In the mathematical theory of random processes, the Markov chain central limit theorem has a conclusion somewhat similar in form to that of the classic central limit theorem (CLT) of probability theory, but the quantity in the role taken by the variance in the classic CLT has a more complicated definition. See also the general form of Bienaymé's identity. == Statement == Suppose that: the sequence X 1 , X 2 , X 3 , … {\textstyle X_{1},X_{2},X_{3},\ldots } of random elements of some set is a Markov chain that has a stationary probability distribution; and the initial distribution of the process, i.e. the distribution of X 1 {\textstyle X_{1}} , is the stationary distribution, so that X 1 , X 2 , X 3 , … {\textstyle X_{1},X_{2},X_{3},\ldots } are identically distributed. In the classic central limit theorem these random variables would be assumed to be independent, but here we have only the weaker assumption that the process has the Markov property; and g {\textstyle g} is some (measurable) real-valued function for which var ( g ( X 1 ) ) < + ∞ . {\textstyle \operatorname {var} (g(X_{1}))<+\infty .} Now let μ = E ( g ( X 1 ) ) , μ ^ n = 1 n ∑ k = 1 n g ( X k ) σ 2 := lim n → ∞ var ( n μ ^ n ) = lim n → ∞ n var ( μ ^ n ) = var ( g ( X 1 ) ) + 2 ∑ k = 1 ∞ cov ( g ( X 1 ) , g ( X 1 + k ) ) . {\displaystyle {\begin{aligned}\mu &=\operatorname {E} (g(X_{1})),\\{\widehat {\mu }}_{n}&={\frac {1}{n}}\sum _{k=1}^{n}g(X_{k})\\\sigma ^{2}&:=\lim _{n\to \infty }\operatorname {var} ({\sqrt {n}}{\widehat {\mu }}_{n})=\lim _{n\to \infty }n\operatorname {var} ({\widehat {\mu }}_{n})=\operatorname {var} (g(X_{1}))+2\sum _{k=1}^{\infty }\operatorname {cov} (g(X_{1}),g(X_{1+k})).\end{aligned}}} Then as n → ∞ , {\textstyle n\to \infty ,} we have n ( μ ^ n − μ ) → D Normal ( 0 , σ 2 ) , {\displaystyle {\sqrt {n}}({\hat {\mu }}_{n}-\mu )\ {\xrightarrow {\mathcal {D}}}\ {\text{Normal}}(0,\sigma ^{2}),} where the decorated arrow indicates convergence in distribution. == Monte Carlo Setting == The Markov chain central limit theorem can be guaranteed for functionals of general state space Markov chains under certain conditions. In particular, this can be done with a focus on Monte Carlo settings. An example of the application in a MCMC (Markov Chain Monte Carlo) setting is the following: Consider a simple hard spheres model on a grid. Suppose X = { 1 , … , n 1 } × { 1 , … , n 2 } ⊆ Z 2 {\displaystyle X=\{1,\ldots ,n_{1}\}\times \{1,\ldots ,n_{2}\}\subseteq Z^{2}} . A proper configuration on X {\displaystyle X} consists of coloring each point either black or white in such a way that no two adjacent points are white. Let χ {\displaystyle \chi } denote the set of all proper configurations on X {\displaystyle X} , N χ ( n 1 , n 2 ) {\displaystyle N_{\chi }(n_{1},n_{2})} be the total number of proper configurations and π be the uniform distribution on χ {\displaystyle \chi } so that each proper configuration is equally likely. Suppose our goal is to calculate the typical number of white points in a proper configuration; that is, if W ( x ) {\displaystyle W(x)} is the number of white points in x ∈ χ {\displaystyle x\in \chi } then we want the value of E π W = ∑ x ∈ χ W ( x ) N χ ( n 1 , n 2 ) {\displaystyle E_{\pi }W=\sum _{x\in \chi }{\frac {W(x)}{N_{\chi }{\bigl (}n_{1},n_{2}{\bigr )}}}} If n 1 {\displaystyle n_{1}} and n 2 {\displaystyle n_{2}} are even moderately large then we will have to resort to an approximation to E π W {\displaystyle E_{\pi }W} . Consider the following Markov chain on χ {\displaystyle \chi } . Fix p ∈ ( 0 , 1 ) {\displaystyle p\in (0,1)} and set X 1 = x 1 {\displaystyle X_{1}=x_{1}} where x 1 ∈ χ {\displaystyle x_{1}\in \chi } is an arbitrary proper configuration. Randomly choose a point ( x , y ) ∈ X {\displaystyle (x,y)\in X} and independently draw U ∼ U n i f o r m ( 0 , 1 ) {\displaystyle U\sim \mathrm {Uniform} (0,1)} . If u ≤ p {\displaystyle u\leq p} and all of the adjacent points are black then color ( x , y ) {\displaystyle (x,y)} white leaving all other points alone. Otherwise, color ( x , y ) {\displaystyle (x,y)} black and leave all other points alone. Call the resulting configuration X 1 {\displaystyle X_{1}} . Continuing in this fashion yields a Harris ergodic Markov chain { X 1 , X 2 , X 3 , … } {\displaystyle \{X_{1},X_{2},X_{3},\ldots \}} having π {\displaystyle \pi } as its invariant distribution. It is now a simple matter to estimate E π W {\displaystyle E_{\pi }W} with w n ¯ = ∑ i = 1 n W ( X i ) / n {\displaystyle {\overline {w_{n}}}=\sum _{i=1}^{n}W(X_{i})/n} . Also, since χ {\displaystyle \chi } is finite (albeit potentially large) it is well known that X {\displaystyle X} will converge exponentially fast to π {\displaystyle \pi } which implies that a CLT holds for w n ¯ {\displaystyle {\overline {w_{n}}}} . == Implications == Not taking into account the additional terms in the variance which stem from correlations (e.g. serial correlations in markov chain monte carlo simulations) can result in the problem of pseudoreplication when computing e.g. the confidence intervals for the sample mean.
Grammar checker
A grammar checker, in computing terms, is a program, or part of a program, that attempts to verify written text for grammatical correctness. Grammar checkers are most often implemented as a feature of a larger program, such as a word processor, but are also available as a stand-alone application that can be activated from within programs that work with editable text. The implementation of a grammar checker makes use of natural language processing. == History == The earliest "grammar checkers" were programs that checked for punctuation and style inconsistencies, rather than a complete range of possible grammatical errors. The first system was called Writer's Workbench, and was a set of writing tools included with Unix systems as far back as the 1970s. The whole Writer's Workbench package included several separate tools to check for various writing problems. The "diction" tool checked for wordy, trite, clichéd or misused phrases in a text. The tool would output a list of questionable phrases and provide suggestions for improving the writing. The "style" tool analyzed the writing style of a given text. It performed a number of readability tests on the text and output the results, and gave some statistical information about the sentences of the text. Aspen Software of Albuquerque, New Mexico released the earliest version of a diction and style checker for personal computers, Grammatik, in 1981. Grammatik was first available for a Radio Shack - TRS-80, and soon had versions for CP/M and the IBM PC. Reference Software International of San Francisco, California, acquired Grammatik in 1985. Development of Grammatik continued, and it became an actual grammar checker that could detect writing errors beyond simple style checking. Other early diction and style checking programs included Punctuation & Style, Correct Grammar, RightWriter and PowerEdit. While all the earliest programs started as simple diction and style checkers, all eventually added various levels of language processing, and developed some level of true grammar checking capability. Until 1992, grammar checkers were sold as add-on programs. There were a large number of different word processing programs available at that time, with WordPerfect and Microsoft Word the top two in market share. In 1992, Microsoft decided to add grammar checking as a feature of Word, and licensed CorrecText, a grammar checker from Houghton Mifflin that had not yet been marketed as a standalone product. WordPerfect answered Microsoft's move by acquiring Reference Software, and the direct descendant of Grammatik is still included with WordPerfect. As of 2019, grammar checkers are built into systems like Google Docs, browser extensions like Grammarly and Qordoba, desktop applications like Ginger, free and open-source software like LanguageTool, and text editor plugins like those available from WebSpellChecker Software. == Technical issues == The earliest writing style programs checked for wordy, trite, clichéd, or misused phrases in a text. This process was based on simple pattern matching. The heart of the program was a list of many hundreds or thousands of phrases that are considered poor writing by many experts. The list of questionable phrases included alternative wording for each phrase. The checking program would simply break text into sentences, check for any matches in the phrase dictionary, flag suspect phrases and show an alternative. These programs could also perform some mechanical checks. For example, they would typically flag doubled words, doubled punctuation, some capitalization errors, and other simple mechanical mistakes. True grammar checking is more complex. While a programming language has a very specific syntax and grammar, this is not so for natural languages. One can write a somewhat complete formal grammar for a natural language, but there are usually so many exceptions in real usage that a formal grammar is of minimal help in writing a grammar checker. One of the most important parts of a natural language grammar checker is a dictionary of all the words in the language, along with the part of speech of each word. The fact that a natural word may be used as any one of several parts of speech (such as "free" being used as an adjective, adverb, noun, or verb) greatly increases the complexity of any grammar checker. A grammar checker will find each sentence in a text, look up each word in the dictionary, and then attempt to parse the sentence into a form that matches a grammar. Using various rules, the program can then detect various errors, such as agreement in tense, number, word order, and so on. It is also possible to detect some stylistic problems with the text. For example, some popular style guides such as The Elements of Style deprecate excessive use of the passive voice. Grammar checkers may attempt to identify passive sentences and suggest an active-voice alternative. The software elements required for grammar checking are closely related to some of the development issues that need to be addressed for speech recognition software. In voice recognition, parsing can be used to help predict which word is most likely intended, based on part of speech and position in the sentence. In grammar checking, the parsing is used to detect words that fail to follow accepted grammar usage. Recently, research has focused on developing algorithms which can recognize grammar errors based on the context of the surrounding words. == Criticism == Grammar checkers are considered a type of foreign language writing aid which non-native speakers can use to proofread their writings as such programs endeavor to identify syntactical errors. However, as with other computerized writing aids such as spell checkers, popular grammar checkers are often criticized when they fail to spot errors and incorrectly flag correct text as erroneous. The linguist Geoffrey K. Pullum argued in 2007 that they were generally so inaccurate as to do more harm than good: "for the most part, accepting the advice of a computer grammar checker on your prose will make it much worse, sometimes hilariously incoherent."
Client-side persistent data
Client-side persistent data or CSPD is a term used in computing for storing data required by web applications to complete internet tasks on the client-side as needed rather than exclusively on the server. As a framework it is one solution to the needs of Occasionally connected computing or OCC. A major challenge for HTTP as a stateless protocol has been asynchronous tasks. The AJAX pattern using XMLHttpRequest was first introduced by Microsoft in the context of the Outlook e-mail product. The first CSPD were the 'cookies' introduced by the Netscape Navigator. ActiveX components which have entries in the Windows registry can also be viewed as a form of client-side persistence.
Curve (tonality)
In image editing, a curve is a remapping of image tonality, specified as a function from input level to output level, used as a way to emphasize colours or other elements in a picture. Curves can usually be applied to all channels together in an image, or to each channel individually. Applying a curve to all channels typically changes the brightness in part of the spectrum. Light parts of a picture can be easily made lighter and dark parts darker to increase contrast. Applying a curve to individual channels can be used to stress a colour. This is particularly efficient in the Lab colour space due to the separation of luminance and chromaticity, but it can also be used in RGB, CMYK or whatever other colour models the software supports.
Wunderlist
Wunderlist is a discontinued cloud-based task management application. It allowed users to create lists to manage their tasks from a smartphone, tablet, computer and smartwatch. Wunderlist was free; additional collaboration features were available in a paid version known as Wunderlist Pro, released April 2013. Wunderlist was created in 2011 by Berlin-based startup 6Wunderkinder (Engl.: 6Prodigies). The company was acquired by Microsoft in June 2015, at which time the app had over 13 million users. In April 2017, Microsoft announced that Wunderlist would eventually be discontinued in favor of Microsoft To Do, a new multi-platform app developed by the Wunderlist team that has direct integration with the company's Office 365 service. On December 6, 2019, Microsoft announced that it would shut down Wunderlist on May 6, 2020. After this date, the application would no longer sync but users could still import their content into Microsoft To Do. == History == In 2009, Wunderlist's CEO Christian Reber called on the social network platform XING for business partners to create a new to-do app. Frank Thelen responded and together Reber and Thelen developed first concepts for Wunderlist. The necessary seed funding was granted by High-Tech Gründerfonds and e42 GmbH. The first version of Wunderlist was launched on November 9, 2010. Initially, the program was created for desktop PCs and platforms such as Windows, Linux and Mac OS X. In December 2011, the app received approval for the iPhone. Subsequently, the developers released a version prepared for the iPad with the name Wunderlist HD. In September 2012, the developers announced a shutdown of their service Wunderkit. Instead they wanted to focus on creating a new version of Wunderlist, which was later on released in December 2012 under the name Wunderlist 2. In September 2013, the company announced it had over 5 million users. In July 2014, a new major update was released under the name of Wunderlist 3, with a new real-time sync architecture. Wunderlist reached 10 million users in December 2014. On June 1, 2015, it was announced that Microsoft had acquired 6Wunderkinder, makers of Wunderlist, for between US$100 million and US$200 million (~$258 million in 2024). Following its acquisition of the app, Microsoft announced in April 2017 a preview of To-Do, a multi-platform task management app developed by the Wunderlist team that was intended to eventually replace Wunderlist and incorporate most of its features. As of January 2019, To-Do had not yet reached feature parity with Wunderlist, with its team citing that the service had to be completely re-written to use Microsoft Azure instead of Amazon Web Services. Frustrated by the perceived lack of roadmap, in September 2019, Reber began to publicly ask Microsoft-related accounts on Twitter whether he could buy Wunderlist back. Shortly afterward, however, Microsoft unveiled updates to To-Do that make it more closely resemble Wunderlist. In December 2019, Microsoft announced that it would fully shut down Wunderlist as of May 6, 2020. The team responsible for creating Wunderlist, led by co-founder Christian Reber, created that Superlist app in early 2024. == Finances == In its initial round of funding, 100,000 euro was invested in 6Wunderkinder by Frank Thelen and others. In December 2010, High-Tech Gründerfonds invested 500,000 euro (approximately US$660,000) in the company. T-Venture also invested an undisclosed amount in the startup. In its Series A round of funding in November 2011, Atomico invested $4.2 million (~$5.76 million in 2024) while High-Tech Gründerfonds invested an undisclosed additional amount. In May 2012, High-Tech Gründerfonds sold off its stake in 6Wunderkinder to Earlybird Venture Capital. In November 2013, $19 million (~$25.2 million in 2024) was raised in a Series B round led by Sequoia Capital with participation from Earlybird and Atomico. == Awards == In 2013, Wunderlist for Mac was named App of the Year. Wunderlist was selected as a Google Play Top Developer in 2013. In 2014, Wunderlist won the "Golden Mi" award from Xiaomi, and also named as one of its Best Apps of 2014 was given a "Google Play Editor's Choice" award, and was named in Google Play's Best Apps of 2014 as well as Apple's Best of 2014.
Real-time computer graphics
Real-time computer graphics or real-time rendering is the sub-field of computer graphics focused on producing and analyzing images in real time. The term can refer to anything from rendering an application's graphical user interface (GUI) to real-time image analysis, but is most often used in reference to interactive 3D computer graphics, typically using a graphics processing unit (GPU). One example of this concept is a video game that rapidly renders changing 3D environments to produce an illusion of motion. Computers have been capable of generating 2D images such as simple lines, images and polygons in real time since their invention. However, quickly rendering detailed 3D objects is a daunting task for traditional Von Neumann architecture-based systems. An early workaround to this problem was the use of sprites, 2D images that could imitate 3D graphics. Different techniques for rendering now exist, such as ray-tracing and rasterization. Using these techniques and advanced hardware, computers can now render images quickly enough to create the illusion of motion while simultaneously accepting user input. This means that the user can respond to rendered images in real time, producing an interactive experience. == Principles of real-time 3D computer graphics == The goal of computer graphics is to generate computer-generated images, or frames, using certain desired metrics. One such metric is the number of frames generated in a given second. Real-time computer graphics systems differ from traditional (i.e., non-real-time) rendering systems in that non-real-time graphics typically rely on ray tracing. In this process, millions or billions of rays are traced from the camera to the world for detailed rendering—this expensive operation can take hours or days to render a single frame. Real-time graphics systems must render each image in less than 1/30th of a second. Ray tracing is far too slow for these systems; instead, they employ the technique of z-buffer triangle rasterization. In this technique, every object is decomposed into individual primitives, usually triangles. Each triangle gets positioned, rotated and scaled on the screen, and rasterizer hardware (or a software emulator) generates pixels inside each triangle. These triangles are then decomposed into atomic units called fragments that are suitable for displaying on a display screen. The fragments are drawn on the screen using a color that is computed in several steps. For example, a texture can be used to "paint" a triangle based on a stored image, and then shadow mapping can alter that triangle's colors based on line-of-sight to light sources. === Video game graphics === Real-time graphics optimizes image quality subject to time and hardware constraints. GPUs and other advances increased the image quality that real-time graphics can produce. GPUs are capable of handling millions of triangles per frame, and modern DirectX/OpenGL class hardware is capable of generating complex effects, such as shadow volumes, motion blurring, and triangle generation, in real-time. The advancement of real-time graphics is evidenced in the progressive improvements between actual gameplay graphics and the pre-rendered cutscenes traditionally found in video games. Cutscenes are typically rendered in real-time—and may be interactive. Although the gap in quality between real-time graphics and traditional off-line graphics is narrowing, offline rendering remains much more accurate. === Advantages === Real-time graphics are typically employed when interactivity (e.g., player feedback) is crucial. When real-time graphics are used in films, the director has complete control of what has to be drawn on each frame, which can sometimes involve lengthy decision-making. Teams of people are typically involved in the making of these decisions. In real-time computer graphics, the user typically operates an input device to influence what is about to be drawn on the display. For example, when the user wants to move a character on the screen, the system updates the character's position before drawing the next frame. Usually, the display's response-time is far slower than the input device—this is justified by the immense difference between the (fast) response time of a human being's motion and the (slow) perspective speed of the human visual system. This difference has other effects too: because input devices must be very fast to keep up with human motion response, advancements in input devices (e.g., the current Wii remote) typically take much longer to achieve than comparable advancements in display devices. Another important factor controlling real-time computer graphics is the combination of physics and animation. These techniques largely dictate what is to be drawn on the screen—especially where to draw objects in the scene. These techniques help realistically imitate real world behavior (the temporal dimension, not the spatial dimensions), adding to the computer graphics' degree of realism. Real-time previewing with graphics software, especially when adjusting lighting effects, can increase work speed. Some parameter adjustments in fractal generating software may be made while viewing changes to the image in real time. == Rendering pipeline == The graphics rendering pipeline ("rendering pipeline" or simply "pipeline") is the foundation of real-time graphics. Its main function is to render a two-dimensional image in relation to a virtual camera, three-dimensional objects (an object that has width, length, and depth), light sources, lighting models, textures and more. === Architecture === The architecture of the real-time rendering pipeline can be divided into conceptual stages: application, geometry and rasterization. === Application stage === The application stage is responsible for generating "scenes", or 3D settings that are drawn to a 2D display. This stage is implemented in software that developers optimize for performance. This stage may perform processing such as collision detection, speed-up techniques, animation and force feedback, in addition to handling user input. Collision detection is an example of an operation that would be performed in the application stage. Collision detection uses algorithms to detect and respond to collisions between (virtual) objects. For example, the application may calculate new positions for the colliding objects and provide feedback via a force feedback device such as a vibrating game controller. The application stage also prepares graphics data for the next stage. This includes texture animation, animation of 3D models, animation via transforms, and geometry morphing. Finally, it produces primitives (points, lines, and triangles) based on scene information and feeds those primitives into the geometry stage of the pipeline. === Geometry stage === The geometry stage manipulates polygons and vertices to compute what to draw, how to draw it and where to draw it. Usually, these operations are performed by specialized hardware or GPUs. Variations across graphics hardware mean that the "geometry stage" may actually be implemented as several consecutive stages. ==== Model and view transformation ==== Before the final model is shown on the output device, the model is transformed onto multiple spaces or coordinate systems. Transformations move and manipulate objects by altering their vertices. Transformation is the general term for the four specific ways that manipulate the shape or position of a point, line or shape. ==== Lighting ==== In order to give the model a more realistic appearance, one or more light sources are usually established during transformation. However, this stage cannot be reached without first transforming the 3D scene into view space. In view space, the observer (camera) is typically placed at the origin. If using a right-handed coordinate system (which is considered standard), the observer looks in the direction of the negative z-axis with the y-axis pointing upwards and the x-axis pointing to the right. ==== Projection ==== Projection is a transformation used to represent a 3D model in a 2D space. The two main types of projection are orthographic projection (also called parallel) and perspective projection. The main characteristic of an orthographic projection is that parallel lines remain parallel after the transformation. Perspective projection utilizes the concept that if the distance between the observer and model increases, the model appears smaller than before. Essentially, perspective projection mimics human sight. ==== Clipping ==== Clipping is the process of removing primitives that are outside of the view box in order to facilitate the rasterizer stage. Once those primitives are removed, the primitives that remain will be drawn into new triangles that reach the next stage. ==== Screen mapping ==== The purpose of screen mapping is to find out the coordinates of the primitives during the clipping stage. ==== Rasterizer stage ==== The rasterizer
Focus recovery based on the linear canonical transform
For digital image processing, the Focus recovery from a defocused image is an ill-posed problem since it loses the component of high frequency. Most of the methods for focus recovery are based on depth estimation theory. The Linear canonical transform (LCT) gives a scalable kernel to fit many well-known optical effects. Using LCTs to approximate an optical system for imaging and inverting this system, theoretically permits recovery of a defocused image. == Depth of field and perceptual focus == In photography, depth of field (DOF) means an effective focal length. It is usually used for stressing an object and deemphasizing the background (and/or the foreground). The important measure related to DOF is the lens aperture. Decreasing the diameter of aperture increases focus and lowers resolution and vice versa. == The Huygens–Fresnel principle and DOF == The Huygens–Fresnel principle describes diffraction of wave propagation between two fields. It belongs to Fourier optics rather than geometric optics. The disturbance of diffraction depends on two circumstance parameters, the size of aperture and the interfiled distance. Consider a source field and a destination field, field 1 and field 0, respectively. P1(x1,y1) is the position in the source field, P0(x0,y0) is the position in the destination field. The Huygens–Fresnel principle gives the diffraction formula for two fields U(x0,y0), U(x1,y1) as following: U ( x 0 , y 0 ) = 1 j λ ∫ ∫ U ( x 1 , y 1 ) e j k r 01 r 01 cos θ d x 1 d y 1 {\displaystyle \mathbf {U} (x_{0},y_{0})={\frac {1}{j\lambda }}\int \!\int \mathbf {U} (x_{1},y_{1}){\frac {e^{jkr_{01}}}{r_{01}}}\cos \theta dx_{1}dy_{1}} where θ denotes the angle between r 01 {\displaystyle r_{01}} and z {\displaystyle z} . Replace cos θ by r 01 z {\displaystyle {\frac {r_{01}}{z}}} and r 01 {\displaystyle r_{01}} by [ ( x 0 − x 1 ) 2 + ( y 0 − y 1 ) 2 + z 2 ] 1 / 2 {\displaystyle [(x_{0}-x_{1})^{2}+(y_{0}-y_{1})^{2}+z^{2}]^{1/2}} we get U ( x 0 , y 0 ) = 1 j λ z ∫ ∫ U ( x 1 , y 1 ) exp ( j k z [ 1 + ( x 0 − x 1 z ) 2 + ( y 0 − y 1 z ) 2 ] 1 / 2 ) 1 + ( x 0 − x 1 z ) 2 + ( y 0 − y 1 z ) 2 d x 1 d y 1 {\displaystyle \mathbf {U} (x_{0},y_{0})={\frac {1}{j\lambda z}}\int \!\int \mathbf {U} (x_{1},y_{1}){\frac {\exp(jkz[1+({\frac {x_{0}-x_{1}}{z}})^{2}+({\frac {y_{0}-y_{1}}{z}})^{2}]^{1/2})}{1+({\frac {x_{0}-x_{1}}{z}})^{2}+({\frac {y_{0}-y_{1}}{z}})^{2}}}dx_{1}dy_{1}} The further distance z or the smaller aperture (x1,y1) causes a greater diffraction. A larger DOF can lead to a more effective focused wave distribution. This seems to be a conflict. Here are the notations: Diffraction In a real imaging environment, the depths of objects comparing to the aperture are usually not enough to lead to serious diffraction. However, a long enough depth of the object can truly blurs the image. Effective Focus Small aperture, small blurring radius, few wave information. Loses details in comparing to a large aperture. In conclusion, diffraction explains a micro behavior whereas DOF shows a macro behavior. Both of them are related to aperture size. == Linear canonical transform == As the meaning of "canonical", the linear canonical transform (LCT) is a scalable transform that connects to many important kernels such as the Fresnel transform, Fraunhofer transform and the fractional Fourier transform. It can be easily controlled by its four parameters, a, b, c, d (3 degrees of freedom). The definition: L M ( f ( u ) ) = ∫ L M ( u , u ′ ) f ( u ′ ) d u ′ {\displaystyle L_{M}(f(u))=\int L_{M}(u,u')f(u')du'} where L M ( u , u ′ ) = { 1 b e − j π / 4 e [ j π ( d b u 2 ) − 2 1 b u u ′ + a b u ′ 2 ] , if b ≠ 0 d e j 2 c d u 2 δ ( u ′ − d u ) , if b = 0 {\displaystyle L_{M}(u,u')={\begin{cases}{\sqrt {\frac {1}{b}}}e^{-j\pi /4}e^{[j\pi ({\frac {d}{b}}u^{2})-2{\frac {1}{b}}uu'+{\frac {a}{b}}u'^{2}]},&{\mbox{if }}b\neq 0\\{\sqrt {d}}e^{{\frac {j}{2}}cdu^{2}}\delta (u'-du),&{\mbox{if }}b=0\end{cases}}} Consider a general imaging system with object distance z0, focal length of the thin lens f and an imaging distance z1. The effect of the propagation in freespace acts as nearly a chirp convolution, that is, the formula of diffraction. Besides, the effect of the propagation in thin lens acts as a chirp multiplication. The parameters are all simplified as paraxial approximations while meeting the freespace propagation. It does not consider aperture size. From the properties of the LCT, it is possible to obtain those 4 parameters for this optical system as: [ 1 − z 1 f λ z 0 − λ z 0 z 1 f + λ z 1 − 1 λ f 1 − z 0 f ] {\displaystyle {\begin{bmatrix}1-{\frac {z_{1}}{f}}\quad &\lambda z_{0}-{\frac {\lambda z_{0}z_{1}}{f}}+\lambda z_{1}\\-{\frac {1}{\lambda f}}\quad &1-{\frac {z_{0}}{f}}\end{bmatrix}}} Once the values of z1, z0 and f are known, the LCT can simulate any optical system.