In formal language theory, the Kleene star (or Kleene operator or Kleene closure) refers to two related unary operations, that can be applied either to an alphabet of symbols or to a formal language, a set of strings (finite sequences of symbols). The Kleene star operator on an alphabet V generates the set V of all finite-length strings over V, that is, finite sequences whose elements belong to V; in mathematics, it is more commonly known as the free monoid construction. The Kleene star operator on a language L generates another language L, the set of all strings that can be obtained as a concatenation of zero or more members of L. In both cases, repetitions are allowed. The Kleene star operators are named after American mathematician Stephen Cole Kleene, who first introduced and widely used it to characterize automata for regular expressions. == Of an alphabet == Given an alphabet V {\displaystyle V} , define V 0 = { ε } {\displaystyle V^{0}=\{\varepsilon \}} (the set consists only of the empty string), V 1 = V , {\displaystyle V^{1}=V,} and define recursively the set V i + 1 = { w v : w ∈ V i and v ∈ V } {\displaystyle V^{i+1}=\{wv:w\in V^{i}{\text{ and }}v\in V\}} for each i > 0 , {\displaystyle i>0,} where w v {\displaystyle wv} denotes the string obtained by appending the single character v {\displaystyle v} to the end of w {\displaystyle w} . Here, V i {\displaystyle V^{i}} can be understood to be the set of all strings of length exactly i {\displaystyle i} , with characters from V {\displaystyle V} . The definition of Kleene star on V {\displaystyle V} is V ∗ = ⋃ i ≥ 0 V i = V 0 ∪ V 1 ∪ V 2 ∪ V 3 ∪ V 4 ∪ ⋯ . {\displaystyle V^{}=\bigcup _{i\geq 0}V^{i}=V^{0}\cup V^{1}\cup V^{2}\cup V^{3}\cup V^{4}\cup \cdots .} == Of a language == Given a language L {\displaystyle L} (any finite or infinite set of strings), define L 0 = { ε } {\displaystyle L^{0}=\{\varepsilon \}} (the language consisting only of the empty string), L 1 = L , {\displaystyle L^{1}=L,} and define recursively the set L i + 1 = { w v : w ∈ L i and v ∈ L } {\displaystyle L^{i+1}=\{wv:w\in L^{i}{\text{ and }}v\in L\}} for each i > 0 , {\displaystyle i>0,} where w v {\displaystyle wv} denotes the string obtained by concatenating w {\displaystyle w} and v {\displaystyle v} . Here, L i {\displaystyle L^{i}} can be understood to be the set of all strings that can be obtained by concatenating exactly i {\displaystyle i} strings from L {\displaystyle L} , allowing repetitions. The definition of Kleene star on L {\displaystyle L} is L ∗ = ⋃ i ≥ 0 L i = L 0 ∪ L 1 ∪ L 2 ∪ L 3 ∪ L 4 ∪ ⋯ . {\displaystyle L^{}=\bigcup _{i\geq 0}L^{i}=L^{0}\cup L^{1}\cup L^{2}\cup L^{3}\cup L^{4}\cup \cdots .} == Kleene plus == In some formal language studies, (e.g. AFL theory) a variation on the Kleene star operation called the Kleene plus is used. The Kleene plus omits the V 0 {\displaystyle V^{0}} or L 0 {\displaystyle L^{0}} term in the above unions. In other words, the Kleene plus on V {\displaystyle V} is V + = ⋃ i ≥ 1 V i = V 1 ∪ V 2 ∪ V 3 ∪ ⋯ , {\displaystyle V^{+}=\bigcup _{i\geq 1}V^{i}=V^{1}\cup V^{2}\cup V^{3}\cup \cdots ,} or V + = V ∗ V . {\displaystyle V^{+}=V^{}V.} == Examples == Example of Kleene star applied to a set of strings: {"ab","c"} = { ε, "ab", "c", "abab", "abc", "cab", "cc", "ababab", "ababc", "abcab", "abcc", "cabab", "cabc", "ccab", "ccc", ...}. Example of Kleene star applied to a set of strings without the prefix property: {"a","ab","b"} = { ε, "a", "ab", "b", "aa", "aab", "aba", "abab", "abb", "ba", "bab", "bb", ...};In this example, the string "aab" can be obtained in two different ways. The Sardinas-Patterson algorithm can be used to check for a given V whether any member of V can be obtained in more than one way. Example of Kleene and Kleene plus applied to a set of characters (following the C programming language convention where a character is denoted by single quotes and a string is denoted by double quotes): {'a', 'b', 'c'} = { ε, "a", "b", "c", "aa", "ab", "ac", "ba", "bb", "bc", "ca", "cb", "cc", "aaa", "aab", ...}. {'a', 'b', 'c'}+ = { "a", "b", "c", "aa", "ab", "ac", "ba", "bb", "bc", "ca", "cb", "cc", "aaa", "aab", ...}. == Properties == If V {\displaystyle V} is any finite or countably infinite set of characters, then V ∗ {\displaystyle V^{}} is a countably infinite set. As a result, each formal language over a finite or countably infinite alphabet Σ {\displaystyle \Sigma } is countable, since it is a subset of the countably infinite set Σ ∗ {\displaystyle \Sigma ^{}} . ( L ∗ ) ∗ = L ∗ {\displaystyle (L^{})^{}=L^{}} , which means that the Kleene star operator is an idempotent unary operator, as ( L ∗ ) i = L ∗ {\displaystyle (L^{})^{i}=L^{}} for every i ≥ 1 {\displaystyle i\geq 1} . V ∗ = { ε } {\displaystyle V^{}=\{\varepsilon \}} , if V {\displaystyle V} is the empty set ∅. For the version of the Kleene star operator on languages, L ∗ = { ε } {\displaystyle L^{}=\{\varepsilon \}} when L {\displaystyle L} is either the empty set ∅ or the singleton set { ε } {\displaystyle \{\varepsilon \}} . == Generalization == Strings form a monoid with concatenation as the binary operation and ε the identity element. In addition to strings, the Kleene star is defined for any monoid. More precisely, let (M, ⋅) be a monoid, and S ⊆ M. Then S is the smallest submonoid of M containing S; that is, S contains the neutral element of M, the set S, and is such that if x,y ∈ S, then x⋅y ∈ S. Furthermore, the Kleene star is generalized by including the -operation (and the union) in the algebraic structure itself by the notion of complete star semiring.
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.
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Avid DS
Avid DS (which was called Avid DS Nitris until early 2008) is a high-end offline and finishing system comprising a non-linear editing system and visual effects software. It was developed by Softimage (this company was owned by Microsoft at the time of DS v1.0's launch before being acquired from Microsoft by Avid Technology, Inc. shortly thereafter) in Montreal. DS was discontinued on September 30, 2013 with support ending on the same date the following year. == Software == DS was called ‘Digital Studio’ in development. It was envisioned to be a complete platform for video/audio work. The first previews of the system were on the SGI platform, but this version was never released. The system was rewritten on Windows NT with different video hardware platforms (Matrox DigiSuite or Play Trinity running on a NetPower system) before the final system was released on Intergraph/StudioZ hardware in January 1998. After its acquisition by Avid, DS was always positioned as a high end video finishing tool. However, many users found it to be uniquely soup-to-nuts in its capabilities. From version 1.0 of the product, it competed with products like Autodesk Smoke, Quantel and Avid Symphony. The toolset in DS offered video timeline editing, an object-oriented vector-based paint tool, 2D layer compositing, sample based audio and starting with version 3.01 of the product, a 3D environment. Originally, a subset of the Softimage|XSI 3D software was planned to become part of the DS toolset, both were built on the same software foundation, but over time the code bases divided between the applications and the integration never happened. While the first version of the DS still lacked a few key features (no 3D, poor keying, no real-time effects), it had some significant features compared to the competing products at the time. It offered a large number of built in effects. Avid OMF import was available, positioning Softimage DS as a strong finishing tool for then typical off-line Avid systems. Lastly the integration of the toolset of Softimage DS was beyond what other product offered. A Softimage DS user could quickly go from editing, to paint, to compositing with a few mouse clicks all inside the same interface. Some of the lacking features were quickly resolved, within months of version 1.0 a new chroma keyer was released. Early versions of the software (up thru 4.0) added additional key features. Development continued with one of the first uncompressed HD editing systems (version 4.01) and an attempt to make the system more friendly to Media Composer editors in version 6. In later versions (v7.5 on beyond) DS was criticized for slow development of compositing tools, mainly lack of a new 3D environment and better tracking tools. Many DS users felt that Avid had not been giving DS the attention that it deserved. On July 7, 2013, Avid sent out an email marking the end of life of the DS product. "To Our Avid DS customers, We are writing to inform you that Avid will be realigning our business strategy to focus on a core suite of products to best leverage our developmental and creative resources. As part of this transition, we will be ceasing future development of Avid DS with a final sale date of September 30th, 2013" == Hardware == Up until version 10.5, DS was sold as a turn-key system; the software was not available without purchasing CPU, I/O and storage hardware from Avid. Beginning with 10.5, customers were able to configure their own systems using widely available components, based on recommended system requirements. In turn-key systems, there were many hardware refreshes over time. StudioZ single stream: Intergraph TDZ-425 with 30 minutes of uncompressed SCSI storage. CPUs at the time were Pentium II/300 MHz. StudioZ dual stream: Intergraph TDZ-2000 GT1 with one hour of fibre channel storage. CPUs on first systems were Pentium II/400 MHz, but last shipping systems had Pentium III/1 GHz. DS was one of the first applications to show that real-time effects could be processed with just the CPUs of the system, not requiring special video cards with real-time effect hardware. Equinox: Developed by Avid, it was one of the first uncompressed HD video cards available. Systems were available on CPUs from Pentium III/1 GHz to Pentium 4/2.8 GHz. Storage was typically SCSI, but fibre channel was also supported. Nitris DNA: Developed by Avid, the Nitris hardware was probably the largest hardware update to the system since it was released. 10-bit HD and SD support was standard. Real-time down and cross convert. This was the only hardware for DS that had on-board effect processing. This allowed a system at the time to play back dual-stream uncompressed HD effects in real-time at 16-bit precision. This was also the first hardware from Avid to support the DNxHD codec. Starting with Pentium 4, Intel Core Xeons were supported. SCSI storage was primarily used. AJA Video Systems: First available as a 4:4:4 option to be used in conjunction with Nitris hardware. Final-generation DS systems used the AJA Video Systems Kona 3 (Xena 2K) card as the only I/O for the system. The last systems shipped with two Intel Core Xeon 6-core processors. SAS is the recommended storage for these systems. == History ==
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