1. The progress report outlines work on a discriminatively trained, multiscale, deformable part model for object detection.
2. Modifications are proposed to optimize the model's function, use lower dimensional but more informative features, and predict bounding boxes.
3. Adding contextual information is also discussed to help rescore detections using surrounding detections from other models.
This document discusses several topics related to Fourier transforms including:
1) Representing polynomials in value representation by evaluating them at roots of unity allows for faster multiplication using the Discrete Fourier Transform (DFT).
2) The DFT reduces the complexity of the Discrete Fourier Transform (DFT) from O(n2) to O(n log n) by formulating it recursively.
3) Converting images from the spatial to frequency domain using techniques like the Discrete Cosine Transform (DCT) allows for image compression by retaining only low frequency components with large coefficients.
Object Detection with Discrmininatively Trained Part based Modelszukun
The document describes an object detection method using deformable part-based models that are discriminatively trained. The models consist of root filters and deformable part filters at multiple resolutions. Latent SVM training is used to learn the filters and deformation costs from weakly labeled images. The method achieved state-of-the-art results on the PASCAL object detection challenge, outperforming other methods in accuracy and speed.
The document discusses using the derivative to determine whether a function is increasing or decreasing over an interval. It provides examples of using the sign of the derivative to determine if a function is increasing or decreasing. It also discusses using the second derivative test to determine if a stationary point is a relative maximum or minimum. Specifically:
- The sign of the derivative indicates whether the function is increasing or decreasing over an interval. Positive derivative means increasing, negative means decreasing.
- Stationary points where the derivative is zero require the second derivative test to determine if it is a relative maximum or minimum. Positive second derivative means a relative minimum, negative means a maximum.
- Examples demonstrate finding stationary points, using the first and second derivative
This document summarizes research on using deformable models for object recognition. It discusses using deformable part models to detect objects by optimizing part locations. Efficient algorithms like dynamic programming and min-convolutions are used for matching. Non-rigid objects are modeled using triangulated polygons that can deform individual triangles. Hierarchical shape models capture shape variations. The document applies these techniques to the PASCAL visual object recognition challenge, achieving state-of-the-art results on 10 of 20 object categories through discriminatively trained, multiscale deformable part models.
1. Geodesic sampling and meshing techniques can be used to generate adaptive triangulations and meshes on Riemannian manifolds based on a metric tensor.
2. Anisotropic metrics can be defined to generate meshes adapted to features like edges in images or curvature on surfaces. Triangles will be elongated along strong features to better approximate functions.
3. Farthest point sampling can be used to generate well-spaced point distributions over manifolds according to a metric, which can then be triangulated using geodesic Delaunay refinement.
The document discusses differential processing on triangular meshes, including defining functions on meshes, local averaging operators, gradient and Laplacian operators, and proving that the normalized Laplacian is symmetric and positive definite using the properties of the gradient and local connectivity of the mesh. Operators like the Laplacian can be used to smooth functions defined on meshes through diffusion.
The course program includes sessions on discrete models in computer vision, message passing algorithms like dynamic programming and tree-reweighted message passing, quadratic pseudo-boolean optimization, transformation and move-making methods, speed and efficiency of algorithms, and a comparison of inference methods. Recent advances like dual decomposition and higher-order models will also be discussed. All materials from the tutorial will be made available online after the conference.
The document provides a review outline for Midterm I in Math 1a. It includes the following topics:
- The Intermediate Value Theorem
- Limits (concept, computation, limits involving infinity)
- Continuity (concept, examples)
- Derivatives (concept, interpretations, implications, computation)
- It also provides learning objectives and outlines for each topic.
This document discusses several topics related to Fourier transforms including:
1) Representing polynomials in value representation by evaluating them at roots of unity allows for faster multiplication using the Discrete Fourier Transform (DFT).
2) The DFT reduces the complexity of the Discrete Fourier Transform (DFT) from O(n2) to O(n log n) by formulating it recursively.
3) Converting images from the spatial to frequency domain using techniques like the Discrete Cosine Transform (DCT) allows for image compression by retaining only low frequency components with large coefficients.
Object Detection with Discrmininatively Trained Part based Modelszukun
The document describes an object detection method using deformable part-based models that are discriminatively trained. The models consist of root filters and deformable part filters at multiple resolutions. Latent SVM training is used to learn the filters and deformation costs from weakly labeled images. The method achieved state-of-the-art results on the PASCAL object detection challenge, outperforming other methods in accuracy and speed.
The document discusses using the derivative to determine whether a function is increasing or decreasing over an interval. It provides examples of using the sign of the derivative to determine if a function is increasing or decreasing. It also discusses using the second derivative test to determine if a stationary point is a relative maximum or minimum. Specifically:
- The sign of the derivative indicates whether the function is increasing or decreasing over an interval. Positive derivative means increasing, negative means decreasing.
- Stationary points where the derivative is zero require the second derivative test to determine if it is a relative maximum or minimum. Positive second derivative means a relative minimum, negative means a maximum.
- Examples demonstrate finding stationary points, using the first and second derivative
This document summarizes research on using deformable models for object recognition. It discusses using deformable part models to detect objects by optimizing part locations. Efficient algorithms like dynamic programming and min-convolutions are used for matching. Non-rigid objects are modeled using triangulated polygons that can deform individual triangles. Hierarchical shape models capture shape variations. The document applies these techniques to the PASCAL visual object recognition challenge, achieving state-of-the-art results on 10 of 20 object categories through discriminatively trained, multiscale deformable part models.
1. Geodesic sampling and meshing techniques can be used to generate adaptive triangulations and meshes on Riemannian manifolds based on a metric tensor.
2. Anisotropic metrics can be defined to generate meshes adapted to features like edges in images or curvature on surfaces. Triangles will be elongated along strong features to better approximate functions.
3. Farthest point sampling can be used to generate well-spaced point distributions over manifolds according to a metric, which can then be triangulated using geodesic Delaunay refinement.
The document discusses differential processing on triangular meshes, including defining functions on meshes, local averaging operators, gradient and Laplacian operators, and proving that the normalized Laplacian is symmetric and positive definite using the properties of the gradient and local connectivity of the mesh. Operators like the Laplacian can be used to smooth functions defined on meshes through diffusion.
The course program includes sessions on discrete models in computer vision, message passing algorithms like dynamic programming and tree-reweighted message passing, quadratic pseudo-boolean optimization, transformation and move-making methods, speed and efficiency of algorithms, and a comparison of inference methods. Recent advances like dual decomposition and higher-order models will also be discussed. All materials from the tutorial will be made available online after the conference.
The document provides a review outline for Midterm I in Math 1a. It includes the following topics:
- The Intermediate Value Theorem
- Limits (concept, computation, limits involving infinity)
- Continuity (concept, examples)
- Derivatives (concept, interpretations, implications, computation)
- It also provides learning objectives and outlines for each topic.
The document discusses univariate and multivariate extreme value theory. It introduces limit probabilities for maxima and maximum domains of attraction in univariate extreme value theory. It also discusses limit distributions of multivariate maxima and the multivariate domain of attraction. The outline previews that the document will cover introduction to limits of maxima, univariate extreme value theory, and multivariate extreme value theory.
This document provides an overview of continuous probability distributions including:
- The probability of an event occurring between two values a and b is defined by the shaded area under the probability density function curve between a and b.
- Key properties of continuous distributions include the mean, median, standard deviation, and cumulative distribution function.
- The normal distribution is discussed as an important continuous distribution with properties like symmetry and effectiveness in modeling real-world data.
- Methods for calculating normal probabilities using tables or Excel are presented with examples.
- An example problem involving a quality control scenario and modeling the number of defective items using a binomial distribution is provided.
This document discusses geodesic data processing on Riemannian manifolds. It defines geodesic distances as the shortest path between two points on the manifold according to the Riemannian metric. Methods are presented for computing geodesic distances and curves, including iterative schemes and fast marching. Applications discussed include shape recognition using geodesic statistics and geodesic meshing.
The Mean Value Theorem is the most important theorem in calculus! It allows us to infer information about a function from information about its derivative. Such as: a function whose derivative is zero must be a constant function.
IJERD (www.ijerd.com) International Journal of Engineering Research and Devel...IJERD Editor
This document presents a theorem that establishes the existence of a fixed point for a mapping under a general contractive condition of integral type. The mapping considered generalizes various types of contractive mappings in an integral setting. The theorem proves that if a self-mapping on a complete metric space satisfies the given integral inequality involving the distance between images of points, where the integral involves a non-negative, summable function, then the mapping has a unique fixed point. Furthermore, the sequence of repeated applications of the mapping to any starting point will converge to this fixed point. The proof involves showing the distance between successive terms in the sequence decreases according to the integral inequality.
The document discusses conditional random fields (CRFs), which are probabilistic models used for structured prediction problems. CRFs define a conditional probability distribution p(y|x) via an exponential family form using feature functions. Maximum likelihood, maximum entropy, and MAP estimation techniques can be used to learn the parameters of a CRF by minimizing the negative conditional log-likelihood of labeled training data. Gradient descent or other numerical optimization methods are then required to perform the actual minimization. CRFs provide a principled probabilistic approach to learning the relationships between inputs x and structured outputs y.
2013-1 Machine Learning Lecture 02 - Andrew Moore: EntropyDongseo University
This document provides an overview of entropy and conditional entropy in information theory. It begins with examples of encoding variables with different probabilities to minimize the number of bits needed. It then defines entropy as the average number of bits needed to encode events from a probability distribution. Several example distributions are provided, along with their entropies. Finally, it defines conditional entropy as the expected entropy of a variable given knowledge of another variable.
1) The document summarizes research into generating conjectures for upper bounds on the domination number of bipartite graphs. 14 conjectures were produced, of which 3 were previously known to be true.
2) For the 11 false conjectures, smallest counterexamples were found to disprove the conjectures. An example of finding a smallest counterexample is shown.
3) The goal of the research was to determine a collection of upper bounds that could accurately predict the domination number of any bipartite graph. Such a collection could help compute domination numbers more efficiently.
This document provides an overview of supervised learning and linear regression. It introduces supervised learning problems using an example of predicting house prices based on living area. Linear regression is discussed as an initial approach to model this relationship. The cost function is defined as the mean squared error between predictions and targets. Gradient descent and stochastic gradient descent are presented as algorithms to minimize this cost function and learn the parameters of the linear regression model.
The document defines the precise definition of a limit. It states that the limit of a function f(x) as x approaches a number x0 is a number L if, for any positive number ε, there exists a corresponding positive number δ such that if 0 < |x - x0| < δ, then |f(x) - L| < ε. It then provides examples of using this definition to evaluate limits, including finding the appropriate δ value algebraically given a function, limit value, and ε.
CVPR2010: higher order models in computer vision: Part 1, 2zukun
This document discusses tractable higher order models in computer vision using random field models. It introduces Markov random fields (MRFs) and factor graphs as graphical models for computer vision problems. Higher order models that include factors over cliques of more than two variables can model problems more accurately but are generally intractable. The document discusses various inference techniques for higher order models such as relaxation, message passing, and decomposition methods. It provides examples of how higher order and global models can be used in problems like segmentation, stereo matching, reconstruction, and denoising.
The document discusses computational methods for Bayesian model choice and model comparison. It introduces Bayes factors and the evidence as central quantities for model comparison. It then describes various computational methods for approximating the evidence, including importance sampling solutions like bridge sampling, harmonic mean approximations using posterior samples, and approximating the evidence using mixture representations.
The document discusses the principle of maximum entropy. It explains that maximum entropy is an approach for making probability assignments where the assigned probability distribution should have the largest entropy or uncertainty possible, subject to whatever information is known. It describes applications of maximum entropy modeling such as part-of-speech tagging and logistic regression. Maximum entropy and maximum likelihood methods are related as they both aim to make distributions as uniform as possible based on available information.
The Mean Value Theorem is the most important theorem in calculus! It allows us to infer information about a function from information about its derivative. Such as: a function whose derivative is zero must be a constant function.
The document discusses an introductory calculus class and provides announcements about homework due dates and a student survey. It also outlines guidelines for written homework assignments, the grading rubric, and examples of what to include and avoid in written work. The document aims to provide students information about course policies and expectations for written assignments.
The document discusses evaluating definite integrals. It begins by reviewing the definition of the definite integral as a limit and properties of integrals such as additivity. It then covers estimating integrals using the Midpoint Rule and properties for comparing integrals. Examples are provided of evaluating definite integrals using known formulas or the Midpoint Rule. The integral is discussed as computing the total change, and an outline of future topics like indefinite integrals and computing area is presented.
The document discusses evaluating definite integrals. It begins by reviewing the definition of the definite integral as a limit. It then discusses estimating integrals using the midpoint rule and properties of integrals such as integrals of nonnegative functions being nonnegative and integrals being "increasing" if one function is greater than another. An example is worked out using the midpoint rule to estimate an integral. The document provides an outline of topics and notation for integrals.
The document provides a review outline for Math 1a Midterm II covering topics including: differentiation using product, quotient, and chain rules; implicit differentiation; logarithmic differentiation; applications such as related rates and optimization; and the shape of curves including the mean value theorem and extreme value theorem. It also lists learning objectives and provides details on key concepts like L'Hopital's rule and the closed interval method for finding extrema.
Integral calculus allows us to calculate quantities like distance traveled, work done, and area under a curve by summing up infinitely many infinitesimally small quantities. The three examples given all involve calculating a quantity that is the product of two factors where one factor varies with respect to the other over an interval. Integral calculus provides a way to find the total of this varying product by breaking it into infinitely many strips and adding them up. Graphically, the definite integral represents the area under a function over an interval, with the area of each strip having a physical meaning relevant to the application.
The document discusses univariate and multivariate extreme value theory. It introduces limit probabilities for maxima and maximum domains of attraction in univariate extreme value theory. It also discusses limit distributions of multivariate maxima and the multivariate domain of attraction. The outline previews that the document will cover introduction to limits of maxima, univariate extreme value theory, and multivariate extreme value theory.
This document provides an overview of continuous probability distributions including:
- The probability of an event occurring between two values a and b is defined by the shaded area under the probability density function curve between a and b.
- Key properties of continuous distributions include the mean, median, standard deviation, and cumulative distribution function.
- The normal distribution is discussed as an important continuous distribution with properties like symmetry and effectiveness in modeling real-world data.
- Methods for calculating normal probabilities using tables or Excel are presented with examples.
- An example problem involving a quality control scenario and modeling the number of defective items using a binomial distribution is provided.
This document discusses geodesic data processing on Riemannian manifolds. It defines geodesic distances as the shortest path between two points on the manifold according to the Riemannian metric. Methods are presented for computing geodesic distances and curves, including iterative schemes and fast marching. Applications discussed include shape recognition using geodesic statistics and geodesic meshing.
The Mean Value Theorem is the most important theorem in calculus! It allows us to infer information about a function from information about its derivative. Such as: a function whose derivative is zero must be a constant function.
IJERD (www.ijerd.com) International Journal of Engineering Research and Devel...IJERD Editor
This document presents a theorem that establishes the existence of a fixed point for a mapping under a general contractive condition of integral type. The mapping considered generalizes various types of contractive mappings in an integral setting. The theorem proves that if a self-mapping on a complete metric space satisfies the given integral inequality involving the distance between images of points, where the integral involves a non-negative, summable function, then the mapping has a unique fixed point. Furthermore, the sequence of repeated applications of the mapping to any starting point will converge to this fixed point. The proof involves showing the distance between successive terms in the sequence decreases according to the integral inequality.
The document discusses conditional random fields (CRFs), which are probabilistic models used for structured prediction problems. CRFs define a conditional probability distribution p(y|x) via an exponential family form using feature functions. Maximum likelihood, maximum entropy, and MAP estimation techniques can be used to learn the parameters of a CRF by minimizing the negative conditional log-likelihood of labeled training data. Gradient descent or other numerical optimization methods are then required to perform the actual minimization. CRFs provide a principled probabilistic approach to learning the relationships between inputs x and structured outputs y.
2013-1 Machine Learning Lecture 02 - Andrew Moore: EntropyDongseo University
This document provides an overview of entropy and conditional entropy in information theory. It begins with examples of encoding variables with different probabilities to minimize the number of bits needed. It then defines entropy as the average number of bits needed to encode events from a probability distribution. Several example distributions are provided, along with their entropies. Finally, it defines conditional entropy as the expected entropy of a variable given knowledge of another variable.
1) The document summarizes research into generating conjectures for upper bounds on the domination number of bipartite graphs. 14 conjectures were produced, of which 3 were previously known to be true.
2) For the 11 false conjectures, smallest counterexamples were found to disprove the conjectures. An example of finding a smallest counterexample is shown.
3) The goal of the research was to determine a collection of upper bounds that could accurately predict the domination number of any bipartite graph. Such a collection could help compute domination numbers more efficiently.
This document provides an overview of supervised learning and linear regression. It introduces supervised learning problems using an example of predicting house prices based on living area. Linear regression is discussed as an initial approach to model this relationship. The cost function is defined as the mean squared error between predictions and targets. Gradient descent and stochastic gradient descent are presented as algorithms to minimize this cost function and learn the parameters of the linear regression model.
The document defines the precise definition of a limit. It states that the limit of a function f(x) as x approaches a number x0 is a number L if, for any positive number ε, there exists a corresponding positive number δ such that if 0 < |x - x0| < δ, then |f(x) - L| < ε. It then provides examples of using this definition to evaluate limits, including finding the appropriate δ value algebraically given a function, limit value, and ε.
CVPR2010: higher order models in computer vision: Part 1, 2zukun
This document discusses tractable higher order models in computer vision using random field models. It introduces Markov random fields (MRFs) and factor graphs as graphical models for computer vision problems. Higher order models that include factors over cliques of more than two variables can model problems more accurately but are generally intractable. The document discusses various inference techniques for higher order models such as relaxation, message passing, and decomposition methods. It provides examples of how higher order and global models can be used in problems like segmentation, stereo matching, reconstruction, and denoising.
The document discusses computational methods for Bayesian model choice and model comparison. It introduces Bayes factors and the evidence as central quantities for model comparison. It then describes various computational methods for approximating the evidence, including importance sampling solutions like bridge sampling, harmonic mean approximations using posterior samples, and approximating the evidence using mixture representations.
The document discusses the principle of maximum entropy. It explains that maximum entropy is an approach for making probability assignments where the assigned probability distribution should have the largest entropy or uncertainty possible, subject to whatever information is known. It describes applications of maximum entropy modeling such as part-of-speech tagging and logistic regression. Maximum entropy and maximum likelihood methods are related as they both aim to make distributions as uniform as possible based on available information.
The Mean Value Theorem is the most important theorem in calculus! It allows us to infer information about a function from information about its derivative. Such as: a function whose derivative is zero must be a constant function.
The document discusses an introductory calculus class and provides announcements about homework due dates and a student survey. It also outlines guidelines for written homework assignments, the grading rubric, and examples of what to include and avoid in written work. The document aims to provide students information about course policies and expectations for written assignments.
The document discusses evaluating definite integrals. It begins by reviewing the definition of the definite integral as a limit and properties of integrals such as additivity. It then covers estimating integrals using the Midpoint Rule and properties for comparing integrals. Examples are provided of evaluating definite integrals using known formulas or the Midpoint Rule. The integral is discussed as computing the total change, and an outline of future topics like indefinite integrals and computing area is presented.
The document discusses evaluating definite integrals. It begins by reviewing the definition of the definite integral as a limit. It then discusses estimating integrals using the midpoint rule and properties of integrals such as integrals of nonnegative functions being nonnegative and integrals being "increasing" if one function is greater than another. An example is worked out using the midpoint rule to estimate an integral. The document provides an outline of topics and notation for integrals.
The document provides a review outline for Math 1a Midterm II covering topics including: differentiation using product, quotient, and chain rules; implicit differentiation; logarithmic differentiation; applications such as related rates and optimization; and the shape of curves including the mean value theorem and extreme value theorem. It also lists learning objectives and provides details on key concepts like L'Hopital's rule and the closed interval method for finding extrema.
Integral calculus allows us to calculate quantities like distance traveled, work done, and area under a curve by summing up infinitely many infinitesimally small quantities. The three examples given all involve calculating a quantity that is the product of two factors where one factor varies with respect to the other over an interval. Integral calculus provides a way to find the total of this varying product by breaking it into infinitely many strips and adding them up. Graphically, the definite integral represents the area under a function over an interval, with the area of each strip having a physical meaning relevant to the application.
We define the definite integral as a limit of Riemann sums, compute some approximations, then investigate the basic additive and comparative properties
ICCV2009: MAP Inference in Discrete Models: Part 2zukun
The course program includes sessions on discrete models in computer vision, message passing algorithms like dynamic programming and tree-reweighted message passing, quadratic pseudo-boolean optimization, transformation and move-making methods, speed and efficiency of algorithms, and a comparison of inference methods. Recent advances like dual decomposition and higher-order models will also be discussed. All materials from the tutorial will be made available online after the conference.
A polynomial interpolation algorithm is developed using the Newton's divided-difference interpolating polynomials. The definition of monotony of a function is then used to define the least degree of the polynomial to make efficient and consistent the interpolation in the discrete given function. The relation between the order of monotony of a particular function and the degree of the interpolating polynomial is justified, analyzing the relation between the derivatives of such function and the truncation error expression. In this algorithm there is not matter about the number and the arrangement of the data points, neither if the points are regularly spaced or not. The algorithm thus defined can be used to make interpolations in functions of one and several dependent variables. The algoritm automatically select the data points nearest to the point where an interpolation is desired, following the criterion of symmetry. Indirectly, the algorithm also select the number of data points, which is a unity higher than the order of the used polynomial, following the criterion of monotony. Finally, the complete algoritm is presented and subroutines in fortran code is exposed as an addendum. Notice that there is not the degree of the interpolating polynomial within the arguments of such subroutines.
This document contains notes from a calculus class. It provides the outline and key points about the Fundamental Theorem of Calculus. It discusses the first and second Fundamental Theorems of Calculus, including proofs and examples. It also provides brief biographies of several important mathematicians that contributed to the development of calculus, including the Fundamental Theorem of Calculus, such as Isaac Newton, Gottfried Leibniz, James Gregory, and Isaac Barrow.
We will define what is a function “formally”, and then
in the next lecture we will use this concept in counting.
We will also study the pigeonhole principle and its applications
This presentation discusses applications of derivatives including: extreme values of functions, the mean value theorem, monotonic functions, and concavity. It defines maximum and minimum values and explains how the mean value theorem states that between two points a and b on a continuous function, there exists a point c where the slope of the tangent line is equal to the slope of the secant line between a and b. Monotonic functions are defined as increasing or decreasing functions based on the mean value theorem. Concavity is defined based on whether the derivative of a function is increasing or decreasing, which determines if the graph is concave up or down.
This document contains lecture notes on the Fundamental Theorem of Calculus. It begins with announcements about the final exam date and current grade distribution. The outline then reviews the Evaluation Theorem and introduces the First and Second Fundamental Theorems of Calculus. It provides examples of how the integral can represent total change in concepts like distance, cost, and mass. Biographies are also included of mathematicians like Gregory, Barrow, Newton, and Leibniz who contributed to the development of calculus.
Detecting Bugs in Binaries Using Decompilation and Data Flow AnalysisSilvio Cesare
The document discusses using static analysis techniques like data flow analysis and decompilation to detect bugs in binary files. It describes decompiling binaries into an intermediate representation and then performing intraprocedural and interprocedural data flow analysis on the representation. This allows detecting bugs involving unsafe functions like getenv() and memory issues like use-after-free and double free errors. The approach involves lifting x86 into a RISC-like intermediate language, inferring stack pointers, and decompiling locals and arguments to perform analysis and optimization.
This document presents a method for estimating the eigenvalues of a covariance matrix when there are few samples. It involves shifting the sampled eigenvalues toward the population values based on theoretical distributions, and balancing the energy across eigenvalues. This simple 3-matrix approach improves estimation and detection performance compared to using the sampled eigenvalues alone. Simulations and hyperspectral data experiments demonstrate the effectiveness of the method.
The document summarizes research on threshold network models, which generate scale-free networks without growth by assigning intrinsic weights to nodes based on a given distribution and connecting nodes based on whether their total weight exceeds a threshold. The model has been extended to spatial networks by incorporating distance between nodes and to include homophily. Analytical results show the degree distribution and other properties depend on the weight distribution and thresholding function used. Several open problems are also discussed.
The document summarizes the Mean Value Theorem and Rolle's Theorem from calculus. The Mean Value Theorem states that for a differentiable function over a closed interval, there exists a point in the interval where the slope of the tangent line equals the average rate of change over the interval. Rolle's Theorem is a specific case of the Mean Value Theorem where the function value is equal at the endpoints of the interval. An example is provided to check if a function satisfies the hypotheses of Rolle's Theorem over an interval.
This document provides an introduction to calculus of variations. It discusses what calculus of variations is and covers the cases of one variable, several variables, and n unknown functions. It also describes Lagrange multipliers and provides a bibliography of references. The goal of calculus of variations is to find functions that optimize functionals, which are functions of other functions, such as finding curves that minimize lengths or surfaces that minimize areas. It involves solving Euler-Lagrange differential equations to find extremal functions.
1) The document presents an algorithm to compute the exact centroid of higher dimensional polyhedra.
2) It then uses this algorithm to approximate the centroid of a polyhedral cone, which represents the version space in kernel machines.
3) Computing the centroid allows building a new learning machine called a Balancing Board Machine, which is shown to have generalization performance comparable to Bayesian Point Machines.
SIFT extracts distinctive invariant features from images to enable object recognition despite variations in scale, rotation, and illumination. The algorithm involves:
1) Constructing scale-space images from differences of Gaussians to identify keypoints.
2) Detecting stable local extrema across scales as candidate keypoints.
3) Filtering out low contrast keypoints and those poorly localized along edges.
4) Assigning orientations based on local gradient directions.
5) Computing descriptors by sampling gradients around keypoints for matching between images.
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Template
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resolution finer resolution models
root filters part filters deformation
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coarse resolution finer resolution
and n part models (Fi, vi, di)
Each component has a root filter F0
5. Feature Pyramid
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z = (p0,..., pn)
p0 : location of root
p1,..., pn : location of parts
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scores minus
deformation costs
Image pyramid HOG feature pyramid
Multiscale model captures features at two-resolutions
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(4)
locatio
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the i-th part is the squared distance between its actual of this
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general the deformation term” is an arbitrary separable time fr
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(6) Recall
−φd (dx1 , dy1 ), filters and(dxconcatenation of HOG
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7. Matching
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• High scoring root locations define detections
- “sliding window approach”
• Efficient computation: dynamic programming +
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distance between
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general the deformation cost is an arbitrary separable tim
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• ffβ(x) = z∈Z(x)ββ··Φ(x, z)! suchconvex f (x ) > 0
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• max(0, 1 − yi fβ (xi )) is convex for negative examples
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eac
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1 To giv
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LD (β) = ||β|| x C max(0, 1 y yi
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we
add
We would like to find ! such that: yi froot(xi ) > 0 detections while the
p1 ,...,pn fro
High-scoring β locations define I
Convex if latent values Φ(x, z) of the parts that yield a!are fixed
fβ (x) = max β · forlocationsdefine convex in high-scoring root
• positive examples can
location is a full object hypothesis. fun
z∈Z(x) By defining an overall score for each root location we Pi,
nimize can detect multiple instances of an object (we assume
max(0, 1 − yi fβ (xi )) is convex for negative examp
there is at most one instance per root location). This
Aft
11. HOG with PCA
0.45617 0.04390 0.02462 0.01339 0.00629 0.00556 0.00456 0.00391 0.00367
0.00353 0.00310 0.00063 0.00030 0.00020 0.00018 0.00018 0.00017 0.00014
0.00013 0.00011 0.00010 0.00010 0.00009 0.00009 0.00008 0.00008 0.00007
0.00006 0.00005 0.00004 0.00004 0.00003 0.00003 0.00003 0.00002 0.00002
6. PCA of HOG features. Each eigenvector is displayed as a 4 by 9 matrix so that each row corresponds t
The first 11 eigenvectors
alization factor and each column to one orientation bin. The eigenvalues are displayed on top of the eigenve
near subspace spanned by the top 11 eigenvectors captures essentially all of the information in a feature v
capture almost all information
how all of the top eigenvectors are either constant along each column or row of the matrix representation.
C be a cell-based feature map computed by aggre- 7 P OST P ROCESSING
g a pixel-level feature map with 9 contrast insensi-
12. 7.3 Contextual Information overla
box, o
We have implemented a simple procedure to rescore positiv
Post-Processing
detections using contextual information.
Let (D1 , . . . , Dk ) be a set of detections obtained using
a syst
with a
k different models (for different object categories) in an diction
image I. Each detection (B, s) ∈ Di is defined by a false p
bounding box B = (x1 , y1 , x2 , y2 ) and a score s. We cision
define the context of I in terms of a k-dimensional vector We
c(I) = (σ(s1 ), .a. regression model to figurethe high-
Learning . , σ(sk )) where si is the score of out each d
est the bounding boxDi , and σ(x) = 1/(1 + exp(−2x))
scoring detection in coordinates on the
is a logistic function for renormalizing the scores. obtain
To rescore a detection (B, s) by an imagewith all
Re-scoring the window in models I we build correc
a 25-dimensional feature vector with the original score
scores of categories detection windows In s
of the detection, the top-left and bottom-right bounding to con
box coordinates, and the image context, cow o
g = (σ(s), x1 , y1 , x2 , y2 , c(I)). (30) detect
box cr
The coordinates x1 , y1 , x2 , y2 ∈ [0, 1] are normalized by catego
the width and height of the image. We use a category- truth b
specific classifier to score this vector to obtain a new