Distinguishable interfaces use computer graphic principles to automatically generate easily distinguishable appearance for computer data. Although the desktop metaphor revolutionized user interfaces, there is evidence that a spatial layout alone does little to help in locating files and other data; distinguishable appearance is also required. Studies have shown that average users have considerable difficulty finding files on their personal computers, even ones that they created the same day. Search engines do not always help, since it has been found that users often know of the existence of a file without being able to specify relevant search terms. On the contrary, people appear to incrementally search for files using some form of context. Recently researchers and web developers have argued that the problem is the lack of distinguishable appearance: in the traditional computer interface most objects and locations appear identical. This problem rarely occurs in the real world, where both objects and locations generally have easily distinguishable appearance. Discriminability was one of the recommendations in the ISO 9241-12 recommendation on presentation of information on visual displays (part of the overall report on Ergonomics of Human System Interaction), however it was assumed in that report that this would be achieved by manual design of graphical symbols. == VisualIDs, semanticons, and identicons == The mass availability of computer graphics supported the introduction of approaches that make better use of the brain's "visual hardware", by providing individual files and other abstract data with distinguishable appearance. This idea initially appeared in strictly academic VisualIDs and Semanticons works, but the web community has explored and rapidly adopted similar ideas, such as the Identicon. The VisualIDs project automatically generated icons for files or other data based on a hash of the data identifier, so the icons had no relation to the content or meaning of the data. It was argued not only that generating meaningful icons is unnecessary (their user study showed rapid learning of the arbitrary icons), but also that basing icons on content is actually incorrect ("contrasting visualization with visual identifiers"). The Semanticons project developed by Setlur et al. demonstrated an algorithm to create icons that reflect the content of files. In this work the name, location and content of a file are parsed and used to retrieve related image(s) from an image database. These are then processed using a Non-photorealistic rendering technique in order to generate graphical icons. Developer Don Park introduced the identicon library for making a visual icon from a hash of a data identifier. This initial public implementation has spawned a large number of implementations for various environments. In particular, identicons are now being used as default visual user identifiers (avatars) for several widely used systems. They are also used as a complement to Gravatars, which are pre-existing avatar images created or chosen by users, instead of automatically generated images. (see #External links). == Current research == While current web practice has followed the semantics-free approach of VisualIDs, recent research has followed the semantics-based approach of Semanticons. Examples include using data mining principles to automatically create "intelligent icons" that reflect the contents of files and creating icons for music files that reflect audio characteristics or affective content.
Exposure Notification
The (Google/Apple) Exposure Notification System (GAEN) is a framework and protocol specification developed by Apple Inc. and Google to facilitate digital contact tracing during the COVID-19 pandemic. When used by health authorities, it augments more traditional contact tracing techniques by automatically logging close approaches among notification system users using Android or iOS smartphones. Exposure Notification is a decentralized reporting protocol built on a combination of Bluetooth Low Energy technology and privacy-preserving cryptography. It is an opt-in feature within COVID-19 apps developed and published by authorized health authorities. Unveiled on April 10, 2020, it was made available on iOS on May 20, 2020, as part of the iOS 13.5 update and on December 14, 2020, as part of the iOS 12.5 update for older iPhones. On Android, it was added to devices via a Google Play Services update, supporting all versions since Android Marshmallow. The Apple/Google protocol is similar to the Decentralized Privacy-Preserving Proximity Tracing (DP-3T) protocol created by the European DP-3T consortium and the Temporary Contact Number (TCN) protocol by Covid Watch, but is implemented at the operating system level, which allows for more efficient operation as a background process. Since May 2020, a variant of the DP-3T protocol is supported by the Exposure Notification Interface. Other protocols are constrained in operation because they are not privileged over normal apps. This leads to issues, particularly on iOS devices where digital contact tracing apps running in the background experience significantly degraded performance. The joint approach is also designed to maintain interoperability between Android and iOS devices, which constitute nearly all of the market. The ACLU stated the approach "appears to mitigate the worst privacy and centralization risks, but there is still room for improvement". In late April, Google and Apple shifted the emphasis of the naming of the system, describing it as an "exposure notification service", rather than "contact tracing" system. == Technical specification == Digital contact tracing protocols typically have two major responsibilities: encounter logging and infection reporting. Exposure Notification only involves encounter logging which is a decentralized architecture. The majority of infection reporting is centralized in individual app implementations. To handle encounter logging, the system uses Bluetooth Low Energy to send tracking messages to nearby devices running the protocol to discover encounters with other people. The tracking messages contain unique identifiers that are encrypted with a secret daily key held by the sending device. These identifiers change every 15–20 minutes as well as Bluetooth MAC address in order to prevent tracking of clients by malicious third parties through observing static identifiers over time. The sender's daily encryption keys are generated using a random number generator. Devices record received messages, retaining them locally for 14 days. If a user tests positive for infection, the last 14 days of their daily encryption keys can be uploaded to a central server, where it is then broadcast to all devices on the network. The method through which daily encryption keys are transmitted to the central server and broadcast is defined by individual app developers. The Google-developed reference implementation calls for a health official to request a one-time verification code (VC) from a verification server, which the user enters into the encounter logging app. This causes the app to obtain a cryptographically signed certificate, which is used to authorize the submission of keys to the central reporting server. The received keys are then provided to the protocol, where each client individually searches for matches in their local encounter history. If a match meeting certain risk parameters is found, the app notifies the user of potential exposure to the infection. Google and Apple intend to use the received signal strength (RSSI) of the beacon messages as a source to infer proximity. RSSI and other signal metadata will also be encrypted to resist deanonymization attacks. === Version 1.0 === To generate encounter identifiers, first a persistent 32-byte private Tracing Key ( t k {\displaystyle tk} ) is generated by a client. From this a 16 byte Daily Tracing Key is derived using the algorithm d t k i = H K D F ( t k , N U L L , 'CT-DTK' | | D i , 16 ) {\displaystyle dtk_{i}=HKDF(tk,NULL,{\text{'CT-DTK'}}||D_{i},16)} , where H K D F ( Key, Salt, Data, OutputLength ) {\displaystyle HKDF({\text{Key, Salt, Data, OutputLength}})} is a HKDF function using SHA-256, and D i {\displaystyle D_{i}} is the day number for the 24-hour window the broadcast is in starting from Unix Epoch Time. These generated keys are later sent to the central reporting server should a user become infected. From the daily tracing key a 16-byte temporary Rolling Proximity Identifier is generated every 10 minutes with the algorithm R P I i , j = Truncate ( H M A C ( d t k i , 'CT-RPI' | | T I N j ) , 16 ) {\displaystyle RPI_{i,j}={\text{Truncate}}(HMAC(dtk_{i},{\text{'CT-RPI'}}||TIN_{j}),16)} , where H M A C ( Key, Data ) {\displaystyle HMAC({\text{Key, Data}})} is a HMAC function using SHA-256, and T I N j {\displaystyle TIN_{j}} is the time interval number, representing a unique index for every 10 minute period in a 24-hour day. The Truncate function returns the first 16 bytes of the HMAC value. When two clients come within proximity of each other they exchange and locally store the current R P I i , j {\displaystyle RPI_{i,j}} as the encounter identifier. Once a registered health authority has confirmed the infection of a user, the user's Daily Tracing Key for the past 14 days is uploaded to the central reporting server. Clients then download this report and individually recalculate every Rolling Proximity Identifier used in the report period, matching it against the user's local encounter log. If a matching entry is found, then contact has been established and the app presents a notification to the user warning them of potential infection. === Version 1.1 === Unlike version 1.0 of the protocol, version 1.1 does not use a persistent tracing key, rather every day a new random 16-byte Temporary Exposure Key ( t e k i {\displaystyle tek_{i}} ) is generated. This is analogous to the daily tracing key from version 1.0. Here i {\displaystyle i} denotes the time is discretized in 10 minute intervals starting from Unix Epoch Time. From this two 128-bit keys are calculated, the Rolling Proximity Identifier Key ( R P I K i {\displaystyle RPIK_{i}} ) and the Associated Encrypted Metadata Key ( A E M K i {\displaystyle AEMK_{i}} ). R P I K i {\displaystyle RPIK_{i}} is calculated with the algorithm R P I K i = H K D F ( t e k i , N U L L , 'EN-RPIK' , 16 ) {\displaystyle RPIK_{i}=HKDF(tek_{i},NULL,{\text{'EN-RPIK'}},16)} , and A E M K i {\displaystyle AEMK_{i}} using the algorithm A E M K i = H K D F ( t e k i , N U L L , 'EN-AEMK' , 16 ) {\displaystyle AEMK_{i}=HKDF(tek_{i},NULL,{\text{'EN-AEMK'}},16)} . From these values a temporary Rolling Proximity Identifier ( R P I i , j {\displaystyle RPI_{i,j}} ) is generated every time the BLE MAC address changes, roughly every 15–20 minutes. The following algorithm is used: R P I i , j = A E S 128 ( R P I K i , 'EN-RPI' | | 0 x 000000000000 | | E N I N j ) {\displaystyle RPI_{i,j}=AES128(RPIK_{i},{\text{'EN-RPI'}}||{\mathtt {0x000000000000}}||ENIN_{j})} , where A E S 128 ( Key, Data ) {\displaystyle AES128({\text{Key, Data}})} is an AES cryptography function with a 128-bit key, the data is one 16-byte block, j {\displaystyle j} denotes the Unix Epoch Time at the moment the roll occurs, and E N I N j {\displaystyle ENIN_{j}} is the corresponding 10-minute interval number. Next, additional Associated Encrypted Metadata is encrypted. What the metadata represents is not specified, likely to allow the later expansion of the protocol. The following algorithm is used: Associated Encrypted Metadata i , j = A E S 128 _ C T R ( A E M K i , R P I i , j , Metadata ) {\displaystyle {\text{Associated Encrypted Metadata}}_{i,j}=AES128\_CTR(AEMK_{i},RPI_{i,j},{\text{Metadata}})} , where A E S 128 _ C T R ( Key, IV, Data ) {\displaystyle AES128\_CTR({\text{Key, IV, Data}})} denotes AES encryption with a 128-bit key in CTR mode. The Rolling Proximity Identifier and the Associated Encrypted Metadata are then combined and broadcast using BLE. Clients exchange and log these payloads. Once a registered health authority has confirmed the infection of a user, the user's Temporary Exposure Keys t e k i {\displaystyle tek_{i}} and their respective interval numbers i {\displaystyle i} for the past 14 days are uploaded to the central reporting server. Clients then download this report and individually recalculate every Rolling Proximity Identifier starting from interval number i {\displaystyle i} ,
List of video editing software
The following is a list of video editing software. The criterion for inclusion in this list is the ability to perform non-linear video editing. Most modern transcoding software supports transcoding a portion of a video clip, which would count as cropping and trimming. However, items in this article have one of the following conditions: Can perform other non-linear video editing function such as montage or compositing Can do the trimming or cropping without transcoding == Free (libre) or open-source == The software listed in this section is either free software or open source, and may or may not be commercial. === Active and stable === === Inactive === == Proprietary (non-commercial) == The software listed in this section is proprietary, and freeware or freemium. === Active === === Discontinued === == Proprietary (commercial) == The software listed in this section is proprietary and commercial. === Active === === Discontinued ===
D/Vision Pro
D/Vision Pro was one of the earliest marketed non-linear editing systems. It was released by TouchVision Systems, Inc. in the mid-1990s. The program was DOS-based and worked on either Intel's 386 or 486 processor. The system used AVI compression and worked with the Action Media II board. The system allowed users to digitize video, audio, and timecode, create an edit decision list (EDL), instantly play back the edited program, and output the finished EDL in a wide variety of formats. These cost-effective editing systems were used by numerous independent filmmakers and in low-budget productions during the mid-late 1990s. D/Vision Pro's low-quality compression led TouchVision (later renamed D/Vision Systems) to abandon it in favor of D/Vision Online, which was purchased by Discreet Logic and renamed edit. In June 2002, Discreet discontinued edit, as they did not want it to interfere with smoke sales which were more profitable. Discreet was later purchased by Autodesk.
Optical granulometry
Optical granulometry is the process of measuring the different grain sizes in a granular material, based on a photograph. Technology has been created to analyze a photograph and create statistics based on what the picture portrays. This information is vital in maintaining machinery in various trades worldwide. Mining companies can use optical granulometry to analyze inactive or moving rock to quantify the size of these fragments. Forestry companies can zero in on wood chip sizes without stopping the production process, and minimize sizing errors. With more photoanalysis technologies being produced, mining companies have shown an increased interest in these types of systems because of their ability to maintain efficiency throughout the mining process. Companies are saving millions of dollars annually because of this new technology, and are cutting back on maintenance costs on equipment. In order for optical granulometry to be completely successful, an accurate photo must be taken – under sufficient lighting, and using proper technology – to obtain quantified results. If these requirements are met, an image analysis system can be implemented. == The process == Software uses four basic steps in determining the average size of material: See the Wikipedia article on Photoanalysis to see how mining, forestry and agricultural companies are using this technology to improve quality control techniques. == Smartphone-based, segmentation-free estimation of grain size distribution == Recently, a methodology has emerged by which soil grain size distribution can be inferred from optical images acquired with commodity smartphones by training convolutional neural networks to predict parameters of the distribution curve directly from the image, without explicit image segmentation . In this approach, a standardized image of a soil surface is captured under controlled conditions, preprocessed to reduce device-specific variability, and passed to a regression model that outputs the parameters of a cumulative distribution function e.g., a two-parameter Weibull curve. The resulting distribution can be used to derive geotechnical descriptors and class boundaries.
Scale space implementation
In the areas of computer vision, image analysis and signal processing, the notion of scale-space representation is used for processing measurement data at multiple scales, and specifically enhance or suppress image features over different ranges of scale (see the article on scale space). A special type of scale-space representation is provided by the Gaussian scale space, where the image data in N dimensions is subjected to smoothing by Gaussian convolution. Most of the theory for Gaussian scale space deals with continuous images, whereas one when implementing this theory will have to face the fact that most measurement data are discrete. Hence, the theoretical problem arises concerning how to discretize the continuous theory while either preserving or well approximating the desirable theoretical properties that lead to the choice of the Gaussian kernel (see the article on scale-space axioms). This article describes basic approaches for this that have been developed in the literature, see also for an in-depth treatment regarding the topic of approximating the Gaussian smoothing operation and the Gaussian derivative computations in scale-space theory, and for a complementary treatment regarding hybrid discretization methods. == Statement of the problem == The Gaussian scale-space representation of an N-dimensional continuous signal, f C ( x 1 , ⋯ , x N , t ) , {\displaystyle f_{C}\left(x_{1},\cdots ,x_{N},t\right),} is obtained by convolving fC with an N-dimensional Gaussian kernel: g N ( x 1 , ⋯ , x N , t ) . {\displaystyle g_{N}\left(x_{1},\cdots ,x_{N},t\right).} In other words: L ( x 1 , ⋯ , x N , t ) = ∫ u 1 = − ∞ ∞ ⋯ ∫ u N = − ∞ ∞ f C ( x 1 − u 1 , ⋯ , x N − u N , t ) ⋅ g N ( u 1 , ⋯ , u N , t ) d u 1 ⋯ d u N . {\displaystyle L\left(x_{1},\cdots ,x_{N},t\right)=\int _{u_{1}=-\infty }^{\infty }\cdots \int _{u_{N}=-\infty }^{\infty }f_{C}\left(x_{1}-u_{1},\cdots ,x_{N}-u_{N},t\right)\cdot g_{N}\left(u_{1},\cdots ,u_{N},t\right)\,du_{1}\cdots du_{N}.} However, for implementation, this definition is impractical, since it is continuous. When applying the scale space concept to a discrete signal fD, different approaches can be taken. This article is a brief summary of some of the most frequently used methods. == Separability == Using the separability property of the Gaussian kernel g N ( x 1 , … , x N , t ) = G ( x 1 , t ) ⋯ G ( x N , t ) {\displaystyle g_{N}\left(x_{1},\dots ,x_{N},t\right)=G\left(x_{1},t\right)\cdots G\left(x_{N},t\right)} the N-dimensional convolution operation can be decomposed into a set of separable smoothing steps with a one-dimensional Gaussian kernel G along each dimension L ( x 1 , ⋯ , x N , t ) = ∫ u 1 = − ∞ ∞ ⋯ ∫ u N = − ∞ ∞ f C ( x 1 − u 1 , ⋯ , x N − u N , t ) G ( u 1 , t ) d u 1 ⋯ G ( u N , t ) d u N , {\displaystyle L(x_{1},\cdots ,x_{N},t)=\int _{u_{1}=-\infty }^{\infty }\cdots \int _{u_{N}=-\infty }^{\infty }f_{C}(x_{1}-u_{1},\cdots ,x_{N}-u_{N},t)G(u_{1},t)\,du_{1}\cdots G(u_{N},t)\,du_{N},} where G ( x , t ) = 1 2 π t e − x 2 2 t {\displaystyle G(x,t)={\frac {1}{\sqrt {2\pi t}}}e^{-{\frac {x^{2}}{2t}}}} and the standard deviation of the Gaussian σ is related to the scale parameter t according to t = σ2. Separability will be assumed in all that follows, even when the kernel is not exactly Gaussian, since separation of the dimensions is the most practical way to implement multidimensional smoothing, especially at larger scales. Therefore, the rest of the article focuses on the one-dimensional case. == The sampled Gaussian kernel == When implementing the one-dimensional smoothing step in practice, the presumably simplest approach is to convolve the discrete signal fD with a sampled Gaussian kernel: L ( x , t ) = ∑ n = − ∞ ∞ f ( x − n ) G ( n , t ) {\displaystyle L(x,t)=\sum _{n=-\infty }^{\infty }f(x-n)\,G(n,t)} where G ( n , t ) = 1 2 π t e − n 2 2 t {\displaystyle G(n,t)={\frac {1}{\sqrt {2\pi t}}}e^{-{\frac {n^{2}}{2t}}}} (with t = σ2) which in turn is truncated at the ends to give a filter with finite impulse response L ( x , t ) = ∑ n = − M M f ( x − n ) G ( n , t ) {\displaystyle L(x,t)=\sum _{n=-M}^{M}f(x-n)\,G(n,t)} for M chosen sufficiently large (see error function) such that 2 ∫ M ∞ G ( u , t ) d u = 2 ∫ M t ∞ G ( v , 1 ) d v < ε . {\displaystyle 2\int _{M}^{\infty }G(u,t)\,du=2\int _{\frac {M}{\sqrt {t}}}^{\infty }G(v,1)\,dv<\varepsilon .} A common choice is to set M to a constant C times the standard deviation of the Gaussian kernel M = C σ + 1 = C t + 1 {\displaystyle M=C\sigma +1=C{\sqrt {t}}+1} where C is often chosen somewhere between 3 and 6. Using the sampled Gaussian kernel can, however, lead to implementation problems, in particular when computing higher-order derivatives at finer scales by applying sampled derivatives of Gaussian kernels. When accuracy and robustness are primary design criteria, alternative implementation approaches should therefore be considered. For small values of ε (10−6 to 10−8) the errors introduced by truncating the Gaussian are usually negligible. For larger values of ε, however, there are many better alternatives to a rectangular window function. For example, for a given number of points, a Hamming window, Blackman window, or Kaiser window will do less damage to the spectral and other properties of the Gaussian than a simple truncation will. Notwithstanding this, since the Gaussian kernel decreases rapidly at the tails, the main recommendation is still to use a sufficiently small value of ε such that the truncation effects are no longer important. == The discrete Gaussian kernel == A more refined approach is to convolve the original signal with the discrete Gaussian kernel T(n, t) L ( x , t ) = ∑ n = − ∞ ∞ f ( x − n ) T ( n , t ) {\displaystyle L(x,t)=\sum _{n=-\infty }^{\infty }f(x-n)\,T(n,t)} where T ( n , t ) = e − t I n ( t ) {\displaystyle T(n,t)=e^{-t}I_{n}(t)} and I n ( t ) {\displaystyle I_{n}(t)} denotes the modified Bessel functions of integer order, n. This is the discrete counterpart of the continuous Gaussian in that it is the solution to the discrete diffusion equation (discrete space, continuous time), just as the continuous Gaussian is the solution to the continuous diffusion equation. This filter can be truncated in the spatial domain as for the sampled Gaussian L ( x , t ) = ∑ n = − M M f ( x − n ) T ( n , t ) {\displaystyle L(x,t)=\sum _{n=-M}^{M}f(x-n)\,T(n,t)} or can be implemented in the Fourier domain using a closed-form expression for its discrete-time Fourier transform: T ^ ( θ , t ) = ∑ n = − ∞ ∞ T ( n , t ) e − i θ n = e t ( cos θ − 1 ) . {\displaystyle {\widehat {T}}(\theta ,t)=\sum _{n=-\infty }^{\infty }T(n,t)\,e^{-i\theta n}=e^{t(\cos \theta -1)}.} With this frequency-domain approach, the scale-space properties transfer exactly to the discrete domain, or with excellent approximation using periodic extension and a suitably long discrete Fourier transform to approximate the discrete-time Fourier transform of the signal being smoothed. Moreover, higher-order derivative approximations can be computed in a straightforward manner (and preserving scale-space properties) by applying small support central difference operators to the discrete scale space representation. As with the sampled Gaussian, a plain truncation of the infinite impulse response will in most cases be a sufficient approximation for small values of ε, while for larger values of ε it is better to use either a decomposition of the discrete Gaussian into a cascade of generalized binomial filters or alternatively to construct a finite approximate kernel by multiplying by a window function. If ε has been chosen too large such that effects of the truncation error begin to appear (for example as spurious extrema or spurious responses to higher-order derivative operators), then the options are to decrease the value of ε such that a larger finite kernel is used, with cutoff where the support is very small, or to use a tapered window. == Recursive filters == Since computational efficiency is often important, low-order recursive filters are often used for scale-space smoothing. For example, Young and van Vliet use a third-order recursive filter with one real pole and a pair of complex poles, applied forward and backward to make a sixth-order symmetric approximation to the Gaussian with low computational complexity for any smoothing scale. By relaxing a few of the axioms, Lindeberg concluded that good smoothing filters would be "normalized Pólya frequency sequences", a family of discrete kernels that includes all filters with real poles at 0 < Z < 1 and/or Z > 1, as well as with real zeros at Z < 0. For symmetry, which leads to approximate directional homogeneity, these filters must be further restricted to pairs of poles and zeros that lead to zero-phase filters. To match the transfer function curvature at zero frequency of the discrete Gaussian, which ensures an approximate semi-group property of additive t, two poles at Z = 1 + 2 t − ( 1 + 2 t ) 2 − 1 {\displaystyle
Reconstruction from projections
The problem of reconstructing a multidimensional signal from its projection is uniquely multidimensional, having no 1-D counterpart. It has applications that range from computer-aided tomography to geophysical signal processing. It is a problem which can be explored from several points of view—as a deconvolution problem, a modeling problem, an estimation problem, or an interpolation problem. == Motivation and applications == Many fields in science and engineering use reconstruction from projections, especially in imaging. It is widely applied geophysical tomography, medical imaging and industrial radiography. For example, in a CT scanner, the 3D structure of the patient’s body being scanned is measured with beams going through the tissue and hitting a detector, giving a flat projection of the body from that angle. Multiple projections are put together to get an image of the position and shape of structures inside in 3D. == Problem statement and basics == A projection is a linear mapping of an M {\displaystyle M} dimensional signal into an N {\displaystyle N} dimensional one, where N ≤ M {\displaystyle N\leq M} . And the objective of reconstruction is to restore the M {\displaystyle M} dimensional signal based on the N {\displaystyle N} dimensional signal. The following case is a 2-D signal projected into 1D signal. The signal in the original coordinate is denoted as d ( u , v ) {\displaystyle d(u,v)} . Now consider a collimated beam of radiation coming from the opposite orientation of v ^ {\displaystyle {\hat {v}}} , producing a projection along u ^ {\displaystyle {\hat {u}}} . v ^ {\displaystyle {\hat {v}}} and u ^ {\displaystyle {\hat {u}}} are normal to each other, and the angle between u {\displaystyle u} and u ^ {\displaystyle {\hat {u}}} is theta. The signal obtained along u ^ {\displaystyle {\hat {u}}} axis is defined to be p θ ( u ^ ) {\displaystyle p_{\theta }({\hat {u}})} . The relationship between the original coordinate and the rotated coordinate is given by [ u ^ v ^ ] = [ cos θ sin θ − sin θ cos θ ] [ u v ] {\displaystyle {\begin{bmatrix}{\hat {u}}\\{\hat {v}}\end{bmatrix}}={\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{bmatrix}}{\begin{bmatrix}u\\v\end{bmatrix}}} or inversely, [ u v ] = [ cos θ − sin θ sin θ cos θ ] [ u ^ v ^ ] {\displaystyle {\begin{bmatrix}u\\v\end{bmatrix}}={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{bmatrix}}{\begin{bmatrix}{\hat {u}}\\{\hat {v}}\end{bmatrix}}} Then we have p θ ( u ^ ) = ∫ − ∞ ∞ d ( u , v ) d v ^ = ∫ − ∞ ∞ d ( u ^ cos ( θ ) − v ^ sin ( θ ) , u ^ sin ( θ ) + v ^ cos ( θ ) ) d v ^ {\displaystyle p_{\theta }({\hat {u}})=\int _{-\infty }^{\infty }d(u,v)\,\mathrm {d} {\hat {v}}=\int _{-\infty }^{\infty }d({\hat {u}}\cos(\theta )-{\hat {v}}\sin(\theta ),{\hat {u}}\sin(\theta )+{\hat {v}}\cos(\theta ))\,\mathrm {d} {\hat {v}}} By varying theta, a large number of projections can be obtained. Given the projection-slice theorem, D ( Ω , θ ) {\displaystyle D(\Omega ,\theta )} ,the slice of the Fourier transform of d ( u , v ) {\displaystyle d(u,v)} at angle theta, is equivalent to P θ ( Ω ) {\displaystyle P_{\theta }(\Omega )} , the Fourier Transform of the projection p θ ( u ^ ) {\displaystyle p_{\theta }({\hat {u}})} . Therefore, the unknown d ( u , v ) {\displaystyle d(u,v)} can be obtained from its Fourier transform by means of the Fourier transform inversion integral d ( u , v ) = 1 4 π 2 ∫ − ∞ ∞ ∫ − ∞ ∞ D ( Ω 1 , Ω 2 ) e j Ω 1 u e j Ω 2 v d Ω 1 , Ω 2 {\displaystyle \mathrm {d} (u,v)={\frac {1}{4\pi ^{2}}}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }D(\Omega _{1},\Omega _{2})e^{j\Omega _{1}u}e^{j\Omega _{2}v}\,\mathrm {d} \Omega _{1},\Omega _{2}} = 1 4 π 2 ∫ 0 ∞ ∫ − π π D ( Ω , θ ) e j Ω u cos ( θ ) e j Ω v s i n θ | Ω | d Ω d θ {\displaystyle ={\frac {1}{4\pi ^{2}}}\int _{0}^{\infty }\int _{-\pi }^{\pi }D(\Omega ,\theta )e^{j\Omega u\cos(\theta )}e^{j\Omega vsin\theta }{\begin{vmatrix}\Omega \end{vmatrix}}\,\mathrm {d} \Omega \mathrm {d} \theta } = 1 4 π 2 ∫ − π π ∫ 0 ∞ P θ ( Ω ) e j Ω ( u cos θ + v sin θ ) | Ω | d Ω d θ {\displaystyle ={\frac {1}{4\pi ^{2}}}\int _{-\pi }^{\pi }\int _{0}^{\infty }P_{\theta }(\Omega )e^{j}\Omega (u\cos \theta +v\sin \theta ){\begin{vmatrix}\Omega \end{vmatrix}}\,\mathrm {d} \Omega \mathrm {d} \theta } = 1 4 π 2 ∫ 0 π ( ∫ − ∞ ∞ P θ ( Ω ) | Ω | {\displaystyle ={\frac {1}{4\pi ^{2}}}\int _{0}^{\pi }(\int _{-\infty }^{\infty }P_{\theta }(\Omega ){\begin{vmatrix}\Omega \end{vmatrix}}} e j Ω u ^ d Ω ) d θ {\displaystyle e^{j\Omega {\hat {u}}}\mathrm {d} \Omega )\mathrm {d} \theta } By taking the inverse Fourier Transform and assuming g ( u ^ ) = F − 1 ( | Ω | 2 ) {\displaystyle g({\hat {u}})={\mathcal {F}}^{-1}({{\begin{vmatrix}\Omega \end{vmatrix}}^{2}})} , we get d ( u , v ) = ∑ i △ θ i [ p θ ( u ^ ) ∗ g θ i ( u ^ ) ] {\displaystyle d(u,v)=\sum _{i}\vartriangle \theta _{i}[p_{\theta }({\hat {u}})g_{\theta i}({\hat {u}})]} == Approaches == In practice, there are a wide variety of methods that are utilized, most of which are reconstruct 3-D information (volume) from 2-D signals (image). Typically used methods are CT, MRI, PET and SPECT. And the filtered back projection based on the principles introduced above are commonly applied. === Computed Tomography (CT) === In CT, a volume is formed by stacking the axial slices. The software cuts the volume in a different plane (usually orthogonal). Commonly, slice data is generated using an X-ray source that rotates around the object. X-ray sensors are positioned on the opposite side of the circle from the X-ray source. === Magnetic resonance imaging (MRI) === In MRI, energy from an oscillating magnetic field is temporarily applied to the patient at the appropriate resonance frequency. The protons (hydrogen atoms) emit a radio frequency signal which is measured by a receiving coil. The radio signal can be made to encode position information by varying the main magnetic field using gradient coils. === Positron emission tomography (PET) === The system detects pairs of gamma rays emitted indirectly by a positron-emitting radionuclide (tracer), which is introduced into the body on a biologically active molecule. Three-dimensional images of tracer concentration within the body are then constructed by computer analysis. In modern PET-CT scanners, three dimensional imaging is often accomplished with the aid of a CT X-ray scan performed on the patient during the same session, in the same machine. === Single-photon emission computed tomography (SPECT) === SPECT imaging is performed by using a gamma camera to acquire multiple 2-D images (projections) from multiple angles. Multiple projections are used to yield a 3-D data set. This data set may then be manipulated to show thin slices along any chosen axis of the body. SPECT is similar to PET in its use of radioactive tracer material and detection of gamma rays, while the tracers used in SPECT emit gamma radiation that is measured more directly.