In computer graphics, the rendering equation is an integral equation that expresses the amount of light leaving a point on a surface as the sum of emitted light and reflected light. It was independently introduced into computer graphics by David Immel et al. and James Kajiya in 1986. The equation is important in the theory of physically based rendering, describing the relationships between the bidirectional reflectance distribution function (BRDF) and the radiometric quantities used in rendering. The rendering equation is defined at every point on every surface in the scene being rendered, including points hidden from the camera. The incoming light quantities on the right side of the equation usually come from the left (outgoing) side at other points in the scene (ray casting can be used to find these other points). The radiosity rendering method solves a discrete approximation of this system of equations. In distributed ray tracing, the integral on the right side of the equation may be evaluated using Monte Carlo integration by randomly sampling possible incoming light directions. Path tracing improves and simplifies this method. The rendering equation can be extended to handle effects such as fluorescence (in which some absorbed energy is re-emitted at different wavelengths) and can support transparent and translucent materials by using a bidirectional scattering distribution function (BSDF) in place of a BRDF. The theory of path tracing sometimes uses a path integral (integral over possible paths from a light source to a point) instead of the integral over possible incoming directions. == Equation form == The rendering equation may be written in the form L o ( x , ω o , λ , t ) = L e ( x , ω o , λ , t ) + L r ( x , ω o , λ , t ) {\displaystyle L_{\text{o}}(\mathbf {x} ,\omega _{\text{o}},\lambda ,t)=L_{\text{e}}(\mathbf {x} ,\omega _{\text{o}},\lambda ,t)+L_{\text{r}}(\mathbf {x} ,\omega _{\text{o}},\lambda ,t)} L r ( x , ω o , λ , t ) = ∫ Ω f r ( x , ω i , ω o , λ , t ) L i ( x , ω i , λ , t ) ( ω i ⋅ n ) d ω i {\displaystyle L_{\text{r}}(\mathbf {x} ,\omega _{\text{o}},\lambda ,t)=\int _{\Omega }f_{\text{r}}(\mathbf {x} ,\omega _{\text{i}},\omega _{\text{o}},\lambda ,t)L_{\text{i}}(\mathbf {x} ,\omega _{\text{i}},\lambda ,t)(\omega _{\text{i}}\cdot \mathbf {n} )\operatorname {d} \omega _{\text{i}}} where L o ( x , ω o , λ , t ) {\displaystyle L_{\text{o}}(\mathbf {x} ,\omega _{\text{o}},\lambda ,t)} is the total spectral radiance of wavelength λ {\displaystyle \lambda } directed outward along direction ω o {\displaystyle \omega _{\text{o}}} at time t {\displaystyle t} , from a particular position x {\displaystyle \mathbf {x} } x {\displaystyle \mathbf {x} } is the location in space ω o {\displaystyle \omega _{\text{o}}} is the direction of the outgoing light λ {\displaystyle \lambda } is a particular wavelength of light t {\displaystyle t} is time L e ( x , ω o , λ , t ) {\displaystyle L_{\text{e}}(\mathbf {x} ,\omega _{\text{o}},\lambda ,t)} is emitted spectral radiance L r ( x , ω o , λ , t ) {\displaystyle L_{\text{r}}(\mathbf {x} ,\omega _{\text{o}},\lambda ,t)} is reflected spectral radiance ∫ Ω … d ω i {\displaystyle \int _{\Omega }\dots \operatorname {d} \omega _{\text{i}}} is an integral over Ω {\displaystyle \Omega } Ω {\displaystyle \Omega } is the unit hemisphere centered around n {\displaystyle \mathbf {n} } containing all possible values for ω i {\displaystyle \omega _{\text{i}}} where ω i ⋅ n > 0 {\displaystyle \omega _{\text{i}}\cdot \mathbf {n} >0} f r ( x , ω i , ω o , λ , t ) {\displaystyle f_{\text{r}}(\mathbf {x} ,\omega _{\text{i}},\omega _{\text{o}},\lambda ,t)} is the bidirectional reflectance distribution function, the proportion of light reflected from ω i {\displaystyle \omega _{\text{i}}} to ω o {\displaystyle \omega _{\text{o}}} at position x {\displaystyle \mathbf {x} } , time t {\displaystyle t} , and at wavelength λ {\displaystyle \lambda } ω i {\displaystyle \omega _{\text{i}}} is the negative direction of the incoming light L i ( x , ω i , λ , t ) {\displaystyle L_{\text{i}}(\mathbf {x} ,\omega _{\text{i}},\lambda ,t)} is spectral radiance of wavelength λ {\displaystyle \lambda } coming inward toward x {\displaystyle \mathbf {x} } from direction ω i {\displaystyle \omega _{\text{i}}} at time t {\displaystyle t} n {\displaystyle \mathbf {n} } is the surface normal at x {\displaystyle \mathbf {x} } ω i ⋅ n {\displaystyle \omega _{\text{i}}\cdot \mathbf {n} } is the weakening factor of outward irradiance due to incident angle, as the light flux is smeared across a surface whose area is larger than the projected area perpendicular to the ray. This is often written as cos θ i {\displaystyle \cos \theta _{i}} . Two noteworthy features are: its linearity—it is composed only of multiplications and additions, and its spatial homogeneity—it is the same in all positions and orientations. These mean a wide range of factorings and rearrangements of the equation are possible. It is a Fredholm integral equation of the second kind, similar to those that arise in quantum field theory. Note this equation's spectral and time dependence — L o {\displaystyle L_{\text{o}}} may be sampled at or integrated over sections of the visible spectrum to obtain, for example, a trichromatic color sample. A pixel value for a single frame in an animation may be obtained by fixing t ; {\displaystyle t;} motion blur can be produced by averaging L o {\displaystyle L_{\text{o}}} over some given time interval (by integrating over the time interval and dividing by the length of the interval). Note that a solution to the rendering equation is the function L o {\displaystyle L_{\text{o}}} . The function L i {\displaystyle L_{\text{i}}} is related to L o {\displaystyle L_{\text{o}}} via a ray-tracing operation: The incoming radiance from some direction at one point is the outgoing radiance at some other point in the opposite direction. == Applications == Solving the rendering equation for any given scene is the primary challenge in realistic rendering. One approach to solving the equation is based on finite element methods, leading to the radiosity algorithm. Another approach using Monte Carlo methods has led to many different algorithms including path tracing, photon mapping, and Metropolis light transport, among others. == Limitations == Although the equation is very general, it does not capture every aspect of light reflection. Some missing aspects include the following: Transmission, which occurs when light is transmitted through the surface, such as when it hits a glass object or a water surface, Subsurface scattering, where the spatial locations for incoming and departing light are different. Surfaces rendered without accounting for subsurface scattering may appear unnaturally opaque — however, it is not necessary to account for this if transmission is included in the equation, since that will effectively include also light scattered under the surface, Polarization, where different light polarizations will sometimes have different reflection distributions, for example when light bounces at a water surface, Phosphorescence, which occurs when light or other electromagnetic radiation is absorbed at one moment and emitted at a later moment, usually with a longer wavelength (unless the absorbed electromagnetic radiation is very intense), Interference, where the wave properties of light are exhibited, Fluorescence, where the absorbed and emitted light have different wavelengths, Non-linear effects, where very intense light can increase the energy level of an electron with more energy than that of a single photon (this can occur if the electron is hit by two photons at the same time), and emission of light with higher frequency than the frequency of the light that hit the surface suddenly becomes possible, and Doppler effect, where light that bounces off an object moving at a very high speed will get its wavelength changed: if the light bounces off an object that is moving towards it, the light will be blueshifted and the photons will be packed more closely so the photon flux will be increased; if it bounces off an object moving away from it, it will be redshifted and the photon flux will be decreased. This effect becomes apparent only at speeds comparable to the speed of light, which is not the case for most rendering applications. For scenes that are either not composed of simple surfaces in a vacuum or for which the travel time for light is an important factor, researchers have generalized the rendering equation to produce a volume rendering equation suitable for volume rendering and a transient rendering equation for use with data from a time-of-flight camera.
Grammatik
Grammatik was the first grammar-checking program for home computers. Aspen Software of Albuquerque, NM, released the earliest version of this diction and style checker for personal computers. It was first released no later than 1981, and was inspired by the Writer's Workbench. Grammatik was first available for the TRS-80, and soon had versions for CP/M and the IBM PC. Reference Software International of San Francisco, California, acquired Grammatik in 1985. Development of Grammatik continued, and it became an actual grammar checker that could detect writing errors beyond simple style checking. Subsequent versions were released for MS-DOS, Windows, Macintosh, and Unix. Grammatik was ultimately acquired by WordPerfect Corporation and is integrated into the WordPerfect word processor.
Hyper-encryption
Hyper-encryption is a form of encryption invented by Michael O. Rabin which uses a high-bandwidth source of public random bits, together with a secret key that is shared by only the sender and recipient(s) of the message. It uses the assumptions of Ueli Maurer's bounded-storage model as the basis of its secrecy. Although everyone can see the data, decryption by adversaries without the secret key is still not feasible, because of the space limitations of storing enough data to mount an attack against the system. Unlike almost all other cryptosystems except the one-time pad, hyper-encryption can be proved to be information-theoretically secure, provided the storage bound cannot be surpassed. Moreover, if the necessary public information cannot be stored at the time of transmission, the plaintext can be shown to be impossible to recover, regardless of the computational capacity available to an adversary in the future, even if they have access to the secret key at that future time. A highly energy-efficient implementation of a hyper-encryption chip was demonstrated by Krishna Palem et al. using the Probabilistic CMOS or PCMOS technology and was shown to be ~205 times more efficient in terms of Energy-Performance-Product.
MDS matrix
An MDS matrix (maximum distance separable) is a matrix representing a function with certain diffusion properties that have useful applications in cryptography. Technically, an m × n {\displaystyle m\times n} matrix A {\displaystyle A} over a finite field K {\displaystyle K} is an MDS matrix if it is the transformation matrix of a linear transformation f ( x ) = A x {\displaystyle f(x)=Ax} from K n {\displaystyle K^{n}} to K m {\displaystyle K^{m}} such that no two different ( m + n ) {\displaystyle (m+n)} -tuples of the form ( x , f ( x ) ) {\displaystyle (x,f(x))} coincide in n {\displaystyle n} or more components. Equivalently, the set of all ( m + n ) {\displaystyle (m+n)} -tuples ( x , f ( x ) ) {\displaystyle (x,f(x))} is an MDS code, i.e., a linear code that reaches the Singleton bound. Let A ~ = ( I n A ) {\displaystyle {\tilde {A}}={\begin{pmatrix}\mathrm {I} _{n}\\\hline \mathrm {A} \end{pmatrix}}} be the matrix obtained by joining the identity matrix I n {\displaystyle \mathrm {I} _{n}} to A {\displaystyle A} . Then a necessary and sufficient condition for a matrix A {\displaystyle A} to be MDS is that every possible n × n {\displaystyle n\times n} submatrix obtained by removing m {\displaystyle m} rows from A ~ {\displaystyle {\tilde {A}}} is non-singular. This is also equivalent to the following: all the sub-determinants of the matrix A {\displaystyle A} are non-zero. Then a binary matrix A {\displaystyle A} (namely over the field with two elements) is never MDS unless it has only one row or only one column with all components 1 {\displaystyle 1} . Reed–Solomon codes have the MDS property and are frequently used to obtain the MDS matrices used in cryptographic algorithms. Serge Vaudenay suggested using MDS matrices in cryptographic primitives to produce what he called multipermutations, not-necessarily linear functions with this same property. These functions have what he called perfect diffusion: changing t {\displaystyle t} of the inputs changes at least m − t + 1 {\displaystyle m-t+1} of the outputs. He showed how to exploit imperfect diffusion to cryptanalyze functions that are not multipermutations. MDS matrices are used for diffusion in such block ciphers as AES, SHARK, Square, Twofish, Anubis, KHAZAD, Manta, Hierocrypt, Kalyna, Camellia and HADESMiMC, and in the stream cipher MUGI and the cryptographic hash function Whirlpool, Poseidon.
Short Weather Cipher
The Short Weather Cipher (German: Wetterkurzschlüssel, abbreviated WKS), also known as the weather short signal book, was a cipher, presented as a codebook, that was used by the radio telegraphists aboard U-boats of the German Navy (Kriegsmarine) during World War II. It was used to condense weather reports into a short 7-letter message, which was enciphered by using the naval Enigma and transmitted by radiomen to intercept stations on shore, where it was deciphered by Enigma and the 7-letter weather report was reconstructed. == History == During World War II, during various times, different versions of the cipher were in operation. The first issue carried the codename Weimar. It was replaced by the edition Eisenach on 20 January 1942. On 10 March 1943, the third edition of the weather key, bearing the codename Naumburg, entered into force. On May 9, 1941, during Operation Primrose, the operation to occupy Åndalsnes and create a diversion south of Trondheim in Norway as part of the Norwegian Campaign, an intact Naval Enigma (M3) cipher machine, a copy of the "Weimar" version of the short weather cipher and a copy of the short signal book (German: Kurzsignalbuch or Kurzsignale for short) was recovered from the submarine U-110, that was captured in the North Atlantic east of Cape Farewell, Greenland. This enabled the cryptanalysts in Bletchley Park to break the encryption of the M3 and to decipher the German submarine radio messages. The Short Weather Cipher was critical in the cryptanalysis of the Naval Enigma M4 and yielded excellent cribs. On 30 October 1942, a copy of the Wetterkurzschlüssel, the short weather cipher, and of the short signal book, the Kurzsignale, were recovered as part of a daring raid on the U-boat U-559, when three Royal Navy sailors, Lieutenant Anthony Fasson, Able Seaman Colin Grazier and NAAFI canteen assistant Tommy Brown, then boarded the abandoned submarine, and recovered the documents after a 90-minute search. They reached the Government Code and Cypher at Bletchley Park after a three-week delay, on 24 November 1942. The documents which cost the lives of Fasson and Grazier proved to be particularly important in breaking the Naval Enigma M4. The version of the short weather cipher recovered was the Eisenach version. Unlike the first version Weimar, the Eisenach did not list the 26 rotor positions that were indicated by a letter, to be used in enciphering weather reports. Thus, Hut 8 cryptanalysts thought that all four rotors were used to encipher weather reports. Testing on the Bombes began to surface weather kisses (identical messages in two cryptosystems). On 13 December 1942, a crib obtained using the Short Weather Cipher gave a key with the Naval Enigma M4 rotatable Umkehrwalze (reversing roller or reflector) in the neutral position, making it equivalent to a standard Enigma and thus making B-Dienst messages potentially breakable on existing bombes. Hut 8 learned that the 4-letter indicators for regular U-boat messages were the same as 3-letter indicators for weather messages the same day, except for one extra letter. This meant that once the key was found for a weather message on any day, the fourth rotor had to be only tested in 26 positions to find the full 4-letter key. By the end of the day on Sunday 13 December, Rodger Winn of the Submarine Tracking Room at Bletchley Park knew that Shark Enigma Cipher was broken. When the third edition of the short signal book was introduced on 10 March 1943, Hut 8 was immediately deprived of cribs. However, by the 19 March, cribs were again being used by Hut 8 personnel, using the method of employing short signal sighting reports. These were reports made by U-boats when contact was made with Kurzsignalheft code book. Hut 8 managed to solve Shark for 90 out of 112 days before the end of June. Kurzsignalheft short sighting reports also used M4 in M3 mode. By the end of June, four-rotor bombes had entered service at Bletchley Park, and by August had been introduced by the US Navy. From September onwards, Shark was generally solved within 24 hours. == Operation == The U-boat encoded weather reports using the Short Weather Cipher, before being enciphered on the Naval Enigma. The shore patrol of the Kriegsmarine, deciphered the message and decoded it, then forwarding it to a central meteorological station, which rebroadcast the data as ship synoptics, after enciphering it with additive tables using a cipher, which was called Germet 3 by Hut 8 personnel. The short weather cipher coded weather reports using a polyphonic single-letter code with X missing. A = +28° ◦ B = +27° ◦ C = +26° ◦ D = +25° ◦ . . . ◦ W = +6° ◦ Y= +5° ◦ Z = +4° ◦ A = +3° ◦ B = +2° ◦ C = +1° ◦ D = 0° ◦ E =−1° ◦ F =−2° ◦ . . . ◦ Z = −21° ◦ In a similar way, water temperature, atmospheric pressure, humidity, wind direction, wind velocity, visibility, degree of cloudiness, geographic latitude, and geographic longitude had to be coded in a prescribed order with the weather report consisted of a single short word. Based on the approximate knowledge of the position of the submarine, the Kriegsmarine telegraphist who received the message could translate the letter "S", according to the above table, which could mean 10 °C or −15 °C, back to the correct temperature. Similarly, the direction and the type of swell was also coded with only a single letter: ----------------------------------------------------- Direction from which | Type of swell the swell comes | low | middle high | high | ----------------------------------------------------- N | a | i | q | NE | b | j | r | E | c | k | s | SE | d | l | t | S | e | m | u | SW | f | n | v | W | g | o | w | NW | h | p | x | No swelling | | | | y Intermittent | | | | z As an example of the cipher, a weather report for 68° North latitude, 20° West longitude (north of Iceland) with atmospheric pressure 972 millibars, temperature minus 5 °C, wind northwest Force 6 (on the Beaufort scale), 3/10 cirrus cloud cover, visibility 5 nautical miles, would be coded as MZNFPED. == Publications == Bauer, Arthur O. (1997), Funkpeilung als alliierte Waffe gegen deutsche U-Boote 1939–1945 [Direction finding as Allied weapon against German submarines from 1939 to 1945] (in German), Diemen, NL: Selbstverlag, ISBN 978-3-00-002142-8 Bauer, Friedrich L. (2007), Decrypted Secrets. Methods and Maxims of Cryptology (4., rev. and extended ed.), Berlin Heidelberg New York: Springer, ISBN 978-3-540-24502-5 Pfeiffer, Paul N. (October 1998), "Breaking the German Weather Ciphers in the Mediterranean Detachment, 849th Signal Intelligence Service", Cryptologia, 22 (4): 354–369, doi:10.1080/0161-119891886975, ISSN 0161-1194 Ulbricht, Heinz (2005), Die Chiffriermaschine Enigma – Trügerische Sicherheit. Ein Beitrag zur Geschichte der Nachrichtendienste [The Enigma cipher machine – Deceptive security. A contribution to the history of the intelligence services], Dissertation, Fachbereich Mathematik und Informatik, Technische Universität Braunschweig (in German)
Decorrelation
Decorrelation is a general term for any process that is used to reduce autocorrelation within a signal, or cross-correlation within a set of signals, while preserving other aspects of the signal. A frequently used method of decorrelation is the use of a matched linear filter to reduce the autocorrelation of a signal as far as possible. Since the minimum possible autocorrelation for a given signal energy is achieved by equalising the power spectrum of the signal to be similar to that of a white noise signal, this is often referred to as signal whitening. == Process == === Signal processing === Most decorrelation algorithms are linear, but there are also non-linear decorrelation algorithms. Many data compression algorithms incorporate a decorrelation stage. For example, many transform coders first apply a fixed linear transformation that would, on average, have the effect of decorrelating a typical signal of the class to be coded, prior to any later processing. This is typically a Karhunen–Loève transform, or a simplified approximation such as the discrete cosine transform. By comparison, sub-band coders do not generally have an explicit decorrelation step, but instead exploit the already-existing reduced correlation within each of the sub-bands of the signal, due to the relative flatness of each sub-band of the power spectrum in many classes of signals. Linear predictive coders can be modelled as an attempt to decorrelate signals by subtracting the best possible linear prediction from the input signal, leaving a whitened residual signal. Decorrelation techniques can also be used for many other purposes, such as reducing crosstalk in a multi-channel signal, or in the design of echo cancellers. In image processing decorrelation techniques can be used to enhance or stretch, colour differences found in each pixel of an image. This is generally termed as 'decorrelation stretching'. === Neuroscience === In neuroscience, decorrelation is used in the analysis of the neural networks in the human visual system. The raw inputs from cone cells and rod cells under go many steps of processing before it is handled by the visual cortex. These steps generally perform decorrelation, both spatial (surround suppression in the retina) and temporal (handling of movement in the lateral geniculate nucleus). === Cryptography === In cryptography, decorrelation is used in cipher design (see Decorrelation theory) and in the design of hardware random number generators.
Omni-Path
Omni-Path Architecture (OPA) is a high-performance communication architecture developed by Intel. It aims for low communication latency, low power consumption and a high throughput. It directly competes with InfiniBand. Intel planned to develop technology based on this architecture for exascale computing. The current owner of Omni-Path is Cornelis Networks. == History == Production of Omni-Path products started in 2015 and delivery of these products started in the first quarter of 2016. In November 2015, adapters based on the 2-port "Wolf River" ASIC were announced, using QSFP28 connectors with channel speeds up to 100 Gbit/s. Simultaneously, switches based on the 48-port "Prairie River" ASIC were announced. First models of that series were available starting in 2015. In April 2016, implementation of the InfiniBand "verbs" interface for the Omni-Path fabric was discussed. In October 2016, IBM, Hewlett Packard Enterprise, Dell, Lenovo, Samsung, Seagate Technology, Micron Technology, Western Digital and SK Hynix announced a joint consortium called Gen-Z to develop an open specification and architecture for non-volatile storage and memory products—including Intel's 3D Xpoint technology—which might in part compete against Omni-Path. Intel offered their Omni-Path products and components via other (hardware) vendors. For example, Dell EMC offered Intel Omni-Path as Dell Networking H-series, following the naming-standard of Dell Networking in 2017. In July 2019, Intel announced it would not continue development of Omni-Path networks and canceled OPA 200 series (200-Gbps variant of Omni-Path). In September 2020, Intel announced that the Omni-Path network products and technology would be spun out into a new venture with Cornelis Networks. Intel would continue to maintain support for legacy Omni-Path products, while Cornelis Networks continues the product line, leveraging existing Intel intellectual property related to Omni-Path architecture. In 2021, Cornelis announced Omni-Path Express, which replaces PSM2-based drivers and middleware, which trace back to PathScale's PSM created in 2003, for the existing Omni-Path hardware, with a native libfabric provider.