of the device's processing power, with the user's consent, while the owner charged their smartphone, to speed up the calculations of the algorithms from Imperial College London. The aim of the tropical cyclone project was to prepare for climate change risks. Other projects aimed to find existing drugs and food molecules that could help people with COVID-19 and other diseases. The performance of 100,000 smartphones would reach the annual output of all research computers at Imperial College in just three months, with a nightly runtime of six hours. The app was developed in 2015 by the Garvan Institute of Medical Research in Sydney and the Vodafone Foundation. In May 2020, the project had over 490,000 registered users.
Autonomous things
Autonomous things, abbreviated AuT, or the Internet of autonomous things, abbreviated as IoAT, is an emerging term for the technological developments that are expected to bring computers into the physical environment as autonomous entities without human direction, freely moving and interacting with humans and other objects. Self-navigating drones are the first AuT technology in (limited) deployment. It is expected that the first mass-deployment of AuT technologies will be the autonomous car, generally expected to be available around 2020. Other currently expected AuT technologies include home robotics (e.g., machines that provide care for the elderly, infirm or young), and military robots (air, land or sea autonomous machines with information-collection or target-attack capabilities). AuT technologies share many common traits, which justify the common notation. They are all based on recent breakthroughs in the domains of (deep) machine learning and artificial intelligence. They all require extensive and prompt regulatory developments to specify the requirements from them and to license and manage their deployment (see the further reading below). And they all require unprecedented levels of safety (e.g., automobile safety) and security, to overcome concerns about the potential negative impact of the new technology. As an example, the autonomous car both addresses the main existing safety issues and creates new issues. It is expected to be much safer than existing vehicles, by eliminating the single most dangerous element – the driver. The US's National Highway Traffic Safety Administration estimates 94 percent of US accidents were the result of human error and poor decision-making, including speeding and impaired driving, and the Center for Internet and Society at Stanford Law School claims that "Some ninety percent of motor vehicle crashes are caused at least in part by human error". So while safety standards like the ISO 26262 specify the required safety, there is still a burden on the industry to demonstrate acceptable safety. While car accidents claim every year 35,000 lives in the US, and 1.25 million worldwide, some believe that even "a car that's 10 times as safe, which means 3,500 people die on the roads each year [in the US alone]" would not be accepted by the public. The acceptable level may be closer to the current figures on aviation accidents and incidents, with under a thousand worldwide deaths in most years – three orders of magnitude lower than cars. This underscores the unprecedented nature of the safety requirements that will need to be met for cars, with similar levels of safety expected for other Autonomous Things.
Passenger drone
A passenger drone is an autonomous aircraft that is designed to carry a small number of passengers to a destination. In 2021, Ehang, a technology company based in Guangzhou, China, developed the Ehang 184, the world's first passenger drone. == History == Unmanned aerial vehicles were first introduced in World War 1, when Britain first developed the Aerial Target, an aircraft controlled remotely through radio signals. A year later in the United States, testing of Kettering Bug, a 12-foot long biplane attached with a bomb and that launched via a “slingshot-like rail”, was also under progress. Both of their unreliable test results and their possibility of endangering friendly troops in deployment caused neither aircraft to be used during the war. Production of UAVs continued after World War I and into World War II and the Vietnam War, where they would be invaluable in assisting with training as well as reconnaissance. Late 20th century also saw the proposition and development of unique methods of travel, including personal jetpacks and even flying cars. While the previously mentioned are not drones, they serve as a precursor and foundation for the passenger drones of today. The first passenger drone was unveiled on January 6 of 2016 at the international Consumer Electronics Show (CES) in Las Vegas. Produced by Ehang, a Chinese company based in Guangzhou, the 184 was a one passenger drone equipped with four propellers that could fly for approximately 23 minutes at a top speed of 63 mph. Since then, many new companies have entered the market, but none yet have been accessible by the public. == Technological development == Since 2013, improvements in designs to wing structures have contributed to the economic feasibility of passenger drones. New structural advancements, such as the flapping-wing propulsion system based on the mechanisms of birds’ wings, are more available as they have proven their capabilities in laboratory testing. As of September 29th, 2015, most market-ready drones are delivery drones with a carrying capacity limited to small packages - with a typical max capacity of under 5 pounds. However, while the technology exists for drones with larger carrying capacities, specifically those capable of carrying multiple humans, the execution of this technology is not yet market accessible. This capacity limit must be addressed for passenger drones; given current designs strive to carry a maximum of 5 people. However, some estimates believe that passengers drones could become a reality, specifically for paid transportation and emergency purposes, as early as 2026. With implementation of this technology, there could be significant effects on ground traffic including reducing gridlock in heavily congested areas and conserving up to 15% of the fuel currently used in heavy traffic patterns. However, extensive growth of the passenger drone market also risks clouding the low-altitude airspace and causing new safety risks. However, this concern is being addressed by recent advancements in the Internet of Drones (IoD) which links drones together to ensure appropriate pathing and reduce mid-air collisions. While this brings additional security issues, including maintaining reliable communication channels in the case of technological failure, researchers hope that this will help reduce crashes that can result in damage to passengers, buildings, and people in and around the airspace. == Notable companies == Ehang is a Chinese company that has developed numerous drones including passenger plane Ehang 184. EHang 184 was their first model, developed as an eight dual rotor wing blade drone that can carry two passengers. The model was retired in 2020 and is replaced by the Ehang 216. Ehang also released a one passenger drone, Ehang 116. Ehang in 2021 unveiled the model VT-30. VT-30 is designed to have eight dual rotor wing blades to complement its fixed wing platform. Flyastro, a Texas-based drone company, developed the Astro ALTA, with two and four person passenger models. The company is known for being the first to develop a solar-powered airplane. The development team initially began with the model, Elroy. It was a two passenger drone with similar design to the ALTA. Once flight was achieved, the model Astro ALTA began development. Joby Aviation is a California based company that has developed a five passenger drone, with one seat for the pilot. The company expects to complete its FAA certification process 2022. Joby in 2020 acquired a 75 million dollar investment from service provider Uber Technologies Inc., leading to Uber Elevate and Expands partnership. Archer Aviation is a California-based company that has developed a two passenger model called Maker. It has fixed wings with twelve rotor wings. Archer is developing five person model. United Airlines has partnered with Archer for commercial sale of the model, Maker. Maker is expected to be released within Los Angeles and Miami by 2024. CityAirbus is a drone project developed by Airbus, a European multinational aerospace company, based in the Netherlands. CityAirbus has developed a four- person passenger drone with fixed wings that include rotor wing blades. Its expected certification for public flight is in 2025. Boeing, an American multinational aviation corporation is developing a passenger drone model called the Passenger Air Vehicle (PAV). The model is a fixed wing with eight rotor blade wings attached onto a platform underneath the base structure. This model can hold two passengers and still is in development. Volocopter is a German aircraft manufacturer that is developing a passenger drone called Volocity. The model consist of eighteen rotor wings above the cockpit on a circular ring. Japan Airlines, an investor of Volocopter plans to have public test in Japan as early as 2023. == Future use == === Potential benefits === Passenger drones can greatly reduce the time for travel. As passenger drones flight paths are not restricted by conventional roads, the travel distance is shortened. Current ventures such as Joby Aviation, after acquiring Uber Air, plan to take advantage of this technology in the form of air taxis. Other potential benefits include the use of passenger drones by emergency services such as search and rescue missions and the delivery of life saving goods. Companies like Ehang have already begun using passenger drones as emergency vehicles as a response to the potential river collapses during the flood season in China. === Concerns === Passenger and air traffic safety remains at the forefront of concerns. Regulations for air traffic centered around passenger drones are still underway and would continue to develop with increasing use cases for passenger drones. Remote security threats on commercial drones such as Man-In-The-Middle (MITM) attack have also exposed the vulnerabilities in current drone systems. Among American adults, 54 percent say that they would feel unsafe flying inside a passenger drone. Passenger drones can be very noisy; a single passenger drone such as Joby Aviation’s all-electric vertical take-off and landing (“eVTOL”) aircraft has an estimated noise production of 70 decibels (dB), a noise level equating to “loud traffic”.
Tensor operator
In pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which apply the notion of the spherical basis and spherical harmonics. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions. The coordinate-free generalization of a tensor operator is known as a representation operator. == The general notion of scalar, vector, and tensor operators == In quantum mechanics, physical observables that are scalars, vectors, and tensors, must be represented by scalar, vector, and tensor operators, respectively. Whether something is a scalar, vector, or tensor depends on how it is viewed by two observers whose coordinate frames are related to each other by a rotation. Alternatively, one may ask how, for a single observer, a physical quantity transforms if the state of the system is rotated. Consider, for example, a system consisting of a molecule of mass M {\displaystyle M} , traveling with a definite center of mass momentum, p z ^ {\displaystyle p{\mathbf {\hat {z}} }} , in the z {\displaystyle z} direction. If we rotate the system by 90 ∘ {\displaystyle 90^{\circ }} about the y {\displaystyle y} axis, the momentum will change to p x ^ {\displaystyle p{\mathbf {\hat {x}} }} , which is in the x {\displaystyle x} direction. The center-of-mass kinetic energy of the molecule will, however, be unchanged at p 2 / 2 M {\displaystyle p^{2}/2M} . The kinetic energy is a scalar and the momentum is a vector, and these two quantities must be represented by a scalar and a vector operator, respectively. By the latter in particular, we mean an operator whose expected values in the initial and the rotated states are p z ^ {\displaystyle p{\mathbf {\hat {z}} }} and p x ^ {\displaystyle p{\mathbf {\hat {x}} }} . The kinetic energy on the other hand must be represented by a scalar operator, whose expected value must be the same in the initial and the rotated states. In the same way, tensor quantities must be represented by tensor operators. An example of a tensor quantity (of rank two) is the electrical quadrupole moment of the above molecule. Likewise, the octupole and hexadecapole moments would be tensors of rank three and four, respectively. Other examples of scalar operators are the total energy operator (more commonly called the Hamiltonian), the potential energy, and the dipole-dipole interaction energy of two atoms. Examples of vector operators are the momentum, the position, the orbital angular momentum, L {\displaystyle {\mathbf {L} }} , and the spin angular momentum, S {\displaystyle {\mathbf {S} }} . (Fine print: Angular momentum is a vector as far as rotations are concerned, but unlike position or momentum it does not change sign under space inversion, and when one wishes to provide this information, it is said to be a pseudovector.) Scalar, vector and tensor operators can also be formed by products of operators. For example, the scalar product L ⋅ S {\displaystyle {\mathbf {L} }\cdot {\mathbf {S} }} of the two vector operators, L {\displaystyle {\mathbf {L} }} and S {\displaystyle {\mathbf {S} }} , is a scalar operator, which figures prominently in discussions of the spin–orbit interaction. Similarly, the quadrupole moment tensor of our example molecule has the nine components Q i j = ∑ α q α ( 3 r α , i r α , j − r α 2 δ i j ) . {\displaystyle Q_{ij}=\sum _{\alpha }q_{\alpha }\left(3r_{\alpha ,i}r_{\alpha ,j}-r_{\alpha }^{2}\delta _{ij}\right).} Here, the indices i {\displaystyle i} and j {\displaystyle j} can independently take on the values 1, 2, and 3 (or x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} ) corresponding to the three Cartesian axes, the index α {\displaystyle \alpha } runs over all particles (electrons and nuclei) in the molecule, q α {\displaystyle q_{\alpha }} is the charge on particle α {\displaystyle \alpha } , and r α , i {\displaystyle r_{\alpha ,i}} is the i {\displaystyle i} -th component of the position of this particle. Each term in the sum is a tensor operator. In particular, the nine products r α , i r α , j {\displaystyle r_{\alpha ,i}r_{\alpha ,j}} together form a second rank tensor, formed by taking the outer product of the vector operator r α {\displaystyle {\mathbf {r} }_{\alpha }} with itself. == Rotations of quantum states == === Quantum rotation operator === The rotation operator about the unit vector n (defining the axis of rotation) through angle θ is U [ R ( θ , n ^ ) ] = exp ( − i θ ℏ n ^ ⋅ J ) {\displaystyle U[R(\theta ,{\hat {\mathbf {n} }})]=\exp \left(-{\frac {i\theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} \right)} where J = (Jx, Jy, Jz) are the rotation generators (also the angular momentum matrices): J x = ℏ 2 ( 0 1 0 1 0 1 0 1 0 ) J y = ℏ 2 ( 0 i 0 − i 0 i 0 − i 0 ) J z = ℏ ( − 1 0 0 0 0 0 0 0 1 ) {\displaystyle J_{x}={\frac {\hbar }{\sqrt {2}}}{\begin{pmatrix}0&1&0\\1&0&1\\0&1&0\end{pmatrix}}\,\quad J_{y}={\frac {\hbar }{\sqrt {2}}}{\begin{pmatrix}0&i&0\\-i&0&i\\0&-i&0\end{pmatrix}}\,\quad J_{z}=\hbar {\begin{pmatrix}-1&0&0\\0&0&0\\0&0&1\end{pmatrix}}} and let R ^ = R ^ ( θ , n ^ ) {\displaystyle {\widehat {R}}={\widehat {R}}(\theta ,{\hat {\mathbf {n} }})} be a rotation matrix. According to the Rodrigues' rotation formula, the rotation operator then amounts to U [ R ( θ , n ^ ) ] = 1 1 − i sin θ ℏ n ^ ⋅ J − 1 − cos θ ℏ 2 ( n ^ ⋅ J ) 2 . {\displaystyle U[R(\theta ,{\hat {\mathbf {n} }})]=1\!\!1-{\frac {i\sin \theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} -{\frac {1-\cos \theta }{\hbar ^{2}}}({\hat {\mathbf {n} }}\cdot \mathbf {J} )^{2}.} An operator Ω ^ {\displaystyle {\widehat {\Omega }}} is invariant under a unitary transformation U if Ω ^ = U † Ω ^ U ; {\displaystyle {\widehat {\Omega }}={U}^{\dagger }{\widehat {\Omega }}U;} in this case for the rotation U ^ ( R ) {\displaystyle {\widehat {U}}(R)} , Ω ^ = U ( R ) † Ω ^ U ( R ) = exp ( i θ ℏ n ^ ⋅ J ) Ω ^ exp ( − i θ ℏ n ^ ⋅ J ) . {\displaystyle {\widehat {\Omega }}={U(R)}^{\dagger }{\widehat {\Omega }}U(R)=\exp \left({\frac {i\theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} \right){\widehat {\Omega }}\exp \left(-{\frac {i\theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} \right).} === Angular momentum eigenkets === The orthonormal basis set for total angular momentum is | j , m ⟩ {\displaystyle |j,m\rangle } , where j is the total angular momentum quantum number and m is the magnetic angular momentum quantum number, which takes values −j, −j + 1, ..., j − 1, j. A general state within the j subspace | ψ ⟩ = ∑ m c j m | j , m ⟩ {\displaystyle |\psi \rangle =\sum _{m}c_{jm}|j,m\rangle } rotates to a new state by: | ψ ¯ ⟩ = U ( R ) | ψ ⟩ = ∑ m c j m U ( R ) | j , m ⟩ {\displaystyle |{\bar {\psi }}\rangle =U(R)|\psi \rangle =\sum _{m}c_{jm}U(R)|j,m\rangle } Using the completeness condition: I = ∑ m ′ | j , m ′ ⟩ ⟨ j , m ′ | {\displaystyle I=\sum _{m'}|j,m'\rangle \langle j,m'|} we have | ψ ¯ ⟩ = I U ( R ) | ψ ⟩ = ∑ m m ′ c j m | j , m ′ ⟩ ⟨ j , m ′ | U ( R ) | j , m ⟩ {\displaystyle |{\bar {\psi }}\rangle =IU(R)|\psi \rangle =\sum _{mm'}c_{jm}|j,m'\rangle \langle j,m'|U(R)|j,m\rangle } Introducing the Wigner D matrix elements: D ( R ) m ′ m ( j ) = ⟨ j , m ′ | U ( R ) | j , m ⟩ {\displaystyle {D(R)}_{m'm}^{(j)}=\langle j,m'|U(R)|j,m\rangle } gives the matrix multiplication: | ψ ¯ ⟩ = ∑ m m ′ c j m D m ′ m ( j ) | j , m ′ ⟩ ⇒ | ψ ¯ ⟩ = D ( j ) | ψ ⟩ {\displaystyle |{\bar {\psi }}\rangle =\sum _{mm'}c_{jm}D_{m'm}^{(j)}|j,m'\rangle \quad \Rightarrow \quad |{\bar {\psi }}\rangle =D^{(j)}|\psi \rangle } For one basis ket: | j , m ¯ ⟩ = ∑ m ′ D ( R ) m ′ m ( j ) | j , m ′ ⟩ {\displaystyle |{\overline {j,m}}\rangle =\sum _{m'}{D(R)}_{m'm}^{(j)}|j,m'\rangle } For the case of orbital angular momentum, the eigenstates | ℓ , m ⟩ {\displaystyle |\ell ,m\rangle } of the orbital angular momentum operator L and solutions of Laplace's equation on a 3d sphere are spherical harmonics: Y ℓ m ( θ , ϕ ) = ⟨ θ , ϕ | ℓ , m ⟩ = ( 2 ℓ + 1 ) 4 π ( ℓ − m ) ! ( ℓ + m ) ! P ℓ m ( cos θ ) e i m ϕ {\displaystyle Y_{\ell }^{m}(\theta ,\phi )=\langle \theta ,\phi |\ell ,m\rangle ={\sqrt {{(2\ell +1) \over 4\pi }{(\ell -m)! \over (\ell +m)!}}}\,P_{\ell }^{m}(\cos {\theta })\,e^{im\phi }} where Pℓm is an associated Legendre polynomial, ℓ is the orbital angular momentum quantum number, and m is the orbital magnetic quantum number which takes the values −ℓ, −ℓ + 1, ... ℓ − 1, ℓ The formalism of spherical harmonics have wide applications in applied mathematics, and are closely related to the formalism of spherical tensors, as shown below. Spherical harmonics are functions of the polar and azimuthal angles, ϕ and θ respectively, which can be conveniently collected into a unit vector n(θ, ϕ) pointing in the direction of those angles, in the Cartesian basis it is: n ^ ( θ , ϕ ) = cos ϕ sin θ e x + s
United States Tech Force
The U.S. Tech Force (also styled as US Tech Force, Tech Force, or Government Tech Force) is a federal hiring initiative launched by the second Donald Trump administration in December 2025. The program, administered by the Office of Personnel Management (OPM), aims to recruit about 1,000 early-career technology professionals into two-year government jobs to modernize federal IT systems, advance artificial intelligence (AI) capabilities, and address technological gaps in government operations. The initiative is an effort to plug capability gaps created by Trump-administration efforts to shrink the federal government, which led to the departure of some 220,000 federal employees, including many in IT. The initiative seeks early-career workers; officials said it would offer competitive salaries and opportunities to work on high-impact government technology projects. Major technology companies—including Amazon, Apple, Microsoft, Nvidia, Meta, Google, and OpenAI—agreed to help identify and refer candidates. Candidates are allowed to take Tech Force positions on leaves of absence and without divesting their stock, raising conflict-of-interest questions. In January 2026, OPM direction Scott Kupor said the deadline for applying to Tech Force was being extended because of "tremendous interest" without saying how many people had actually applied. Also in December 2025, news broke that the administration is planning another novel use of private-sector workers: hiring cybersecurity firms for offensive cyber operations.
Speech recognition
Speech recognition (automatic speech recognition (ASR), computer speech recognition, or speech-to-text (STT)) is a sub-field of computational linguistics concerned with methods and technologies that translate spoken language into text or other interpretable forms. Speech recognition applications include voice user interfaces, where the user speaks to a device, which "listens" and processes the audio. Common voice applications include interpreting commands for calling, call routing, home automation, and aircraft control. These applications are called direct voice input. Productivity applications include searching audio recordings, creating transcripts, and dictation. Speech recognition can be used to analyse speaker characteristics, such as identifying native language using pronunciation assessment. Voice recognition (speaker identification) refers to identifying the speaker, rather than speech contents. Recognizing the speaker can simplify the task of translating speech in systems trained on a specific person's voice. It can also be used to authenticate the speaker as part of a security process. == History == Applications for speech recognition developed over many decades, with progress accelerated due to advances in deep learning and the use of big data. These advances are reflected in an increase in academic papers, and greater system adoption. Key areas of growth include vocabulary size, more accurate recognition for unfamiliar speakers (speaker independence), and faster processing speed. === Pre-1970 === 1952 – Bell Labs researchers, Stephen Balashek, R. Biddulph, and K. H. Davis, built Audrey for single-speaker digit recognition. Their system located the formants in the power spectrum of each utterance. 1960 – Gunnar Fant developed and published the source–filter model of speech production. 1962 – IBM's 16-word "Shoebox" machine's speech recognition debuted at the 1962 World's Fair. 1966 – Linear predictive coding, a speech coding method, was proposed by Fumitada Itakura of Nagoya University and Shuzo Saito of Nippon Telegraph and Telephone. 1969 – Funding at Bell Labs came to a halt for several years after the company's head engineer, John R. Pierce, wrote an open letter criticizing speech recognition research. This defunding lasted until Pierce retired and James L. Flanagan took over. Raj Reddy was the first person to work on continuous speech recognition, as a graduate student at Stanford University in the late 1960s. Previous systems required users to pause after each word. Reddy's system issued spoken commands for playing chess. Around this time, Soviet researchers invented the dynamic time warping (DTW) algorithm and used it to create a recognizer capable of operating on a 200-word vocabulary. DTW processed speech by dividing it into short frames (e.g. 10 ms segments) and treating each frame as a unit. Speaker independence, however, remained unsolved. === 1970–1990 === 1971 – DARPA funded a five-year speech recognition research project, Speech Understanding Research, seeking a minimum vocabulary size of 1,000 words. The project considered speech understanding a key to achieving progress in speech recognition, which was later disproved. BBN, IBM, Carnegie Mellon (CMU), and Stanford Research Institute participated. 1972 – The IEEE Acoustics, Speech, and Signal Processing group held a conference in Newton, Massachusetts. 1976 – The first ICASSP was held in Philadelphia, which became a major venue for publishing on speech recognition. During the late 1960s, Leonard Baum developed the mathematics of Markov chains at the Institute for Defense Analysis. A decade later, at CMU, Raj Reddy's students James Baker and Janet M. Baker began using the hidden Markov model (HMM) for speech recognition. James Baker had learned about HMMs while at the Institute for Defense Analysis. HMMs enabled researchers to combine sources of knowledge, such as acoustics, language, and syntax, in a unified probabilistic model. By the mid-1980s, Fred Jelinek's team at IBM created a voice-activated typewriter called Tangora, which could handle a 20,000-word vocabulary. Jelinek's statistical approach placed less emphasis on emulating human brain processes in favor of statistical modelling. (Jelinek's group independently discovered the application of HMMs to speech.) This was controversial among linguists since HMMs are too simplistic to account for many features of human languages. However, the HMM proved to be a highly useful way for modelling speech and replaced dynamic time warping as the dominant speech recognition algorithm in the 1980s. 1982 – Dragon Systems, founded by James and Janet M. Baker, was one of IBM's few competitors. === Practical speech recognition === The 1980s also saw the introduction of the n-gram language model. 1987 – The back-off model enabled language models to use multiple-length n-grams, and CSELT used HMM to recognize languages (in software and hardware, e.g. RIPAC). At the end of the DARPA program in 1976, the best computer available to researchers was the PDP-10 with 4 MB of RAM. It could take up to 100 minutes to decode 30 seconds of speech. Practical products included: 1984 – the Apricot Portable was released with up to 4096 words support, of which only 64 could be held in RAM at a time. 1987 – a recognizer from Kurzweil Applied Intelligence 1990 – Dragon Dictate, a consumer product released in 1990. AT&T deployed the Voice Recognition Call Processing service in 1992 to route telephone calls without a human operator. The technology was developed by Lawrence Rabiner and others at Bell Labs. By the early 1990s, the vocabulary of the typical commercial speech recognition system had exceeded the average human vocabulary. Reddy's former student, Xuedong Huang, developed the Sphinx-II system at CMU. Sphinx-II was the first to do speaker-independent, large vocabulary, continuous speech recognition, and it won DARPA's 1992 evaluation. Handling continuous speech with a large vocabulary was a major milestone. Huang later founded the speech recognition group at Microsoft in 1993. Reddy's student Kai-Fu Lee joined Apple, where, in 1992, he helped develop the Casper speech interface prototype. Lernout & Hauspie, a Belgium-based speech recognition company, acquired other companies, including Kurzweil Applied Intelligence in 1997 and Dragon Systems in 2000. L&H was used in Windows XP. L&H was an industry leader until an accounting scandal destroyed it in 2001. L&H speech technology was bought by ScanSoft, which became Nuance in 2005. Apple licensed Nuance software for its digital assistant Siri. ==== 2000s ==== In the 2000s, DARPA sponsored two speech recognition programs: Effective Affordable Reusable Speech-to-Text (EARS) in 2002, followed by Global Autonomous Language Exploitation (GALE) in 2005. Four teams participated in EARS: IBM; a team led by BBN with LIMSI and the University of Pittsburgh; Cambridge University; and a team composed of ICSI, SRI, and the University of Washington. EARS funded the collection of the Switchboard telephone speech corpus, which contained 260 hours of recorded conversations from over 500 speakers. The GALE program focused on Arabic and Mandarin broadcast news. Google's first effort at speech recognition came in 2007 after recruiting Nuance researchers. Its first product, GOOG-411, was a telephone-based directory service. Since at least 2006, the U.S. National Security Agency has employed keyword spotting, allowing analysts to index large volumes of recorded conversations and identify speech containing "interesting" keywords. Other government research programs focused on intelligence applications, such as DARPA's EARS program and IARPA's Babel program. In the early 2000s, speech recognition was dominated by hidden Markov models combined with feed-forward artificial neural networks (ANN). Later, speech recognition was taken over by long short-term memory (LSTM), a recurrent neural network (RNN) published by Sepp Hochreiter & Jürgen Schmidhuber in 1997. LSTM RNNs avoid the vanishing gradient problem and can learn "Very Deep Learning" tasks that require memories of events that happened thousands of discrete time steps earlier, which is important for speech. Around 2007, LSTMs trained with Connectionist Temporal Classification (CTC) began to outperform. In 2015, Google reported a 49 percent error‑rate reduction in its speech recognition via CTC‑trained LSTM. Transformers, a type of neural network based solely on attention, were adopted in computer vision and language modelling, and then to speech recognition. Deep feed-forward (non-recurrent) networks for acoustic modelling were introduced in 2009 by Geoffrey Hinton and his students at the University of Toronto, and by Li Deng and colleagues at Microsoft Research. In contrast to the prioer incremental improvements, deep learning decreased error rates by 30%. Both shallow and deep forms (e.g., recurrent nets) of ANNs had been explored since the 1980s. Howev