The document summarizes linear dynamical models and tracking using the Kalman filter. It discusses prediction using the previous state estimate, correction using the new measurement, and modeling the system and measurements as Gaussian processes. The key steps of prediction using the dynamic model and correction by updating the state estimate based on the new measurement are derived for a linear system with a one-dimensional state vector.
1) The document discusses query suggestion techniques using hitting time on graphs to model relationships between queries, reformulations, and URLs.
2) It presents algorithms for calculating the hitting time between nodes in a graph and using this to determine the likelihood of queries and URLs being related.
3) Experimental results on benchmark datasets show the hitting time approach achieves good performance for query suggestion compared to other methods.
The document discusses probabilistic reasoning in intelligent systems using Bayesian networks. It covers the following topics:
1. Updating beliefs in a network by propagating probabilities between connected nodes using conditional probability tables.
2. Computing the posterior probability at a node given evidence elsewhere in the network by multiplying the prior at the node by the likelihood of the evidence.
3. Updating beliefs in chains, trees, and polytrees by propagating probabilities along the edges of the graph structure.
The document describes a Hamiltonian with terms including Ji,j|ωiωj| and Ei|ωiωi| that depends on parameters ∆/J and ω. It studies the behavior of the system as ∆/J increases from 0 to greater than 6, including plots of the momentum distribution |P(k)|2 that show it spreading out over more values of k/k1. The dependence of the system on other parameters like α, s1, and s2 is also examined through additional plots.
Approximative Bayesian Computation (ABC) methods allow approximating intractable likelihoods in Bayesian inference. ABC rejection sampling simulates parameters from the prior and keeps those where simulated data is close to observed data. ABC Markov chain Monte Carlo creates a Markov chain over the parameters where proposed moves are accepted if simulated data is similar to observed. Population Monte Carlo and ABC-MCMC improve on rejection sampling by using sequential importance sampling and MCMC moves to propose parameters in high density regions.
Scientific Computing with Python Webinar 9/18/2009:Curve FittingEnthought, Inc.
This webinar will provide an overview of the tools that SciPy and NumPy provide for regression analysis including linear and non-linear least-squares and a brief look at handling other error metrics. We will also demonstrate simple GUI tools that can make some problems easier and provide a quick overview of the new Scikits package statsmodels whose API is maturing in a separate package but should be incorporated into SciPy in the future.
This document describes a clustering procedure and nonparametric mixture estimation. It introduces a mixture density model where the goal is to efficiently estimate the mixture weights (αi) and component densities (fi). A two-stage clustering algorithm is proposed: 1) perform clustering on covariates (X) to estimate labels (Ik), and 2) estimate component densities (fi) using kernel density estimation within each cluster. The performance of this approach depends on the clustering method's misclassification error. A toy example with two components having disjoint support densities for X is provided to illustrate the model.
This document introduces the concept of conditional expectation and stochastic calculus. It defines conditional expectation as the projection of a random variable X onto the sub-σ-algebra generated by another random variable or process Y. It must minimize the mean squared error between X and the projected variable. Properties like linearity and monotonicity are proven. Conditional expectation allows incorporating observable information to make optimal guesses about unobserved variables. Martingales, which generalize random walks, also play an important role in stochastic calculus.
The document provides an overview of probability theory and random variables including:
1) It defines probability as a measure of the chance of obtaining a particular outcome from an event. Common properties of probability such as mutually exclusive events and conditional probability are also covered.
2) Random variables are introduced as rules that assign real numbers to possible outcomes of an experiment. Both discrete and continuous random variables are defined.
3) Key concepts related to random variables are summarized including the cumulative distribution function, probability density function, expected value, variance, and common distributions like the uniform, binomial, and Gaussian distributions.
4) Finally, random processes are defined as sets of random variables indexed by time, with properties like the mean
1) The document discusses query suggestion techniques using hitting time on graphs to model relationships between queries, reformulations, and URLs.
2) It presents algorithms for calculating the hitting time between nodes in a graph and using this to determine the likelihood of queries and URLs being related.
3) Experimental results on benchmark datasets show the hitting time approach achieves good performance for query suggestion compared to other methods.
The document discusses probabilistic reasoning in intelligent systems using Bayesian networks. It covers the following topics:
1. Updating beliefs in a network by propagating probabilities between connected nodes using conditional probability tables.
2. Computing the posterior probability at a node given evidence elsewhere in the network by multiplying the prior at the node by the likelihood of the evidence.
3. Updating beliefs in chains, trees, and polytrees by propagating probabilities along the edges of the graph structure.
The document describes a Hamiltonian with terms including Ji,j|ωiωj| and Ei|ωiωi| that depends on parameters ∆/J and ω. It studies the behavior of the system as ∆/J increases from 0 to greater than 6, including plots of the momentum distribution |P(k)|2 that show it spreading out over more values of k/k1. The dependence of the system on other parameters like α, s1, and s2 is also examined through additional plots.
Approximative Bayesian Computation (ABC) methods allow approximating intractable likelihoods in Bayesian inference. ABC rejection sampling simulates parameters from the prior and keeps those where simulated data is close to observed data. ABC Markov chain Monte Carlo creates a Markov chain over the parameters where proposed moves are accepted if simulated data is similar to observed. Population Monte Carlo and ABC-MCMC improve on rejection sampling by using sequential importance sampling and MCMC moves to propose parameters in high density regions.
Scientific Computing with Python Webinar 9/18/2009:Curve FittingEnthought, Inc.
This webinar will provide an overview of the tools that SciPy and NumPy provide for regression analysis including linear and non-linear least-squares and a brief look at handling other error metrics. We will also demonstrate simple GUI tools that can make some problems easier and provide a quick overview of the new Scikits package statsmodels whose API is maturing in a separate package but should be incorporated into SciPy in the future.
This document describes a clustering procedure and nonparametric mixture estimation. It introduces a mixture density model where the goal is to efficiently estimate the mixture weights (αi) and component densities (fi). A two-stage clustering algorithm is proposed: 1) perform clustering on covariates (X) to estimate labels (Ik), and 2) estimate component densities (fi) using kernel density estimation within each cluster. The performance of this approach depends on the clustering method's misclassification error. A toy example with two components having disjoint support densities for X is provided to illustrate the model.
This document introduces the concept of conditional expectation and stochastic calculus. It defines conditional expectation as the projection of a random variable X onto the sub-σ-algebra generated by another random variable or process Y. It must minimize the mean squared error between X and the projected variable. Properties like linearity and monotonicity are proven. Conditional expectation allows incorporating observable information to make optimal guesses about unobserved variables. Martingales, which generalize random walks, also play an important role in stochastic calculus.
The document provides an overview of probability theory and random variables including:
1) It defines probability as a measure of the chance of obtaining a particular outcome from an event. Common properties of probability such as mutually exclusive events and conditional probability are also covered.
2) Random variables are introduced as rules that assign real numbers to possible outcomes of an experiment. Both discrete and continuous random variables are defined.
3) Key concepts related to random variables are summarized including the cumulative distribution function, probability density function, expected value, variance, and common distributions like the uniform, binomial, and Gaussian distributions.
4) Finally, random processes are defined as sets of random variables indexed by time, with properties like the mean
This document provides an introduction to stochastic calculus. It begins with a review of key probability concepts such as the Lebesgue integral, change of measure, and the Radon-Nikodym derivative. It then discusses information and σ-algebras, including filtrations and adapted processes. Conditional expectation is explained. The document concludes by introducing random walks and their connection to Brownian motion through the scaled random walk process. Key concepts such as martingales and quadratic variation are defined.
This document derives several important trigonometric identities by considering a right triangle with angle θ and using the Pythagorean theorem and definitions of trigonometric functions. It shows that sin2θ + cos2θ = 1, which can be rearranged to obtain other important identities relating sin, cos, tan, sec, and cosec.
The document discusses trigonometric functions on the unit circle. It defines trig ratios for angles in each of the four quadrants using right triangles formed with the point (x,y) and the origin. Key identities presented are:
1) tanθ = sinθ/cosθ
2) sin2θ + cos2θ = 1
The signs of the trig functions depend on the quadrant, with trig ratios being positive in Quadrant I and changing appropriately in other quadrants based on the signs of x and y.
There are three possible ROC's:
1. Outside all poles (a, b, c)
2. Between innermost and outermost pole
3. Inside all poles
So the possible ROC's are:
1. Outside circle through a, b, c
2. Annular region between a, c
3. Inside circle through a, b, c
a b c Re
The z-Transform
Important z-Transform Pairs
Important z-Transform Pairs
1. Unit Impulse: δ(n)
1, if n = 0
δ(n) = 0, otherwise
1
X(z) =
2.
On the solvability of a system of forward-backward linear equations with unbo...Nikita V. Artamonov
The document discusses a system of forward-backward linear evolution equations (FBEE) with unbounded operator coefficients. It introduces the necessary mathematical framework including a triple of Banach spaces and associated operators. It then defines the system of FBEE, discusses mild solutions, and relates it to a differential operator Riccati equation. The main result is a theorem stating that under certain assumptions on the operators, including accretivity of A, the Riccati equation has a unique mild solution.
1. Gibbs sampling is a technique for drawing samples from probability distributions by iteratively sampling each variable conditioned on the current values of the other variables. It can be used to sample from Markov random fields and Bayesian networks.
2. An Ising model is a Markov random field with binary variables on a grid that are correlated with their neighbors. Gibbs sampling in an Ising model samples each variable based on its neighbors' current values.
3. Boltzmann machines generalize the Ising model to arbitrary graph structures between variables. Restricted Boltzmann machines and Hopfield networks are specific types of Boltzmann machines.
1. The document describes Anchor Graph Hashing (AGH), a method for learning binary codes for approximate nearest neighbor search using graphs.
2. AGH constructs an anchor graph from a set of anchor points and learns binary codes by solving a graph partitioning problem on the anchor graph.
3. AGH has time and space complexities that are sublinear in the number of data points for training and efficient computation for out-of-sample extensions.
1. This document provides an overview of key probability and statistics concepts covered on actuarial exams P and FM.
2. It covers topics like probability spaces, random variables, expectations, distributions, and functions including CDFs, PDFs, moments, and transformations.
3. Formulas and properties are presented for concepts like independence, conditional probability, multivariate distributions, the central limit theorem, and more.
On Foundations of Parameter Estimation for Generalized Partial Linear Models ...SSA KPI
1) The document discusses estimation methods for generalized linear models (GLMs) and generalized partial linear models (GPLMs). 2) GPLMs extend GLMs by adding a single nonparametric component to the linear predictor. 3) Parameter estimation for GPLMs is performed by maximizing a penalized likelihood function, where the penalty term controls the tradeoff between model fit and smoothness of the nonparametric component. 4) An iterative algorithm such as Newton-Raphson is used to solve the penalized maximum likelihood estimation problem.
This document discusses unconditionally stable finite-difference time-domain (FDTD) methods for solving Maxwell's equations numerically. It outlines FDTD algorithms such as Yee's method from 1966 which discretize the equations on a staggered grid. It also discusses the von Neumann stability analysis and compares implicit Crank-Nicolson and alternating-direction implicit methods to conventional explicit FDTD methods. The document notes the advantages of unconditionally stable methods but also mentions potential disadvantages.
New Mathematical Tools for the Financial SectorSSA KPI
AACIMP 2010 Summer School lecture by Gerhard Wilhelm Weber. "Applied Mathematics" stream. "Modern Operational Research and Its Mathematical Methods with a Focus on Financial Mathematics" course. Part 5.
More info at http://summerschool.ssa.org.ua
This document discusses Approximate Bayesian Computation (ABC), a simulation-based method for conducting Bayesian inference when the likelihood function is intractable or impossible to evaluate directly. ABC produces an approximation of the posterior distribution by simulating data under different parameter values and accepting simulations that match the observed data. The document provides background on how ABC originated from population genetics models and outlines some of the advances in ABC, including how it can be used as an inference machine to estimate parameters from simulated data.
The document describes methods for tomographic focusing using polarimetric SAR (PolSAR) data, including:
1) A hybrid spectral approach using CAPON and weighted signal subspace fitting to estimate volume boundaries and ground topography from tropical forest data.
2) A single-baseline PolInSAR technique using an RVOG coherence model to retrieve ground elevation and volume coherence from the data.
3) Experimental results applying these methods to P-band PolSAR data collected over tropical forests in Paracou, France.
An Introduction to HSIC for Independence TestingYuchi Matsuoka
This document introduces Hilbert-Schmidt Independence Criterion (HSIC) for testing independence between random variables. HSIC embeds probability distributions into reproducing kernel Hilbert spaces and computes the distance between joint and product distributions using the Maximum Mean Discrepancy. It presents HSIC as a completely nonparametric measure of dependence that is applicable to high dimensional data. The document outlines how to compute HSIC from samples and discusses its relationship to U-statistics, providing an independence test using HSIC with permutations.
This document discusses approximate Bayesian computation (ABC). ABC allows Bayesian inference when the likelihood function is intractable or impossible to evaluate directly. It introduces ABC, describes how it originated from population genetics models, and outlines some of its limitations and advances, including various related computational methods like ABC with empirical likelihoods. The document also examines how ABC relates to other simulation-based statistical methods and considers perspectives on how Bayesian ABC can be.
Certificates of appreciation and trophiesSteve Vorster
South African Police Services from multiple regions honored Peermont resorts for their sustainable and crime prevention programs. Peermont received 15 letters of appreciation and awards from police services over several years for their efforts combating crime, including programs in informal settlements helping women and children. Peermont leadership was also recognized for their work with police.
Este documento define una sucesión geométrica como una secuencia de números donde la razón entre términos consecutivos es constante. Explica que la fórmula para el término general de una sucesión geométrica es an = a1 * rn-1, donde a1 es el primer término, n es el número de términos, y r es la razón constante. También muestra cómo calcular la suma de los términos de una sucesión geométrica y provee un ejemplo para hallar el término general de una sucesión dada.
This document provides an introduction to stochastic calculus. It begins with a review of key probability concepts such as the Lebesgue integral, change of measure, and the Radon-Nikodym derivative. It then discusses information and σ-algebras, including filtrations and adapted processes. Conditional expectation is explained. The document concludes by introducing random walks and their connection to Brownian motion through the scaled random walk process. Key concepts such as martingales and quadratic variation are defined.
This document derives several important trigonometric identities by considering a right triangle with angle θ and using the Pythagorean theorem and definitions of trigonometric functions. It shows that sin2θ + cos2θ = 1, which can be rearranged to obtain other important identities relating sin, cos, tan, sec, and cosec.
The document discusses trigonometric functions on the unit circle. It defines trig ratios for angles in each of the four quadrants using right triangles formed with the point (x,y) and the origin. Key identities presented are:
1) tanθ = sinθ/cosθ
2) sin2θ + cos2θ = 1
The signs of the trig functions depend on the quadrant, with trig ratios being positive in Quadrant I and changing appropriately in other quadrants based on the signs of x and y.
There are three possible ROC's:
1. Outside all poles (a, b, c)
2. Between innermost and outermost pole
3. Inside all poles
So the possible ROC's are:
1. Outside circle through a, b, c
2. Annular region between a, c
3. Inside circle through a, b, c
a b c Re
The z-Transform
Important z-Transform Pairs
Important z-Transform Pairs
1. Unit Impulse: δ(n)
1, if n = 0
δ(n) = 0, otherwise
1
X(z) =
2.
On the solvability of a system of forward-backward linear equations with unbo...Nikita V. Artamonov
The document discusses a system of forward-backward linear evolution equations (FBEE) with unbounded operator coefficients. It introduces the necessary mathematical framework including a triple of Banach spaces and associated operators. It then defines the system of FBEE, discusses mild solutions, and relates it to a differential operator Riccati equation. The main result is a theorem stating that under certain assumptions on the operators, including accretivity of A, the Riccati equation has a unique mild solution.
1. Gibbs sampling is a technique for drawing samples from probability distributions by iteratively sampling each variable conditioned on the current values of the other variables. It can be used to sample from Markov random fields and Bayesian networks.
2. An Ising model is a Markov random field with binary variables on a grid that are correlated with their neighbors. Gibbs sampling in an Ising model samples each variable based on its neighbors' current values.
3. Boltzmann machines generalize the Ising model to arbitrary graph structures between variables. Restricted Boltzmann machines and Hopfield networks are specific types of Boltzmann machines.
1. The document describes Anchor Graph Hashing (AGH), a method for learning binary codes for approximate nearest neighbor search using graphs.
2. AGH constructs an anchor graph from a set of anchor points and learns binary codes by solving a graph partitioning problem on the anchor graph.
3. AGH has time and space complexities that are sublinear in the number of data points for training and efficient computation for out-of-sample extensions.
1. This document provides an overview of key probability and statistics concepts covered on actuarial exams P and FM.
2. It covers topics like probability spaces, random variables, expectations, distributions, and functions including CDFs, PDFs, moments, and transformations.
3. Formulas and properties are presented for concepts like independence, conditional probability, multivariate distributions, the central limit theorem, and more.
On Foundations of Parameter Estimation for Generalized Partial Linear Models ...SSA KPI
1) The document discusses estimation methods for generalized linear models (GLMs) and generalized partial linear models (GPLMs). 2) GPLMs extend GLMs by adding a single nonparametric component to the linear predictor. 3) Parameter estimation for GPLMs is performed by maximizing a penalized likelihood function, where the penalty term controls the tradeoff between model fit and smoothness of the nonparametric component. 4) An iterative algorithm such as Newton-Raphson is used to solve the penalized maximum likelihood estimation problem.
This document discusses unconditionally stable finite-difference time-domain (FDTD) methods for solving Maxwell's equations numerically. It outlines FDTD algorithms such as Yee's method from 1966 which discretize the equations on a staggered grid. It also discusses the von Neumann stability analysis and compares implicit Crank-Nicolson and alternating-direction implicit methods to conventional explicit FDTD methods. The document notes the advantages of unconditionally stable methods but also mentions potential disadvantages.
New Mathematical Tools for the Financial SectorSSA KPI
AACIMP 2010 Summer School lecture by Gerhard Wilhelm Weber. "Applied Mathematics" stream. "Modern Operational Research and Its Mathematical Methods with a Focus on Financial Mathematics" course. Part 5.
More info at http://summerschool.ssa.org.ua
This document discusses Approximate Bayesian Computation (ABC), a simulation-based method for conducting Bayesian inference when the likelihood function is intractable or impossible to evaluate directly. ABC produces an approximation of the posterior distribution by simulating data under different parameter values and accepting simulations that match the observed data. The document provides background on how ABC originated from population genetics models and outlines some of the advances in ABC, including how it can be used as an inference machine to estimate parameters from simulated data.
The document describes methods for tomographic focusing using polarimetric SAR (PolSAR) data, including:
1) A hybrid spectral approach using CAPON and weighted signal subspace fitting to estimate volume boundaries and ground topography from tropical forest data.
2) A single-baseline PolInSAR technique using an RVOG coherence model to retrieve ground elevation and volume coherence from the data.
3) Experimental results applying these methods to P-band PolSAR data collected over tropical forests in Paracou, France.
An Introduction to HSIC for Independence TestingYuchi Matsuoka
This document introduces Hilbert-Schmidt Independence Criterion (HSIC) for testing independence between random variables. HSIC embeds probability distributions into reproducing kernel Hilbert spaces and computes the distance between joint and product distributions using the Maximum Mean Discrepancy. It presents HSIC as a completely nonparametric measure of dependence that is applicable to high dimensional data. The document outlines how to compute HSIC from samples and discusses its relationship to U-statistics, providing an independence test using HSIC with permutations.
This document discusses approximate Bayesian computation (ABC). ABC allows Bayesian inference when the likelihood function is intractable or impossible to evaluate directly. It introduces ABC, describes how it originated from population genetics models, and outlines some of its limitations and advances, including various related computational methods like ABC with empirical likelihoods. The document also examines how ABC relates to other simulation-based statistical methods and considers perspectives on how Bayesian ABC can be.
Certificates of appreciation and trophiesSteve Vorster
South African Police Services from multiple regions honored Peermont resorts for their sustainable and crime prevention programs. Peermont received 15 letters of appreciation and awards from police services over several years for their efforts combating crime, including programs in informal settlements helping women and children. Peermont leadership was also recognized for their work with police.
Este documento define una sucesión geométrica como una secuencia de números donde la razón entre términos consecutivos es constante. Explica que la fórmula para el término general de una sucesión geométrica es an = a1 * rn-1, donde a1 es el primer término, n es el número de términos, y r es la razón constante. También muestra cómo calcular la suma de los términos de una sucesión geométrica y provee un ejemplo para hallar el término general de una sucesión dada.
John deere 9550 and 9550 SH self-p rolled combine parts catalogPartCatalogs Net
This document provides parts catalog information for the John Deere 9550 and 9550SH self-propelled combines. It includes an index of sections covering different components of the combines, as well as notes on identifying information like serial numbers. Key parts like the engine, grain tank, and threshing/cleaning components are also referenced.
The document summarizes linear dynamical models and tracking using the Kalman filter. It discusses prediction using the previous state estimate, correction using the new measurement, and representing the state as a Gaussian distribution. Key steps include predicting the next state using the dynamic model, then correcting the prediction using the new measurement via Bayes' rule to get an updated state estimate. Calculations involve multiplying and summing Gaussian probability densities.
Un sismo es un movimiento repentino de la corteza terrestre causado por la liberación de energía en las fallas geológicas bajo la superficie de la Tierra. Los sismos se originan debido a la acumulación de energía y su repentina liberación a lo largo de las fallas, lo que causa vibraciones en la corteza. Ante un sismo, es importante protegerse debajo de una mesa o marco de puerta y alejarse de objetos que puedan caer.
1) The document provides details of an assignment and lessons from a math notebook. It includes a warm-up with fraction, decimal, and order of operations questions.
2) The lesson explains that to divide by a decimal, you move the decimal point to the right in the divisor.
3) Practice problems with dividing decimals are provided as examples. An word problem asks how many $0.75 pens can be bought with $12, and another sets up dividing $13.93 by 0.07.
This document appears to be a portfolio created by Mike Summers containing descriptions of and processes for various visual media projects completed for a Portfolio Communications course. The portfolio includes a magazine cover, Prezi presentation, photography project, photo montage, business identity, infographic, HTML/CSS webpage, webpage mockup, and brochure. For each project, Mike summarizes the assignment, programs used, and his creative process in developing the visual content and layout.
Throughout history, women have fought for equal treatment and rights. In Egypt, significant progress has been made, with women now commonly attending university. However, traditional views still persist in some communities, where women face harassment and the expectation that their primary roles are in the home. Overall though, Egyptian women today are well-educated and participate fully in the workforce across many fields, with equal pay and opportunities, demonstrating that women have rightfully earned their place in society.
Automotive part production capability - Bluestar Mould GroupHuy Dickens
Bluestar Mould Group (BSM Group) has more than 20 year experience in precision mould making and plastic injection moulding, especially in Automotive industry. We have become a reliable strategic partner of many companies over the world: FORD, BMW, SKODA, TRW, PHILIPS, TRUCK-LITE...
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Homepage: http://bluestar-mould.com
Serie de textos que se han reunido como una antología a partir de varias fuentes bibliográficas. Son reflexiones acerca de distintos aspectos de la investigación en educación especial.
The document discusses dimensionality reduction techniques for reducing high-dimensional data to fewer dimensions. It categorizes dimensionality reduction into feature extraction and feature selection. Feature extraction transforms features to generate new ones, while feature selection selects the best original features. The document then discusses several feature selection algorithms from different categories (filter, wrapper, hybrid) and evaluates their performance on cancer datasets. It finds that linear support vector machines using mRMR feature selection provided the best results.
Cervical cancer rates have dramatically declined in the United States due to widespread Pap smear screening and the ability to treat precancerous lesions before they develop into cancer. The introduction of the Pap test in the 1940s allowed early detection and helped reduce cervical cancer incidence and mortality rates by over 60% between 1955 and 1992. New automated screening systems using digital imaging and computational analysis now further aid in screening and may help expand screening to rural areas through remote image analysis.
DLT stands for Direct Linear Transformation. It is an algorithm that estimates the camera matrix P by minimizing the algebraic error between measured image points xi and projected 3D points PXi. Specifically, DLT finds P by solving the equation Ap=0, where A is constructed from point correspondences and p contains the entries of P. This minimizes the sum of squared algebraic distances between the points. For affine cameras, the algebraic and geometric distances are equivalent. DLT provides an initial estimate of P that can be refined using nonlinear optimization techniques.
The document discusses camera models used in computer vision. It begins by defining a camera as a mapping from the 3D world to a 2D image. The basic pinhole camera model is then described, including the camera center, image plane, principal axis, and principal point. Central projection using homogeneous coordinates is shown. The camera calibration matrix K is introduced, which relates the camera coordinate system to pixel coordinates. Finally, the full camera matrix P is defined, which combines camera intrinsics K, rotation R, and translation -C to map 3D world points to 2D image points.
This document discusses singular value decomposition (SVD) and its applications. SVD decomposes a matrix into three component matrices that reveal useful properties about the matrix's structure and rank. SVD can be used to find the best-fitting line to a set of points by minimizing the sum of squared distances between points and the line. The solution involves computing the SVD of a transformed matrix and taking the right singular vector corresponding to the second largest singular value.
The document discusses estimating 2D homography from point correspondences between two images using the Direct Linear Transformation algorithm. It describes how each point correspondence provides two linear equations relating the entries of the homography matrix. At least four point correspondences are needed to compute the homography using DLT. The document also discusses issues like degenerate configurations, data normalization, robust estimation techniques like RANSAC to deal with outlier correspondences.
The document discusses projective geometry in 3D space (P3). It defines how points, planes, and lines are represented using homogeneous coordinates. Under projective transformations, incidence relations between points and planes are preserved. Three non-coplanar points uniquely define a plane, and three planes intersect at a point. The hierarchy of transformations from projective to Euclidean is described, along with the invariants each preserve. The plane at infinity π∞ and absolute conic Ω∞ allow measurement of affine and metric properties within a projective frame.
The document discusses projective geometry and its applications in computer vision. It begins by introducing planar geometry and algebraic geometry. It then describes the 2D projective plane and how points and lines can be represented using homogeneous coordinates. Ideal points and the line at infinity are discussed. Projective transformations including homographies are explained. Conic sections and how they transform under projectivities are covered. The key concepts of duality and various subgroups of projective transformations are summarized. Examples of projective transformations and corrections are provided.
The document discusses probabilistic segmentation using mixture models and the expectation-maximization (EM) algorithm. It addresses image segmentation and line fitting applications.
For image segmentation, the missing data is an (n x g) matrix of indicator variables showing which pixel belongs to which segment. The E-step computes the probability each pixel belongs to each segment. The M-step re-estimates the mixture model parameters to maximize the complete data log-likelihood.
For line fitting, the missing data is similarly an (n x g) matrix showing which point belongs to which line. The E-step computes the probability each point was drawn from each line. The M-step then re-estimates the line parameters.
The document discusses segmentation and is from the Computer Science and Engineering department at the Indian Institute of Technology in Kharagpur. It contains 29 pages of content about segmentation but provides no other context or summaries of the information within.
The trifocal tensor encapsulates the projective geometry relations between three views. It depends only on the relative pose between the three cameras and their internal parameters. The trifocal tensor can uniquely determine point and line correspondences between the three views and can be used to transfer points from a correspondence in two views to the corresponding point in the third view. It consists of three 3x3 matrices that relate image lines between the views and can induce homographies between views from lines in one of the images.
The document discusses two-view geometry and epipolar geometry in computer vision. It contains the following key points in 3 sentences:
Epipolar geometry describes the intrinsic projective geometry between two views of a scene and is defined by the fundamental matrix F, which is a 3x3 matrix that maps a point in one image to an epipolar line in the other image. The epipolar line is the intersection of the epipolar plane containing the baseline between cameras and the second image plane. Special motions like pure translation result in all epipolar lines intersecting at the epipole, which is the image of the camera center from the other view.
Camera calibration involves determining the internal camera parameters like focal length, image center, distortion, and scaling factors that affect the imaging process. These parameters are important for applications like 3D reconstruction and robotics that require understanding the relationship between 3D world points and their 2D projections in an image. The document describes estimating internal parameters by taking images of a calibration target with known 3D positions and solving for the camera projection matrix P that relates 3D scene points to their 2D image coordinates.
This document discusses probabilistic segmentation using mixture models. It explains that a mixture model represents the probability of generating a pixel measurement vector as a weighted sum of component densities. The likelihood for all observations is calculated as the product of probabilities for each data point. Missing data problems are also discussed, where the incomplete data likelihood is calculated as the product of probabilities for each incomplete data observation.
The document discusses segmentation and is from the Computer Science and Engineering department at the Indian Institute of Technology in Kharagpur. It contains 29 pages of content about segmentation but provides no other context or summaries of the information within.
The document discusses least squares minimization and solving systems of linear equations. It begins by introducing overdetermined systems with more equations than unknowns and describes finding the least squares solution that minimizes the residual. It then presents the algorithm which uses the singular value decomposition to solve the normal equations and find the pseudo-inverse. It also covers solving homogeneous systems of equations by minimizing the residual subject to the constraint that the solution vector has unit length.
1. L INEAR DYNAMICAL M ODELS
IIT Kharagpur
Computer Science and Engineering,
Indian Institute of Technology
Kharagpur.
,
1 / 23
2. Problems addressed in Tracking
P REDICTION :
P Xi | Y0 = y0 , . . . , Yi−1 = yi−1
DATA A SSOCIATION :
The prediction of the object’s state is used
to identify the measurements in the current
frame.
C ORRECTION :
P Xi | Y0 = y0 , . . . , Yi−1 = yi−1 , Yi = yi
,
2 / 23
3. Independence Assumptions
Only the immediate past matters.
P (Xi | X1 , . . . , Xi−1 ) = P (Xi | Xi−1 )
Conditional independence of measurements.
P Yi , Yj , . . . , Yk | Xi = P (Yi | Xi ) P Yj , . . . , Yk | Xi
,
3 / 23
4. Tracking as Inference
P y0 | X0 P (X0 )
P X0 | Y0 = y0 =
P y0
P y0 | X0 P (X0 )
=
P y0 | X0 P (X0 ) dX0
∝ P y0 | X0 P (X0 )
,
4 / 23
5. Prediction
P Xi | y0 , . . . , yi−1
= P Xi , Xi−1 | y0 , . . . , yi−1 dXi−1
,
5 / 23
6. Prediction
P Xi | y0 , . . . , yi−1
= P Xi , Xi−1 | y0 , . . . , yi−1 dXi−1
= P Xi | Xi−1 , y0 , . . . , yi−1 P Xi−1 | y0 , . . . yi−1 dXi−1
,
5 / 23
7. Prediction
P Xi | y0 , . . . , yi−1
= P Xi , Xi−1 | y0 , . . . , yi−1 dXi−1
= P Xi | Xi−1 , y0 , . . . , yi−1 P Xi−1 | y0 , . . . yi−1 dXi−1
= P (Xi | Xi−1 ) P Xi−1 | y0 , . . . yi−1 dXi−1
,
5 / 23
8. Correction
P Xi | y0 , . . . , yi−1 , yi
P Xi , y0 , . . . , yi−1 , yi
=
P y0 , . . . , yi−1 , yi
,
6 / 23
9. Correction
P Xi | y0 , . . . , yi−1 , yi
P Xi , y0 , . . . , yi−1 , yi
=
P y0 , . . . , yi−1 , yi
P yi | Xi , y0 , . . . , yi−1 P Xi | y0 , . . . , yi−1 P y0 , . . . , yi−1
=
P y0 , . . . , yi−1 , yi
,
6 / 23
10. Correction
P Xi | y0 , . . . , yi−1 , yi
P Xi , y0 , . . . , yi−1 , yi
=
P y0 , . . . , yi−1 , yi
P yi | Xi , y0 , . . . , yi−1 P Xi | y0 , . . . , yi−1 P y0 , . . . , yi−1
=
P y0 , . . . , yi−1 , yi
P y0 , . . . , yi−1
= P yi | Xi P Xi | y0 , . . . , yi−1
P y0 , . . . , yi−1 , yi
,
6 / 23
11. Correction
P Xi | y0 , . . . , yi−1 , yi
P Xi , y0 , . . . , yi−1 , yi
=
P y0 , . . . , yi−1 , yi
P yi | Xi , y0 , . . . , yi−1 P Xi | y0 , . . . , yi−1 P y0 , . . . , yi−1
=
P y0 , . . . , yi−1 , yi
P y0 , . . . , yi−1
= P yi | Xi P Xi | y0 , . . . , yi−1
P y0 , . . . , yi−1 , yi
P yi | Xi P Xi | y0 , . . . , yi−1
=
P yi | Xi P Xi | y0 , . . . , yi−1 dXi
,
6 / 23
13. Kalman Filtering
The dynamic model:
xi ∼ N di xi−1 , σ2i
d
Gaussian Distributions
(Normal Distributions)
N mi xi , σ2 i
yi ∼ m
Tracking implies maintaining a representation of:
P Xi | y0 , . . . , yi−1
P Xi−1 | y0 , . . . , yi−1 , yi
,
8 / 23
14. Notation
What we have to estimate:
−
Xi σi− P Xi | y0 , . . . , yi−1
+
Xi σi+ P Xi | y0 , . . . , yi−1 , yi
What we know:
+ +
Xi−1 σi−1 P Xi−1 | y0 , . . . , yi−1
,
9 / 23
15. Tricks with the integrals
A new notation:
2
x−µ
g x ; µ, v = exp −
2v
,
10 / 23
16. Tricks with the integrals
A new notation:
2
x−µ
g x ; µ, v = exp −
2v
Some convenient transformations:
g x ; µ, v = g x − µ ; 0, v
,
10 / 23
17. Tricks with the integrals
A new notation:
2
x−µ
g x ; µ, v = exp −
2v
Some convenient transformations:
g x ; µ, v = g x − µ ; 0, v
g(m ; n, v) = g(n ; m, v)
,
10 / 23
18. Tricks with the integrals
A new notation:
2
x−µ
g x ; µ, v = exp −
2v
Some convenient transformations:
g x ; µ, v = g x − µ ; 0, v
g(m ; n, v) = g(n ; m, v)
µ v
g ax ; µ, v = gx ; a , a2
,
10 / 23
19. Tricks with Integrals
∞
g x − u ; µ, va g u ; 0, vb du ∝
−∞
g x ; µ, v2 + v2
a b
,
11 / 23
20. Tricks with Integrals
∞
g x − u ; µ, va g u ; 0, vb du ∝
−∞
g x ; µ, v2 + v2
a b
ad + cb bd
g(x ; a, b) g(x ; c, d) = g x ; , f (a, b, c, d)
b+d b+d
,
11 / 23
21. Prediction P Xi | y0 , . . . , yi−1
∞
= P (Xi | Xi−1 ) P Xi−1 | y0 , . . . , yi−1 dXi−1
−∞
,
12 / 23
22. Prediction P Xi | y0 , . . . , yi−1
∞
= P (Xi | Xi−1 ) P Xi−1 | y0 , . . . , yi−1 dXi−1
−∞
+ + 2
∝ g Xi ; di Xi−1 , σ2i
d g Xi−1 ; Xi−1 , σi−1 dXi−1
,
12 / 23
23. Prediction P Xi | y0 , . . . , yi−1
∞
= P (Xi | Xi−1 ) P Xi−1 | y0 , . . . , yi−1 dXi−1
−∞
+ + 2
∝ g Xi ; di Xi−1 , σ2i
d g Xi−1 ; Xi−1 , σi−1 dXi−1
+ + 2
∝ g Xi −di Xi−1 ; 0, σ2i g Xi−1 −Xi−1 ; 0, σi−1
d dXi−1
,
12 / 23
24. Prediction P Xi | y0 , . . . , yi−1
∞
= P (Xi | Xi−1 ) P Xi−1 | y0 , . . . , yi−1 dXi−1
−∞
+ + 2
∝ g Xi ; di Xi−1 , σ2i
d g Xi−1 ; Xi−1 , σi−1 dXi−1
+ + 2
∝ g Xi −di Xi−1 ; 0, σ2i g Xi−1 −Xi−1 ; 0, σi−1
d dXi−1
+ + 2
∝ g Xi −di u + Xi−1 ; 0, σ2i g u ; 0, σi−1
d du
,
12 / 23
25. Prediction P Xi | y0 , . . . , yi−1
∞
= P (Xi | Xi−1 ) P Xi−1 | y0 , . . . , yi−1 dXi−1
−∞
+ + 2
∝ g Xi ; di Xi−1 , σ2i
d g Xi−1 ; Xi−1 , σi−1 dXi−1
+ + 2
∝ g Xi −di Xi−1 ; 0, σ2i g Xi−1 −Xi−1 ; 0, σi−1
d dXi−1
+ + 2
∝ g Xi −di u + Xi−1 ; 0, σ2i g u ; 0, σi−1
d du
+ + 2
∝ g Xi −di u ; di Xi−1 , σ2i g u ; 0, σi−1
d du
,
12 / 23
26. Prediction P Xi | y0 , . . . , yi−1
∞
= P (Xi | Xi−1 ) P Xi−1 | y0 , . . . , yi−1 dXi−1
−∞
+ + 2
∝ g Xi ; di Xi−1 , σ2i
d g Xi−1 ; Xi−1 , σi−1 dXi−1
+ + 2
∝ g Xi −di Xi−1 ; 0, σ2i g Xi−1 −Xi−1 ; 0, σi−1
d dXi−1
+ + 2
∝ g Xi −di u + Xi−1 ; 0, σ2i g u ; 0, σi−1
d du
+ + 2
∝ g Xi −di u ; di Xi−1 , σ2i g u ; 0, σi−1
d du
+ + 2
∝ g Xi −v ; di Xi−1 , σ2i g v ; 0, di σi−1
d dv
,
12 / 23
27. Prediction 1-D state vector
P Xi | y0 , . . . , yi−1
+ + 2
∝ g Xi −v ; di Xi−1 , σ2i g v ; 0, di σi−1
d dv
+ + 2
∝ g Xi ; di X0 , σ2i + di σi−1
d
,
13 / 23
28. Prediction 1-D state vector
P Xi | y0 , . . . , yi−1
+ + 2
∝ g Xi −v ; di Xi−1 , σ2i g v ; 0, di σi−1
d dv
+ + 2
∝ g Xi ; di X0 , σ2i + di σi−1
d
− +
Xi = di Xi−1
2 2
+
σi−1
−
= σ2i + di σi−1
d
,
13 / 23
29. Correction 1-D state vector
P yi | Xi P Xi | y0 , . . . , yi−1
P Xi | y0 , . . . , yi−1 , yi =
P yi | Xi P Xi | y0 , . . . , yi−1 dXi
∝ P yi | X i P Xi | y0 , . . . , yi−1
We know P Xi | y0 , . . . , yi−1
−
we know Xi and σi−
,
14 / 23
30. Correction 1-D state vector
− 2
P Xi | y0 , . . . , yi−1 , yi ∝ g yi ; mi Xi , σ2 i g Xi ; Xi , σi−
m
,
15 / 23
31. Correction 1-D state vector
− 2
P Xi | y0 , . . . , yi−1 , yi ∝ g yi ; mi Xi , σ2 i g Xi ; Xi , σi−
m
− 2
= g mi Xi ; yi , σ2 i g Xi ; Xi , σi−
m
,
15 / 23
32. Correction 1-D state vector
− 2
P Xi | y0 , . . . , yi−1 , yi ∝ g yi ; mi Xi , σ2 i g Xi ; Xi , σi−
m
− 2
= g mi Xi ; yi , σ2 i g Xi ; Xi , σi−
m
y i σ2 i
m − 2
= gXi ; , 2 g Xi ; Xi , σi−
m m i i
,
15 / 23
33. Correction 1-D state vector
− 2
P Xi | y0 , . . . , yi−1 , yi ∝ g yi ; mi Xi , σ2 i g Xi ; Xi , σi−
m
− 2
= g mi Xi ; yi , σ2 i g Xi ; Xi , σi−
m
y i σ2 i
m − 2
= gXi ; , 2 g Xi ; Xi , σi−
m m i i
− 2 2
Xi σ2 + mi y σ −
mi
σ2 i σi−
m
X+ =
i
σi+ =
i
i
σ2 + m2 σ − 2 2
σ2 i + m2 σi−
mi i i m i
,
15 / 23
34. A general state vector Kalman Filtering
DYNAMIC M ODEL
xi ∼ N Di xi−1 , Σdi
yi ∼ N Mi xi , Σmi
S TART A SSUMPTIONS x− and Σ− are known.
0 0
U PDATE E QUATIONS
P REDICTION : C ORRECTION :
x − = Di x +
i i−1 Ki = Σ− Mi Mi Σ− Mi + Σmi
−1
i i
Σ− = Σdi + Di Σ+ Di
i i−1 x+ = x− + Ki yi − Mi x−
i i i
Σ+ = [ I d − Ki Mi ] Σ−
i i
,
16 / 23
35. Forward-Backward Smoothing
F ORWARD -BACKWARD F ILTER : P Xi | y0 , . . . , yN
P Xi , yi+1 , . . . , yN | y0 , . . . , yi P y0 , . . . , yi
=
P y0 , . . . , yN
P yi+1 , . . . , yN | Xi , y0 , . . . , yi P Xi | y0 , . . . , yi P y0 , . . . , yi
=
P y0 , . . . , yN
P yi+1 , . . . , yN | Xi P Xi | y0 , . . . , yi P y0 , . . . , yi
=
P y0 , . . . , yN
= P Xi | yi+1 , . . . , yN P Xi | y0 , . . . , yi α
,
17 / 23
36. Forward-Backward Smoothing
F ORWARD -BACKWARD F ILTER : P Xi | y0 , . . . , yN
= P Xi | yi+1 , . . . , yN P Xi | y0 , . . . , yi α
where
P yi+1 , . . . , yN P y0 , . . . , yi
α=
P (Xi ) P y0 , . . . , yN
,
18 / 23
37. Combining the Forward-Backward
Dynamics
Forward dynamics: P Xi | y0 , . . . , yi
Backward dynamics: P Xi | yi+1 , . . . , yN
Forward-backward dynamics: P Xi | y0 , . . . , yN
N OTATION :
f ,+ f ,+
Forward dynamics: Xi and Σi
b,−
Backward dynamics: Measurement is Xb with mean Xi
i and Σb,−
i
∗
Forward-Backward dynamics: Xi and Σ∗
i
,
19 / 23
38. If we consider the backward dynamics as measurements, then the forward
dynamics would give the state just before the measurement comes in:
F ORWARD -BACKWARD SMOOTHING
K ALMAN U PDATE EQUATIONS ( CORRECTION )
( CORRECTION )
f ,+ f ,+ −1
−1
Ki∗ = Σi Σi + Σb,−
i
Ki = Σ− Mi Mi Σ− Mi + Σmi
i i
∗ f ,+ b,− f ,+
Xi = Xi + Ki X i − Xi
+ − −
xi = xi + Ki yi − Mi xi
f ,+
Σ∗ = [ I − Ki ] Σi
Σ+ = [ I d − Ki Mi ] Σ−
i i
i
,
20 / 23
41. Data Association
N EAREST N EIGHBOURS
The r th region offers a measurement yir
We choose the region with the best value of
P Yi = yir | y0 , . . . , yi−1
= P Yi = yir | Xi , y0 , . . . , yi−1 P Xi | y0 , . . . , yi−1 dXi
= P Yi = yir | Xi P Xi | y0 , . . . , yi−1 dXi
The Kalman filter is used to compute P Yi = yir | y0 , . . . , yi−1
,
23 / 23