Known-item search is a specialization of information exploration which represents the activities carried out by searchers who have a particular item in mind. In the context of library catalogs, known‐item search means a search for an item for which the author or title is known. Although the concept of known-item search originated in library science, it is now applied in the context of web search and other online search activities. Known-item search is distinguished from exploratory search, in which a searcher is unfamiliar with the domain of their search goal, unsure about the ways to achieve their goal, and/or unsure about what their goal is.
Swap chain
In computer graphics, a swap chain (also swapchain) is a series of virtual framebuffers used by the graphics card and graphics API for frame rate stabilization, stutter reduction, and several other purposes. Because of these benefits, many graphics APIs require the use of a swap chain. The swap chain usually exists in graphics memory, but it can exist in system memory as well. A swap chain with two buffers is a kind of double buffer. == Function == In every swap chain there are at least two buffers. The first framebuffer, the screenbuffer, is the buffer that is rendered to the output of the video card. The remaining buffers are known as backbuffers. Each time a new frame is displayed, the first backbuffer in the swap chain takes the place of the screenbuffer, this is called presentation or swapping. A variety of other actions may be taken on the previous screenbuffer and other backbuffers (if they exist). The screenbuffer may be simply overwritten or returned to the back of the swap chain for further processing. The action taken is decided by the client application and is API dependent. == Direct3D == Microsoft Direct3D implements a SwapChain class. Each host device has at least one swap chain assigned to it, and others may be created by the client application. The API provides three methods of swapping: copy, discard, and flip. When the SwapChain is set to flip, the screenbuffer is copied onto the last backbuffer, then all the existing backbuffers are copied forward in the chain. When copy is set, each backbuffer is copied forward, but the screenbuffer is not wrapped to the last buffer, leaving it unchanged. Flip does not work when there is only one backbuffer, as the screenbuffer is copied over the only backbuffer before it can be presented. In discard mode, the driver selects the best method. == Comparison with triple buffering == Outside the context of Direct3D, triple buffering refers to the technique of allowing an application to draw to whichever back buffer was least recently updated. This allows the application to always proceed with rendering, regardless of the pace at which frames are being drawn by the application or the pace at which frames are being sent to the display. Triple buffering may result in a frame being discarded without being displayed if two or more newer frames are completely rendered in the time it takes for one frame to be sent to the display. By contrast, Direct3D swap chains are a strict first-in, first-out queue, so every frame that is drawn by the application will be displayed even if newer frames are available. Direct3D does not implement a most-recent buffer swapping strategy, and Microsoft's documentation calls a Direct3D swap chain of three buffers "triple buffering". Triple buffering as described above is superior for interactive purposes such as gaming, but Direct3D swap chains of more than three buffers can be better for tasks such as presenting frames of a video where the time taken to decode each frame may be highly variable.
Physics-informed neural networks
In machine learning, physics-informed neural networks (PINNs), also referred to as theory-trained neural networks (TTNs), are a type of universal function approximator that can embed the knowledge of any physical laws that govern a given data-set in the learning process, and can be described by partial differential equations (PDEs). Low data availability for some biological and engineering problems limit the robustness of conventional machine learning models used for these applications. The prior knowledge of general physical laws acts in the training of neural networks (NNs) as a regularization agent that limits the space of admissible solutions, increasing the generalizability of the function approximation. This way, embedding this prior information into a neural network results in enhancing the information content of the available data, facilitating the learning algorithm to capture the right solution and to generalize well even with a low amount of training examples. Because they process continuous spatial and time coordinates and output continuous PDE solutions, they can be categorized as neural fields. == Function approximation == Most of the physical laws that govern the dynamics of a system can be described by partial differential equations. For example, the Navier–Stokes equations are a set of partial differential equations derived from the conservation laws (i.e., conservation of mass, momentum, and energy) that govern fluid mechanics. The solution of the Navier–Stokes equations with appropriate initial and boundary conditions allows the quantification of flow dynamics in a precisely defined geometry. However, these equations cannot be solved exactly and therefore numerical methods must be used (such as finite differences, finite elements and finite volumes). In this setting, these governing equations must be solved while accounting for prior assumptions, linearization, and adequate time and space discretization. Recently, solving the governing partial differential equations of physical phenomena using deep learning has emerged as a new field of scientific machine learning (SciML), leveraging the universal approximation theorem and high expressivity of neural networks. In general, deep neural networks could approximate any high-dimensional function given that sufficient training data are supplied. However, such networks do not consider the physical characteristics underlying the problem, and the level of approximation accuracy provided by them is still heavily dependent on careful specifications of the problem geometry as well as the initial and boundary conditions. Without this preliminary information, the solution is not unique and may lose physical correctness. To remedy this, Physics-Informed Neural Networks (PINNs) leverage governing physical equations in neural network training. Namely, PINNs are designed to be trained to satisfy the given training data as well as the imposed governing equations. In this fashion, a neural network can be guided with training datasets that do not necessarily need to be large or complete. An accurate solution of partial differential equations can potentially be found without knowing the boundary conditions. Therefore, with some knowledge about the physical characteristics of the problem and some form of training data (even sparse and incomplete), PINNs may be used for finding an optimal solution with high fidelity. PINNs can be applied to a wide range of problems in computational science, and are a pioneering technology leading to the development of new classes of numerical solvers for PDEs. PINNs can be thought of as a mesh-free alternative to traditional approaches (e.g., CFD for fluid dynamics), and new data-driven approaches for model inversion and system identification. Notably, a trained PINN network can be used to predict values on simulation grids of different resolutions without needing to be retrained. Additionally, the derivatives used in the partial differential equations can be computed using automatic differentiation (AD), which is assessed to be superior to numerical or symbolic differentiation. == Modeling and computation == A general nonlinear partial differential equation can be written as: u t + N [ u ; λ ] = 0 , x ∈ Ω , t ∈ [ 0 , T ] {\displaystyle u_{t}+{\mathcal {N}}[u;\lambda ]=0,\quad x\in \Omega ,\quad t\in [0,T]} where u ( t , x ) {\displaystyle u(t,x)} denotes the solution, N [ ⋅ ; λ ] {\displaystyle {\mathcal {N}}[\cdot ;\lambda ]} is a nonlinear operator parameterized by λ {\displaystyle \lambda } , and Ω {\displaystyle \Omega } is a subset of R D {\displaystyle \mathbb {R} ^{D}} . This general form of governing equations summarizes a wide range of problems in mathematical physics, such as conservative laws, diffusion process, advection-diffusion systems, and kinetic equations. Given noisy measurements of a generic dynamic system described by the equation above, PINNs can be designed to solve two classes of problems: data-driven solutions of partial differential equations data-driven discovery of partial differential equations === Data-driven solution of partial differential equations === The data-driven solution of PDE computes the hidden state u ( t , x ) {\displaystyle u(t,x)} of the system given boundary data and/or measurements z {\displaystyle z} , and fixed model parameters λ {\displaystyle \lambda } . We solve: u t + N [ u ] = 0 , x ∈ Ω , t ∈ [ 0 , T ] {\displaystyle u_{t}+{\mathcal {N}}[u]=0,\quad x\in \Omega ,\quad t\in [0,T]} . by defining the residual f ( t , x ) {\displaystyle f(t,x)} as: f := u t + N [ u ] {\displaystyle f:=u_{t}+{\mathcal {N}}[u]} , and approximating u ( t , x ) {\displaystyle u(t,x)} by a deep neural network. This network can be differentiated using automatic differentiation. The parameters of u ( t , x ) {\displaystyle u(t,x)} and f ( t , x ) {\displaystyle f(t,x)} can be then learned by minimizing the following loss function L tot {\displaystyle L_{\text{tot}}} : L tot = L u + L f {\displaystyle L_{\text{tot}}=L_{u}+L_{f}} where: L u = ‖ u − z ‖ Γ {\displaystyle L_{u}=\Vert u-z\Vert _{\Gamma }} is the error between the PINN u ( t , x ) {\displaystyle u(t,x)} and the set of boundary conditions and measured data on the set of points Γ {\displaystyle \Gamma } where the boundary conditions and data are defined. L f = ‖ f ‖ Γ {\displaystyle L_{f}=\Vert f\Vert _{\Gamma }} is the mean-squared error of the residual function. This second term encourages the PINN to learn the structural information expressed by the PDE during the training process. This approach has been used to yield computationally efficient physics-informed surrogate models with applications in the forecasting of physical processes, model predictive control, multi-physics and multi-scale modeling, and simulation. It has been shown to converge to the solution of the PDE. === Data-driven discovery of partial differential equations === Given noisy and incomplete measurements z {\displaystyle z} of the state of the system, the data-driven discovery of PDEs results in computing the unknown state u ( t , x ) {\displaystyle u(t,x)} and learning model parameters λ {\displaystyle \lambda } that best describe the observed data: u t + N [ u ; λ ] = 0 , x ∈ Ω , t ∈ [ 0 , T ] {\displaystyle u_{t}+{\mathcal {N}}[u;\lambda ]=0,\quad x\in \Omega ,\quad t\in [0,T]} By defining f ( t , x ) {\displaystyle f(t,x)} as: f := u t + N [ u ; λ ] = 0 {\displaystyle f:=u_{t}+{\mathcal {N}}[u;\lambda ]=0} , and approximating u ( t , x ) {\displaystyle u(t,x)} by a deep neural network, f ( t , x ) {\displaystyle f(t,x)} results in a PINN. This network can be derived using automatic differentiation. The parameters of u ( t , x ) {\displaystyle u(t,x)} and f ( t , x ) {\displaystyle f(t,x)} , together with the parameter λ {\displaystyle \lambda } of the differential operator can be then learned by minimizing the following loss function L tot {\displaystyle L_{\text{tot}}} : L tot = L u + L f {\displaystyle L_{\text{tot}}=L_{u}+L_{f}} where: L u = ‖ u − z ‖ Γ {\displaystyle L_{u}=\Vert u-z\Vert _{\Gamma }} , with u {\displaystyle u} and z {\displaystyle z} state solutions and measurements at sparse location Γ {\displaystyle \Gamma } , respectively. L f = ‖ f ‖ Γ {\displaystyle L_{f}=\Vert f\Vert _{\Gamma }} is the residual function. This second term requires the structured information represented by the partial differential equations to be satisfied in the training process. This strategy allows for discovering dynamic models described by nonlinear PDEs assembling computationally efficient and fully differentiable surrogate models that may find application in predictive forecasting, control, and data assimilation. == Extensions and applications == === For piece-wise function approximation === PINNs are unable to approximate PDEs that have strong non-linearity or sharp gradients (such as those that commonly occur in practical fluid flow problems). Piecewise approximation has been an old practic
Document mosaicing
Document mosaicing is a process that stitches multiple, overlapping snapshot images of a document together to produce one large, high resolution composite. The document is slid under a stationary, over-the-desk camera by hand until all parts of the document are snapshotted by the camera's field of view. As the document slid under the camera, all motion of the document is coarsely tracked by the vision system. The document is periodically snapshotted such that the successive snapshots are overlap by about 50%. The system then finds the overlapped pairs and stitches them together repeatedly until all pairs are stitched together as one piece of document. The document mosaicing can be divided into four main processes. Tracking Feature detecting Correspondences establishing Images mosaicing. == Tracking (simple correlation process) == In this process, the motion of the document slid under the camera is coarsely tracked by the system. Tracking is performed by a process called simple correlation process. In the first frame of snapshots, a small patch is extracted from the center of the image as a correlation template. The correlation process is performed in the four times size of the patch area of the next frame. The motion of the paper is indicated by the peak in the correlation function. The peak in the correlation function indicates the motion of the paper. The template is resampled from this frame and the tracking continues until the template reaches the edge of the document. After the template reaches the edge of the document, another snapshot is taken and the tracking process performs repeatedly until the whole document is imaged. The snapshots are stored in an ordered list to facilitate pairing the overlapped images in later processes. == Feature detecting for efficient matching == Feature detection is the process of finding the transformation that aligns one image with another. There are two main approaches for feature detection. Feature-based approach : Motion parameters are estimated from point correspondences. This approach is suitable for the case that there is plenty supply of stable and detectable features. Featureless approach : When the motion between the two images is small, the motion parameters are estimated using optical flow. On the other hand, when the motion between the two images is large, the motion parameters are estimated using generalised cross-correlation. However, this approach requires a computationally expensive resources. Each image is segmented into a hierarchy of columns, lines, and words to match the organised sets of features across images. Skew angle estimation and columns, lines and words finding are the examples of feature detection operations. === Skew angle estimation === Firstly, the angle that the rows of text make with the image raster lines (skew angle) is estimated. It is assumed to lie in the range of ±20°. A small patch of text in the image is selected randomly and then rotated in the range of ±20° until the variance of the pixel intensities of the patch summed along the raster lines is maximised. To ensure that the found skew angle is accurate, the document mosaic system performs calculation at many image patches and derive the final estimation by finding the average of the individual angles weighted by the variance of the pixel intensities of each patch. === Columns, lines and words finding === In this operation, the de-skewed document is intuitively segmented into a hierarchy of columns, lines and words. The sensitivity to illumination and page coloration of the de-skewed document can be removed by applying a Sobel operator to the de-skewed image and thresholding the output to obtain the binary gradient, de-skewed image. The operation can be roughly separated into 3 steps: column segmentation, line segmentation and word segmentation. Columns are easily segmented from the binary gradient, de-skewed images by summing pixels vertically. Baselines of each row are segmented in the same way as the column segmentation process but horizontally. Finally, individual words are segmented by applying the vertical process at each segmented row. These segmentations are important because the document mosaic is created by matching the lower right corners of words in overlapping images pair. Moreover, the segmentation operation can organize the list of images in the context of a hierarchy of rows and column reliably. The segmentation operation involves a considerable amount of summing in the binary gradient, de-skewed images, which done by construct a matrix of partial sums whose elements are given by p i y = ∑ u = 1 i ∑ v = 1 j b u v {\displaystyle p_{iy}=\sum _{u=1}^{i}\sum _{v=1}^{j}b_{uv}} The matrix of partial sums is calculated in one pass through the binary gradient, de-skewed image. ∑ u = u 1 u 2 ∑ v = v 1 v 2 b u v = p u 2 v 2 + p u 1 v 1 − p u 1 v 2 − p u 2 v 1 {\displaystyle \sum _{u=u_{1}}^{u_{2}}\sum _{v=v_{1}}^{v_{2}}b_{uv}=p_{u_{2}v_{2}}+p_{u_{1}v_{1}}-p_{u_{1}v_{2}}-p_{u_{2}v_{1}}} == Correspondences establishing == The two images are now organized in hierarchy of linked lists in following structure : image=list of columns row=list of words column=list of row word=length (in pixels) At the bottom of the structure, the length of each word is recorded for establishing correspondence between two images to reduce to search only the corresponding structures for the groups of words with the matching lengths. === Seed match finding === A seed match finding is done by comparing each row in image1 with each row in image2. The two rows are then compared to each other by every word. If the length (in pixel) of the two words (one from image1 and one from image2) and their immediate neighbours agree with each other within a predefined tolerance threshold (5 pixels, for example), then they are assumed to match. The row of each image is assumed a match if there are three or more word matches between the two rows. The seed match finding operation is terminated when two pairs of consecutive row match are found. === Match list building === After finishing a seed match finding operation, the next process is to build the match list to generate the correspondences points of the two images. The process is done by searching the matching pairs of rows away from the seed row. == Images mosaicing == Given the list of corresponding points of the two images, finding the transformation of the overlapping portion of the images is the next process. Assuming a pinhole camera model, the transformation between pixels (u,v) of image 1 and pixels (u0, v0) of image 2 is demonstrated by a plane-to-plane projectivity. [ s u ′ s v ′ s ] = [ p 11 p 12 p 13 p 21 p 22 p 23 p 31 p 32 1 ] [ u v 1 ] E q .1 {\displaystyle \left[{\begin{array}{c}su'\\sv'\\s\end{array}}\right]=\left[{\begin{array}{ccc}p_{11}&p_{12}&p_{13}\\p_{21}&p_{22}&p_{23}\\p_{31}&p_{32}&1\end{array}}\right]\left[{\begin{array}{c}u\\v\\1\end{array}}\right]\qquad Eq.1} The parameters of the projectivity is found from four pairs of matching points. RANSAC regression technique is used to reject outlying matches and estimate the projectivity from the remaining good matches. The projectivity is fine-tuned using correlation at the corners of the overlapping portion to obtain four correspondences to sub-pixel accuracy. Therefore, image1 is then transformed into image2's coordinate system using Eq.1. The typical result of the process is shown in Figure 5. === Many images coping === Finally, the whole page composition is built up by mapping all the images into the coordinate system of an "anchor" image, which is normally the one nearest the page center. The transformations to the anchor frame are calculated by concatenating the pair-wise transformations found earlier. The raw document mosaic is shown in Figure 6. However, there might be a problem of non-consecutive images that are overlap. This problem can be solved by performing Hierarchical sub-mosaics. As shown in Figure 7, image1 and image2 are registered, as are image3 and image4, creating two sub-mosaics. These two sub-mosaics are later stitched together in another mosaicing process. == Applied areas == There are various areas that the technique of document mosaicing can be applied to such as : Text segmentation of images of documents Document Recognition Interaction with paper on the digital desk Video mosaics for virtual environments Image registration techniques == Relevant research papers == Huang, T.S.; Netravali, A.N. (1994). "Motion and structure from feature correspondences: A review". Proceedings of the IEEE. 82 (2): 252–268. doi:10.1109/5.265351. D.G. Lowe. [1] Perceptual Organization and Visual Recognition. Kluwer Academic Publishers, Boston, 1985. Irani, M.; Peleg, S. (1991). "Improving resolution by image registration". CVGIP: Graphical Models and Image Processing. 53 (3): 231–239. doi:10.1016/1049-9652(91)90045-L. S2CID 4834546. Shivakumara, P.; Kumar, G. Hemantha; Guru, D. S.; Nagabhushan, P. (2006). "
Mistral Vibe
Mistral Vibe or Vibe (Le Chat until May 2026), is a chatbot that uses generative artificial intelligence developed in France by Mistral AI. Mistral Vibe is available in iOS and Android. Its services are operated on a freemium model. == History == In February 2024, Mistral AI released Le Chat. In January 2025, Mistral AI made a content deal with Agence France-Presse (AFP) that lets Le Chat query AFP's entire archive dating back to 1983. On 6 February 2025, a mobile app for Le Chat was released for iOS and Android, and a subscription tier, Pro, was introduced at a cost of $14.99 per month. In July 2025, Mistral AI released Voxtral, an open-source language model that understands and generates audio. Mistral introduced a voice mode for chatting that uses Voxtral, and projects, which allows grouping chats and files. In September 2025, Le Chat introduced the capability to remember previous conversations. In May 2026, Mistral AI announced the rebrand from Le Chat to Mistral Vibe and new features were introduced at the same time.
WiPay
WiPay is a Caribbean-based payment technology company that specializes in electronic payments for businesses. WiPay was founded in 2016 by Aldwyn Wayne Jr., a Trinidadian businessman and graduate of Georgia Tech Institute. In September 2019, WiPay partnered with MasterCard. As a result, WiPay became the only licensed Payment Facilitator (PAYFAC) on both the MasterCard and Visa networks in the region.
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