State complexity is an area of theoretical computer science dealing with the size of abstract automata, such as different kinds of finite automata. The classical result in the area is that simulating an n {\displaystyle n} -state nondeterministic finite automaton by a deterministic finite automaton requires exactly 2 n {\displaystyle 2^{n}} states in the worst case. == Transformation between variants of finite automata == Finite automata can be deterministic and nondeterministic, one-way (DFA, NFA) and two-way (2DFA, 2NFA). Other related classes are unambiguous (UFA), self-verifying (SVFA) and alternating (AFA) finite automata. These automata can also be two-way (2UFA, 2SVFA, 2AFA). All these machines can accept exactly the regular languages. However, the size of different types of automata necessary to accept the same language (measured in the number of their states) may be different. For any two types of finite automata, the state complexity tradeoff between them is an integer function f {\displaystyle f} where f ( n ) {\displaystyle f(n)} is the least number of states in automata of the second type sufficient to recognize every language recognized by an n {\displaystyle n} -state automaton of the first type. The following results are known. NFA to DFA: 2 n {\displaystyle 2^{n}} states. This is the subset construction by Rabin and Scott, proved optimal by Lupanov. UFA to DFA: 2 n {\displaystyle 2^{n}} states, see Leung, An earlier lower bound by Schmidt was smaller. NFA to UFA: 2 n − 1 {\displaystyle 2^{n}-1} states, see Leung. There was an earlier smaller lower bound by Schmidt. SVFA to DFA: Θ ( 3 n / 3 ) {\displaystyle \Theta (3^{n/3})} states, see Jirásková and Pighizzini 2DFA to DFA: n ( n n − ( n − 1 ) n ) {\displaystyle n(n^{n}-(n-1)^{n})} states, see Kapoutsis. Earlier construction by Shepherdson used more states, and an earlier lower bound by Moore was smaller. 2DFA to NFA: ( 2 n n + 1 ) = O ( 4 n n ) {\displaystyle {\binom {2n}{n+1}}=O({\frac {4^{n}}{\sqrt {n}}})} , see Kapoutsis. Earlier construction by Birget used more states. 2NFA to NFA: ( 2 n n + 1 ) {\displaystyle {\binom {2n}{n+1}}} , see Kapoutsis. 2NFA to NFA accepting the complement: O ( 4 n ) {\displaystyle O(4^{n})} states, see Vardi. AFA to DFA: 2 2 n {\displaystyle 2^{2^{n}}} states, see Chandra, Kozen and Stockmeyer. AFA to NFA: 2 n {\displaystyle 2^{n}} states, see Fellah, Jürgensen and Yu. 2AFA to DFA: 2 n 2 n {\displaystyle 2^{n2^{n}}} , see Ladner, Lipton and Stockmeyer. 2AFA to NFA: 2 Θ ( n log n ) {\displaystyle 2^{\Theta (n\log n)}} , see Geffert and Okhotin. === The 2DFA vs. 2NFA problem and logarithmic space === It is an open problem whether all 2NFAs can be converted to 2DFAs with polynomially many states, i.e. whether there is a polynomial p ( n ) {\displaystyle p(n)} such that for every n {\displaystyle n} -state 2NFA there exists a p ( n ) {\displaystyle p(n)} -state 2DFA. The problem was raised by Sakoda and Sipser, who compared it to the P vs. NP problem in the computational complexity theory. Berman and Lingas discovered a formal relation between this problem and the L vs. NL open problem. This relation was further elaborated by Kapoutsis. == State complexity of operations for finite automata == Given a binary regularity-preserving operation on languages ∘ {\displaystyle \circ } and a family of automata X (DFA, NFA, etc.), the state complexity of ∘ {\displaystyle \circ } is an integer function f ( m , n ) {\displaystyle f(m,n)} such that for each m-state X-automaton A and n-state X-automaton B there is an f ( m , n ) {\displaystyle f(m,n)} -state X-automaton for L ( A ) ∘ L ( B ) {\displaystyle L(A)\circ L(B)} , and for all integers m, n there is an m-state X-automaton A and an n-state X-automaton B such that every X-automaton for L ( A ) ∘ L ( B ) {\displaystyle L(A)\circ L(B)} must have at least f ( m , n ) {\displaystyle f(m,n)} states. Analogous definition applies for operations with any number of arguments. The first results on state complexity of operations for DFAs were published by Maslov and by Yu, Zhuang and Salomaa. Holzer and Kutrib pioneered the state complexity of operations on NFA. The known results for basic operations are listed below. === Union === If language L 1 {\displaystyle L_{1}} requires m states and language L 2 {\displaystyle L_{2}} requires n states, how many states does L 1 ∪ L 2 {\displaystyle L_{1}\cup L_{2}} require? DFA: m n {\displaystyle mn} states, see Maslov and Yu, Zhuang and Salomaa. NFA: m + n + 1 {\displaystyle m+n+1} states, see Holzer and Kutrib. UFA: at least min ( n , m ) Ω ( log ( min ( n , m ) ) ) {\displaystyle \min(n,m)^{\Omega (\log(\min(n,m)))}} ; between m n + m + n {\displaystyle mn+m+n} and m + n m 2 0.79 m {\displaystyle m+nm2^{0.79m}} states, see Jirásek, Jirásková and Šebej. SVFA: m n {\displaystyle mn} states, see Jirásek, Jirásková and Szabari. 2DFA: between m + n {\displaystyle m+n} and 4 m + n + 4 {\displaystyle 4m+n+4} states, see Kunc and Okhotin. 2NFA: m + n {\displaystyle m+n} states, see Kunc and Okhotin. === Intersection === How many states does L 1 ∩ L 2 {\displaystyle L_{1}\cap L_{2}} require? DFA: m n {\displaystyle mn} states, see Maslov and Yu, Zhuang and Salomaa. NFA: m n {\displaystyle mn} states, see Holzer and Kutrib. UFA: m n {\displaystyle mn} states, see Jirásek, Jirásková and Šebej. SVFA: m n {\displaystyle mn} states, see Jirásek, Jirásková and Szabari. 2DFA: between m + n {\displaystyle m+n} and m + n + 1 {\displaystyle m+n+1} states, see Kunc and Okhotin. 2NFA: between m + n {\displaystyle m+n} and m + n + 1 {\displaystyle m+n+1} states, see Kunc and Okhotin. === Complementation === If language L requires n states then how many states does its complement require? DFA: n {\displaystyle n} states, by exchanging accepting and rejecting states. NFA: 2 n {\displaystyle 2^{n}} states, see Birget. or Jirásková UFA: at least n Ω ~ ( log n ) {\displaystyle n^{{\tilde {\Omega }}(\log n)}} states, see Göös, Kiefer and Yuan, (this follows an earlier bound by Raskin); and at most n + 1 ⋅ 2 0.5 n {\displaystyle {\sqrt {n+1}}\cdot 2^{0.5n}} states, see Indzhev and Kiefer. SVFA: n {\displaystyle n} states, by exchanging accepting and rejecting states. 2DFA: at least n {\displaystyle n} and at most 4 n {\displaystyle 4n} states, see Geffert, Mereghetti and Pighizzini. === Concatenation === How many states does L 1 L 2 = { w 1 w 2 ∣ w 1 ∈ L 1 , w 2 ∈ L 2 } {\displaystyle L_{1}L_{2}=\{w_{1}w_{2}\mid w_{1}\in L_{1},w_{2}\in L_{2}\}} require? DFA: m ⋅ 2 n − 2 n − 1 {\displaystyle m\cdot 2^{n}-2^{n-1}} states, see Maslov and Yu, Zhuang and Salomaa. NFA: m + n {\displaystyle m+n} states, see Holzer and Kutrib. UFA: 3 4 2 m + n − 1 {\displaystyle {\frac {3}{4}}2^{m+n}-1} states, see Jirásek, Jirásková and Šebej. SVFA: Θ ( 3 n / 3 2 m ) {\displaystyle \Theta (3^{n/3}2^{m})} states, see Jirásek, Jirásková and Szabari. 2DFA: at least 2 Ω ( n ) log m {\displaystyle {\frac {2^{\Omega (n)}}{\log m}}} and at most 2 m m + 1 ⋅ 2 n n + 1 {\displaystyle 2m^{m+1}\cdot 2^{n^{n+1}}} states, see Jirásková and Okhotin. === Kleene star === DFA: 3 4 2 n {\displaystyle {\frac {3}{4}}2^{n}} states, see Maslov and Yu, Zhuang and Salomaa. NFA: n + 1 {\displaystyle n+1} states, see Holzer and Kutrib. UFA: 3 4 2 n {\displaystyle {\frac {3}{4}}2^{n}} states, see Jirásek, Jirásková and Šebej. SVFA: 3 4 2 n {\displaystyle {\frac {3}{4}}2^{n}} states, see Jirásek, Jirásková and Szabari. 2DFA: at least 1 n 2 n 2 − 1 {\displaystyle {\frac {1}{n}}2^{{\frac {n}{2}}-1}} and at most 2 O ( n n + 1 ) {\displaystyle 2^{O(n^{n+1})}} states, see Jirásková and Okhotin. === Reversal === DFA: 2 n {\displaystyle 2^{n}} states, see Mirkin, Leiss, and Yu, Zhuang and Salomaa. NFA: n + 1 {\displaystyle n+1} states, see Holzer and Kutrib. UFA: n {\displaystyle n} states. SVFA: 2 n + 1 {\displaystyle 2n+1} states, see Jirásek, Jirásková and Szabari. 2DFA: between n + 1 {\displaystyle n+1} and n + 2 {\displaystyle n+2} states, see Jirásková and Okhotin. == Finite automata over a unary alphabet == State complexity of finite automata with a one-letter (unary) alphabet, pioneered by Chrobak, is different from the multi-letter case. Let g ( n ) = e Θ ( n ln n ) {\displaystyle g(n)=e^{\Theta ({\sqrt {n\ln n}})}} be Landau's function. === Transformation between models === For a one-letter alphabet, transformations between different types of finite automata are sometimes more efficient than in the general case. NFA to DFA: g ( n ) + O ( n 2 ) {\displaystyle g(n)+O(n^{2})} states, see Chrobak. 2DFA to DFA: g ( n ) + O ( n ) {\displaystyle g(n)+O(n)} states, see Chrobak and Kunc and Okhotin. 2NFA to DFA: O ( g ( n ) ) {\displaystyle O(g(n))} states, see Mereghetti and Pighizzini. and Geffert, Mereghetti and Pighizzini. NFA to 2DFA: at most O ( n 2 ) {\displaystyle O(n^{2})} states, see Chrobak. 2NFA to 2DFA: at most n O ( log n ) {\displaystyle n^{O(\log n)}} states, proved by implementing the method of Savitch's theorem, see
Iterative reconstruction
Iterative reconstruction refers to iterative algorithms used to reconstruct 2D and 3D images in certain imaging techniques. For example, in computed tomography an image must be reconstructed from projections of an object. Here, iterative reconstruction techniques are usually a better, but computationally more expensive alternative to the common filtered back projection (FBP) method, which directly calculates the image in a single reconstruction step. In recent research works, scientists have shown that extremely fast computations and massive parallelism is possible for iterative reconstruction, which makes iterative reconstruction practical for commercialization. == Basic concepts == The reconstruction of an image from the acquired data is an inverse problem. Often, it is not possible to exactly solve the inverse problem directly. In this case, a direct algorithm has to approximate the solution, which might cause visible reconstruction artifacts in the image. Iterative algorithms approach the correct solution using multiple iteration steps, which allows to obtain a better reconstruction at the cost of a higher computation time. There are a large variety of algorithms, but each starts with an assumed image, computes projections from the image, compares the original projection data and updates the image based upon the difference between the calculated and the actual projections. === Algebraic reconstruction === The Algebraic Reconstruction Technique (ART) was the first iterative reconstruction technique used for computed tomography by Hounsfield. === Iterative Sparse Asymptotic Minimum Variance === The iterative sparse asymptotic minimum variance algorithm is an iterative, parameter-free superresolution tomographic reconstruction method inspired by compressed sensing, with applications in synthetic-aperture radar, computed tomography scan, and magnetic resonance imaging (MRI). === Statistical reconstruction === There are typically five components to statistical iterative image reconstruction algorithms, e.g. An object model that expresses the unknown continuous-space function f ( r ) {\displaystyle f(r)} that is to be reconstructed in terms of a finite series with unknown coefficients that must be estimated from the data. A system model that relates the unknown object to the "ideal" measurements that would be recorded in the absence of measurement noise. Often this is a linear model of the form A x + ϵ {\displaystyle \mathbf {A} x+\epsilon } , where ϵ {\displaystyle \epsilon } represents the noise. A statistical model that describes how the noisy measurements vary around their ideal values. Often Gaussian noise or Poisson statistics are assumed. Because Poisson statistics are closer to reality, it is more widely used. A cost function that is to be minimized to estimate the image coefficient vector. Often this cost function includes some form of regularization. Sometimes the regularization is based on Markov random fields. An algorithm, usually iterative, for minimizing the cost function, including some initial estimate of the image and some stopping criterion for terminating the iterations. === Learned Iterative Reconstruction === In learned iterative reconstruction, the updating algorithm is learned from training data using techniques from machine learning such as convolutional neural networks, while still incorporating the image formation model. This typically gives faster and higher quality reconstructions and has been applied to CT and MRI reconstruction. == Advantages == The advantages of the iterative approach include improved insensitivity to noise and capability of reconstructing an optimal image in the case of incomplete data. The method has been applied in emission tomography modalities like SPECT and PET, where there is significant attenuation along ray paths and noise statistics are relatively poor. Statistical, likelihood-based approaches: Statistical, likelihood-based iterative expectation-maximization algorithms are now the preferred method of reconstruction. Such algorithms compute estimates of the likely distribution of annihilation events that led to the measured data, based on statistical principle, often providing better noise profiles and resistance to the streak artifacts common with FBP. Since the density of radioactive tracer is a function in a function space, therefore of extremely high-dimensions, methods which regularize the maximum-likelihood solution turning it towards penalized or maximum a-posteriori methods can have significant advantages for low counts. Examples such as Ulf Grenander's Sieve estimator or Bayes penalty methods, or via I.J. Good's roughness method may yield superior performance to expectation-maximization-based methods which involve a Poisson likelihood function only. As another example, it is considered superior when one does not have a large set of projections available, when the projections are not distributed uniformly in angle, or when the projections are sparse or missing at certain orientations. These scenarios may occur in intraoperative CT, in cardiac CT, or when metal artifacts require the exclusion of some portions of the projection data. In Magnetic Resonance Imaging it can be used to reconstruct images from data acquired with multiple receive coils and with sampling patterns different from the conventional Cartesian grid and allows the use of improved regularization techniques (e.g. total variation) or an extended modeling of physical processes to improve the reconstruction. For example, with iterative algorithms it is possible to reconstruct images from data acquired in a very short time as required for real-time MRI (rt-MRI). In Cryo Electron Tomography, where the limited number of projections are acquired due to the hardware limitations and to avoid the biological specimen damage, it can be used along with compressive sensing techniques or regularization functions (e.g. Huber function) to improve the reconstruction for better interpretation. Here is an example that illustrates the benefits of iterative image reconstruction for cardiac MRI.
Thompson sampling
Thompson sampling, named after William R. Thompson, is a heuristic for choosing actions that address the exploration–exploitation dilemma in the multi-armed bandit problem. It consists of choosing the action that maximizes the expected reward with respect to a randomly drawn belief. == Description == Consider a set of contexts X {\displaystyle {\mathcal {X}}} , a set of actions A {\displaystyle {\mathcal {A}}} , and rewards in R {\displaystyle \mathbb {R} } . The aim of the player is to play actions under the various contexts, such as to maximize the cumulative rewards. Specifically, in each round, the player obtains a context x ∈ X {\displaystyle x\in {\mathcal {X}}} , plays an action a ∈ A {\displaystyle a\in {\mathcal {A}}} and receives a reward r ∈ R {\displaystyle r\in \mathbb {R} } following a distribution that depends on the context and the issued action. The elements of Thompson sampling are as follows: a likelihood function P ( r | θ , a , x ) {\displaystyle P(r|\theta ,a,x)} ; a set Θ {\displaystyle \Theta } of parameters θ {\displaystyle \theta } of the distribution of r {\displaystyle r} ; a prior distribution P ( θ ) {\displaystyle P(\theta )} on these parameters; past observations triplets D = { ( x ; a ; r ) } {\displaystyle {\mathcal {D}}=\{(x;a;r)\}} ; a posterior distribution P ( θ | D ) ∝ P ( D | θ ) P ( θ ) {\displaystyle P(\theta |{\mathcal {D}})\propto P({\mathcal {D}}|\theta )P(\theta )} , where P ( D | θ ) {\displaystyle P({\mathcal {D}}|\theta )} is the likelihood function. Thompson sampling consists of playing the action a ∗ ∈ A {\displaystyle a^{\ast }\in {\mathcal {A}}} according to the probability that it maximizes the expected reward; action a ∗ {\displaystyle a^{\ast }} is chosen with probability ∫ I [ E ( r | a ∗ , x , θ ) = max a ′ E ( r | a ′ , x , θ ) ] P ( θ | D ) d θ , {\displaystyle \int \mathbb {I} \left[\mathbb {E} (r|a^{\ast },x,\theta )=\max _{a'}\mathbb {E} (r|a',x,\theta )\right]P(\theta |{\mathcal {D}})d\theta ,} where I {\displaystyle \mathbb {I} } is the indicator function. In practice, the rule is implemented by sampling. In each round, parameters θ ∗ {\displaystyle \theta ^{\ast }} are sampled from the posterior P ( θ | D ) {\displaystyle P(\theta |{\mathcal {D}})} , and an action a ∗ {\displaystyle a^{\ast }} chosen that maximizes E [ r | θ ∗ , a ∗ , x ] {\displaystyle \mathbb {E} [r|\theta ^{\ast },a^{\ast },x]} , i.e. the expected reward given the sampled parameters, the action, and the current context. Conceptually, this means that the player instantiates their beliefs randomly in each round according to the posterior distribution, and then acts optimally according to them. In most practical applications, it is computationally onerous to maintain and sample from a posterior distribution over models. As such, Thompson sampling is often used in conjunction with approximate sampling techniques. == History == Thompson sampling was originally described by Thompson in 1933. It was subsequently rediscovered numerous times independently in the context of multi-armed bandit problems. A first proof of convergence for the bandit case has been shown in 1997. The first application to Markov decision processes was in 2000. A related approach (see Bayesian control rule) was published in 2010. In 2010 it was also shown that Thompson sampling is instantaneously self-correcting. Asymptotic convergence results for contextual bandits were published in 2011. Thompson Sampling has been widely used in many online learning problems including A/B testing in website design and online advertising, and accelerated learning in decentralized decision making. A Double Thompson Sampling (D-TS) algorithm has been proposed for dueling bandits, a variant of traditional MAB, where feedback comes in the form of pairwise comparison. == Relationship to other approaches == === Probability matching === Probability matching is a decision strategy in which predictions of class membership are proportional to the class base rates. Thus, if in the training set positive examples are observed 60% of the time, and negative examples are observed 40% of the time, the observer using a probability-matching strategy will predict (for unlabeled examples) a class label of "positive" on 60% of instances, and a class label of "negative" on 40% of instances. === Bayesian control rule === A generalization of Thompson sampling to arbitrary dynamical environments and causal structures, known as Bayesian control rule, has been shown to be the optimal solution to the adaptive coding problem with actions and observations. In this formulation, an agent is conceptualized as a mixture over a set of behaviours. As the agent interacts with its environment, it learns the causal properties and adopts the behaviour that minimizes the relative entropy to the behaviour with the best prediction of the environment's behaviour. If these behaviours have been chosen according to the maximum expected utility principle, then the asymptotic behaviour of the Bayesian control rule matches the asymptotic behaviour of the perfectly rational agent. The setup is as follows. Let a 1 , a 2 , … , a T {\displaystyle a_{1},a_{2},\ldots ,a_{T}} be the actions issued by an agent up to time T {\displaystyle T} , and let o 1 , o 2 , … , o T {\displaystyle o_{1},o_{2},\ldots ,o_{T}} be the observations gathered by the agent up to time T {\displaystyle T} . Then, the agent issues the action a T + 1 {\displaystyle a_{T+1}} with probability: P ( a T + 1 | a ^ 1 : T , o 1 : T ) , {\displaystyle P(a_{T+1}|{\hat {a}}_{1:T},o_{1:T}),} where the "hat"-notation a ^ t {\displaystyle {\hat {a}}_{t}} denotes the fact that a t {\displaystyle a_{t}} is a causal intervention (see Causality), and not an ordinary observation. If the agent holds beliefs θ ∈ Θ {\displaystyle \theta \in \Theta } over its behaviors, then the Bayesian control rule becomes P ( a T + 1 | a ^ 1 : T , o 1 : T ) = ∫ Θ P ( a T + 1 | θ , a ^ 1 : T , o 1 : T ) P ( θ | a ^ 1 : T , o 1 : T ) d θ {\displaystyle P(a_{T+1}|{\hat {a}}_{1:T},o_{1:T})=\int _{\Theta }P(a_{T+1}|\theta ,{\hat {a}}_{1:T},o_{1:T})P(\theta |{\hat {a}}_{1:T},o_{1:T})\,d\theta } , where P ( θ | a ^ 1 : T , o 1 : T ) {\displaystyle P(\theta |{\hat {a}}_{1:T},o_{1:T})} is the posterior distribution over the parameter θ {\displaystyle \theta } given actions a 1 : T {\displaystyle a_{1:T}} and observations o 1 : T {\displaystyle o_{1:T}} . In practice, the Bayesian control amounts to sampling, at each time step, a parameter θ ∗ {\displaystyle \theta ^{\ast }} from the posterior distribution P ( θ | a ^ 1 : T , o 1 : T ) {\displaystyle P(\theta |{\hat {a}}_{1:T},o_{1:T})} , where the posterior distribution is computed using Bayes' rule by only considering the (causal) likelihoods of the observations o 1 , o 2 , … , o T {\displaystyle o_{1},o_{2},\ldots ,o_{T}} and ignoring the (causal) likelihoods of the actions a 1 , a 2 , … , a T {\displaystyle a_{1},a_{2},\ldots ,a_{T}} , and then by sampling the action a T + 1 ∗ {\displaystyle a_{T+1}^{\ast }} from the action distribution P ( a T + 1 | θ ∗ , a ^ 1 : T , o 1 : T ) {\displaystyle P(a_{T+1}|\theta ^{\ast },{\hat {a}}_{1:T},o_{1:T})} . === Upper-confidence-bound (UCB) algorithms === Thompson sampling and upper-confidence bound algorithms share a fundamental property that underlies many of their theoretical guarantees. Roughly speaking, both algorithms allocate exploratory effort to actions that might be optimal and are in this sense "optimistic". Leveraging this property, one can translate regret bounds established for UCB algorithms to Bayesian regret bounds for Thompson sampling or unify regret analysis across both these algorithms and many classes of problems.
Mark I Perceptron
The Mark I Perceptron was a pioneering supervised image classification learning system developed by Frank Rosenblatt in 1958. It was the first implementation of an artificial intelligence (AI) machine. It differs from the Perceptron which is a software architecture proposed in 1943 by Warren McCulloch and Walter Pitts, which was also employed in Mark I, and enhancements of which have continued to be an integral part of cutting edge AI technologies like the Transformer. == Architecture == The Mark I Perceptron was organized into three layers: A set of sensory units which receive optical input A set of association units, each of which fire based on input from multiple sensory units A set of response units, which fire based on input from multiple association units The connection between sensory units and association units were random. The working of association units was very similar to the response units. Different versions of the Mark I used different numbers of units in each of the layers. == Capabilities == In his 1957 proposal for funding for development of the "Cornell Photoperceptron", Rosenblatt claimed:"Devices of this sort are expected ultimately to be capable of concept formation, language translation, collation of military intelligence, and the solution of problems through inductive logic."With the first version of the Mark I Perceptron as early as 1958, Rosenblatt demonstrated a simple binary classification experiment, namely distinguishing between sheets of paper marked on the right versus those marked on the left side. One of the later experiments distinguished a square from a circle printed on paper. The shapes were perfect and their sizes fixed; the only variation was in their position and orientation. The Mark I Perceptron achieved 99.8% accuracy on a test dataset with 500 neurons in a single layer. The size of the training dataset was 10,000 example images. It took 3 seconds for the training pipeline to go through a single image. Higher accuracy was observed with thick outline figures compared to solid figures, likely because outline figures reduced overfitting. Another experiment distinguished between a square and a diamond for which 100% accuracy was achieved with only 60 training images, with a Perceptron having 1,000 neurons in a single layer. The time taken to process each training input for this larger perceptron was 15 seconds. The only variation was in position of the image, since rotation would have been ambiguous. In that same experiment, it could distinguish between the letters X and E with 100% accuracy when trained with only 20 images (10 images of each letter). Variations in the images included both position and rotation by up to 30 degrees. When variation in rotation was increased to any angle (both in training and test datasets), the accuracy reduced to 90% with 60 training images (30 images of each letter). For distinguishing between the letters E and F, a more challenging problem due to their similarity, the same 1,000 neuron perceptron achieved an accuracy of more than 80% with 60 training images. Variation was only in the position of the image, with no rotation.
The Last Question
"The Last Question" is a science fiction short story by American writer Isaac Asimov. It first appeared in the November 1956 issue of Science Fiction Quarterly; and in the anthologies in the collections Nine Tomorrows (1959), The Best of Isaac Asimov (1973), Robot Dreams (1986), The Best Science Fiction of Isaac Asimov (1986), the retrospective Opus 100 (1969), and Isaac Asimov: The Complete Stories, Vol. 1 (1990). While he also considered it one of his best works, "The Last Question" was Asimov's favorite short story of his own authorship, and is one of a loosely connected series of stories concerning a fictional computer called Multivac. Through successive generations, humanity questions Multivac on the subject of entropy. The story blends science fiction, theology, and philosophy. It has been recognized as a counterpoint to Fredric Brown's short short story "Answer", published two years earlier. == History == In conceiving Multivac, Asimov was extrapolating the trend towards centralization that characterized computation technology planning in the 1950s to an ultimate centrally managed global computer. After seeing a planetarium adaptation of his work, Asimov "privately" concluded that the story was his best science fiction yet written. He placed it just higher than "The Ugly Little Boy" (September 1958) and "The Bicentennial Man" (1976). The story asks the question of humanity's fate, and human existence as a whole, highlighting Asimov's focus on important aspects of our future like population growth and environmental issues. "The Last Question" ranks with "Nightfall" (1941) as one of Asimov's best-known and most acclaimed short stories. He wrote in 1973 that he appreciated how easy the story was to write after he had the idea. He was so often approached by fans who remembered the story but not the title, that in one instance he gave the answer, correctly, before the fan had even described the story. == Plot summary == By the year 2061, Multivac, a self-adjusting and self-correcting computer, has allowed mankind to reach beyond the planetary confines of Earth and harness solar energy. Two technicians, Adell and Lupov, celebrate Multivac's role in this development. Over drinks, they discuss that the sun will expire due to the second law of thermodynamics, which states that entropy inevitably increases. When Adell asks Multivac whether this can be reversed, the computer responds that it has insufficient data to answer. In several episodes over ten trillion years, increasingly advanced humans pose the same question to the computers of their time. Each time the computer gives the same response. At the heat death of the universe, the last disembodied consciousness of Man asks the question a final time of a computer that resides in hyperspace before merging with it. After collecting the last data from the dead universe, the computer continues to process it alone and finds an answer to the last question. Having no one to tell it to, it proceeds to demonstrate by saying "LET THERE BE LIGHT!" == Themes == === Philosophy === Although science and religion are frequently presented as having an oppositional relationship, "The Last Question" explores some biblical contexts ("Let there be light"). In Asimov's story, aspects like the great meaning of existence are culminated through both technology and human knowledge. The evolution from Multivac to AC also emulates a sort of cycle of existence. === Dystopian happy ending === Multivac's purpose was conceptualized with a desire for knowledge, promoting the idea that more knowledge will lead to a better and more fruitful future for humanity. However, the computer's answers regarding the future suggest an inevitable exhaustion of the Sun, and this thirst for knowledge becomes an obsession with the future. The story's end displays a dichotomy between annihilation and peace. == Dramatic adaptations == === Planetarium shows === "The Last Question" was first adapted for the Abrams Planetarium at Michigan State University (in 1966), featuring the voice of Leonard Nimoy, as Asimov wrote in his autobiography In Joy Still Felt (1980). It was adapted for the Strasenburgh Planetarium in Rochester, New York (in 1969), under the direction of Ian C. McLennan. It was adapted for the Edmonton Space Sciences Centre in Edmonton, Alberta (early 1970s), under the direction of John Hault. It was adapted for the Gates Planetarium at the Denver Museum of Natural History in 1973 under the direction of Mark B. Peterson It subsequently played at the: Fels Planetarium of the Franklin Institute in Philadelphia in 1973 Planetarium of the Reading School District in Reading, Pennsylvania in 1974 Buhl Planetarium, Pittsburgh in 1974 The Space Transit Planetarium of the Museum of Science in Miami during 1977 Vanderbilt Planetarium in Centerport New York, in 1978, read by singer-songwriter and Long Island resident Harry Chapin. Hansen Planetarium in Salt Lake City, Utah (in 1980 and 1989) A reading of the story was played on BBC Radio 7 in 2008 and 2009. Gates Planetarium in Denver, Colorado (in early 2020) In 1989 Asimov updated the star show adaptation to add in quasars and black holes. The story was adapted as a comic book by Don Thompson and drawn by John Estes in the third issue of ORBiT.
Structured-light 3D scanner
A structured-light 3D scanner is a device used to capture the three-dimensional shape of an object by projecting light patterns, such as grids or stripes, onto its surface. The deformation of these patterns is recorded by cameras and processed using specialized algorithms to generate a detailed 3D model. Structured-light 3D scanning is widely employed in fields such as industrial design, quality control, cultural heritage preservation, augmented reality gaming, and medical imaging. Compared to laser-based 3D scanning, structured-light scanners use non-coherent light sources, such as LEDs or projectors, which enable faster data acquisition and eliminate potential safety concerns associated with lasers. However, the accuracy of structured-light scanning can be influenced by external factors, including ambient lighting conditions and the reflective properties of the scanned object. == Principle == Projecting a narrow band of light onto a three-dimensional surface creates a line of illumination that appears distorted when viewed from perspectives other than that of the projector. This distortion can be analyzed to reconstruct the geometry of the surface, a technique known as light sectioning. Projecting patterns composed of multiple stripes or arbitrary fringes simultaneously enables the acquisition of numerous data points at once, improving scanning speed. While various structured light projection techniques exist, parallel stripe patterns are among the most commonly used. By analyzing the displacement of these stripes, the three-dimensional coordinates of surface details can be accurately determined. === Generation of light patterns === Two major methods of stripe pattern generation have been established: Laser interference and projection. The laser interference method works with two wide planar laser beam fronts. Their interference results in regular, equidistant line patterns. Different pattern sizes can be obtained by changing the angle between these beams. The method allows for the exact and easy generation of very fine patterns with unlimited depth of field. Disadvantages are high cost of implementation, difficulties providing the ideal beam geometry, and laser typical effects like speckle noise and the possible self interference with beam parts reflected from objects. Typically, there is no means of modulating individual stripes, such as with Gray codes. The projection method uses incoherent light and basically works like a video projector. Patterns are usually generated by passing light through a digital spatial light modulator, typically based on one of the three currently most widespread digital projection technologies, transmissive liquid crystal, reflective liquid crystal on silicon (LCOS) or digital light processing (DLP; moving micro mirror) modulators, which have various comparative advantages and disadvantages for this application. Other methods of projection could be and have been used, however. Patterns generated by digital display projectors have small discontinuities due to the pixel boundaries in the displays. Sufficiently small boundaries however can practically be neglected as they are evened out by the slightest defocus. A typical measuring assembly consists of one projector and at least one camera. For many applications, two cameras on opposite sides of the projector have been established as useful. Invisible (or imperceptible) structured light uses structured light without interfering with other computer vision tasks for which the projected pattern will be confusing. Example methods include the use of infrared light or of extremely high framerates alternating between two exact opposite patterns. === Calibration === Geometric distortions by optics and perspective must be compensated by a calibration of the measuring equipment, using special calibration patterns and surfaces. A mathematical model is used for describing the imaging properties of projector and cameras. Essentially based on the simple geometric properties of a pinhole camera, the model also has to take into account the geometric distortions and optical aberration of projector and camera lenses. The parameters of the camera as well as its orientation in space can be determined by a series of calibration measurements, using photogrammetric bundle adjustment. === Analysis of stripe patterns === There are several depth cues contained in the observed stripe patterns. The displacement of any single stripe can directly be converted into 3D coordinates. For this purpose, the individual stripe has to be identified, which can for example be accomplished by tracing or counting stripes (pattern recognition method). Another common method projects alternating stripe patterns, resulting in binary Gray code sequences identifying the number of each individual stripe hitting the object. An important depth cue also results from the varying stripe widths along the object surface. Stripe width is a function of the steepness of a surface part, i.e. the first derivative of the elevation. Stripe frequency and phase deliver similar cues and can be analyzed by a Fourier transform. Finally, the wavelet transform has recently been discussed for the same purpose. In many practical implementations, series of measurements combining pattern recognition, Gray codes and Fourier transform are obtained for a complete and unambiguous reconstruction of shapes. Another method also belonging to the area of fringe projection has been demonstrated, utilizing the depth of field of the camera. It is also possible to use projected patterns primarily as a means of structure insertion into scenes, for an essentially photogrammetric acquisition. === Precision and range === The optical resolution of fringe projection methods depends on the width of the stripes used and their optical quality. It is also limited by the wavelength of light. An extreme reduction of stripe width proves inefficient due to limitations in depth of field, camera resolution and display resolution. Therefore, the phase shift method has been widely established: A number of at least 3, typically about 10 exposures are taken with slightly shifted stripes. The first theoretical deductions of this method relied on stripes with a sine wave shaped intensity modulation, but the methods work with "rectangular" modulated stripes, as delivered from LCD or DLP displays as well. By phase shifting, surface detail of e.g. 1/10 the stripe pitch can be resolved. Current optical stripe pattern profilometry hence allows for detail resolutions down to the wavelength of light, below 1 micrometer in practice or, with larger stripe patterns, to approx. 1/10 of the stripe width. Concerning level accuracy, interpolating over several pixels of the acquired camera image can yield a reliable height resolution and also accuracy, down to 1/50 pixel. Arbitrarily large objects can be measured with accordingly large stripe patterns and setups. Practical applications are documented involving objects several meters in size. Typical accuracy figures are: Planarity of a 2-foot (0.61 m) wide surface, to 10 micrometres (0.00039 in). Shape of a motor combustion chamber to 2 micrometres (7.9×10−5 in) (elevation), yielding a volume accuracy 10 times better than with volumetric dosing. Shape of an object 2 inches (51 mm) large, to about 1 micrometre (3.9×10−5 in) Radius of a blade edge of e.g. 10 micrometres (0.00039 in), to ±0.4 μm === Navigation === As the method can measure shapes from only one perspective at a time, complete 3D shapes have to be combined from different measurements in different angles. This can be accomplished by attaching marker points to the object and combining perspectives afterwards by matching these markers. The process can be automated, by mounting the object on a motorized turntable on robotic inspection cell, or CNC positioning device. Markers can as well be applied on a positioning device instead of the object itself. The 3D data gathered can be used to retrieve CAD (computer aided design) data and models from existing components (reverse engineering), hand formed samples or sculptures, natural objects or artifacts. === Challenges === As with all optical methods, reflective or transparent surfaces raise difficulties. Reflections cause light to be reflected either away from the camera or right into its optics. In both cases, the dynamic range of the camera can be exceeded. Transparent or semi-transparent surfaces also cause major difficulties. In these cases, coating the surfaces with a thin opaque lacquer just for measuring purposes is a common practice. A recent method handles highly reflective and specular objects by inserting a 1-dimensional diffuser between the light source (e.g., projector) and the object to be scanned. Alternative optical techniques have been proposed for handling perfectly transparent and specular objects. Double reflections and inter-reflections can cause the stripe pattern to be overlaid with unwanted ligh
International Aerial Robotics Competition
The International Aerial Robotics Competition (IARC) is a university-based robotics competition held on the campus of the Georgia Institute of Technology, currently hosted by RoboNation. Since 1991, collegiate teams with the backing of industry and government have fielded autonomous flying robots in an attempt to perform missions requiring robotic behaviors not previously exhibited by a flying machine. The term “aerial robotics” was coined by competition creator Robert Michelson in 1990 to describe a new class of small highly intelligent flying machines. Successive years of competition saw these aerial robots grow from vehicles that could barely maintain themselves in the air, to automatons which are self-stable, self-navigating, and able to interact with their environment. The goal of the competition has been to provide a reason for the state-of-the-art of aerial robotics to move forward. Challenges have been geared towards producing advances. From 1991 through 2009, six missions were proposed. Each involved fully autonomous robotic behavior undemonstrated at the time. In October 2013 a seventh mission was proposed. It was the first to involve interaction between aerial robots and multiple ground robots. In 2016, the competition and its creator were recognized during the Georgia legislative session in the form of a senate resolution as the longest running aerial robotics competition in the world. == History == === First mission === The initial mission to move a metallic disc from one side of an arena to the other was seen by many as almost impossible. The college teams improved their entries over the next two years when the competition saw its first autonomous takeoff, flight, and landing by a team from the Georgia Institute of Technology. In 1995, a team from Stanford University was able to acquire a single disk and move it from one side of the arena to the other in a fully autonomous flight—half. === Second mission === The competition mission was toughened and made less abstract by requiring teams to search for a toxic waste dump, map the location of partially buried randomly oriented toxic waste drums, identify the contents of each drum from the hazard labels on the outside of each drum, and bring a sample back from one of the drums. In 1996, a team from the Massachusetts Institute of Technology and Boston University, with backing from Draper Labs, created a small fully autonomous flying robot that repeatedly and correctly mapped the location of all five of the toxic waste drums, and correctly identified the contents of two from the air, completing approximately seventy five percent of the mission. The following year, an aerial robot developed by a team from Carnegie Mellon University completed the entire mission. === Third mission === The third mission began in 1998. It was a search and rescue mission requiring fully autonomous robots to take off, fly to a disaster area and search amid fires, broken water mains, clouds of toxic gas, and rubble. The scenario was recreated at the U.S. Department of Energy's Hazardous Material Management and Emergency Response (HAMMER) training facility. Because of the realism of the scenario, animatrons were used instead of human actors to simulate survivors incapable of extracting themselves from the disaster area. An aerial robot from Germany's Technische Universität Berlin was able to detect and avoid all of the obstacles, identify all the dead on the ground and the survivors (distinguishing between the two based on movement), and relay pictures of the survivors along with their locations back to first responders who would attempt a rescue. This mission was completed in 2000. === Fourth mission === The fourth mission was initiated in 2001. It involved three scenarios requiring the same autonomous behavior: a hostage rescue mission where a submarine 3 kilometers off the coast must send an aerial robot to find a coastal city, identify the embassy where hostages are being held, locate valid openings in the embassy building, enter (or send in a sensor probe/subvehicle) and relay pictures of the hostages 3 km to the submarine prior to mounting an amphibious assault on the embassy to free the hostages; the discovery of an ancient mausoleum where a virus had killed the archaeological team, who had radioed that an important and undocumented tapestry was hanging inside, with 15 minutes to send an autonomous aerial robot to find the mausoleum, enter it (or send in a sensor probe/subvehicle) and relay pictures of the tapestry back prior to the destruction of the mausoleum and its contents; and an explosion at a nuclear reactor facility where scientists must send in an aerial robot to find the operating reactor building, enter the building (or send in a sensor probe/subvehicle) and relay pictures of the control panels to determine if a melt-down is imminent. All three missions involved the same elements of ingress, locating, identification, entry, and relaying pictures within 15 minutes. It was conducted at the U.S. Army's Fort Benning Soldier Battle Lab using the McKenna MOUT (Military Operations on Urban Terrain) site. The fourth mission was completed in 2008 with 27 teams who had demonstrated each of the required aerial robotic behaviors, except being able to demonstrate these behaviors in under 15 minutes—a feat considered by the judges to be inevitable given more time, and therefore no longer a significant challenge. Thus the fourth mission was terminated, $80,000 in awards distributed, and the fifth mission established. === Fifth mission === The fifth mission picked up where the fourth mission left off by demonstrating the fully autonomous aerial robotic behaviors necessary to rapidly negotiate the confined internal spaces of a structure once it has been penetrated by an air vehicle. The nuclear reactor complex explosion scenario of the fourth mission was used as the backdrop for the fifth mission. The fifth mission required a fully autonomous aerial vehicle to penetrate the structure and negotiate the more complex interior space containing hallways, small rooms, obstacles, and dead ends in order to search for a designated target without the aid of global-positioning navigational aids, and relay pictures back to a monitoring station some distance from the structure. The First Symposium on Indoor Flight Issues was held in conjunction with this 2009 IARC event. === Sixth mission === The sixth mission began in 2010 as an extension of the fifth mission theme of autonomous indoor flight behavior, however it demanded more advanced behaviors than were possible by any aerial robot extant in 2010. This espionage mission involved covertly stealing a flash drive from a particular room in a building and depositing an identical drive to avoid detection of the theft. The 2010 Symposium on Indoor Flight Issues was held concurrently at the University of Puerto Rico - Mayagüez during the 20th anniversary competition. === Seventh mission === The seventh mission began in 2014 demanding more advanced behaviors than were possible by any aerial robot extant in 2014. A single autonomous aerial robot had to herd up to 10 autonomous ground robot targets across one designated end of a 20m x 20m (65.62 feet x 65.62 feet) arena in under 10 minutes. The arena had neither walls for SLAM mapping nor GPS availability. Techniques such as optical flow or optical odometry were possible solutions to navigation within the arena. Collisions with obstacle ground robots ended the run with no score. The autonomous aerial robots interacted with the ground robots in the following way: if an aerial robot touched the ground robot on top, the ground robot would turn clockwise 45°. If the aerial robot blocked its forward motion by landing in front of it, the ground robot would reverse direction. Ground robots that feely escaped the arena, counted against the aerial robot's overall score, so the autonomous aerial robots had to decide which ground robots were in imminent danger of crossing any boundary except the designated one, and redirect them toward the designated boundary.Zhejiang University was the overall winner of Mission 7, of 52 teams from 12 nations entered as competitors. === Eighth mission === In 2018, the 8th mission was announced. Mission 8 focused on non-electronic human-machine interaction for the first time, with four aerial robots assisting humans to complete tasks that one person could not independently accomplish. The gist of mission 8 involved a swarm of autonomous aerial robots working with a human to achieve a task in the presence of hostile "Sentry aerial robots" which were trying to impede the human. In 2018, the inaugural year of mission 8, the American Venue was held on the campus of the Georgia Institute of Technology in Atlanta, Georgia, and the Asia/Pacific Venue was conducted at Beihang University in Beijing China. The following year, Mission 8 was successfully completed in Kunming China at the Yunnan Innovation