PDE surfaces are used in geometric modelling and computer graphics for creating smooth surfaces conforming to a given boundary configuration. PDE surfaces use partial differential equations to generate a surface which usually satisfy a mathematical boundary value problem. PDE surfaces were first introduced into the area of geometric modelling and computer graphics by two British mathematicians, Malcolm Bloor and Michael Wilson. == Technical details == The PDE method involves generating a surface for some boundary by means of solving an elliptic partial differential equation of the form ( ∂ 2 ∂ u 2 + a 2 ∂ 2 ∂ v 2 ) 2 X ( u , v ) = 0. {\displaystyle \left({\frac {\partial ^{2}}{\partial u^{2}}}+a^{2}{\frac {\partial ^{2}}{\partial v^{2}}}\right)^{2}X(u,v)=0.} Here X ( u , v ) {\displaystyle X(u,v)} is a function parameterised by the two parameters u {\displaystyle u} and v {\displaystyle v} such that X ( u , v ) = ( x ( u , v ) , y ( u , v ) , z ( u , v ) ) {\displaystyle X(u,v)=(x(u,v),y(u,v),z(u,v))} where x {\displaystyle x} , y {\displaystyle y} and z {\displaystyle z} are the usual cartesian coordinate space. The boundary conditions on the function X ( u , v ) {\displaystyle X(u,v)} and its normal derivatives ∂ X / ∂ n {\displaystyle \partial {X}/\partial {n}} are imposed at the edges of the surface patch. With the above formulation it is notable that the elliptic partial differential operator in the above PDE represents a smoothing process in which the value of the function at any point on the surface is, in some sense, a weighted average of the surrounding values. In this way, a surface is obtained as a smooth transition between the chosen set of boundary conditions. The parameter a {\displaystyle a} is a special design parameter which controls the relative smoothing of the surface in the u {\displaystyle u} and v {\displaystyle v} directions. When a = 1 {\displaystyle a=1} , the PDE is the biharmonic equation: X u u u u + 2 X u u v v + X v v v v = 0 {\displaystyle X_{uuuu}+2X_{uuvv}+X_{vvvv}=0} . The biharmonic equation is the equation produced by applying the Euler-Lagrange equation to the simplified thin plate energy functional X u u 2 + 2 X u v 2 + X v v 2 {\displaystyle X_{uu}^{2}+2X_{uv}^{2}+X_{vv}^{2}} . So solving the PDE with a = 1 {\displaystyle a=1} is equivalent to minimizing the thin plate energy functional subject to the same boundary conditions. == Applications == PDE surfaces can be used in many application areas. These include computer-aided design, interactive design, parametric design, computer animation, computer-aided physical analysis and design optimisation. == Related publications == M.I.G. Bloor and M.J. Wilson, Generating Blend Surfaces using Partial Differential Equations, Computer Aided Design, 21(3), 165–171, (1989). H. Ugail, M.I.G. Bloor, and M.J. Wilson, Techniques for Interactive Design Using the PDE Method, ACM Transactions on Graphics, 18(2), 195–212, (1999). J. Huband, W. Li and R. Smith, An Explicit Representation of Bloor-Wilson PDE Surface Model by using Canonical Basis for Hermite Interpolation, Mathematical Engineering in Industry, 7(4), 421-33 (1999). H. Du and H. Qin, Direct Manipulation and Interactive Sculpting of PDE surfaces, Computer Graphics Forum, 19(3), C261-C270, (2000). H. Ugail, Spine Based Shape Parameterisations for PDE surfaces, Computing, 72, 195–204, (2004). L. You, P. Comninos, J.J. Zhang, PDE Blending Surfaces with C2 Continuity, Computers and Graphics, 28(6), 895–906, (2004).
Automated negotiation
Automated negotiation is a form of interaction in systems that are composed of multiple autonomous agents, in which the aim is to reach agreements through an iterative process of making offers. Automated negotiation can be employed for many tasks human negotiators regularly engage in, such as bargaining and joint decision making. The main topics in automated negotiation revolve around the design of protocols and negotiating strategies. == History == Through digitization, the beginning of the 21st century has seen a growing interest in the automation of negotiation and e-negotiation systems, for example in the setting of e-commerce. This interest is fueled by the promise of automated agents being able to negotiate on behalf of human negotiators, and to find better outcomes than human negotiators. == Examples == Examples of automated negotiation include: Online dispute resolution, in which disagreements between parties are settled. Sponsored search auction, where bids are placed on advertisement keywords. Content negotiation, in which user agents negotiate over HTTP about how to best represent a web resource. Negotiation support systems, in which negotiation decision-making activities are supported by an information system.
Wasserstein GAN
The Wasserstein Generative Adversarial Network (WGAN) is a variant of generative adversarial network (GAN) proposed in 2017 that aims to "improve the stability of learning, get rid of problems like mode collapse, and provide meaningful learning curves useful for debugging and hyperparameter searches". Compared with the original GAN discriminator, the Wasserstein GAN discriminator provides a better learning signal to the generator. This allows the training to be more stable when generator is learning distributions in very high dimensional spaces. == Motivation == === The GAN game === The original GAN method is based on the GAN game, a zero-sum game with 2 players: generator and discriminator. The game is defined over a probability space ( Ω , B , μ r e f ) {\displaystyle (\Omega ,{\mathcal {B}},\mu _{ref})} , The generator's strategy set is the set of all probability measures μ G {\displaystyle \mu _{G}} on ( Ω , B ) {\displaystyle (\Omega ,{\mathcal {B}})} , and the discriminator's strategy set is the set of measurable functions D : Ω → [ 0 , 1 ] {\displaystyle D:\Omega \to [0,1]} . The objective of the game is L ( μ G , D ) := E x ∼ μ r e f [ ln D ( x ) ] + E x ∼ μ G [ ln ( 1 − D ( x ) ) ] . {\displaystyle L(\mu _{G},D):=\mathbb {E} _{x\sim \mu _{ref}}[\ln D(x)]+\mathbb {E} _{x\sim \mu _{G}}[\ln(1-D(x))].} The generator aims to minimize it, and the discriminator aims to maximize it. A basic theorem of the GAN game states that Repeat the GAN game many times, each time with the generator moving first, and the discriminator moving second. Each time the generator μ G {\displaystyle \mu _{G}} changes, the discriminator must adapt by approaching the ideal D ∗ ( x ) = d μ r e f d ( μ r e f + μ G ) . {\displaystyle D^{}(x)={\frac {d\mu _{ref}}{d(\mu _{ref}+\mu _{G})}}.} Since we are really interested in μ r e f {\displaystyle \mu _{ref}} , the discriminator function D {\displaystyle D} is by itself rather uninteresting. It merely keeps track of the likelihood ratio between the generator distribution and the reference distribution. At equilibrium, the discriminator is just outputting 1 2 {\displaystyle {\frac {1}{2}}} constantly, having given up trying to perceive any difference. Concretely, in the GAN game, let us fix a generator μ G {\displaystyle \mu _{G}} , and improve the discriminator step-by-step, with μ D , t {\displaystyle \mu _{D,t}} being the discriminator at step t {\displaystyle t} . Then we (ideally) have L ( μ G , μ D , 1 ) ≤ L ( μ G , μ D , 2 ) ≤ ⋯ ≤ max μ D L ( μ G , μ D ) = 2 D J S ( μ r e f ‖ μ G ) − 2 ln 2 , {\displaystyle L(\mu _{G},\mu _{D,1})\leq L(\mu _{G},\mu _{D,2})\leq \cdots \leq \max _{\mu _{D}}L(\mu _{G},\mu _{D})=2D_{JS}(\mu _{ref}\|\mu _{G})-2\ln 2,} so we see that the discriminator is actually lower-bounding D J S ( μ r e f ‖ μ G ) {\displaystyle D_{JS}(\mu _{ref}\|\mu _{G})} . === Wasserstein distance === Thus, we see that the point of the discriminator is mainly as a critic to provide feedback for the generator, about "how far it is from perfection", where "far" is defined as Jensen–Shannon divergence. Naturally, this brings the possibility of using a different criteria of farness. There are many possible divergences to choose from, such as the f-divergence family, which would give the f-GAN. The Wasserstein GAN is obtained by using the Wasserstein metric, which satisfies a "dual representation theorem" that renders it highly efficient to compute: A proof can be found in the main page on Wasserstein metric. == Definition == By the Kantorovich-Rubenstein duality, the definition of Wasserstein GAN is clear:A Wasserstein GAN game is defined by a probability space ( Ω , B , μ r e f ) {\displaystyle (\Omega ,{\mathcal {B}},\mu _{ref})} , where Ω {\displaystyle \Omega } is a metric space, and a constant K > 0 {\displaystyle K>0} . There are 2 players: generator and discriminator (also called "critic"). The generator's strategy set is the set of all probability measures μ G {\displaystyle \mu _{G}} on ( Ω , B ) {\displaystyle (\Omega ,{\mathcal {B}})} . The discriminator's strategy set is the set of measurable functions of type D : Ω → R {\displaystyle D:\Omega \to \mathbb {R} } with bounded Lipschitz-norm: ‖ D ‖ L ≤ K {\displaystyle \|D\|_{L}\leq K} . The Wasserstein GAN game is a zero-sum game, with objective function L W G A N ( μ G , D ) := E x ∼ μ G [ D ( x ) ] − E x ∼ μ r e f [ D ( x ) ] . {\displaystyle L_{WGAN}(\mu _{G},D):=\mathbb {E} _{x\sim \mu _{G}}[D(x)]-\mathbb {E} _{x\sim \mu _{ref}}[D(x)].} The generator goes first, and the discriminator goes second. The generator aims to minimize the objective, and the discriminator aims to maximize the objective: min μ G max D L W G A N ( μ G , D ) . {\displaystyle \min _{\mu _{G}}\max _{D}L_{WGAN}(\mu _{G},D).} By the Kantorovich-Rubenstein duality, for any generator strategy μ G {\displaystyle \mu _{G}} , the optimal reply by the discriminator is D ∗ {\displaystyle D^{}} , such that L W G A N ( μ G , D ∗ ) = K ⋅ W 1 ( μ G , μ r e f ) . {\displaystyle L_{WGAN}(\mu _{G},D^{})=K\cdot W_{1}(\mu _{G},\mu _{ref}).} Consequently, if the discriminator is good, the generator would be constantly pushed to minimize W 1 ( μ G , μ r e f ) {\displaystyle W_{1}(\mu _{G},\mu _{ref})} , and the optimal strategy for the generator is just μ G = μ r e f {\displaystyle \mu _{G}=\mu _{ref}} , as it should. == Comparison with GAN == In the Wasserstein GAN game, the discriminator provides a better gradient than in the GAN game. Consider for example a game on the real line where both μ G {\displaystyle \mu _{G}} and μ r e f {\displaystyle \mu _{ref}} are Gaussian. Then the optimal Wasserstein critic D W G A N {\displaystyle D_{WGAN}} and the optimal GAN discriminator D {\displaystyle D} are plotted as below: For fixed discriminator, the generator needs to minimize the following objectives: For GAN, E x ∼ μ G [ ln ( 1 − D ( x ) ) ] {\displaystyle \mathbb {E} _{x\sim \mu _{G}}[\ln(1-D(x))]} . For Wasserstein GAN, E x ∼ μ G [ D W G A N ( x ) ] {\displaystyle \mathbb {E} _{x\sim \mu _{G}}[D_{WGAN}(x)]} . Let μ G {\displaystyle \mu _{G}} be parametrized by θ {\displaystyle \theta } , then we can perform stochastic gradient descent by using two unbiased estimators of the gradient: ∇ θ E x ∼ μ G [ ln ( 1 − D ( x ) ) ] = E x ∼ μ G [ ln ( 1 − D ( x ) ) ⋅ ∇ θ ln ρ μ G ( x ) ] {\displaystyle \nabla _{\theta }\mathbb {E} _{x\sim \mu _{G}}[\ln(1-D(x))]=\mathbb {E} _{x\sim \mu _{G}}[\ln(1-D(x))\cdot \nabla _{\theta }\ln \rho _{\mu _{G}}(x)]} ∇ θ E x ∼ μ G [ D W G A N ( x ) ] = E x ∼ μ G [ D W G A N ( x ) ⋅ ∇ θ ln ρ μ G ( x ) ] {\displaystyle \nabla _{\theta }\mathbb {E} _{x\sim \mu _{G}}[D_{WGAN}(x)]=\mathbb {E} _{x\sim \mu _{G}}[D_{WGAN}(x)\cdot \nabla _{\theta }\ln \rho _{\mu _{G}}(x)]} where we used the reparameterization trick. As shown, the generator in GAN is motivated to let its μ G {\displaystyle \mu _{G}} "slide down the peak" of ln ( 1 − D ( x ) ) {\displaystyle \ln(1-D(x))} . Similarly for the generator in Wasserstein GAN. For Wasserstein GAN, D W G A N {\displaystyle D_{WGAN}} has gradient 1 almost everywhere, while for GAN, ln ( 1 − D ) {\displaystyle \ln(1-D)} has flat gradient in the middle, and steep gradient elsewhere. As a result, the variance for the estimator in GAN is usually much larger than that in Wasserstein GAN. See also Figure 3 of. The problem with D J S {\displaystyle D_{JS}} is much more severe in actual machine learning situations. Consider training a GAN to generate ImageNet, a collection of photos of size 256-by-256. The space of all such photos is R 256 2 {\displaystyle \mathbb {R} ^{256^{2}}} , and the distribution of ImageNet pictures, μ r e f {\displaystyle \mu _{ref}} , concentrates on a manifold of much lower dimension in it. Consequently, any generator strategy μ G {\displaystyle \mu _{G}} would almost surely be entirely disjoint from μ r e f {\displaystyle \mu _{ref}} , making D J S ( μ G ‖ μ r e f ) = + ∞ {\displaystyle D_{JS}(\mu _{G}\|\mu _{ref})=+\infty } . Thus, a good discriminator can almost perfectly distinguish μ r e f {\displaystyle \mu _{ref}} from μ G {\displaystyle \mu _{G}} , as well as any μ G ′ {\displaystyle \mu _{G}'} close to μ G {\displaystyle \mu _{G}} . Thus, the gradient ∇ μ G L ( μ G , D ) ≈ 0 {\displaystyle \nabla _{\mu _{G}}L(\mu _{G},D)\approx 0} , creating no learning signal for the generator. Detailed theorems can be found in. == Training Wasserstein GANs == Training the generator in Wasserstein GAN is just gradient descent, the same as in GAN (or most deep learning methods), but training the discriminator is different, as the discriminator is now restricted to have bounded Lipschitz norm. There are several methods for this. === Upper-bounding the Lipschitz norm === Let the discriminator function D {\displaystyle D} to be implemented by a multilayer perceptron: D = D n ∘ D n − 1 ∘ ⋯ ∘ D 1 {\displaystyle D=D_{n}\circ D_{n-1}\circ \cdots \circ D_{1}} where D i ( x ) = h ( W i x ) {\displaystyle D_{i}(x)=h(W_
Probabilistic automaton
In mathematics and computer science, the probabilistic automaton (PA) is a generalization of the nondeterministic finite automaton; it includes the probability of a given transition into the transition function, turning it into a transition matrix. Thus, the probabilistic automaton also generalizes the concepts of a Markov chain and of a subshift of finite type. The languages recognized by probabilistic automata are called stochastic languages; these include the regular languages as a subset. The number of stochastic languages is uncountable. The concept was introduced by Michael O. Rabin in 1963; a certain special case is sometimes known as the Rabin automaton (not to be confused with the subclass of ω-automata also referred to as Rabin automata). In recent years, a variant has been formulated in terms of quantum probabilities, the quantum finite automaton. == Informal Description == For a given initial state and input character, a deterministic finite automaton (DFA) has exactly one next state, and a nondeterministic finite automaton (NFA) has a set of next states. A probabilistic automaton (PA) instead has a weighted set (or vector) of next states, where the weights must sum to 1 and therefore can be interpreted as probabilities (making it a stochastic vector). The notions states and acceptance must also be modified to reflect the introduction of these weights. The state of the machine as a given step must now also be represented by a stochastic vector of states, and a state accepted if its total probability of being in an acceptance state exceeds some cut-off. A PA is in some sense a half-way step from deterministic to non-deterministic, as it allows a set of next states but with restrictions on their weights. However, this is somewhat misleading, as the PA utilizes the notion of the real numbers to define the weights, which is absent in the definition of both DFAs and NFAs. This additional freedom enables them to decide languages that are not regular, such as the p-adic languages with irrational parameters. As such, PAs are more powerful than both DFAs and NFAs (which are famously equally powerful). == Formal Definition == The probabilistic automaton may be defined as an extension of a nondeterministic finite automaton ( Q , Σ , δ , q 0 , F ) {\displaystyle (Q,\Sigma ,\delta ,q_{0},F)} , together with two probabilities: the probability P {\displaystyle P} of a particular state transition taking place, and with the initial state q 0 {\displaystyle q_{0}} replaced by a stochastic vector giving the probability of the automaton being in a given initial state. For the ordinary non-deterministic finite automaton, one has a finite set of states Q {\displaystyle Q} a finite set of input symbols Σ {\displaystyle \Sigma } a transition function δ : Q × Σ → ℘ ( Q ) {\displaystyle \delta :Q\times \Sigma \to \wp (Q)} a set of states F {\displaystyle F} distinguished as accepting (or final) states F ⊆ Q {\displaystyle F\subseteq Q} . Here, ℘ ( Q ) {\displaystyle \wp (Q)} denotes the power set of Q {\displaystyle Q} . By use of currying, the transition function δ : Q × Σ → ℘ ( Q ) {\displaystyle \delta :Q\times \Sigma \to \wp (Q)} of a non-deterministic finite automaton can be written as a membership function δ : Q × Σ × Q → { 0 , 1 } {\displaystyle \delta :Q\times \Sigma \times Q\to \{0,1\}} so that δ ( q , a , q ′ ) = 1 {\displaystyle \delta (q,a,q^{\prime })=1} if q ′ ∈ δ ( q , a ) {\displaystyle q^{\prime }\in \delta (q,a)} and 0 {\displaystyle 0} otherwise. The curried transition function can be understood to be a matrix with matrix entries [ θ a ] q q ′ = δ ( q , a , q ′ ) {\displaystyle \left[\theta _{a}\right]_{qq^{\prime }}=\delta (q,a,q^{\prime })} The matrix θ a {\displaystyle \theta _{a}} is then a square matrix, whose entries are zero or one, indicating whether a transition q → a q ′ {\displaystyle q{\stackrel {a}{\rightarrow }}q^{\prime }} is allowed by the NFA. Such a transition matrix is always defined for a non-deterministic finite automaton. The probabilistic automaton replaces these matrices by a family of right stochastic matrices P a {\displaystyle P_{a}} , for each symbol a in the alphabet Σ {\displaystyle \Sigma } so that the probability of a transition is given by [ P a ] q q ′ {\displaystyle \left[P_{a}\right]_{qq^{\prime }}} A state change from some state to any state must occur with probability one, of course, and so one must have ∑ q ′ [ P a ] q q ′ = 1 {\displaystyle \sum _{q^{\prime }}\left[P_{a}\right]_{qq^{\prime }}=1} for all input letters a {\displaystyle a} and internal states q {\displaystyle q} . The initial state of a probabilistic automaton is given by a row vector v {\displaystyle v} , whose components are the probabilities of the individual initial states q {\displaystyle q} , that add to 1: ∑ q [ v ] q = 1 {\displaystyle \sum _{q}\left[v\right]_{q}=1} The transition matrix acts on the right, so that the state of the probabilistic automaton, after consuming the input string a b c {\displaystyle abc} , would be v P a P b P c {\displaystyle vP_{a}P_{b}P_{c}} In particular, the state of a probabilistic automaton is always a stochastic vector, since the product of any two stochastic matrices is a stochastic matrix, and the product of a stochastic vector and a stochastic matrix is again a stochastic vector. This vector is sometimes called the distribution of states, emphasizing that it is a discrete probability distribution. Formally, the definition of a probabilistic automaton does not require the mechanics of the non-deterministic automaton, which may be dispensed with. Formally, a probabilistic automaton PA is defined as the tuple ( Q , Σ , P , v , F ) {\displaystyle (Q,\Sigma ,P,v,F)} . A Rabin automaton is one for which the initial distribution v {\displaystyle v} is a coordinate vector; that is, has zero for all but one entries, and the remaining entry being one. == Stochastic languages == The set of languages recognized by probabilistic automata are called stochastic languages. They include the regular languages as a subset. Let F = Q accept ⊆ Q {\displaystyle F=Q_{\text{accept}}\subseteq Q} be the set of "accepting" or "final" states of the automaton. By abuse of notation, Q accept {\displaystyle Q_{\text{accept}}} can also be understood to be the column vector that is the membership function for Q accept {\displaystyle Q_{\text{accept}}} ; that is, it has a 1 at the places corresponding to elements in Q accept {\displaystyle Q_{\text{accept}}} , and a zero otherwise. This vector may be contracted with the internal state probability, to form a scalar. The language recognized by a specific automaton is then defined as L η = { s ∈ Σ ∗ | v P s Q accept > η } {\displaystyle L_{\eta }=\{s\in \Sigma ^{}\vert vP_{s}Q_{\text{accept}}>\eta \}} where Σ ∗ {\displaystyle \Sigma ^{}} is the set of all strings in the alphabet Σ {\displaystyle \Sigma } (so that is the Kleene star). The language depends on the value of the cut-point η {\displaystyle \eta } , normally taken to be in the range 0 ≤ η < 1 {\displaystyle 0\leq \eta <1} . A language is called η-stochastic if and only if there exists some PA that recognizes the language, for fixed η {\displaystyle \eta } . A language is called stochastic if and only if there is some 0 ≤ η < 1 {\displaystyle 0\leq \eta <1} for which L η {\displaystyle L_{\eta }} is η-stochastic. A cut-point is said to be an isolated cut-point if and only if there exists a δ > 0 {\displaystyle \delta >0} such that | v P ( s ) Q accept − η | ≥ δ {\displaystyle \vert vP(s)Q_{\text{accept}}-\eta \vert \geq \delta } for all s ∈ Σ ∗ {\displaystyle s\in \Sigma ^{}} == Properties == Every regular language is stochastic, and more strongly, every regular language is η-stochastic. A weak converse is that every 0-stochastic language is regular; however, the general converse does not hold: there are stochastic languages that are not regular. Every η-stochastic language is stochastic, for some 0 < η < 1 {\displaystyle 0<\eta <1} . Every stochastic language is representable by a Rabin automaton. If η {\displaystyle \eta } is an isolated cut-point, then L η {\displaystyle L_{\eta }} is a regular language. == p-adic languages == The p-adic languages provide an example of a stochastic language that is not regular, and also show that the number of stochastic languages is uncountable. A p-adic language is defined as the set of strings L η ( p ) = { n 1 n 2 n 3 … | 0 ≤ n k < p and 0. n 1 n 2 n 3 … > η } {\displaystyle L_{\eta }(p)=\{n_{1}n_{2}n_{3}\ldots \vert 0\leq n_{k}
Top 10 AI Sales Assistants Compared (2026)
Looking for the best AI sales assistant? An AI sales assistant is software that uses machine learning to help you get more done — it can save you hours every week by automating repetitive work. Most options offer a generous free tier, with paid plans unlocking higher limits, faster processing, and team features. Whether you are a beginner or a pro, the right AI sales assistant slots into your workflow and pays for itself fast. This guide breaks down the top picks, their pros and cons, and who each one is best for.
Layers (digital image editing)
Layers are used in digital image editing to separate different elements of an image. A layer can be compared to a transparency on which imaging effects or images are applied and placed over or under an image. Today they are an integral feature of image editors. In the early days of computing, memory was at a premium and the idea of using multi-layered images was considered infeasible in personal computer applications as the tradeoffs were image size and color depth. As the price of memory fell it became feasible to apply the concept of layering to raster images. The first software known to apply the concept of layers was LALF, which was released in 1989 for the NEC PC-9801. LALF's terminology for layers is "cells", after the concept of drawing animation frames over-top of a stencil. Layers were introduced in Western markets by Fauve Matisse (later Macromedia xRes), and then available in Adobe Photoshop 3.0, in 1994, which lead to widespread adoption. In vector image editors that support animation, layers are used to further enable manipulation along a common timeline for the animation; in SVG images, the equivalent to layers are "groups". == Layer types == There are different kinds of layers, and not all of them exist in all programs. They represent a part of a picture, either as pixels or as modification instructions. They are stacked on top of each other, and depending on the order, determine the appearance of the final picture. In graphics software, layers are the different levels at which one can place an object or image file. In the program, layers can be stacked, merged, or defined when creating a digital image. Layers can be partially obscured allowing portions of images within a layer to be hidden or shown in a translucent manner within another image. Layers can also be used to combine two or more images into a single digital image. For the purpose of editing, working with layers allows for applying changes to just one specific layer. == Layer (basic) == The standard layer available to most programs consists of a rectangular, semitransparent picture which may be superimposed over other layers. Some programs require that layers cover the same area as the final canvas, but others offer layers of multiple sizes. Each layer may bear individual settings, such as opacity, blending modes, dynamic filters, and potentially hundreds of other properties. == Layer mask == A layer mask is linked to a layer and hides part of the layer from the picture. What is painted black on the layer mask will not be visible in the final picture. What is grey will be more or less transparent depending on the shade of grey. As the layer mask can be both edited and moved around independently of both the background layer and the layer it applies to, it gives the user the ability to test a lot of different combinations of overlay. == Adjustment layer == An adjustment layer typically applies a common effect like brightness or saturation to other layers. However, as the effect is stored in a separate layer, it is easy to try it out and switch between different alternatives, without changing the original layer. In addition, an adjustment layer can easily be edited, just like a layer mask, so an effect can be applied to just part of the image.
Christopher K. I. Williams
Christopher Kenneth Ingle Williams (born 1960) is a professor at the School of Informatics, University of Edinburgh, working in Artificial intelligence, and particularly the areas of Machine learning and Computer vision. == Education == Williams received a BA in Physics and Theoretical Physics from the University of Cambridge in 1982, followed by Part III Mathematics (1983). He did a MSc in Water Resources at the University of Newcastle-Upon-Tyne, then worked in Lesotho on low-cost sanitation. In 1988, he studied at the Department of Computer Science of the University of Toronto under the supervision of Geoffrey Hinton. He obtained his MSc and PhD both in computer science, in 1990 and 1994, respectively. == Career and research == In 1994, Williams moved to Aston University as a Research Fellow. He became a Lecturer in August 1995. He moved to the University of Edinburgh in July 1998 and became Reader in 2000. He obtained a Personal Chair in Machine Learning in 2005 in the School of Informatics. Williams has been a Fellow of the European Laboratory for Learning and Intelligent Systems (ELLIS) since 2019. Williams' research interests are in machine learning and computer vision. He has worked on new models for understanding time-series and images, and for finding structure in data. He is best known for his work on Gaussian processes and for the book Gaussian Processes for Machine Learning, co-authored with Carl Rasmussen. The book received the 2009 DeGroot Prize of the International Society for Bayesian Analysis. Williams was an organizer of the PASCAL Visual Object Classes (VOC) project (2005–2012) along with Mark Everingham, Luc van Gool, John Winn, and Andrew Zisserman. == Awards and honours == In 2021 Williams was elected a Fellow of the Royal Society of Edinburgh (FRSE).