The charge-based formulation of the boundary element method (BEM) is a dimensionality reduction numerical technique that is used to model quasistatic electromagnetic phenomena in highly complex conducting media (targeting, e.g., the human brain) with a very large (up to approximately 1 billion) number of unknowns. The charge-based BEM solves an integral equation of the potential theory written in terms of the induced surface charge density. This formulation is naturally combined with fast multipole method (FMM) acceleration, and the entire method is known as charge-based BEM-FMM. The combination of BEM and FMM is a common technique in different areas of computational electromagnetics and, in the context of bioelectromagnetism, it provides improvements over the finite element method. == Historical development == Along with more common electric potential-based BEM, the quasistatic charge-based BEM, derived in terms of the single-layer (charge) density, for a single-compartment medium has been known in the potential theory since the beginning of the 20th century. For multi-compartment conducting media, the surface charge density formulation first appeared in discretized form (for faceted interfaces) in the 1964 paper by Gelernter and Swihart. A subsequent continuous form, including time-dependent and dielectric effects, appeared in the 1967 paper by Barnard, Duck, and Lynn. The charge-based BEM has also been formulated for conducting, dielectric, and magnetic media, and used in different applications. In 2009, Greengard et al. successfully applied the charge-based BEM with fast multipole acceleration to molecular electrostatics of dielectrics. A similar approach to realistic modeling of the human brain with multiple conducting compartments was first described by Makarov et al. in 2018. Along with this, the BEM-based multilevel fast multipole method has been widely used in radar and antenna studies at microwave frequencies as well as in acoustics. == Physical background - surface charges in biological media == The charge-based BEM is based on the concept of an impressed (or primary) electric field E i {\displaystyle \mathbf {E} ^{i}} and a secondary electric field E s {\displaystyle \mathbf {E} ^{s}} . The impressed field is usually known a priori or is trivial to find. For the human brain, the impressed electric field can be classified as one of the following: A conservative field E i {\displaystyle \mathbf {E} ^{i}} derived from an impressed density of EEG or MEG current sources in a homogeneous infinite medium with the conductivity σ {\displaystyle \sigma } at the source location; An instantaneous solenoidal field E i {\displaystyle \mathbf {E} ^{i}} of an induction coil obtained from Faraday's law of induction in a homogeneous infinite medium (air), when transcranial magnetic stimulation (TMS) problems are concerned; A surface field E i {\displaystyle \mathbf {E} ^{i}} derived from an impressed surface current density J i = σ E i {\displaystyle \mathbf {J} ^{i}=\sigma \mathbf {E} ^{i}} of current electrodes injecting electric current at a boundary of a compartment with conductivity σ {\displaystyle \sigma } when transcranial direct-current stimulation (tDCS) or deep brain stimulation (DBS) are concerned; A conservative field E i {\displaystyle \mathbf {E} ^{i}} of charges deposited on voltage electrodes for tDCS or DBS. This specific problem requires a coupled treatment since these charges will depend on the environment; In application to multiscale modeling, a field E i {\displaystyle \mathbf {E} ^{i}} obtained from any other macroscopic numerical solution in a small (mesoscale or microscale) spatial domain within the brain. For example, a constant field can be used. When the impressed field is "turned on", free charges located within a conducting volume D immediately begin to redistribute and accumulate at the boundaries (interfaces) of regions of different conductivity in D. A surface charge density ρ ( r ) {\displaystyle \rho (\mathbf {r} )} appears on the conductivity interfaces. This charge density induces a secondary conservative electric field E s {\displaystyle \mathbf {E} ^{s}} following Coulomb's law. One example is a human under a direct current powerline with the known field E i {\displaystyle \mathbf {E} ^{i}} directed down. The superior surface of the human's conducting body will be charged negatively while its inferior portion is charged positively. These surface charges create a secondary electric field that effectively cancels or blocks the primary field everywhere in the body so that no current will flow within the body under DC steady state conditions. Another example is a human head with electrodes attached. At any conductivity interface with a normal vector n {\displaystyle \mathbf {n} } pointing from an "inside" (-) compartment of conductivity σ − {\displaystyle \sigma ^{-}} to an "outside" (+) compartment of conductivity σ + {\displaystyle \sigma ^{+}} , Kirchhoff's current law requires continuity of the normal component of the electric current density. This leads to the interfacial boundary condition in the form for every facet at a triangulated interface. As long as σ ± {\displaystyle \sigma ^{\pm }} are different from each other, the two normal components of the electric field, E ± ⋅ n {\displaystyle \mathbf {E} ^{\pm }\cdot \mathbf {n} } , must also be different. Such a jump across the interface is only possible when a sheet of surface charge exists at that interface. Thus, if an electric current or voltage is applied, the surface charge density follows. The goal of the numerical analysis is to find the unknown surface charge distribution and thus the total electric field E = E i + E s {\displaystyle \mathbf {E} =\mathbf {E} ^{i}+\mathbf {E} ^{s}} (and the total electric potential if required) anywhere in space. == System of equations for surface charges == Below, a derivation is given based on Gauss's law and Coulomb's law. All conductivity interfaces, denoted by S, are discretized into planar triangular facets t m {\displaystyle t_{m}} with centers r m {\displaystyle \mathbf {r} _{m}} . Assume that an m-th facet with the normal vector n m {\displaystyle \mathbf {n} _{m}} and area A m {\displaystyle A_{m}} carries a uniform surface charge density ρ m {\displaystyle \rho _{m}} . If a volumetric tetrahedral mesh were present, the charged facets would belong to tetrahedra with different conductivity values. We first compute the electric field E m + {\displaystyle \mathbf {E} _{m}^{+}} at the point r m + δ n m {\displaystyle \mathbf {r} _{m}+\delta \mathbf {n} _{m}} , for δ → 0 + {\displaystyle \delta \rightarrow 0^{+}} i.e., just outside facet 𝑚 at its center. This field contains three contributions: The continuous impressed electric field E i {\displaystyle \mathbf {E} ^{i}} itself; An electric field of the m-th charged facet itself. Very close to the facet, it can be approximated as the electric field of an infinite sheet of uniform surface charge ρ m {\displaystyle \rho _{m}} . By Gauss's law, it is given by + ρ m / 2 ε 0 ⋅ n m {\displaystyle +\rho _{m}/2\varepsilon _{0}\cdot \mathbf {n} _{m}} where ε 0 {\displaystyle \varepsilon _{0}} is a background electrical permittivity; An electric field generated by all other facets t n {\displaystyle t_{n}} , which we approximate as point charges of charge A n ρ n {\displaystyle A_{n}\rho _{n}} at each center r n {\displaystyle \mathbf {r} _{n}} . A similar treatment holds for the electric field E m − {\displaystyle \mathbf {E} _{m}^{-}} just inside facet 𝑚, but the electric field of the flat sheet of charge changes its sign. Using Coulomb's law to calculate the contribution of facets different from t m {\displaystyle t_{m}} , we find From this equation, we see that the normal component of the electric field indeed undergoes a jump through the charged interface. This is equivalent to a jump relation of the potential theory. As a second step, the two expressions for E m ± {\displaystyle \mathbf {E} _{m}^{\pm }} are substituted into the interfacial boundary condition σ − E m − ⋅ n m = σ + E m + ⋅ n m {\displaystyle \sigma ^{-}\mathbf {E} _{m}^{-}\cdot \mathbf {n} _{m}=\sigma ^{+}\mathbf {E} _{m}^{+}\cdot \mathbf {n} _{m}} , applied to every facet 𝑚. This operation leads to a system of linear equations for unknown charge densities ρ m {\displaystyle \rho _{m}} which solves the problem: where K m = σ − − σ + σ − + σ + {\displaystyle K_{m}={\frac {\sigma ^{-}-\sigma ^{+}}{\sigma ^{-}+\sigma ^{+}}}} is the electric conductivity contrast at the m-th facet. The normalization constant ε 0 {\displaystyle \varepsilon _{0}} will cancel out after the solution is substituted in the expression for E s {\displaystyle \mathbf {E} ^{s}} and becomes redundant. == Application of fast multipole method == For modern characterizations of brain topologies with ever-increasing levels of complexity, the above system of equations for ρ m {\displaystyle \rho _{m}} is very large; it is t
MoltenVK
MoltenVK is a software library which allows Vulkan applications to run on top of Metal on Apple's macOS, iOS, and tvOS operating systems. It is the first software component to be released for the Vulkan Portability Initiative, a project to have a subset of Vulkan run on platforms lacking native Vulkan drivers. There are some limitations compared with a native Vulkan implementation. == History == MoltenVK was first released as a proprietary and commercially licensed product by The Brenwill Workshop on July 27, 2016. On July 31, 2017, Khronos announced the formation of the Vulkan Portability Technical Subgroup. === Open source === On February 26, 2018, Khronos announced that Vulkan became available on macOS and iOS products through the MoltenVK library. Valve announced that Dota 2 will run on macOS using the Vulkan API with the aid of MoltenVK, and that they had made an arrangement with developer The Brenwill Workshop Ltd to release MoltenVK as open-source software under the Apache License version 2.0. On May 30, 2018, Qt was updated with Vulkan for Qt on macOS using MoltenVK. On May 31, 2018, optional Vulkan support for Dota 2 on macOS was released. Benchmarks for the game were available the following day, showing better performance using Vulkan and MoltenVK compared to OpenGL. On July 20, 2018, Wine was updated with Vulkan support on macOS using MoltenVK. On 29 July 2018, the first app using MoltenVK was accepted onto the App Store, after initially being rejected. On 6 August 2018, Google open-sourced Filament, a crossplatform real-time physically based rendering engine with MoltenVK for macOS/iOS. On November 28, 2018, Valve released Artifact, their first Vulkan-only game on macOS using MoltenVK. === Version 1.0 === On 29 January 2019, MoltenVK 1.0.32 was released with early prototype of Vulkan Portability Extensions. RPCS3 and Dolphin emulators were updated with Vulkan support on macOS using MoltenVK. On 13 April 2019, MoltenVK 1.0.34 was released with support for tessellation. On July 30, 2019, MoltenVK 1.0.36 was released targeting Metal 3.0. On July 31, 2020, MoltenVK 1.0.44 was released, adding support for the tvOS platform. On January 23, 2020, MoltenVK was updated to support for some of the new features of Vulkan 1.2, as of Vulkan SDK 1.2.121. === Version 1.1 === On October 1, 2020, MoltenVK 1.1.0 was released, adding full support for Vulkan 1.1, as of Vulkan SDK 1.2.154. On 9 December 2020, MoltenVK 1.1.1 was released, providing support for Vulkan on Apple silicon GPUs and support for the Mac Catalyst platform for porting iOS/iPadOS apps to macOS. === Version 1.2 === On October 18, 2022, MoltenVK 1.2.0 was released, adding full support for Vulkan 1.2 as of Vulkan SDK 1.3.231. In January 2023, MoltenVK 1.2.2 added support for Vulkan as of SDK 1.3.239, while this version of Vulkan SDK fixed some issues with the interconnectivity with Metal API, while version 1.2.3 supported some additional extensions. === Version 1.3 === On May 1, 2025, MoltenVK 1.3 was released with support for Vulkan 1.3. === Version 1.4 === On August 20, 2025, MoltenVK 1.4 was released with support for Vulkan 1.4.
Neural style transfer
Neural style transfer (NST) software algorithms are able to manipulate digital images, or videos, in order to adopt the appearance or visual style of another image. NST algorithms are characterized by their use of deep neural networks for the sake of image transformation. Common uses for NST are the creation of artificial artwork from photographs, for example by transferring the appearance of famous paintings to user-supplied photographs. Several notable mobile apps use NST techniques for this purpose, including DeepArt and Prisma. This method has been used by artists and designers around the globe to develop new artwork based on existent style(s). == History == NST is an example of image stylization, a problem studied for over two decades within the field of non-photorealistic rendering. The first two example-based style transfer algorithms were image analogies and image quilting. Both of these methods were based on patch-based texture synthesis algorithms. Given a training pair of images–a photo and an artwork depicting that photo–a transformation could be learned and then applied to create new artwork from a new photo, by analogy. If no training photo was available, it would need to be produced by processing the input artwork; image quilting did not require this processing step, though it was demonstrated on only one style. NST was first published in the paper "A Neural Algorithm of Artistic Style" by Leon Gatys et al., originally released to ArXiv 2015, and subsequently accepted by the peer-reviewed CVPR conference in 2016. The original paper used a VGG-19 architecture that has been pre-trained to perform object recognition using the ImageNet dataset. In 2017, Google AI introduced a method that allows a single deep convolutional style transfer network to learn multiple styles at the same time. This algorithm permits style interpolation in real-time, even when done on video media. == Mathematics == This section closely follows the original paper. === Overview === The idea of Neural Style Transfer (NST) is to take two images—a content image p → {\displaystyle {\vec {p}}} and a style image a → {\displaystyle {\vec {a}}} —and generate a third image x → {\displaystyle {\vec {x}}} that minimizes a weighted combination of two loss functions: a content loss L content ( p → , x → ) {\displaystyle {\mathcal {L}}_{\text{content }}({\vec {p}},{\vec {x}})} and a style loss L style ( a → , x → ) {\displaystyle {\mathcal {L}}_{\text{style }}({\vec {a}},{\vec {x}})} . The total loss is a linear sum of the two: L NST ( p → , a → , x → ) = α L content ( p → , x → ) + β L style ( a → , x → ) {\displaystyle {\mathcal {L}}_{\text{NST}}({\vec {p}},{\vec {a}},{\vec {x}})=\alpha {\mathcal {L}}_{\text{content}}({\vec {p}},{\vec {x}})+\beta {\mathcal {L}}_{\text{style}}({\vec {a}},{\vec {x}})} By jointly minimizing the content and style losses, NST generates an image that blends the content of the content image with the style of the style image. Both the content loss and the style loss measures the similarity of two images. The content similarity is the weighted sum of squared-differences between the neural activations of a single convolutional neural network (CNN) on two images. The style similarity is the weighted sum of Gram matrices within each layer (see below for details). The original paper used a VGG-19 CNN, but the method works for any CNN. === Symbols === Let x → {\textstyle {\vec {x}}} be an image input to a CNN. Let F l ∈ R N l × M l {\textstyle F^{l}\in \mathbb {R} ^{N_{l}\times M_{l}}} be the matrix of filter responses in layer l {\textstyle l} to the image x → {\textstyle {\vec {x}}} , where: N l {\textstyle N_{l}} is the number of filters in layer l {\textstyle l} ; M l {\textstyle M_{l}} is the height times the width (i.e. number of pixels) of each filter in layer l {\textstyle l} ; F i j l ( x → ) {\textstyle F_{ij}^{l}({\vec {x}})} is the activation of the i th {\textstyle i^{\text{th}}} filter at position j {\textstyle j} in layer l {\textstyle l} . A given input image x → {\textstyle {\vec {x}}} is encoded in each layer of the CNN by the filter responses to that image, with higher layers encoding more global features, but losing details on local features. === Content loss === Let p → {\textstyle {\vec {p}}} be an original image. Let x → {\textstyle {\vec {x}}} be an image that is generated to match the content of p → {\textstyle {\vec {p}}} . Let P l {\textstyle P^{l}} be the matrix of filter responses in layer l {\textstyle l} to the image p → {\textstyle {\vec {p}}} . The content loss is defined as the squared-error loss between the feature representations of the generated image and the content image at a chosen layer l {\displaystyle l} of a CNN: L content ( p → , x → , l ) = 1 2 ∑ i , j ( A i j l ( x → ) − A i j l ( p → ) ) 2 {\displaystyle {\mathcal {L}}_{\text{content }}({\vec {p}},{\vec {x}},l)={\frac {1}{2}}\sum _{i,j}\left(A_{ij}^{l}({\vec {x}})-A_{ij}^{l}({\vec {p}})\right)^{2}} where A i j l ( x → ) {\displaystyle A_{ij}^{l}({\vec {x}})} and A i j l ( p → ) {\displaystyle A_{ij}^{l}({\vec {p}})} are the activations of the i th {\displaystyle i^{\text{th}}} filter at position j {\displaystyle j} in layer l {\displaystyle l} for the generated and content images, respectively. Minimizing this loss encourages the generated image to have similar content to the content image, as captured by the feature activations in the chosen layer. The total content loss is a linear sum of the content losses of each layer: L content ( p → , x → ) = ∑ l v l L content ( p → , x → , l ) {\displaystyle {\mathcal {L}}_{\text{content }}({\vec {p}},{\vec {x}})=\sum _{l}v_{l}{\mathcal {L}}_{\text{content }}({\vec {p}},{\vec {x}},l)} , where the v l {\displaystyle v_{l}} are positive real numbers chosen as hyperparameters. === Style loss === The style loss is based on the Gram matrices of the generated and style images, which capture the correlations between different filter responses at different layers of the CNN: L style ( a → , x → ) = ∑ l = 0 L w l E l , {\displaystyle {\mathcal {L}}_{\text{style }}({\vec {a}},{\vec {x}})=\sum _{l=0}^{L}w_{l}E_{l},} where E l = 1 4 N l 2 M l 2 ∑ i , j ( G i j l ( x → ) − G i j l ( a → ) ) 2 . {\displaystyle E_{l}={\frac {1}{4N_{l}^{2}M_{l}^{2}}}\sum _{i,j}\left(G_{ij}^{l}({\vec {x}})-G_{ij}^{l}({\vec {a}})\right)^{2}.} Here, G i j l ( x → ) {\displaystyle G_{ij}^{l}({\vec {x}})} and G i j l ( a → ) {\displaystyle G_{ij}^{l}({\vec {a}})} are the entries of the Gram matrices for the generated and style images at layer l {\displaystyle l} . Explicitly, G i j l ( x → ) = ∑ k F i k l ( x → ) F j k l ( x → ) {\displaystyle G_{ij}^{l}({\vec {x}})=\sum _{k}F_{ik}^{l}({\vec {x}})F_{jk}^{l}({\vec {x}})} Minimizing this loss encourages the generated image to have similar style characteristics to the style image, as captured by the correlations between feature responses in each layer. The idea is that activation pattern correlations between filters in a single layer captures the "style" on the order of the receptive fields at that layer. Similarly to the previous case, the w l {\displaystyle w_{l}} are positive real numbers chosen as hyperparameters. === Hyperparameters === In the original paper, they used a particular choice of hyperparameters. The style loss is computed by w l = 0.2 {\displaystyle w_{l}=0.2} for the outputs of layers conv1_1, conv2_1, conv3_1, conv4_1, conv5_1 in the VGG-19 network, and zero otherwise. The content loss is computed by w l = 1 {\displaystyle w_{l}=1} for conv4_2, and zero otherwise. The ratio α / β ∈ [ 5 , 50 ] × 10 − 4 {\displaystyle \alpha /\beta \in [5,50]\times 10^{-4}} . === Training === Image x → {\displaystyle {\vec {x}}} is initially approximated by adding a small amount of white noise to input image p → {\displaystyle {\vec {p}}} and feeding it through the CNN. Then we successively backpropagate this loss through the network with the CNN weights fixed in order to update the pixels of x → {\displaystyle {\vec {x}}} . After several thousand epochs of training, an x → {\displaystyle {\vec {x}}} (hopefully) emerges that matches the style of a → {\displaystyle {\vec {a}}} and the content of p → {\displaystyle {\vec {p}}} . As of 2017, when implemented on a GPU, it takes a few minutes to converge. == Extensions == In some practical implementations, it is noted that the resulting image has too much high-frequency artifact, which can be suppressed by adding the total variation to the total loss. Compared to VGGNet, AlexNet does not work well for neural style transfer. NST has also been extended to videos. Subsequent work improved the speed of NST for images by using special-purpose normalizations. In a paper by Fei-Fei Li et al. adopted a different regularized loss metric and accelerated method for training to produce results in real-time (three orders of magnitude faster than Gatys). Their idea was to use not the pixel-based loss defined above but rather a 'perceptual loss' measuring t
Image moment
In image processing, computer vision and related fields, an image moment is a certain particular weighted average (moment) of the image pixels' intensities, or a function of such moments, usually chosen to have some attractive property or interpretation. Image moments are useful to describe objects after segmentation. Simple properties of the image which are found via image moments include area (or total intensity), its centroid, and information about its orientation. == Raw moments == For a 2D continuous function f(x,y) the moment (sometimes called "raw moment") of order (p + q) is defined as M p q = ∫ − ∞ ∞ ∫ − ∞ ∞ x p y q f ( x , y ) d x d y {\displaystyle M_{pq}=\int \limits _{-\infty }^{\infty }\int \limits _{-\infty }^{\infty }x^{p}y^{q}f(x,y)\,dx\,dy} for p,q = 0,1,2,... Adapting this to scalar (grayscale) image with pixel intensities I(x,y), raw image moments Mij are calculated by M i j = ∑ x ∑ y x i y j I ( x , y ) {\displaystyle M_{ij}=\sum _{x}\sum _{y}x^{i}y^{j}I(x,y)\,\!} In some cases, this may be calculated by considering the image as a probability density function, i.e., by dividing the above by ∑ x ∑ y I ( x , y ) {\displaystyle \sum _{x}\sum _{y}I(x,y)\,\!} A uniqueness theorem states that if f(x,y) is piecewise continuous and has nonzero values only in a finite part of the xy plane, moments of all orders exist, and the moment sequence (Mpq) is uniquely determined by f(x,y). Conversely, (Mpq) uniquely determines f(x,y). In practice, the image is summarized with functions of a few lower order moments. === Examples === Simple image properties derived via raw moments include: Area (for binary images) or sum of grey level (for greytone images): M 00 {\displaystyle M_{00}} Centroid: { x ¯ , y ¯ } = { M 10 M 00 , M 01 M 00 } {\displaystyle \{{\bar {x}},\ {\bar {y}}\}=\left\{{\frac {M_{10}}{M_{00}}},{\frac {M_{01}}{M_{00}}}\right\}} == Central moments == Central moments are defined as μ p q = ∫ − ∞ ∞ ∫ − ∞ ∞ ( x − x ¯ ) p ( y − y ¯ ) q f ( x , y ) d x d y {\displaystyle \mu _{pq}=\int \limits _{-\infty }^{\infty }\int \limits _{-\infty }^{\infty }(x-{\bar {x}})^{p}(y-{\bar {y}})^{q}f(x,y)\,dx\,dy} where x ¯ = M 10 M 00 {\displaystyle {\bar {x}}={\frac {M_{10}}{M_{00}}}} and y ¯ = M 01 M 00 {\displaystyle {\bar {y}}={\frac {M_{01}}{M_{00}}}} are the components of the centroid. If ƒ(x, y) is a digital image, then the previous equation becomes μ p q = ∑ x ∑ y ( x − x ¯ ) p ( y − y ¯ ) q f ( x , y ) {\displaystyle \mu _{pq}=\sum _{x}\sum _{y}(x-{\bar {x}})^{p}(y-{\bar {y}})^{q}f(x,y)} The central moments of order up to 3 are: μ 00 = M 00 , μ 01 = 0 , μ 10 = 0 , μ 11 = M 11 − x ¯ M 01 = M 11 − y ¯ M 10 , μ 20 = M 20 − x ¯ M 10 , μ 02 = M 02 − y ¯ M 01 , μ 21 = M 21 − 2 x ¯ M 11 − y ¯ M 20 + 2 x ¯ 2 M 01 , μ 12 = M 12 − 2 y ¯ M 11 − x ¯ M 02 + 2 y ¯ 2 M 10 , μ 30 = M 30 − 3 x ¯ M 20 + 2 x ¯ 2 M 10 , μ 03 = M 03 − 3 y ¯ M 02 + 2 y ¯ 2 M 01 . {\displaystyle {\begin{aligned}\mu _{00}&=M_{00},&\mu _{01}&=0,\\\mu _{10}&=0,&\mu _{11}&=M_{11}-{\bar {x}}M_{01}=M_{11}-{\bar {y}}M_{10},\\\mu _{20}&=M_{20}-{\bar {x}}M_{10},&\mu _{02}&=M_{02}-{\bar {y}}M_{01},\\\mu _{21}&=M_{21}-2{\bar {x}}M_{11}-{\bar {y}}M_{20}+2{\bar {x}}^{2}M_{01},&\mu _{12}&=M_{12}-2{\bar {y}}M_{11}-{\bar {x}}M_{02}+2{\bar {y}}^{2}M_{10},\\\mu _{30}&=M_{30}-3{\bar {x}}M_{20}+2{\bar {x}}^{2}M_{10},&\mu _{03}&=M_{03}-3{\bar {y}}M_{02}+2{\bar {y}}^{2}M_{01}.\end{aligned}}} It can be shown that: μ p q = ∑ m p ∑ n q ( p m ) ( q n ) ( − x ¯ ) ( p − m ) ( − y ¯ ) ( q − n ) M m n {\displaystyle \mu _{pq}=\sum _{m}^{p}\sum _{n}^{q}{p \choose m}{q \choose n}(-{\bar {x}})^{(p-m)}(-{\bar {y}})^{(q-n)}M_{mn}} Central moments are translational invariant. === Examples === Information about image orientation can be derived by first using the second order central moments to construct a covariance matrix. μ 20 ′ = μ 20 / μ 00 = M 20 / M 00 − x ¯ 2 μ 02 ′ = μ 02 / μ 00 = M 02 / M 00 − y ¯ 2 μ 11 ′ = μ 11 / μ 00 = M 11 / M 00 − x ¯ y ¯ {\displaystyle {\begin{aligned}\mu '_{20}&=\mu _{20}/\mu _{00}=M_{20}/M_{00}-{\bar {x}}^{2}\\\mu '_{02}&=\mu _{02}/\mu _{00}=M_{02}/M_{00}-{\bar {y}}^{2}\\\mu '_{11}&=\mu _{11}/\mu _{00}=M_{11}/M_{00}-{\bar {x}}{\bar {y}}\end{aligned}}} The covariance matrix of the image I ( x , y ) {\displaystyle I(x,y)} is now cov [ I ( x , y ) ] = [ μ 20 ′ μ 11 ′ μ 11 ′ μ 02 ′ ] . {\displaystyle \operatorname {cov} [I(x,y)]={\begin{bmatrix}\mu '_{20}&\mu '_{11}\\\mu '_{11}&\mu '_{02}\end{bmatrix}}.} The eigenvectors of this matrix correspond to the major and minor axes of the image intensity, so the orientation can thus be extracted from the angle of the eigenvector associated with the largest eigenvalue towards the axis closest to this eigenvector. It can be shown that this angle Θ is given by the following formula: Θ = 1 2 arctan ( 2 μ 11 ′ μ 20 ′ − μ 02 ′ ) {\displaystyle \Theta ={\frac {1}{2}}\arctan \left({\frac {2\mu '_{11}}{\mu '_{20}-\mu '_{02}}}\right)} The above formula holds as long as: μ 20 ′ − μ 02 ′ ≠ 0 {\displaystyle \mu '_{20}-\mu '_{02}\neq 0} The eigenvalues of the covariance matrix can easily be shown to be λ i = μ 20 ′ + μ 02 ′ 2 ± 4 μ ′ 11 2 + ( μ ′ 20 − μ ′ 02 ) 2 2 , {\displaystyle \lambda _{i}={\frac {\mu '_{20}+\mu '_{02}}{2}}\pm {\frac {\sqrt {4{\mu '}_{11}^{2}+({\mu '}_{20}-{\mu '}_{02})^{2}}}{2}},} and are proportional to the squared length of the eigenvector axes. The relative difference in magnitude of the eigenvalues are thus an indication of the eccentricity of the image, or how elongated it is. The eccentricity is 1 − λ 2 λ 1 . {\displaystyle {\sqrt {1-{\frac {\lambda _{2}}{\lambda _{1}}}}}.} == Moment invariants == Moments are well-known for their application in image analysis, since they can be used to derive invariants with respect to specific transformation classes. The term invariant moments is often abused in this context. However, while moment invariants are invariants that are formed from moments, the only moments that are invariants themselves are the central moments. Note that the invariants detailed below are exactly invariant only in the continuous domain. In a discrete domain, neither scaling nor rotation are well defined: a discrete image transformed in such a way is generally an approximation, and the transformation is not reversible. These invariants therefore are only approximately invariant when describing a shape in a discrete image. === Translation invariants === The central moments μi j of any order are, by construction, invariant with respect to translations. === Scale invariants === Invariants ηi j with respect to both translation and scale can be constructed from central moments by dividing through a properly scaled zero-th central moment: η i j = μ i j μ 00 ( 1 + i + j 2 ) {\displaystyle \eta _{ij}={\frac {\mu _{ij}}{\mu _{00}^{\left(1+{\frac {i+j}{2}}\right)}}}\,\!} where i + j ≥ 2. Note that translational invariance directly follows by only using central moments. === Rotation invariants === As shown in the work of Hu, invariants with respect to translation, scale, and rotation can be constructed: I 1 = η 20 + η 02 {\displaystyle I_{1}=\eta _{20}+\eta _{02}} I 2 = ( η 20 − η 02 ) 2 + 4 η 11 2 {\displaystyle I_{2}=(\eta _{20}-\eta _{02})^{2}+4\eta _{11}^{2}} I 3 = ( η 30 − 3 η 12 ) 2 + ( 3 η 21 − η 03 ) 2 {\displaystyle I_{3}=(\eta _{30}-3\eta _{12})^{2}+(3\eta _{21}-\eta _{03})^{2}} I 4 = ( η 30 + η 12 ) 2 + ( η 21 + η 03 ) 2 {\displaystyle I_{4}=(\eta _{30}+\eta _{12})^{2}+(\eta _{21}+\eta _{03})^{2}} I 5 = ( η 30 − 3 η 12 ) ( η 30 + η 12 ) [ ( η 30 + η 12 ) 2 − 3 ( η 21 + η 03 ) 2 ] + ( 3 η 21 − η 03 ) ( η 21 + η 03 ) [ 3 ( η 30 + η 12 ) 2 − ( η 21 + η 03 ) 2 ] {\displaystyle I_{5}=(\eta _{30}-3\eta _{12})(\eta _{30}+\eta _{12})[(\eta _{30}+\eta _{12})^{2}-3(\eta _{21}+\eta _{03})^{2}]+(3\eta _{21}-\eta _{03})(\eta _{21}+\eta _{03})[3(\eta _{30}+\eta _{12})^{2}-(\eta _{21}+\eta _{03})^{2}]} I 6 = ( η 20 − η 02 ) [ ( η 30 + η 12 ) 2 − ( η 21 + η 03 ) 2 ] + 4 η 11 ( η 30 + η 12 ) ( η 21 + η 03 ) {\displaystyle I_{6}=(\eta _{20}-\eta _{02})[(\eta _{30}+\eta _{12})^{2}-(\eta _{21}+\eta _{03})^{2}]+4\eta _{11}(\eta _{30}+\eta _{12})(\eta _{21}+\eta _{03})} I 7 = ( 3 η 21 − η 03 ) ( η 30 + η 12 ) [ ( η 30 + η 12 ) 2 − 3 ( η 21 + η 03 ) 2 ] − ( η 30 − 3 η 12 ) ( η 21 + η 03 ) [ 3 ( η 30 + η 12 ) 2 − ( η 21 + η 03 ) 2 ] . {\displaystyle I_{7}=(3\eta _{21}-\eta _{03})(\eta _{30}+\eta _{12})[(\eta _{30}+\eta _{12})^{2}-3(\eta _{21}+\eta _{03})^{2}]-(\eta _{30}-3\eta _{12})(\eta _{21}+\eta _{03})[3(\eta _{30}+\eta _{12})^{2}-(\eta _{21}+\eta _{03})^{2}].} These are well-known as Hu moment invariants. The first one, I1, is analogous to the moment of inertia around the image's centroid, where the pixels' intensities are analogous to physical density. The first six, I1 ... I6, are reflection symmetric, i.e. they are unchanged if the image is changed to a mirror image. The last one, I7, is reflection antisymmetric (changes sign under reflection), which enables it to distinguish mirror images of otherwise identical im
ACL Data Collection Initiative
The ACL Data Collection Initiative (ACL/DCI) was a project established in 1989 by the Association for Computational Linguistics (ACL) to create and distribute large text and speech corpora for computational linguistics research. The initiative aimed to address the growing need for substantial text databases that could support research in areas such as natural language processing, speech recognition, and computational linguistics. By 1993, the initiative’s activities had effectively ceased, with its functions and datasets absorbed by the Linguistic Data Consortium (LDC), which was founded in 1992. == Objectives == The ACL/DCI had several key objectives: To acquire a large and diverse text corpus from various sources To transform the collected texts into a common format based on the Standard Generalized Markup Language (SGML) To make the corpus available for scientific research at low cost with minimal restrictions To provide a common database that would allow researchers to replicate or extend published results To reduce duplication of effort among researchers in obtaining and preparing text data These objectives were designed to address the growing demand for very large amounts of text arising from applications in recognition and analysis of text and speech. Its core objective was to "oversee the acquisition and preparation of a large text corpus to be made available for scientific research at cost and without royalties". == History == By the late 1980s, researchers in computational linguistics and speech recognition faced a significant problem: the lack of large-scale, accessible text corpora for developing statistical models and testing algorithms. Existing generally available text databases were too small to meet the needs of developing applications in text and speech recognition. The initiative was formed to meet this need by collecting, standardizing, and distributing large quantities of text data with minimal restrictions for scientific research. As stated by Liberman (1990), "research workers have been severely hampered by the lack of appropriate materials, and specially by the lack of a large enough body of text on which published results can be replicated or extended by others." The ACL/DCI committee was established in February 1989. The committee included members from academic and industrial research laboratories in the United States and Europe. The initiative was chaired by Mark Liberman from the University of Pennsylvania (formerly of AT&T Bell Laboratories). Other committee members included representatives from organizations such as Bellcore, IBM T.J. Watson Research Center, Cambridge University, Virginia Polytechnic Institute & State University, Northeastern University, University of Pennsylvania, SRI International, MCC, Xerox PARC, ISSCO, and University of Pisa. The project operated initially without dedicated funding, relying on volunteer efforts from committee members and their affiliated institutions. Key supporters included AT&T Bell Labs, Bellcore, IBM, Xerox, and the University of Pennsylvania, which allowed the use of their computing facilities for ACL/DCI-related work. Previously running on volunteer effort pro bono, in 1991, it obtained funding from General Electric and the National Science Foundation (IRI-9113530). == Data == As of 1990, the ACL/DCI had collected hundreds of millions of words of diverse text. The collection included: Wall Street Journal articles (25 to 50 million words); Canadian Hansard (parliamentary records) in parallel English and French versions: cleaned-up English Hansard donated by the IBM alignment models group (100 million words), and original Bilingual Hansard (from a different time period) obtained directly (200 million words). Collins English Dictionary (1979 edition), both as fulltext (3 million words) and as various "database" versions, constructed using "typographers' tape" donated by Collins, which were computer tapes containing the structured digital data used to typeset and print the 1979 edition of the dictionary; Emails from ARPANET newsletters for the ACM Special Interest Group on Information Retrieval Forum (IRLIST) and AIList Digest issues distributed over the ARPANET (AILIST) (5 million words), both collected by Edward A. Fox at VIPSU; Articles on networking (2 million words); U.S. Department of Agriculture Extension Service Fact Sheets (>1 million words); 200,000 scientific abstracts of about 1,500 words each from the Department of Energy (25 million words); Archives of the Challenger Investigation Commission, including transcripts of depositions and hearings (2.5 million words); Books from the Library of America, including works by Mark Twain, Eugene O'Neill, Ralph Waldo Emerson, Herman Melville, W.E.B. DuBois, Willa Cather, and Benjamin Franklin (130 books, 20 million words); Public domain books like the King James Bible, Tristram Shandy, The Federalist Papers; Several million words of transcribed radiologists' reports, donated by Francis Ganong at Kurzweil Applied Intelligence Inc (about 5 million words); The Child Language Data Exchange corpus of child language acquisition transcripts; U.S. Department of Justice Justice Retrieval and Inquiry System (JURIS) materials; The Swiss Civil Code in parallel German, French and Italian; Economic reports from the Union Bank of Switzerland, in parallel English, German, French and Italian; About 12K words of administrative policy manuals and 14K words of administrative memos, contributed by Geoff Pullum of U.C.S.C.; Material from various ACM journals and the ACL journal Computational Linguistics; The CSLI publications series: 50-100 reports (8K words each) and 5-10 books (80K words each). The initiative started with North American English text but expanded to include Canadian French and planned to include Japanese, Chinese, and other Asian languages. At least 5 million words from the collection were tagged under the Penn Treebank project, and those tags were distributed by DCI as well. After DCI was absorbed by the LDC, the datasets were curated under LDC. == Format == The ACL/DCI corpus was coded in a standard form based on SGML (Standard Generalized Markup Language, ISO 8879), consistent with the recommendations of the Text Encoding Initiative (TEI), of which the DCI was an affiliated project. The TEI was a joint project of the ACL, the Association for Computers and the Humanities, and the Association for Literary and Linguistic Computing, aiming to provide a common interchange format for literary and linguistic data. The initiative planned to add annotations reflecting consensually approved linguistic features like part of speech and various aspects of syntactic and semantic structure over time. == Examples == As an example of the use of ACL/DCI, consider the Wall Street Journal (WSJ) corpus for speech recognition research. The WSJ corpus was used as the basis for the DARPA Spoken Language System (SLS) community's Continuous Speech Recognition (CSR) Corpus. The WSJ corpus became a standard benchmark for evaluating speech recognition systems and has been used in numerous research papers. The WSJ CSR Corpus provided DARPA with its first general-purpose English, large vocabulary, natural language, high perplexity corpus containing speech (400 hours) and text (47 million words) during 1987–89. The text corpus was 313 MB in size. The text was preprocessed to remove ambiguity in the word sequence that a reader might choose, ensuring that the unread text used to train language models was representative of the spoken test material. The preprocessing included converting numbers into orthographics, expanding abbreviations, resolving apostrophes and quotation marks, and marking punctuation. As another example, the Yarowsky algorithm used bitext data from DCI to train a simple word-sense disambiguation model that was competitive with advanced models trained on smaller datasets. == Distribution == Materials from the ACL/DCI collection were distributed to research groups on a non-commercial basis. By 1990, about 25 research groups and individual researchers had received tapes containing various portions of the collected material. To obtain the data, researchers had to sign an agreement not to redistribute the data or make direct commercial use of it. However, commercial application of "analytical materials" derived from the text, such as statistical tables or grammar rules, was explicitly permitted. The initiative first distributed data via 12-inch reels of 9-track tape, then via CD-ROMs. Each such tape could contain 30 million words compressed via the Lempel-Ziv algorithms. The first CD-ROM distribution was in 1991, funded by Dragon Systems Inc. It contained Collins English Dictionary, WSJ, scientific abstracts provided by the U.S. Department of Energy, and the Penn Treebank.
Nextcloud
Nextcloud is a modular workspace platform designed to provide teams and businesses with a comprehensive environment for digital collaboration. Beyond central data management, it integrates office suites like Collabora Online and EuroOffice office suites. for seamless, cooperative workflows. The platform features built-in tools for chat, videoconferencing, and a privacy-focused AI assistant capable of running entirely on local LLMs. Supported by a rich ecosystem of apps, it can be hosted in the cloud or on premises and can scale up to millions of users. It has been translated into over 100 languages. == Features == Nextcloud files are stored in conventional directory structures, accessible via WebDAV if necessary. A SQLite, MySQL/MariaDB or PostgreSQL database is required to provide additional functionality like permissions, shares, and comments. Nextcloud can synchronize with local clients running Windows (Windows 8.1 and above), macOS (10.14 or later), Linux and FreeBSD. Nextcloud permits user and group administration locally or via different backends like OpenID or LDAP. Content can be shared inside the system by defining granular read/write permissions between users and groups. Nextcloud users can create public URLs when sharing files. Logging of file-related actions, as well as disallowing access based on file access rules is also available. Security options like brute-force protection and multi-factor authentication using TOTP, WebAuthn, Oauth2, and OpenID Connect are available. Nextcloud has planned new features such as monitoring capabilities, full-text search and Kerberos authentication, as well as audio/video conferencing, expanded federation and smaller user interface improvements. == History == In April 2016 Frank Karlitschek and most core contributors left ownCloud Inc. These included some of ownCloud's staff according to sources near to the ownCloud community. Karlitschek and many of these contributors went on to fork ownCloud, creating Nextcloud. The fork was preceded by a blog post of Karlitschek announcing his departure and raising questions about the management of the ownCloud, its community, and priorities between growth, money, and sustainability. There have been no official statements about the reason for the fork. However, Karlitschek mentioned the fork several times in a talk at the 2018 FOSDEM conference and in two appearances on the FLOSS Weekly podcast, emphasizing cultural mismatch between open source developers and business oriented people not used to the open source community. On June 2, within 12 hours of the announcement of the fork, the American entity "ownCloud Inc." announced that it is shutting down with immediate effect, stating that "[...] main lenders in the US have cancelled our credit. Following American law, we are forced to close the doors of ownCloud, Inc. with immediate effect and terminate the contracts of 8 employees." ownCloud Inc. accused Karlitschek of poaching developers, while Nextcloud developers such as Arthur Schiwon stated that he "decided to quit because not everything in the ownCloud Inc. company world evolved as I imagined". ownCloud GmbH continued operations, secured financing from new investors and took over the business of ownCloud Inc. In April 2018 Informationstechnikzentrum Bund (ITZBund) reported Nextcloud won the tender for "Bundescloud" (Germany government cloud) project. In August 2019 it was announced that the governments of France, Sweden and the Netherlands would use Nextcloud for file transfer. In January 2020 Nextcloud 18 "Nextcloud Hub" was released. The major change was direct integration with an Office suite (OnlyOffice) and Nextcloud announced that their goal was to compete with Office 365 and Google Docs. A partnership with Ionos was revealed – its hosting location in Germany and compliance with GDPR should support the goal of data sovereignty. In spring 2020 remote work and web conferencing usage increased due to the COVID-19 pandemic and Nextcloud released version 19 with chat and videoconferencing Talk app integrated into the application core. Communication with an optional "high performance back-end" allows self-hosting of web conferences with more than 10 participants. Collabora Online was introduced as another integrated office suite. In August 2021 Nextcloud was chosen as a collaboration platform for European cloud software GAIA-X. In a September 2021 European Commission report it was mentioned as "the most widely deployed Open Source content collaboration platform" Following the 2025 United States tariffs against the European Union, fear of overreliance on US cloud providers such as Microsoft 365 and Google Workspace increased, with Nextcloud being one of the foremost contenders to replace them. Some governmental organisations including the European Data Protection Supervisor and the German state of Schleswig-Holstein have since switched from Microsoft's Sharepoint to Nextcloud. According to Nextcloud, during the first 5 months of 2025, customer interest in the software had tripled.
Niki.ai
Niki was an artificial intelligence company headquartered in Bangalore, Karnataka. It was founded in May 2015 by IIT Kharagpur graduates Sachin Jaiswal, Keshav Prawasi, Shishir Modi, and Nitin Babel. The Niki android app was launched for a limited beta in June 2015, then released for public during YourStory's TechSparks 2015, and is a Tech30 company. The company raised an undisclosed amount in seed funding from Unilazer Ventures, a Mumbai-based VC firm founded by Ronnie Screwvala, in October 2015. This was followed by another seed funding round by Ratan Tata in May 2016. The company then raised US$2 million in Series A round of funding from SAP.iO, existing investors and some US and German-based investors, among others. Niki.ai shut down in October 2021 as per media reports. Website not working. == Product == The product is an artificial intelligence-powered chatbot which works as an intelligent personal assistant, named Niki. Leveraging natural language processing and machine learning, Niki presents a chat-based natural language user interface to the users where they can interact with Niki in their natural language. Niki understands how users chat in India, deciphers the words, in the context of product/services that they would like to purchase, and comes up with apt recommendations. Initially, it was only available on the Android platform as a mobile app. The company has expanded its operations to the Facebook Messenger and Apple iOS platforms. The company aims to soon be present on more messaging platforms like Slack and WhatsApp. The company currently provides 20+ services to over 2 million consumers, covering a wide spectrum ranging from utility services like mobile recharge, bill payments, travel services like cabs, buses, hotels and entertainment services like movies and events. Services such as flights and healthcare are also planned. == Partnerships == In September 2017, Infosys Finacle joined with Niki.ai to provide chat-based service to banking customers. In August 2017, Niki partnered with LazyPay to enable a 'buy now, pay later' feature for its users.