Great Pyramid of Giza and Golden Section Transform PreviewJason Li
The document discusses connections between the Great Pyramid of Giza and the Reverse Order Low Golden Section Transform (RLGST). It provides code examples for performing the 2D RLGST and its inverse on an image. The RLGST is based on the golden ratio, Fibonacci numbers, and the Wythoff sequence. There may be mathematical relationships between the dimensions of the Great Pyramid and properties of the RLGST that could reveal insights into ancient engineering. References are provided on the RLGST, wavelets, Fibonacci words, and the Great Pyramid.
Is the Macroeconomy Locally Unstable and Why Should We Care?ADEMU_Project
The document discusses evidence that macroeconomic dynamics may exhibit locally unstable behavior like limit cycles rather than stable convergence to a steady state. It presents a reduced-form model allowing for nonlinearities, estimated on total hours and other labor market variables. The estimates provide intriguing evidence of limit cycle dynamics, with simulations showing attracting cycles. This suggests macroeconomic fluctuations could be endogenous rather than just responses to shocks, with implications for stabilization policy.
The document discusses drawing 2D primitives such as lines, circles, and polygons in a raster graphics system. It covers:
- Representations of lines, circles, and polygons using implicit, explicit, and parametric formulas
- Scan conversion algorithms to draw these primitives by mapping them to pixels, including basic and midpoint line algorithms, a circle midpoint algorithm, and flood fill and scan conversion approaches for polygon fill
- Components of an interactive graphics system including the application model, program, and graphics system that interfaces with display hardware like CRT and FED displays
This document provides a summary of spatial data modeling and analysis techniques. It begins with an outline of the topics to be covered, including additive statistical models for spatial data, spatial covariance functions, the multivariate normal distribution, kriging for prediction and uncertainty, and the likelihood function for parameter estimation. It then introduces the key concepts and equations for modeling spatial processes as Gaussian random fields with specified covariance functions. Examples are given of commonly used covariance functions and the types of random surfaces they generate. Kriging is described as a best linear unbiased prediction technique that uses a spatial covariance function and observations to make predictions at unknown locations. The document concludes with examples of parameter estimation via maximum likelihood and using the fitted model to make predictions and conditional simulations
Singularities in the one control problem. S.I.S.S.A., Trieste August 16, 2007.Igor Moiseev
Singularities in the one control problem. S.I.S.S.A., Trieste August 16, 2007.
The geometry of strokes arises in the control problems of Reeds–Shepp car, Dubins’ car, modeling of vision and some others. The main problem is to characterize the shortest paths and minimal distances on the plane, equipped with the structure of geometry of strokes.
This problem is formulated as an optimal control problem in 3-space with 2 dimensional control and a quadratic integral cost. Here is studied the symmetries of the sub-Riemannian structure, extremals of the optimal control problem, the Maxwell stratum, conjugate points and boundary value problem for the corresponding Hamiltonian system.
This document discusses optimal control problems for stochastic sequential machines (SSMs). It begins by introducing SSMs and defining their components. It then formulates the optimal control problem for processes represented by SSMs, proving the principle of optimality. Using dynamic programming, it derives the Bellman equation to find the optimal control solution. In conclusions, it shows that the Bellman equation and principle of optimality apply to obtaining the optimal control for processes modeled as SSMs.
Phase diagram at finite T & Mu in strong coupling limit of lattice QCDBenjamin Jaedon Choi
This document summarizes the derivation of an effective free energy for QCD at strong coupling using a mean field approximation with 1 flavor staggered fermion. Key steps include:
1) Performing a path integral over spatial link variables to obtain quark propagators.
2) Introducing auxiliary bosonic fields using a Hubbard-Stratonovich transformation to obtain a bilinear form in quark fields.
3) Applying a mean field approximation to the auxiliary fields.
4) Exactly integrating over temporal links, quark and auxiliary baryon fields to obtain an effective free energy in terms of the auxiliary meson field.
5) Analyzing the effective free energy to determine the QCD phase diagram as functions of temperature and
Great Pyramid of Giza and Golden Section Transform PreviewJason Li
The document discusses connections between the Great Pyramid of Giza and the Reverse Order Low Golden Section Transform (RLGST). It provides code examples for performing the 2D RLGST and its inverse on an image. The RLGST is based on the golden ratio, Fibonacci numbers, and the Wythoff sequence. There may be mathematical relationships between the dimensions of the Great Pyramid and properties of the RLGST that could reveal insights into ancient engineering. References are provided on the RLGST, wavelets, Fibonacci words, and the Great Pyramid.
Is the Macroeconomy Locally Unstable and Why Should We Care?ADEMU_Project
The document discusses evidence that macroeconomic dynamics may exhibit locally unstable behavior like limit cycles rather than stable convergence to a steady state. It presents a reduced-form model allowing for nonlinearities, estimated on total hours and other labor market variables. The estimates provide intriguing evidence of limit cycle dynamics, with simulations showing attracting cycles. This suggests macroeconomic fluctuations could be endogenous rather than just responses to shocks, with implications for stabilization policy.
The document discusses drawing 2D primitives such as lines, circles, and polygons in a raster graphics system. It covers:
- Representations of lines, circles, and polygons using implicit, explicit, and parametric formulas
- Scan conversion algorithms to draw these primitives by mapping them to pixels, including basic and midpoint line algorithms, a circle midpoint algorithm, and flood fill and scan conversion approaches for polygon fill
- Components of an interactive graphics system including the application model, program, and graphics system that interfaces with display hardware like CRT and FED displays
This document provides a summary of spatial data modeling and analysis techniques. It begins with an outline of the topics to be covered, including additive statistical models for spatial data, spatial covariance functions, the multivariate normal distribution, kriging for prediction and uncertainty, and the likelihood function for parameter estimation. It then introduces the key concepts and equations for modeling spatial processes as Gaussian random fields with specified covariance functions. Examples are given of commonly used covariance functions and the types of random surfaces they generate. Kriging is described as a best linear unbiased prediction technique that uses a spatial covariance function and observations to make predictions at unknown locations. The document concludes with examples of parameter estimation via maximum likelihood and using the fitted model to make predictions and conditional simulations
Singularities in the one control problem. S.I.S.S.A., Trieste August 16, 2007.Igor Moiseev
Singularities in the one control problem. S.I.S.S.A., Trieste August 16, 2007.
The geometry of strokes arises in the control problems of Reeds–Shepp car, Dubins’ car, modeling of vision and some others. The main problem is to characterize the shortest paths and minimal distances on the plane, equipped with the structure of geometry of strokes.
This problem is formulated as an optimal control problem in 3-space with 2 dimensional control and a quadratic integral cost. Here is studied the symmetries of the sub-Riemannian structure, extremals of the optimal control problem, the Maxwell stratum, conjugate points and boundary value problem for the corresponding Hamiltonian system.
This document discusses optimal control problems for stochastic sequential machines (SSMs). It begins by introducing SSMs and defining their components. It then formulates the optimal control problem for processes represented by SSMs, proving the principle of optimality. Using dynamic programming, it derives the Bellman equation to find the optimal control solution. In conclusions, it shows that the Bellman equation and principle of optimality apply to obtaining the optimal control for processes modeled as SSMs.
Phase diagram at finite T & Mu in strong coupling limit of lattice QCDBenjamin Jaedon Choi
This document summarizes the derivation of an effective free energy for QCD at strong coupling using a mean field approximation with 1 flavor staggered fermion. Key steps include:
1) Performing a path integral over spatial link variables to obtain quark propagators.
2) Introducing auxiliary bosonic fields using a Hubbard-Stratonovich transformation to obtain a bilinear form in quark fields.
3) Applying a mean field approximation to the auxiliary fields.
4) Exactly integrating over temporal links, quark and auxiliary baryon fields to obtain an effective free energy in terms of the auxiliary meson field.
5) Analyzing the effective free energy to determine the QCD phase diagram as functions of temperature and
1) The document describes a model for calculating the uncertainty of measurement in calibrating roughness standards. It breaks the measurement process down into a series of functions that propagate uncertainty.
2) It examines various sources of uncertainty at each step, including the calibration standard, positioning errors, noise, and properties of the surface and measurement device.
3) Numerical values or equations are provided for estimating the uncertainty from various effects, such as the calibration certificate, repeatability of measurements, and statistical variations in the surface. These sources of uncertainty are combined to determine the overall measurement uncertainty.
The document describes various computer graphics output primitives and algorithms for drawing them, including lines, circles, and filled areas. It discusses line drawing algorithms like DDA, Bresenham's, and midpoint circle algorithms. These algorithms use incremental integer calculations to efficiently rasterize primitives by determining the next pixel coordinates without performing floating point calculations at each step. The midpoint circle algorithm in particular uses a "circle function" and incremental updates to its value to determine whether the next pixel is inside or outside the circle boundary.
The document discusses local linear approximations, which provide a linear function that closely approximates a given non-linear function near a specific point. It defines the local linear approximation at a point x0 as f(x0) + f'(x0)(x - x0). Graphs and examples are provided to illustrate how the local linear approximation can be used to estimate function values close to x0. The concept of differentials is also introduced to estimate small changes in a function using its derivative. Examples demonstrate using differentials to approximate changes and estimate errors in computations involving measured values.
Theories and Engineering Technics of 2D-to-3D Back-Projection ProblemSeongcheol Baek
The slides introduce mathematical basics of 3d-to-2d image projection, 2d-to-3d back-projection problem, and its engineering technics, such as convex optimization problem, principal component analysis (PCV), singular value decomposition (SVD), etc.
Boris Blagov. Financial Crises and Time-Varying Risk Premia in a Small Open E...Eesti Pank
This document presents a Markov-switching DSGE model for Estonia that allows for time-varying risk premia. The model includes domestic consumption, inflation, and interest rates that are linked to foreign variables through a fixed exchange rate. Shocks follow Markov-switching processes where volatility varies between states. The model is solved using a forward-looking approach and estimated via Bayesian methods. Estimated parameters provide evidence of regime-dependent behavior and time-varying volatility consistent with financial crises in Estonia.
This document provides an overview of key concepts in multivariable calculus including:
- Three-dimensional coordinate systems and vectors in space. Operations on vectors such as addition, scalar multiplication, dot products, and cross products.
- Lines, planes, and quadric surfaces in space. Multiple integrals, integration in vector fields including line integrals, work, and flux.
- Coordinate transformations between rectangular and cylindrical coordinates. Green's theorem and its application to calculating line integrals and surface areas.
This document provides formulas for taking derivatives and integrals of common functions. There are 22 differentiation formulas that give the derivatives of functions like constants, sums and differences of functions, composite functions, polynomials, trigonometric functions and their inverses, exponential functions, and logarithmic functions. There are also 19 integration formulas that give the integrals of polynomials, rational functions, trigonometric functions and their inverses, exponential functions, and logarithmic functions.
1) 2D geometric transformations include translations, scaling, and rotations. They can be represented by transformation matrices.
2) Translation moves an object by adding offsets to x and y coordinates. It can be represented by a 3x3 matrix with 1s on the diagonal and offsets as the last column.
3) Scaling enlarges or shrinks an object by multiplying x and y coordinates by scaling factors. It can be represented by a 2x2 diagonal matrix with scaling factors.
4) Rotation rotates an object by applying a trigonometric transformation to x and y coordinates. It can be represented by a 2x2 rotation matrix containing cosine and sine of the rotation angle.
There are several numerical methods described for approximating derivatives and integrals:
1) Forward difference formula approximates the derivative as the slope of the secant line through two nearby points.
2) Three-point formulas approximate the derivative using the slopes of secant lines through three evenly spaced points, reducing the error term to O(h^2).
3) Trapezoidal rule approximates the integral of a function as the area of trapezoids formed between the function graph and the x-axis at evenly spaced points, with error term O(h^2).
The document discusses 2D geometric transformations including translation, rotation, and scaling. It explains how each transformation can be represented by a matrix and how point coordinates are transformed. It introduces homogeneous coordinates to allow multiple transformations to be combined into a single matrix multiplication by expanding points into 3D vectors. This allows complex sequences of transformations to be applied efficiently in one step.
Model reduction design for continuous systems with finite frequency specificationsIJECEIAES
This paper is concerned with the problem of model reduction design for continuous systems in Takagi-Sugeno fuzzy model. Through the defined FF H gain performance, sufficient conditions are derived to design model reduction and to assure the fuzzy error system to be asymptotically stable with a FF H gain performance index. The explicit conditions of fuzzy model reduction are developed by solving linear matrix inequalities. Finally, a numerical example is given to illustrate the effectiveness of the proposed method. 1 1
This document provides an outline and review for the final exam in Math 1a Integration. It covers key topics like the Riemann integral, properties of the integral, comparison properties, the Fundamental Theorem of Calculus, and integration by substitution. Examples are provided to illustrate computing integrals using properties, the relationship between antiderivatives and derivatives defined by integrals, and applying the Fundamental Theorem of Calculus to find derivatives.
Finite Element Analysis of the Beams Under Thermal LoadingMohammad Tawfik
This document presents the derivation of a finite element model for analyzing beams under thermal loading. It describes:
1) The displacement functions used in the model, including a 4-term polynomial for transverse displacement and 2-term polynomial for in-plane displacement.
2) Deriving the element matrices using the principle of virtual work, accounting for external and thermal loading.
3) The procedures and results of applying the model to analyze a panel subjected to thermal loading.
1. The document discusses the basis functions used in 1D and 2D finite element analysis.
2. For 1D elements, it presents the linear, quadratic and cubic basis functions derived from imposing interpolation conditions on linear, quadratic and cubic shape functions respectively.
3. For 2D elements, it derives the linear and quadratic basis functions for triangular and rectangular elements by transforming the element coordinates and imposing interpolation conditions on linear and quadratic Ansatz shape functions.
This document discusses image transformation, which represents an image as a series of summations of unitary matrices. It describes how 1D signals can be represented as linear combinations of orthogonal basis functions through Fourier analysis. Similarly, images can be transformed and represented as linear combinations of orthonormal basis images using unitary transformations. The transformation decomposes an image into coefficient values that can be used for processing and analysis.
Vector mechanics for engineers statics 7th chapter 5 Nahla Hazem
This problem involves locating the centroid of a plane area shown in multiple problems. The solution provides the area (A) of each section, the x and y coordinates, the moment of area about the x-axis (xA), and the moment of area about the y-axis (yA). It then calculates the x and y coordinates of the centroid by taking the sum of the xA and yA values, respectively, and dividing each by the total area.
The document discusses different computer graphics display systems and algorithms for drawing lines. It describes raster scan and random scan display systems. Raster scan systems sweep an electron beam across the screen row by row to draw the image based on values stored in a frame buffer. It also covers the digital differential analyzer (DDA) and Bresenham's algorithms for drawing lines on a digital display. DDA calculates increments to move the line incrementally pixel by pixel, while Bresenham's uses a decision parameter to efficiently draw lines on a raster display. An example demonstrates applying each algorithm to draw a line between two points.
The document describes algorithms for 2D transformations in computer graphics using C programming language. It discusses functions for translation, rotation, and scaling of 2D objects.
The algorithms take input for the type of transformation, parameters for the transformation, and coordinates of the 2D object. Translation simply adds the translation offsets to the x and y coordinates. Rotation rotates the object around the x-axis or y-axis based on the angle input. Scaling multiplies the x and y coordinates by respective scaling factors.
The transformed object is displayed on the graphics screen. Thus basic 2D transformations like translation, rotation, scaling are implemented through these algorithms in C.
This document provides an overview of Laplace transforms. Key points include:
- Laplace transforms convert differential equations from the time domain to the algebraic s-domain, making them easier to solve. The process involves taking the Laplace transform of each term in the differential equation.
- Common Laplace transforms of functions are presented. Properties such as linearity, differentiation, integration, and convolution are also covered.
- Partial fraction expansion is used to break complex fractions in the s-domain into simpler forms with individual terms that can be inverted using tables of transforms.
- Solving differential equations using Laplace transforms follows a standard process of taking the Laplace transform of each term, rewriting the equation in the s-domain, solving
This document discusses information theory and related concepts such as entropy, Kullback-Leibler divergence, mutual information, independent component analysis, clustering algorithms, change point detection, kernel density estimation, and nonparametric regression. It provides mathematical definitions and formulas for these concepts. Figures are included to illustrate clustering and change point detection methods. The document contains information that could be useful for understanding techniques in machine learning, signal processing, and statistics.
1) The document describes a model for calculating the uncertainty of measurement in calibrating roughness standards. It breaks the measurement process down into a series of functions that propagate uncertainty.
2) It examines various sources of uncertainty at each step, including the calibration standard, positioning errors, noise, and properties of the surface and measurement device.
3) Numerical values or equations are provided for estimating the uncertainty from various effects, such as the calibration certificate, repeatability of measurements, and statistical variations in the surface. These sources of uncertainty are combined to determine the overall measurement uncertainty.
The document describes various computer graphics output primitives and algorithms for drawing them, including lines, circles, and filled areas. It discusses line drawing algorithms like DDA, Bresenham's, and midpoint circle algorithms. These algorithms use incremental integer calculations to efficiently rasterize primitives by determining the next pixel coordinates without performing floating point calculations at each step. The midpoint circle algorithm in particular uses a "circle function" and incremental updates to its value to determine whether the next pixel is inside or outside the circle boundary.
The document discusses local linear approximations, which provide a linear function that closely approximates a given non-linear function near a specific point. It defines the local linear approximation at a point x0 as f(x0) + f'(x0)(x - x0). Graphs and examples are provided to illustrate how the local linear approximation can be used to estimate function values close to x0. The concept of differentials is also introduced to estimate small changes in a function using its derivative. Examples demonstrate using differentials to approximate changes and estimate errors in computations involving measured values.
Theories and Engineering Technics of 2D-to-3D Back-Projection ProblemSeongcheol Baek
The slides introduce mathematical basics of 3d-to-2d image projection, 2d-to-3d back-projection problem, and its engineering technics, such as convex optimization problem, principal component analysis (PCV), singular value decomposition (SVD), etc.
Boris Blagov. Financial Crises and Time-Varying Risk Premia in a Small Open E...Eesti Pank
This document presents a Markov-switching DSGE model for Estonia that allows for time-varying risk premia. The model includes domestic consumption, inflation, and interest rates that are linked to foreign variables through a fixed exchange rate. Shocks follow Markov-switching processes where volatility varies between states. The model is solved using a forward-looking approach and estimated via Bayesian methods. Estimated parameters provide evidence of regime-dependent behavior and time-varying volatility consistent with financial crises in Estonia.
This document provides an overview of key concepts in multivariable calculus including:
- Three-dimensional coordinate systems and vectors in space. Operations on vectors such as addition, scalar multiplication, dot products, and cross products.
- Lines, planes, and quadric surfaces in space. Multiple integrals, integration in vector fields including line integrals, work, and flux.
- Coordinate transformations between rectangular and cylindrical coordinates. Green's theorem and its application to calculating line integrals and surface areas.
This document provides formulas for taking derivatives and integrals of common functions. There are 22 differentiation formulas that give the derivatives of functions like constants, sums and differences of functions, composite functions, polynomials, trigonometric functions and their inverses, exponential functions, and logarithmic functions. There are also 19 integration formulas that give the integrals of polynomials, rational functions, trigonometric functions and their inverses, exponential functions, and logarithmic functions.
1) 2D geometric transformations include translations, scaling, and rotations. They can be represented by transformation matrices.
2) Translation moves an object by adding offsets to x and y coordinates. It can be represented by a 3x3 matrix with 1s on the diagonal and offsets as the last column.
3) Scaling enlarges or shrinks an object by multiplying x and y coordinates by scaling factors. It can be represented by a 2x2 diagonal matrix with scaling factors.
4) Rotation rotates an object by applying a trigonometric transformation to x and y coordinates. It can be represented by a 2x2 rotation matrix containing cosine and sine of the rotation angle.
There are several numerical methods described for approximating derivatives and integrals:
1) Forward difference formula approximates the derivative as the slope of the secant line through two nearby points.
2) Three-point formulas approximate the derivative using the slopes of secant lines through three evenly spaced points, reducing the error term to O(h^2).
3) Trapezoidal rule approximates the integral of a function as the area of trapezoids formed between the function graph and the x-axis at evenly spaced points, with error term O(h^2).
The document discusses 2D geometric transformations including translation, rotation, and scaling. It explains how each transformation can be represented by a matrix and how point coordinates are transformed. It introduces homogeneous coordinates to allow multiple transformations to be combined into a single matrix multiplication by expanding points into 3D vectors. This allows complex sequences of transformations to be applied efficiently in one step.
Model reduction design for continuous systems with finite frequency specificationsIJECEIAES
This paper is concerned with the problem of model reduction design for continuous systems in Takagi-Sugeno fuzzy model. Through the defined FF H gain performance, sufficient conditions are derived to design model reduction and to assure the fuzzy error system to be asymptotically stable with a FF H gain performance index. The explicit conditions of fuzzy model reduction are developed by solving linear matrix inequalities. Finally, a numerical example is given to illustrate the effectiveness of the proposed method. 1 1
This document provides an outline and review for the final exam in Math 1a Integration. It covers key topics like the Riemann integral, properties of the integral, comparison properties, the Fundamental Theorem of Calculus, and integration by substitution. Examples are provided to illustrate computing integrals using properties, the relationship between antiderivatives and derivatives defined by integrals, and applying the Fundamental Theorem of Calculus to find derivatives.
Finite Element Analysis of the Beams Under Thermal LoadingMohammad Tawfik
This document presents the derivation of a finite element model for analyzing beams under thermal loading. It describes:
1) The displacement functions used in the model, including a 4-term polynomial for transverse displacement and 2-term polynomial for in-plane displacement.
2) Deriving the element matrices using the principle of virtual work, accounting for external and thermal loading.
3) The procedures and results of applying the model to analyze a panel subjected to thermal loading.
1. The document discusses the basis functions used in 1D and 2D finite element analysis.
2. For 1D elements, it presents the linear, quadratic and cubic basis functions derived from imposing interpolation conditions on linear, quadratic and cubic shape functions respectively.
3. For 2D elements, it derives the linear and quadratic basis functions for triangular and rectangular elements by transforming the element coordinates and imposing interpolation conditions on linear and quadratic Ansatz shape functions.
This document discusses image transformation, which represents an image as a series of summations of unitary matrices. It describes how 1D signals can be represented as linear combinations of orthogonal basis functions through Fourier analysis. Similarly, images can be transformed and represented as linear combinations of orthonormal basis images using unitary transformations. The transformation decomposes an image into coefficient values that can be used for processing and analysis.
Vector mechanics for engineers statics 7th chapter 5 Nahla Hazem
This problem involves locating the centroid of a plane area shown in multiple problems. The solution provides the area (A) of each section, the x and y coordinates, the moment of area about the x-axis (xA), and the moment of area about the y-axis (yA). It then calculates the x and y coordinates of the centroid by taking the sum of the xA and yA values, respectively, and dividing each by the total area.
The document discusses different computer graphics display systems and algorithms for drawing lines. It describes raster scan and random scan display systems. Raster scan systems sweep an electron beam across the screen row by row to draw the image based on values stored in a frame buffer. It also covers the digital differential analyzer (DDA) and Bresenham's algorithms for drawing lines on a digital display. DDA calculates increments to move the line incrementally pixel by pixel, while Bresenham's uses a decision parameter to efficiently draw lines on a raster display. An example demonstrates applying each algorithm to draw a line between two points.
The document describes algorithms for 2D transformations in computer graphics using C programming language. It discusses functions for translation, rotation, and scaling of 2D objects.
The algorithms take input for the type of transformation, parameters for the transformation, and coordinates of the 2D object. Translation simply adds the translation offsets to the x and y coordinates. Rotation rotates the object around the x-axis or y-axis based on the angle input. Scaling multiplies the x and y coordinates by respective scaling factors.
The transformed object is displayed on the graphics screen. Thus basic 2D transformations like translation, rotation, scaling are implemented through these algorithms in C.
This document provides an overview of Laplace transforms. Key points include:
- Laplace transforms convert differential equations from the time domain to the algebraic s-domain, making them easier to solve. The process involves taking the Laplace transform of each term in the differential equation.
- Common Laplace transforms of functions are presented. Properties such as linearity, differentiation, integration, and convolution are also covered.
- Partial fraction expansion is used to break complex fractions in the s-domain into simpler forms with individual terms that can be inverted using tables of transforms.
- Solving differential equations using Laplace transforms follows a standard process of taking the Laplace transform of each term, rewriting the equation in the s-domain, solving
This document discusses information theory and related concepts such as entropy, Kullback-Leibler divergence, mutual information, independent component analysis, clustering algorithms, change point detection, kernel density estimation, and nonparametric regression. It provides mathematical definitions and formulas for these concepts. Figures are included to illustrate clustering and change point detection methods. The document contains information that could be useful for understanding techniques in machine learning, signal processing, and statistics.
This MATLAB code applies the Roberts edge detection filter to an input image. It first reads in the image and converts it to grayscale. It then initializes an output matrix to store the filtered image. Next, it defines the Roberts filter masks and loops through the image, calculating the gradient approximations in the x and y directions using the masks. It calculates the magnitude of the gradient vector and stores it in the output matrix. Finally, it displays the original and filtered images, with the filtered image highlighting edges detected by the Roberts filter.
1) The Fourier transform is useful for designing filters by allowing systems to be described in the frequency domain. Important properties include linearity, time shifts, differentiation, and convolution.
2) Convolution becomes simple multiplication in the frequency domain. To solve a differential/convolution equation using Fourier transforms, take the Fourier transform of the inputs, multiply them, and take the inverse Fourier transform of the result.
3) An example shows designing a low-pass filter by taking the inverse Fourier transform of a rectangular function, producing an ideal low-pass response without time-domain oscillations. Approximating this with a causal function provides some low-pass filtering characteristics.
Computational Tools and Techniques for Numerical Macro-Financial ModelingVictor Zhorin
A set of numerical tools used to create and analyze non-linear macroeconomic models with financial sector is discussed. New methods and results for computing Hansen-Scheinkman-Borovicka shock-price and shock-exposure elasticities for variety of models are presented. Spectral approximation technology (chebfun):
numerical computation in Chebyshev functions piece-wise smooth functions
breakpoints detection
rootfinding
functions with singularities
fast adaptive quadratures continuous QR, SVD, least-squares linear operators
solution of linear and non-linear ODE
Frechet derivatives via automatic differentiation PDEs in one space variable plus time
Stochastic processes:
(quazi) Monte-Carlo simulations, Polynomial Expansion (gPC), finite-differences (FD) non-linear IRF
Boroviˇcka-Hansen-Sc[heinkman shock-exposure and shock-price elasticities Malliavin derivatives
Many states:
Dimensionality Curse Cure
low-rank tensor decomposition
sparse Smolyak grids
Modern Control - Lec 05 - Analysis and Design of Control Systems using Freque...Amr E. Mohamed
The document discusses frequency response analysis and Bode plots. It begins with an introduction to frequency response and how the steady state response of a linear time-invariant system to a sinusoidal input is another sinusoid at the same frequency with a different magnitude and phase. The complex ratio of the output to input is called the frequency response. It then discusses Bode plots which show the magnitude and phase of the frequency response on logarithmic scales. Key features of components in open-loop transfer functions and how they affect the Bode plot shapes are explained. An example demonstrates drawing the Bode plots for a sample transfer function.
The document contains 20 MATLAB programs demonstrating various plotting and data visualization techniques including:
- Plotting sinusoidal waves and applying half/full wave rectification
- Converting between polar and Cartesian coordinates and plotting circles
- Generating and plotting cylinder surfaces
- Using EZ functions like ezplot, ezsurf, and ezpolar to plot functions
- Taking derivatives and plotting functions and their derivatives
- Plotting noise signals and calculating rate of change over time
- Generating 2D and 3D surfaces from gridded data
- Plotting polar plots and converting to Cartesian coordinates
- Applying FFT and filtering to decompose a signal into frequency components
- Using pie charts and 3D pie plots to
This document provides chapter summaries and example problems from a solutions manual for a textbook on electromagnetism. It includes 20 chapters that cover topics like vector calculus, electrostatics, magnetostatics, and Maxwell's equations. The document also provides notes on mathematical expressions and references for additional resources related to the textbook.
1. The document discusses various 2D and 3D transformations including translation, scaling, rotation, reflection, and shearing.
2. It explains the basic transformation types including rigid-body, affine, and free-form and provides examples of each.
3. Key concepts covered include the homogeneous coordinate system, composition of transformations using matrices, and expressing transformations like translation, scaling, and rotation using matrices.
1) The document proposes a method for correcting rolling shutter distortion and stabilizing video without correspondence using an iterative optimization approach.
2) Key steps include estimating the transformation matrix between frames using intensity gradients, optimizing the matrix parameters to minimize reconstruction error, and applying the transformations for distortion correction and motion stabilization.
3) Experimental results demonstrate the method can effectively correct rolling shutter distortion and stabilize shaky video on benchmark datasets.
This document is a Master's thesis that applies the technique of differential reduction to the pole (DRTP) to aeromagnetic anomaly data from Iran. DRTP transforms magnetic anomalies measured at various latitudes and orientations to how they would appear if measured at the magnetic pole. The thesis applies this technique using MATLAB programs to iteratively calculate the DRTP correction. It then uses the corrected DRTP map of Iran's magnetic anomalies to better delineate geological structures like fold belts and delineate boundaries between volcanic and metamorphic zones.
The document provides an introduction to Laplace transforms. Key points:
- Laplace transforms are a mathematical tool that converts differential equations in the time domain to algebraic equations in the complex frequency (s) domain, making problems easier to solve.
- Common transforms include impulse, step, ramp, and exponential functions.
- Properties and theorems allow transforming derivatives, integrals, shifts, and scaling.
- Tables provide standard transforms to convert between time and s domains.
- Solving problems involves taking the Laplace transform of equations, using properties to solve for the transform in s domain, then applying the inverse transform.
- Partial fraction expansions break complex fractions into simpler forms for applying transforms.
Cálculo ii howard anton - capítulo 16 [tópicos do cálculo vetorial]Henrique Covatti
This document contains a chapter from a textbook on vector calculus. It includes 33 multi-part exercises involving concepts like divergence, curl, line integrals, and parameterizing curves. The exercises provide calculations and proofs related to vector fields and vector operations in three dimensions.
- The code defines a class called PrintLoops that inherits from IRVisitor. It overrides the visit method to print the name of any For nodes visited.
- A print_loops function takes a statement and uses a PrintLoops visitor to print the name of all for loops in the statement.
This document summarizes Haar wavelet transforms and Daubechies wavelet transforms, including their forward and inverse transforms. It provides MATLAB code implementations for 1D and 2D transforms using Haar, Daub4, Daub6, Daub5/3 and Daub4 wavelets. The transforms can be applied for multiple levels (levels 1 through k) to decompose signals into approximation and detail coefficients.
This document summarizes Haar wavelet transforms and Daubechies wavelet transforms, including their forward and inverse transforms. It provides MATLAB code implementations for 1D and 2D transforms using Haar, Daub4, Daub6, Daub5/3 and Daub4 wavelets. The transforms can be applied for multiple levels (layers) to decompose signals into different frequency subbands.
Problem Solving by Computer Finite Element MethodPeter Herbert
This document discusses using finite element methods and the cotangent Laplacian to solve partial differential equations numerically. It begins by explaining how to generate simplicial meshes by dividing a region into basic pieces. It then introduces the cotangent Laplacian, which approximates the Laplacian operator, and how it is calculated based on angles in triangles. Finally, it demonstrates applying the cotangent Laplacian to solve sample Dirichlet and Neumann boundary value problems and compares the approximate solutions to exact solutions, showing convergence as the mesh is refined.
Similar to Image Compression Comparison Using Golden Section Transform, CDF 5/3 (Le Gall 5/3) and CDF 9/7 Wavelet Transform By Matlab (20)
GraphRAG for Life Science to increase LLM accuracyTomaz Bratanic
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2. Introduction
In mathematics, Golden Section Transform is a
new class of discrete orthogonal transform. Its
theory involves golden ratio, Fibonacci number,
Fourier analysis, wavelet theory, stochastic
process, integer partition, matrix analysis, group
theory, combinatorics on words, its applied fields
include digital signal (image/audio/video)
processing, denoising, watermarking,
compression, coding and so on. see here.
3. Simple Compression Scheme
Source image is
divided into 8 × 8
blocks of pixels
Apply 2D transform
(LGST, HGST,
CDF 5/3, CDF 9/7)
to each 8 × 8 block
4. Set compression ratio, keep largest
abs(elements) in each block, and set all the
others to zero
Apply inverse 2D transform to each 8 × 8 block
to reconstruct the image
Compute PSNR and compare the results.
9. Matlab Code
gstdemo.m % main file
lgst2d.m % 4-level 2d low golden section transform
ilgst2d.m % 4-level inverse 2d low golden section transform
lword.m % Type-L golden section decomposion of fibonacci number
legall53td.m % 3-level 2d CDF 5/3 transform lifting scheme
ilegall53td.m % 3-level inverse 2d CDF 5/3 transform
hgst2d.m % 2-level 2d high golden section transform
ihgst2d.m % 2-level inverse 2d high golden section transform
hword.m % Type-H golden section decomposion of fibonacci number
cdf97td.m % 3-level 2d CDF 9/7 transform lifting scheme
icdf97td.m % 3-level inverse 2d CDF 9/7 transform
keep.m % keep largest abs(values) in matrix
psnr.m % compute MSE and PSNR
all the codes can be downloaded here:
http://goldensectiontransform.org/
10. legall53td.m
function H = legall53td(X)
% Author: Jun Li, more info@ http://goldensectiontransform.org/
%
% function H = legall53td(X,nlevel)
% 3-level 2d LeGall 5/3 wavelet transform of 8*8 image matrix
nlevel = 3; % 3-level transform for each 8*8 image block
[xx,yy] = size(X);
H=X;
for i=1:nlevel
for j=1:xx
[ss,dd] = legall531d(H(j,1:yy)); % row transform
H(j,1:yy) = [ss,dd];
end
for k=1:yy
[ss,dd] = legall531d(H(1:xx,k)'); % column transform
H(1:xx,k) = [ss,dd]';
end
xx = xx/2;
yy = yy/2;
end
%% 1d LeGall 5/3 lifting scheme % symw_ext->...12321...
function [ss,dd] = legall531d(S)
N = length(S);
ga = -1/2;
gb = 1/4;
gc = sqrt(2);
s0 = S(1:2:N-1); % S(1),S(3),S(5),S(7)...
d0 = S(2:2:N); % S(2),S(4),S(6),S(8)...
d1 = d0 + ga*(s0 + [s0(2:N/2) s0(N/2)]);
s1 = s0 + gb*(d1 + [d1(1) d1(1:N/2-1)]);
ss = gc*s1;
dd = d1/gc;
11. cdf97td.m function H = cdf97td(X)
% Author: Jun Li, more info@ http://goldensectiontransform.org/
%%
function H = cdf97td(X,nlevel)
% 3-level 2d CDF 9/7 wavelet transform of 8*8 image matrix
nlevel = 3; % 3-level transform for each 8*8 image block
[xx,yy] = size(X);
H=X;
for i=1:nlevel
for j=1:xx
[ss,dd] = cdf971d(H(j,1:yy)); % row transform
H(j,1:yy) = [ss,dd];
end
for k=1:yy
[ss,dd] = cdf971d(H(1:xx,k)'); % column transform
H(1:xx,k) = [ss,dd]';
end
xx = xx/2;
yy = yy/2;
end
%% 1d CDF 9/7 wavelet lifting scheme % symw_ext->...12321...
function [ss,dd] = cdf971d(S)
N = length(S);
fa = -1.586134342;
fb = -0.05298011854;
fc = 0.8829110762;
fd = 0.4435068522;
fz = 1.149604398;
s0 = S(1:2:N-1); % S(1),S(3),S(5),S(7)...
d0 = S(2:2:N); % S(2),S(4),S(6),S(8)...
d1 = d0 + fa*(s0 + [s0(2:N/2) s0(N/2)]);
s1 = s0 + fb*(d1 + [d1(1) d1(1:N/2-1)]);
d2 = d1 + fc*(s1 + [s1(2:N/2) s1(N/2)]);
s2 = s1 + fd*(d2 + [d2(1) d2(1:N/2-1)]);
ss = fz*s2;
dd = d2/fz;
12. lgst2d.m
function H = lgst2d(X)
% Author: Jun Li, more info@ http://goldensectiontransform.org/
% function H = lgst2d(X,nlevel)
% 4-level 2d low golden section transform of 8*8 image matrix
nlevel = 4; % 4-level transform for each 8*8 image block
global lj; % only used by function lgst1d(S) below.
global FBL; % only used by function lgst1d(S) below.
[xx,yy] = size(X);
ind = floor(log(xx*sqrt(5)+1/2)/log((sqrt(5)+1)/2)); % determine index
FBL = filter(1,[1 -1 -1],[1 zeros(1,ind-1)]);
% FBL = Fibonacci sequence -> [1 1 2 3 5 8...];
H=X;
for lj=1:nlevel
for j=1:xx
[ss,dd] = lgst1d(H(j,1:yy)); % row transform
H(j,1:yy) = [ss,dd];
end
for k=1:yy
[ss,dd] = lgst1d(H(1:xx,k)'); % column transform
H(1:xx,k) = [ss,dd]';
end
xx = FBL(end-lj); % round((sqrt(5)-1)/2*xx); 8*8 block: xx=8->5->3->2
yy = FBL(end-lj); % round((sqrt(5)-1)/2*yy); 8*8 block: yy=8->5->3->2
end
%% 1d low golden section transform
function [ss,dd] = lgst1d(S)
global lj;
global FBL;
index = 0;
h = 1;
lform = lword(length(S));
for i=1:length(lform)
index = index + lform(i);
if lform(i) == 1
ss(i) = S(index);
else % lform(i) == 2
ss(i) = (sqrt(FBL(lj))*S(index-1)+sqrt(FBL(lj+1))*S(index))/sqrt(FBL(lj+2));
dd(h) = (sqrt(FBL(lj+1))*S(index-1)-sqrt(FBL(lj))*S(index))/sqrt(FBL(lj+2));
h = h+1;
end
end
13. lword.m
function lform = lword(n)
% Author: Jun Li, more info@ http://goldensectiontransform.org/
% Type-L golden section decomposion of fibonacci number n,
% e.g. lword(8) = [1 2 2 1 2];
if n == 1
lform = [1];
elseif n == 2
lform = [2];
else
next = round((sqrt(5)-1)/2*n);
lform = [lword(n-next),lword(next)];
end
14. hgst2d.m
function H = hgst2d(X)
% Author: Jun Li, more info@ http://goldensectiontransform.org/
% function H = hgst2d(X,nlevel)
% 2-level 2d high golden section transform of 8*8 image matrix
nlevel = 2; % 2-level transform for each 8*8 image block
global hj; % only used by function hgst1d(S) below.
global FBH; % only used by function hgst1d(S) below.
[xx,yy] = size(X);
ind = floor(log(xx*sqrt(5)+1/2)/log((sqrt(5)+1)/2)); % determine index
FBH = filter(1,[1 -1 -1],[1 zeros(1,ind-1)]);
% FBH = Fibonacci sequence -> [1 1 2 3 5 8...];
H=X;
for hj=1:nlevel
for j=1:xx
[ss,dd] = hgst1d(H(j,1:yy)); % row transform
H(j,1:yy) = [ss,dd];
end
for k=1:yy
[ss,dd] = hgst1d(H(1:xx,k)'); % column transform
H(1:xx,k) = [ss,dd]';
end
xx = FBH(end-2*hj); % round((3-sqrt(5))/2*xx); 8*8 block: xx=8->3
yy = FBH(end-2*hj); % round((3-sqrt(5))/2*yy); 8*8 block: yy=8->3
end
%% 1d high golden section transform
function [ss,dd] = hgst1d(S)
global hj;
global FBH;
index = 0;
g = 1;
h = 1;
hform = hword(length(S));
for i=1:length(hform)
index = index + hform(i);
if hform(i) == 2
ss(i) = (sqrt(FBH(2*hj-1))*S(index-1)+sqrt(FBH(2*hj))*S(index))/sqrt(FBH(2*hj+1));
dd(2*i-g) = (sqrt(FBH(2*hj))*S(index-1)-sqrt(FBH(2*hj-1))*S(index))/sqrt(FBH(2*hj+1));
g = g+1;
else % hform(i) == 3
ss(i) = (sqrt(FBH(2*hj))*S(index-2)+sqrt(FBH(2*hj-1))*S(index-1)+sqrt(FBH(2*hj))*S(index))/sqrt(FBH(2*hj+2));
dd(i+h-1) = (sqrt(FBH(2*hj-1))*S(index-2)-2*sqrt(FBH(2*hj))*S(index-1)+sqrt(FBH(2*hj-1))*S(index))/sqrt(2*FBH(2*hj+2));
dd(i+h) = (S(index-2)-S(index))/sqrt(2);
h = h+1;
end
end
15. hword.m
function hform = hword(n)
% Author: Jun Li, more info@ http://goldensectiontransform.org/
% Type-H golden section decomposion of fibonacci number n,
% e.g. hword(8) = [3 2 3];
if n == 2
hform = [2];
elseif n == 3
hform = [3];
else
next = round((sqrt(5)-1)/2*n);
hform = [hword(n-next),hword(next)];
end
16. keep.m
function X = keep(X)
% keep the largest abs(elements) of X,
% global RATIO set in gstdemo.m is in [0,1].
global RATIO;
N = floor(prod(size(X))*RATIO);
[MM,i] = sort(abs(X(:)));
X(i(1:end-N)) = 0;
17. psnr.m
function [MSE,PSNR] = psnr(X,Y)
% Compute MSE and PSNR.
MSE = sum((X(:)-Y(:)).^2)/prod(size(X));
if MSE == 0
PSNR = Inf;
else
PSNR = 10*log10(255^2/MSE);
end
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27. Links
Jun Li's Golden Section Transform Home Page
http://goldensectiontransform.org/
http://www.maths.surrey.ac.uk/hosted-sites/
R.Knott/Fibonacci/fibrab.html
http://en.wikipedia.org/wiki/Fibonacci_word
http://mathworld.wolfram.com/RabbitSequence.html
The infinite Fibonacci word, https://oeis.org/A005614
Lower Wythoff sequence, http://oeis.org/A000201
Upper Wythoff sequence, http://oeis.org/A001950
https://oeis.org/A072042