In machine learning, a neural scaling law is an empirical scaling law that describes how neural network performance changes as key factors are scaled up or down. These factors typically include the number of parameters, training dataset size, and training cost. Some models also exhibit performance gains by scaling inference through increased test-time compute (TTC), extending neural scaling laws beyond training to the deployment phase. == Introduction == In general, a deep learning model can be characterized by four parameters: model size, training dataset size, training cost, and the post-training error rate (e.g., the test set error rate). Each of these variables can be defined as a real number, usually written as N , D , C , L {\displaystyle N,D,C,L} (respectively: parameter count, dataset size, computing cost, and loss). A neural scaling law is a theoretical or empirical statistical law between these parameters. There are also other parameters with other scaling laws. === Size of the model === In most cases, the model's size is simply the number of parameters. However, one complication arises with the use of sparse models, such as mixture-of-expert models. With sparse models, during inference, only a fraction of their parameters are used. In comparison, most other kinds of neural networks, such as transformer models, always use all their parameters during inference. === Size of the training dataset === The size of the training dataset is usually quantified by the number of data points within it. Larger training datasets are typically preferred, as they provide a richer and more diverse source of information from which the model can learn. This can lead to improved generalization performance when the model is applied to new, unseen data. However, increasing the size of the training dataset also increases the computational resources and time required for model training. With the "pretrain, then finetune" method used for most large language models, there are two kinds of training dataset: the pretraining dataset and the finetuning dataset. Their sizes have different effects on model performance. Generally, the finetuning dataset is less than 1% the size of pretraining dataset. In some cases, a small amount of high quality data suffices for finetuning, and more data does not necessarily improve performance. Many scaling laws, due to their inherent diminishing returns nature, value data based on a submodular set function which was shown in a paper on this topic. === Cost of training === Training cost is typically measured in terms of time (how long it takes to train the model) and computational resources (how much processing power and memory are required). It is important to note that the cost of training can be significantly reduced with efficient training algorithms, optimized software libraries, and parallel computing on specialized hardware such as GPUs or TPUs. The cost of training a neural network model is a function of several factors, including model size, training dataset size, the training algorithm complexity, and the computational resources available. In particular, doubling the training dataset size does not necessarily double the cost of training, because one may train the model for several times over the same dataset (each being an "epoch"). === Performance === The performance of a neural network model is evaluated based on its ability to accurately predict the output given some input data. Common metrics for evaluating model performance include: Negative log-likelihood per token (logarithm of perplexity) for language modeling; Accuracy, precision, recall, and F1 score for classification tasks; Mean squared error (MSE) or mean absolute error (MAE) for regression tasks; Elo rating in a competition against other models, such as gameplay or preference by a human judge. Performance can be improved by using more data, larger models, different training algorithms, regularizing the model to prevent overfitting, and early stopping using a validation set. When the performance is a number bounded within the range of [ 0 , 1 ] {\displaystyle [0,1]} , such as accuracy, precision, etc., it often scales as a sigmoid function of cost, as seen in the figures. == Examples == === (Hestness, Narang, et al, 2017) === The 2017 paper is a common reference point for neural scaling laws fitted by statistical analysis on experimental data. Previous works before the 2000s, as cited in the paper, were either theoretical or orders of magnitude smaller in scale. Whereas previous works generally found the scaling exponent to scale like L ∝ D − α {\displaystyle L\propto D^{-\alpha }} , with α ∈ { 0.5 , 1 , 2 } {\displaystyle \alpha \in \{0.5,1,2\}} , the paper found that α ∈ [ 0.07 , 0.35 ] {\displaystyle \alpha \in [0.07,0.35]} . Of the factors they varied, only task can change the exponent α {\displaystyle \alpha } . Changing the architecture optimizers, regularizers, and loss functions, would only change the proportionality factor, not the exponent. For example, for the same task, one architecture might have L = 1000 D − 0.3 {\displaystyle L=1000D^{-0.3}} while another might have L = 500 D − 0.3 {\displaystyle L=500D^{-0.3}} . They also found that for a given architecture, the number of parameters necessary to reach lowest levels of loss, given a fixed dataset size, grows like N ∝ D β {\displaystyle N\propto D^{\beta }} for another exponent β {\displaystyle \beta } . They studied machine translation with LSTM ( α ∼ 0.13 {\displaystyle \alpha \sim 0.13} ), generative language modelling with LSTM ( α ∈ [ 0.06 , 0.09 ] , β ≈ 0.7 {\displaystyle \alpha \in [0.06,0.09],\beta \approx 0.7} ), ImageNet classification with ResNet ( α ∈ [ 0.3 , 0.5 ] , β ≈ 0.6 {\displaystyle \alpha \in [0.3,0.5],\beta \approx 0.6} ), and speech recognition with two hybrid (LSTMs complemented by either CNNs or an attention decoder) architectures ( α ≈ 0.3 {\displaystyle \alpha \approx 0.3} ). === (Henighan, Kaplan, et al, 2020) === A 2020 analysis studied statistical relations between C , N , D , L {\displaystyle C,N,D,L} over a wide range of values and found similar scaling laws, over the range of N ∈ [ 10 3 , 10 9 ] {\displaystyle N\in [10^{3},10^{9}]} , C ∈ [ 10 12 , 10 21 ] {\displaystyle C\in [10^{12},10^{21}]} , and over multiple modalities (text, video, image, text to image, etc.). In particular, the scaling laws it found are (Table 1 of ): For each modality, they fixed one of the two C , N {\displaystyle C,N} , and varying the other one ( D {\displaystyle D} is varied along using D = C / 6 N {\displaystyle D=C/6N} ), the achievable test loss satisfies L = L 0 + ( x 0 x ) α {\displaystyle L=L_{0}+\left({\frac {x_{0}}{x}}\right)^{\alpha }} where x {\displaystyle x} is the varied variable, and L 0 , x 0 , α {\displaystyle L_{0},x_{0},\alpha } are parameters to be found by statistical fitting. The parameter α {\displaystyle \alpha } is the most important one. When N {\displaystyle N} is the varied variable, α {\displaystyle \alpha } ranges from 0.037 {\displaystyle 0.037} to 0.24 {\displaystyle 0.24} depending on the model modality. This corresponds to the α = 0.34 {\displaystyle \alpha =0.34} from the Chinchilla scaling paper. When C {\displaystyle C} is the varied variable, α {\displaystyle \alpha } ranges from 0.048 {\displaystyle 0.048} to 0.19 {\displaystyle 0.19} depending on the model modality. This corresponds to the β = 0.28 {\displaystyle \beta =0.28} from the Chinchilla scaling paper. Given fixed computing budget, optimal model parameter count is consistently around N o p t ( C ) = ( C 5 × 10 − 12 petaFLOP-day ) 0.7 = 9.0 × 10 − 7 C 0.7 {\displaystyle N_{opt}(C)=\left({\frac {C}{5\times 10^{-12}{\text{petaFLOP-day}}}}\right)^{0.7}=9.0\times 10^{-7}C^{0.7}} The parameter 9.0 × 10 − 7 {\displaystyle 9.0\times 10^{-7}} varies by a factor of up to 10 for different modalities. The exponent parameter 0.7 {\displaystyle 0.7} varies from 0.64 {\displaystyle 0.64} to 0.75 {\displaystyle 0.75} for different modalities. This exponent corresponds to the ≈ 0.5 {\displaystyle \approx 0.5} from the Chinchilla scaling paper. It's "strongly suggested" (but not statistically checked) that D o p t ( C ) ∝ N o p t ( C ) 0.4 ∝ C 0.28 {\displaystyle D_{opt}(C)\propto N_{opt}(C)^{0.4}\propto C^{0.28}} . This exponent corresponds to the ≈ 0.5 {\displaystyle \approx 0.5} from the Chinchilla scaling paper. The scaling law of L = L 0 + ( C 0 / C ) 0.048 {\displaystyle L=L_{0}+(C_{0}/C)^{0.048}} was confirmed during the training of GPT-3 (Figure 3.1 ). === Chinchilla scaling (Hoffmann, et al, 2022) === One particular scaling law ("Chinchilla scaling") states that, for a large language model (LLM) autoregressively trained for one epoch, with a cosine learning rate schedule, we have: { C = C 0 N D L = A N α + B D β + L 0 {\displaystyle {\begin{cases}C=C_{0}ND\\L={\frac {A}{N^{\alpha }}}+{\frac {B}{D^{\beta }}}+L_{0}\end{cases}}} where the variables are C {\displaystyle C} is the cost o
Super-resolution imaging
Super-resolution imaging (SR) is a class of techniques that improve the resolution of an imaging system. In optical SR the diffraction limit of systems is transcended, while in geometrical SR the resolution of digital imaging sensors is enhanced. In some radar and sonar imaging applications (e.g. magnetic resonance imaging (MRI), high-resolution computed tomography), subspace decomposition-based methods (e.g. MUSIC) and compressed sensing-based algorithms (e.g., SAMV) are employed to achieve SR over standard periodogram algorithm. Super-resolution imaging techniques are used in general image processing and in super-resolution microscopy. == Super-resolution principles == Several concepts are fundamental to super-resolution imaging: Diffraction limit: the capacity of an optical instrument to reproduce the details of an object in an image has limits that are imposed by laws of physics: the diffraction equations in the wave theory of light, or the uncertainty principle for photons in quantum mechanics. Information transfer can never be increased beyond this boundary, but packets outside the limits can be cleverly swapped for (or multiplexed with) some inside it. Super-resolution microscopy does not so much “break” as “circumvent” the diffraction limit. New procedures probing electro-magnetic disturbances at the molecular level (in the so-called near field) remain fully consistent with Maxwell's equations. Spatial frequency domain: A succinct expression of the diffraction limit is given in the spatial frequency domain. In Fourier optics light distributions are expressed as superpositions of a series of grating light patterns in a range of fringe widths - these widths represent the spatial frequencies. It is generally taught that diffraction theory stipulates an upper limit, the cut-off spatial-frequency, beyond which pattern elements fail to be transferred into the optical image, i.e., are not resolved. But in fact what is set by diffraction theory is the width of the passband, not a fixed upper limit. No laws of physics are broken when a spatial frequency band beyond the cut-off spatial frequency is swapped for one inside it: this has long been implemented in dark-field microscopy. Nor are information-theoretical rules broken when superimposing several bands, disentangling them in the received image needs assumptions of object invariance during multiple exposures, i.e., the substitution of one kind of uncertainty for another. Information: When the term super-resolution is used in techniques based on the inference of object details using a statistical treatment of the image within standard resolution limits (for example, averaging multiple exposures), it involves an exchange of one kind of information (extracting signal from noise) for another (the assumption that the target has remained invariant). Recent breakthroughs incorporate quantum-transformer hybrids into super-resolution, such as QUIET‑SR, a 2025 model that employs shifted quantum window attention within a transformer to enhance image detail while respecting diffraction and information-theory limits Similarly, frequency-integrated transformers (e.g., FIT) enrich super-resolution by explicitly combining spatial and frequency-domain information via FFT-based attention, improving reconstruction across scales Resolution and localization: True resolution involves the distinction of whether a target, e.g. a star or a spectral line, is single or double, ordinarily requiring separable peaks in the image. When a target is known to be single, its location can be determined with higher precision than the image width by finding the centroid (center of gravity) of its image light distribution. The word ultra-resolution had been proposed for this process but it did not catch on, and the high-precision localization procedure is typically referred to as super-resolution. == Techniques == === Optical or diffractive super-resolution === Substituting spatial-frequency bands: Though the bandwidth allowable by diffraction is fixed, it can be positioned anywhere in the spatial-frequency spectrum. Dark-field illumination in microscopy is an example. See also aperture synthesis. ==== Multiplexing spatial-frequency bands ==== An image is formed using the normal passband of the optical device. Then, some known light structure (for example, a set of light fringes) is superimposed on the target. The image now contains components resulting from the combination of the target and the superimposed light structure, e.g. moiré fringes, and carries information about target detail which simple unstructured illumination does not. The “superresolved” components, however, need disentangling to be revealed. For an example, see structured illumination (figure to left). ==== Multiple parameter use within traditional diffraction limit ==== If a target has no special polarization or wavelength properties, two polarization states or non-overlapping wavelength regions can be used to encode target details, one in a spatial-frequency band inside the cut-off limit the other beyond it. Both would use normal passband transmission but are then separately decoded to reconstitute target structure with extended resolution. ==== Probing near-field electromagnetic disturbance ==== Super-resolution microscopy is generally discussed within the realm of conventional optical imagery. However, modern technology allows the probing of electromagnetic disturbance within molecular distances of the source, which has superior resolution properties. See also evanescent waves and the development of the new super lens. === Geometrical or image-processing super-resolution === ==== Multi-exposure image noise reduction ==== When an image is degraded by noise, the resolution may be improved by averaging multiple exposures. See example on the right. ==== Single-frame deblurring ==== Known defects in a given imaging situation, such as defocus or aberrations, can sometimes be mitigated in whole or in part by suitable spatial-frequency filtering of even a single image. Such procedures all stay within the diffraction-mandated passband, and do not extend it. ==== Sub-pixel image localization ==== The location of a single source can be determined by computing the "center of gravity" (centroid) of the light distribution extending over several adjacent pixels (see figure on the left). Provided that there is enough light, this can be achieved with arbitrary precision, very much better than pixel width of the detecting apparatus and the resolution limit for the decision of whether the source is single or double. This technique, which requires the presupposition that all the light comes from a single source, is at the basis of what has become known as super-resolution microscopy, e.g. stochastic optical reconstruction microscopy (STORM), where fluorescent probes attached to molecules give nanoscale distance information. It is also the mechanism underlying visual hyperacuity. ==== Bayesian induction beyond traditional diffraction limit ==== Some object features, though beyond the diffraction limit, may be known to be associated with other object features that are within the limits and hence contained in the image. Then conclusions can be drawn, using statistical methods, from the available image data about the presence of the full object. The classical example is Toraldo di Francia's proposition of judging whether an image is that of a single or double star by determining whether its width exceeds the spread from a single star. This can be achieved at separations well below the classical resolution bounds, and requires the prior limitation to the choice "single or double?" The approach can take the form of extrapolating the image in the frequency domain, by assuming that the object is an analytic function, and that we can exactly know the function values in some interval. This method is severely limited by the ever-present noise in digital imaging systems, but it can work for radar, astronomy, microscopy or magnetic resonance imaging. More recently, a fast single image super-resolution algorithm based on a closed-form solution to ℓ 2 − ℓ 2 {\displaystyle \ell _{2}-\ell _{2}} problems has been proposed and demonstrated to accelerate most of the existing Bayesian super-resolution methods significantly. == Aliasing == Geometrical SR reconstruction algorithms are possible if and only if the input low resolution images have been under-sampled and therefore contain aliasing. Because of this aliasing, the high-frequency content of the desired reconstruction image is embedded in the low-frequency content of each of the observed images. Given a sufficient number of observation images, and if the set of observations vary in their phase (i.e. if the images of the scene are shifted by a sub-pixel amount), then the phase information can be used to separate the aliased high-frequency content from the true low-frequency content, and the full-resolution image can be accurate
WebPlus
Serif WebPlus was a website design program for Microsoft Windows, developed by the software company, Serif. It allows users to design, create and upload their website onto the internet without any knowledge of HTML or other web technologies. Much like Microsoft Word, WebPlus uses WYSIWYG drag and drop editing to add and position text, images and links as they would appear on the finished web page. Once a user has designed their site, WebPlus can preview the site in a web browser before uploading the site using the in-built FTP. The software comes with a variety of pre-designed sample websites containing Filler text like Lorem ipsum, which can be used as a template for quickly designing a site. It also provides drawing tools for creating and editing buttons and web graphics. == Free WebPlus Starter Edition == Previously Serif had made available feature limited Starter Editions of their software, based on older versions, which could be obtained and used free of charge. For WebPlus the final free edition was based on version X5 and this was released in September 2012. This continued to be available from Serif's server until it was withdrawn around March 2016. WebPlus was then only available as a paid-for version X8. == Program Withdrawal == In March 2016, Serif announced that WebPlus X8 would be the final version, and that there were no current plans to design an application to replace it. Sales of WebPlus X8 by Serif were ended around December 2016. In early 2018, Serif announced that Serif Web Resources, hosted on Serif servers and required to implement some advanced web-site functionality in WebPlus created sites, would no longer work after 31 August 2018. In 2018, Serif also shutdown the servers that generated the "Plus" software registration numbers on-line from the product version and the individual generated installation number. Serif revealed the alternative was to use a universal master registration number, which is 881887. This is known to work with post 2003 Serif "Plus" software (e.g. verified to work with PagePlus v5.02). However, later Serif "Plus" software still registers itself automatically if within a certain recent period of a previous Serif software registration on the same PC. == Supported platforms == WebPlus was developed for Microsoft Windows "Win32" graphical desktop interface and is fully compatible with Windows XP, Windows Vista (32/64bit), Windows 7 (32/64bit) and Windows 8. == Features == Web hosting to upload websites to the internet with the address www.sitename.webplus.net and email [email protected]. E-Commerce tool to create online stores with providers such as PayPal. Form wizard generates online forms to collect information from website visitors. Add blogs, forums, hit counters, online polls and content management systems to websites using Smart Objects. Google Maps tool embeds maps and optional navigation markers within a website. Site navigation bars adopt a website's structure providing a tool for navigating around the website. Photo gallery groups a collection of images together and displays them as an animated slideshow. Search engine optimization (SEO) tools optimise a websites search ranking with the likes of Google, Yahoo! and Bing. Collect website metrics such as page popularity and number of website hits using Google Analytics. WebPlus X5 introduced a button studio for creating button graphics. Restrict access to specific pages on a website with a secure member's area. WebPlus automatically converts images and graphics into a web targeted format, optimising them for fast download. Embed YouTube videos within a web page. Add animated effects to a website with Animated GIFs, Animated Marquees or by importing Flash videos. Stream news and information feeds to a website using RSS and podcasts. Automated Site Checker analyses and corrects potential problems with a website. AdSense tool incorporates Google AdSense advertisements into a website In-built FTP transfers files onto a web server, uploading a website to the internet. In-built Basic Photo Editor the PhotoLab can make automatic adjustments and "Quick Fix's" to photos. From X5, WebPlus offers image editing and filters, through its PhotoLab and also provides a dedicated background-removal tool in the form of Cutout Studio. Display images, Flash videos and web pages using animated Lightboxes. Filter Effects can be applied to the graphical objects, giving convincing, realistic effects such as glass, metallic, plastic and other 2D/3D filters. WebPlus also provides QuickShapes for creating button and web graphics. These predefined shapes can be quickly modified with sliders to adjust certain parameters, for example creating rounded rectangles, etc. Shapes include: rectangles, ellipses, stars, spirals, cogs, petals, etc.
Avizo (software)
Avizo (pronounce: 'a-VEE-zo') is a general-purpose commercial software application for scientific and industrial data visualization and analysis. Avizo is developed by Thermo Fisher Scientific and was originally designed and developed by the Visualization and Data Analysis Group at Zuse Institute Berlin (ZIB) under the name Amira. Avizo was commercially released in November 2007. For the history of its development, see the Wikipedia article about Amira. == Overview == Avizo is a software application which enables users to perform interactive visualization and computation on 3D data sets. The Avizo interface is modelled on the visual programming. Users manipulate data and module components, organized in an interactive graph representation (called Pool), or in a Tree view. Data and modules can be interactively connected together, and controlled with several parameters, creating a visual processing network whose output is displayed in a 3D viewer. With this interface, complex data can be interactively explored and analyzed by applying a controlled sequence of computation and display processes resulting in a meaningful visual representation and associated derived data. == Application areas == Avizo has been designed to support different types of applications and workflows from 2D and 3D image data processing to simulations. It is a versatile and customizable visualization tool used in many fields: Scientific visualization Materials Research Tomography, Microscopy, etc. Nondestructive testing, Industrial Inspection, and Visual Inspection Computer-aided Engineering and simulation data post-processing Porous medium analysis Civil Engineering Seismic Exploration, Reservoir Engineering, Microseismic Monitoring, Borehole Imaging Geology, Digital Rock Physics (DRP), Earth Sciences Archaeology Food technology and agricultural science Physics, Chemistry Climatology, Oceanography, Environmental Studies Astrophysics == Features == Data import: 2D and 3D image stack and volume data: from microscopes (electron, optical), X-ray tomography (CT, micro-/nano-CT, synchrotron), neutron tomography and other acquisition devices (MRI, radiography, GPR) Geometric models (such as point sets, line sets, surfaces, grids) Numerical simulation data (such as Computational fluid dynamics or Finite element analysis data) Molecular data Time series and animations Seismic data Well logs 4D Multivariate Climate Models 2D/3D data visualization: Volume rendering Digital Volume Correlation Visualization of sections, through various slicing and clipping methods Isosurface rendering Polygonal meshes Scalar fields, Vector fields, Tensor representations, Flow visualization (Illuminated Streamlines, Stream Ribbons) Image processing: 2D/3D Alignment of image slices, Image registration Image filtering Mathematical Morphology (erode, dilate, open, close, tophat) Watershed Transform, Distance Transform Image segmentation 3D models reconstruction: Polygonal surface generation from segmented objects Generation of tetrahedral grids Surface reconstruction from point clouds Skeletonization (reconstruction of dendritic, porous or fracture network) Surface model simplification Quantification and analysis: Measurements and statistics Analysis spreadsheet and charting Material properties computation, based on 3D images: Absolute permeability Thermal conductivity Molecular diffusivity Electrical resistivity/formation factor 3D image-based meshing for CFD and FEA: From 3D imaging modalities (CT, micro-CT, MRI, etc.) Surface and volume meshes generation Export to FEA and CFD solvers for simulation Post-processing for simulation analysis Presentation, automation: MovieMaker, Multiscreen, Video wall, collaboration, and VR support TCL Scripting, C++ extension API Avizo is based on Open Inventor 3D graphics toolkits (FEI Visualization Sciences Group).
Shearlet
In applied mathematical analysis, shearlets are a multiscale framework which allows efficient encoding of anisotropic features in multivariate problem classes. Originally, shearlets were introduced in 2006 for the analysis and sparse approximation of functions f ∈ L 2 ( R 2 ) {\displaystyle f\in L^{2}(\mathbb {R} ^{2})} . They are a natural extension of wavelets, to accommodate the fact that multivariate functions are typically governed by anisotropic features such as edges in images, since wavelets, as isotropic objects, are not capable of capturing such phenomena. Shearlets are constructed by parabolic scaling, shearing, and translation applied to a few generating functions. At fine scales, they are essentially supported within skinny and directional ridges following the parabolic scaling law, which reads length² ≈ width. Similar to wavelets, shearlets arise from the affine group and allow a unified treatment of the continuum and digital situation leading to faithful implementations. Although they do not constitute an orthonormal basis for L 2 ( R 2 ) {\displaystyle L^{2}(\mathbb {R} ^{2})} , they still form a frame allowing stable expansions of arbitrary functions f ∈ L 2 ( R 2 ) {\displaystyle f\in L^{2}(\mathbb {R} ^{2})} . One of the most important properties of shearlets is their ability to provide optimally sparse approximations (in the sense of optimality in ) for cartoon-like functions f {\displaystyle f} . In imaging sciences, cartoon-like functions serve as a model for anisotropic features and are compactly supported in [ 0 , 1 ] 2 {\displaystyle [0,1]^{2}} while being C 2 {\displaystyle C^{2}} apart from a closed piecewise C 2 {\displaystyle C^{2}} singularity curve with bounded curvature. The decay rate of the L 2 {\displaystyle L^{2}} -error of the N {\displaystyle N} -term shearlet approximation obtained by taking the N {\displaystyle N} largest coefficients from the shearlet expansion is in fact optimal up to a log-factor: ‖ f − f N ‖ L 2 2 ≤ C N − 2 ( log N ) 3 , N → ∞ , {\displaystyle \|f-f_{N}\|_{L^{2}}^{2}\leq CN^{-2}(\log N)^{3},\quad N\to \infty ,} where the constant C {\displaystyle C} depends only on the maximum curvature of the singularity curve and the maximum magnitudes of f {\displaystyle f} , f ′ {\displaystyle f'} and f ″ . {\displaystyle f''.} This approximation rate significantly improves the best N {\displaystyle N} -term approximation rate of wavelets providing only O ( N − 1 ) {\displaystyle O(N^{-1})} for such class of functions. Shearlets are to date the only directional representation system that provides sparse approximation of anisotropic features while providing a unified treatment of the continuum and digital realm that allows faithful implementation. Extensions of shearlet systems to L 2 ( R d ) , d ≥ 2 {\displaystyle L^{2}(\mathbb {R} ^{d}),d\geq 2} are also available. A comprehensive presentation of the theory and applications of shearlets can be found in. == Definition == === Continuous shearlet systems === The construction of continuous shearlet systems is based on parabolic scaling matrices A a = [ a 0 0 a 1 / 2 ] , a > 0 {\displaystyle A_{a}={\begin{bmatrix}a&0\\0&a^{1/2}\end{bmatrix}},\quad a>0} as a means to change the resolution, on shear matrices S s = [ 1 s 0 1 ] , s ∈ R {\displaystyle S_{s}={\begin{bmatrix}1&s\\0&1\end{bmatrix}},\quad s\in \mathbb {R} } as a means to change the orientation, and finally on translations to change the positioning. In comparison to curvelets, shearlets use shearings instead of rotations, the advantage being that the shear operator S s {\displaystyle S_{s}} leaves the integer lattice invariant in case s ∈ Z {\displaystyle s\in \mathbb {Z} } , i.e., S s Z 2 ⊆ Z 2 . {\displaystyle S_{s}\mathbb {Z} ^{2}\subseteq \mathbb {Z} ^{2}.} This indeed allows a unified treatment of the continuum and digital realm, thereby guaranteeing a faithful digital implementation. For ψ ∈ L 2 ( R 2 ) {\displaystyle \psi \in L^{2}(\mathbb {R} ^{2})} the continuous shearlet system generated by ψ {\displaystyle \psi } is then defined as SH c o n t ( ψ ) = { ψ a , s , t = a 3 / 4 ψ ( S s A a ( ⋅ − t ) ) ∣ a > 0 , s ∈ R , t ∈ R 2 } , {\displaystyle \operatorname {SH} _{\mathrm {cont} }(\psi )=\{\psi _{a,s,t}=a^{3/4}\psi (S_{s}A_{a}(\cdot -t))\mid a>0,s\in \mathbb {R} ,t\in \mathbb {R} ^{2}\},} and the corresponding continuous shearlet transform is given by the map f ↦ S H ψ f ( a , s , t ) = ⟨ f , ψ a , s , t ⟩ , f ∈ L 2 ( R 2 ) , ( a , s , t ) ∈ R > 0 × R × R 2 . {\displaystyle f\mapsto {\mathcal {SH}}_{\psi }f(a,s,t)=\langle f,\psi _{a,s,t}\rangle ,\quad f\in L^{2}(\mathbb {R} ^{2}),\quad (a,s,t)\in \mathbb {R} _{>0}\times \mathbb {R} \times \mathbb {R} ^{2}.} === Discrete shearlet systems === A discrete version of shearlet systems can be directly obtained from SH c o n t ( ψ ) {\displaystyle \operatorname {SH} _{\mathrm {cont} }(\psi )} by discretizing the parameter set R > 0 × R × R 2 . {\displaystyle \mathbb {R} _{>0}\times \mathbb {R} \times \mathbb {R} ^{2}.} There are numerous approaches for this but the most popular one is given by { ( 2 j , k , A 2 j − 1 S k − 1 m ) ∣ j ∈ Z , k ∈ Z , m ∈ Z 2 } ⊆ R > 0 × R × R 2 . {\displaystyle \{(2^{j},k,A_{2^{j}}^{-1}S_{k}^{-1}m)\mid j\in \mathbb {Z} ,k\in \mathbb {Z} ,m\in \mathbb {Z} ^{2}\}\subseteq \mathbb {R} _{>0}\times \mathbb {R} \times \mathbb {R} ^{2}.} From this, the discrete shearlet system associated with the shearlet generator ψ {\displaystyle \psi } is defined by SH ( ψ ) = { ψ j , k , m = 2 3 j / 4 ψ ( S k A 2 j ⋅ − m ) ∣ j ∈ Z , k ∈ Z , m ∈ Z 2 } , {\displaystyle \operatorname {SH} (\psi )=\{\psi _{j,k,m}=2^{3j/4}\psi (S_{k}A_{2^{j}}\cdot {}-m)\mid j\in \mathbb {Z} ,k\in \mathbb {Z} ,m\in \mathbb {Z} ^{2}\},} and the associated discrete shearlet transform is defined by f ↦ S H ψ f ( j , k , m ) = ⟨ f , ψ j , k , m ⟩ , f ∈ L 2 ( R 2 ) , ( j , k , m ) ∈ Z × Z × Z 2 . {\displaystyle f\mapsto {\mathcal {SH}}_{\psi }f(j,k,m)=\langle f,\psi _{j,k,m}\rangle ,\quad f\in L^{2}(\mathbb {R} ^{2}),\quad (j,k,m)\in \mathbb {Z} \times \mathbb {Z} \times \mathbb {Z} ^{2}.} == Examples == Let ψ 1 ∈ L 2 ( R ) {\displaystyle \psi _{1}\in L^{2}(\mathbb {R} )} be a function satisfying the discrete Calderón condition, i.e., ∑ j ∈ Z | ψ ^ 1 ( 2 − j ξ ) | 2 = 1 , for a.e. ξ ∈ R , {\displaystyle \sum _{j\in \mathbb {Z} }|{\hat {\psi }}_{1}(2^{-j}\xi )|^{2}=1,{\text{for a.e. }}\xi \in \mathbb {R} ,} with ψ ^ 1 ∈ C ∞ ( R ) {\displaystyle {\hat {\psi }}_{1}\in C^{\infty }(\mathbb {R} )} and supp ψ ^ 1 ⊆ [ − 1 2 , − 1 16 ] ∪ [ 1 16 , 1 2 ] , {\displaystyle \operatorname {supp} {\hat {\psi }}_{1}\subseteq [-{\tfrac {1}{2}},-{\tfrac {1}{16}}]\cup [{\tfrac {1}{16}},{\tfrac {1}{2}}],} where ψ ^ 1 {\displaystyle {\hat {\psi }}_{1}} denotes the Fourier transform of ψ 1 . {\displaystyle \psi _{1}.} For instance, one can choose ψ 1 {\displaystyle \psi _{1}} to be a Meyer wavelet. Furthermore, let ψ 2 ∈ L 2 ( R ) {\displaystyle \psi _{2}\in L^{2}(\mathbb {R} )} be such that ψ ^ 2 ∈ C ∞ ( R ) , {\displaystyle {\hat {\psi }}_{2}\in C^{\infty }(\mathbb {R} ),} supp ψ ^ 2 ⊆ [ − 1 , 1 ] {\displaystyle \operatorname {supp} {\hat {\psi }}_{2}\subseteq [-1,1]} and ∑ k = − 1 1 | ψ ^ 2 ( ξ + k ) | 2 = 1 , for a.e. ξ ∈ [ − 1 , 1 ] . {\displaystyle \sum _{k=-1}^{1}|{\hat {\psi }}_{2}(\xi +k)|^{2}=1,{\text{for a.e. }}\xi \in \left[-1,1\right].} One typically chooses ψ ^ 2 {\displaystyle {\hat {\psi }}_{2}} to be a smooth bump function. Then ψ ∈ L 2 ( R 2 ) {\displaystyle \psi \in L^{2}(\mathbb {R} ^{2})} given by ψ ^ ( ξ ) = ψ ^ 1 ( ξ 1 ) ψ ^ 2 ( ξ 2 ξ 1 ) , ξ = ( ξ 1 , ξ 2 ) ∈ R 2 , {\displaystyle {\hat {\psi }}(\xi )={\hat {\psi }}_{1}(\xi _{1}){\hat {\psi }}_{2}\left({\tfrac {\xi _{2}}{\xi _{1}}}\right),\quad \xi =(\xi _{1},\xi _{2})\in \mathbb {R} ^{2},} is called a classical shearlet. It can be shown that the corresponding discrete shearlet system SH ( ψ ) {\displaystyle \operatorname {SH} (\psi )} constitutes a Parseval frame for L 2 ( R 2 ) {\displaystyle L^{2}(\mathbb {R} ^{2})} consisting of bandlimited functions. Another example are compactly supported shearlet systems, where a compactly supported function ψ ∈ L 2 ( R 2 ) {\displaystyle \psi \in L^{2}(\mathbb {R} ^{2})} can be chosen so that SH ( ψ ) {\displaystyle \operatorname {SH} (\psi )} forms a frame for L 2 ( R 2 ) {\displaystyle L^{2}(\mathbb {R} ^{2})} . In this case, all shearlet elements in SH ( ψ ) {\displaystyle \operatorname {SH} (\psi )} are compactly supported providing superior spatial localization compared to the classical shearlets, which are bandlimited. Although a compactly supported shearlet system does not generally form a Parseval frame, any function f ∈ L 2 ( R 2 ) {\displaystyle f\in L^{2}(\mathbb {R} ^{2})} can be represented by the shearlet expansion due to its frame property. == Cone-adapted shearlets == One drawback of shearlets defined as above is the directional bias of shearlet elements associated with large shearing parameters. This effect is already r
MetaMask
MetaMask is a software cryptocurrency wallet developed by ConsenSys for interacting with the Ethereum blockchain and other EVM-compatible networks. It enables users to manage Ethereum accounts and connect to decentralized applications (dApps) via a browser extension or mobile app. As of early 2026, MetaMask reports over 100 million users worldwide. == Overview == MetaMask allows users to store and manage private keys, send and receive Ethereum-based cryptocurrencies and tokens (including ERC-20 and ERC-721 standards), broadcast transactions, and interact with dApps. dApps connect to the wallet via JavaScript interfaces, prompting users to approve signatures or transactions. The wallet features MetaMask Swaps, an in-app token swap aggregator sourcing liquidity from multiple decentralized exchanges (DEXs), with a service fee of 0.875%. In 2025, MetaMask introduced the MetaMask Rewards program (initially mobile-only), where users earn points for activities such as swaps, bridging, and referrals. Season 1 (October 2025 – January 2026) distributed over $30 million in Linea tokens and other perks to participants. == History == MetaMask launched in 2016 as open-source software under the MIT license. It initially supported browser extensions for Chrome and Firefox. Mobile versions were in closed beta from 2019 and publicly released for iOS and Android in September 2020. In August 2020, the license changed to a custom proprietary one. MetaMask Swaps launched on desktop in October 2020 and on mobile in March 2021. The Rewards program launched in late 2025 with Linea integration. == Criticism == MetaMask has faced criticism over privacy, including default analytics settings that share some user data (which can be disabled). Its reliance on Infura (acquired by ConsenSys in 2019) has raised concerns about centralization in Ethereum infrastructure. The wallet regularly issues warnings about phishing scams and fake airdrops impersonating MetaMask.
Audio-visual speech recognition
Audio visual speech recognition (AVSR) is a technique that uses image processing capabilities in lip reading to aid speech recognition systems in recognizing indeterministic phones or giving preponderance among near probability decisions. Each system of lip reading and speech recognition works separately, then their results are mixed at the stage of feature fusion. As the name suggests, it has two parts. First one is the audio part and second one is the visual part. In audio part we use features like log mel spectrogram, mfcc etc. from the raw audio samples and we build a model to get feature vector out of it . For visual part generally we use some variant of convolutional neural network to compress the image to a feature vector after that we concatenate these two vectors (audio and visual ) and try to predict the target object.