- The document provides an introduction to Fourier theory, which describes how any function can be represented as a summation of sine and cosine terms of increasing frequency. This transforms the function from the spatial domain to the frequency domain.
- Key advantages of the Fourier transform include making large filtering operations faster and collecting information in a way that can separate signal from noise. It allows moving between the spatial and frequency domains using forward and inverse transforms.
- Common applications include removing periodic noise from an image by identifying and removing corresponding bright spots in the frequency domain representation.
In this paper we introduce the concept of connectedness in fuzzy rough topological spaces.
We also investigate some properties of connectedness in fuzzy rough topological spaces.
This document discusses continuity and one-sided limits of functions. It defines continuity at a point and on an open interval. Functions can have removable or nonremovable discontinuities. One-sided limits are used to investigate continuity on closed intervals. The intermediate value theorem guarantees that a continuous function takes on all intermediate values between its values at the endpoints of a closed interval. Examples demonstrate determining continuity and applying properties of continuity and the intermediate value theorem.
The document discusses the Mean Value Theorem, which states that if a function f(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists some value c in (a,b) such that:
f(b) - f(a) = f'(c)(b - a)
In other words, there is at least one point where the slope of the tangent line equals the slope of the secant line between points a and b. The document provides examples and illustrations to demonstrate how to apply the Mean Value Theorem.
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
The document discusses transformations in geometry. It defines a geometric transformation as a bijective mapping between two geometries that maps points to points and lines to lines. Reflections, rotations, and translations are provided as examples of geometric transformations in Euclidean plane geometry. It is shown that reflections, rotations, and translations are isometries that preserve distance, angle measure, and area. The composition of transformations is also discussed, and it is shown that the composition of isometries is again an isometry.
- Rolle's theorem states that if a function f(x) is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), with f(a) = f(b), then there exists at least one value c in (a,b) where the derivative f'(c) = 0.
- The mean value theorem states that if a function f(x) is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), then there exists a value c in (a,b) such that the average rate of change of f over the interval [a,b] equals the instantaneous rate of change
- The document outlines key concepts in fuzzy set theory including membership functions, set-theoretic operations, and T-norms and T-conorms (fuzzy intersection and union). It discusses basic definitions like fuzzy complements and extensions to two dimensions.
- Membership functions can take various shapes like triangular, trapezoidal, bell-shaped, and Gaussian and are used to characterize fuzzy sets.
- T-norms and T-conorms provide a way to model fuzzy intersection and union operations and must satisfy properties like monotonicity and commutativity. Common examples include minimum, product, and maximum functions.
1. The document defines extrema as the minimum and maximum values of a function over an interval. On a closed interval, a continuous function is guaranteed by the Extreme Value Theorem to have both a minimum and maximum value.
2. To find the extrema of a function over a closed interval, one finds the critical numbers where the derivative is zero or undefined, evaluates the function at the endpoints and critical numbers, and the minimum and maximum values are the lowest and highest outputs.
3. The document provides examples of finding critical numbers and using them to determine the minimum and maximum values of functions over given intervals. It emphasizes the steps of finding critical numbers, evaluating the function, and identifying the extrema.
In this paper we introduce the concept of connectedness in fuzzy rough topological spaces.
We also investigate some properties of connectedness in fuzzy rough topological spaces.
This document discusses continuity and one-sided limits of functions. It defines continuity at a point and on an open interval. Functions can have removable or nonremovable discontinuities. One-sided limits are used to investigate continuity on closed intervals. The intermediate value theorem guarantees that a continuous function takes on all intermediate values between its values at the endpoints of a closed interval. Examples demonstrate determining continuity and applying properties of continuity and the intermediate value theorem.
The document discusses the Mean Value Theorem, which states that if a function f(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists some value c in (a,b) such that:
f(b) - f(a) = f'(c)(b - a)
In other words, there is at least one point where the slope of the tangent line equals the slope of the secant line between points a and b. The document provides examples and illustrations to demonstrate how to apply the Mean Value Theorem.
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
The document discusses transformations in geometry. It defines a geometric transformation as a bijective mapping between two geometries that maps points to points and lines to lines. Reflections, rotations, and translations are provided as examples of geometric transformations in Euclidean plane geometry. It is shown that reflections, rotations, and translations are isometries that preserve distance, angle measure, and area. The composition of transformations is also discussed, and it is shown that the composition of isometries is again an isometry.
- Rolle's theorem states that if a function f(x) is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), with f(a) = f(b), then there exists at least one value c in (a,b) where the derivative f'(c) = 0.
- The mean value theorem states that if a function f(x) is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), then there exists a value c in (a,b) such that the average rate of change of f over the interval [a,b] equals the instantaneous rate of change
- The document outlines key concepts in fuzzy set theory including membership functions, set-theoretic operations, and T-norms and T-conorms (fuzzy intersection and union). It discusses basic definitions like fuzzy complements and extensions to two dimensions.
- Membership functions can take various shapes like triangular, trapezoidal, bell-shaped, and Gaussian and are used to characterize fuzzy sets.
- T-norms and T-conorms provide a way to model fuzzy intersection and union operations and must satisfy properties like monotonicity and commutativity. Common examples include minimum, product, and maximum functions.
1. The document defines extrema as the minimum and maximum values of a function over an interval. On a closed interval, a continuous function is guaranteed by the Extreme Value Theorem to have both a minimum and maximum value.
2. To find the extrema of a function over a closed interval, one finds the critical numbers where the derivative is zero or undefined, evaluates the function at the endpoints and critical numbers, and the minimum and maximum values are the lowest and highest outputs.
3. The document provides examples of finding critical numbers and using them to determine the minimum and maximum values of functions over given intervals. It emphasizes the steps of finding critical numbers, evaluating the function, and identifying the extrema.
Rolle's theorem states that if a function is continuous on a closed interval and differentiable on the open interval with equal values at the endpoints, then the derivative is 0 for at least one value in the interval. The mean value theorems - Lagrange's and Cauchy's - generalize this idea, relating the average rate of change over an interval to the instantaneous rate at a point within the interval. Examples are provided to illustrate the theorems and exceptions that can occur when their conditions are not fully met.
The document discusses fuzzy set theory and fuzzy logic. It introduces fuzzy sets and membership functions, which allow for gradual or "fuzzy" boundaries between values. Membership functions assign a value between 0 and 1 to indicate the degree to which an element belongs to a fuzzy set. Several common types of membership functions are presented, including triangular, trapezoidal, Gaussian, and generalized bell-shaped functions. The key set-theoretic operations of union, intersection, and complement are also defined for fuzzy sets using max-min or min-max composition rules.
The document discusses the Mean Value Theorem and its applications. It states that according to the Mean Value Theorem, the slope of the tangent line at some point in an interval must match the slope of the secant line between the endpoints. It provides examples of using the Mean Value Theorem to find a tangent line where the derivative is equal to the slope of the secant line, and to determine if a truck exceeded the speed limit based on its speeds passing two police cars.
This document discusses Rolle's Theorem from calculus. Rolle's Theorem states that if a function f is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), and if f(a) = f(b), then there exists at least one value c in the interval (a,b) where the derivative of f is equal to 0. The document provides an example of applying Rolle's Theorem to show that the derivative of the function f(x) = x^2 - 3x + 2 is equal to 0 at some point between the two x-intercepts of the function.
The document provides an overview of the Mean Value Theorem and Rolle's Theorem. It discusses that the Mean Value Theorem states that for any function continuous on a closed interval, there exists a point where the slope of the tangent line equals the slope of the secant line through the endpoints. Rolle's Theorem is a special case where if a function is continuous on a closed interval and differentiable on the open interval, if the function is equal at the endpoints, the derivative at some interior point is zero. Graphical interpretations are also provided to illustrate these theorems.
This document discusses continuity and the Intermediate Value Theorem (IVT) in mathematics. It defines continuity, examines examples of continuous and discontinuous functions, and establishes theorems about continuity. The IVT states that if a function is continuous on a closed interval and takes on intermediate values within its range, there exists at least one value in the domain where the function value is intermediate. An example proves the existence of the square root of two using the IVT and bisection method.
New approach to modified schur cohn criterion for stability analysis of a dis...Ritesh Keshri
This document describes a new approach to the modified Schur-Cohn criterion for stability analysis of discrete time invariant systems. The approach formulates the problem in the form of an array, similar to Chen-Chang array and Jury's criterion. The array provides values for the coefficients of the inverse polynomials at each step of iteration, as well as the ratios αi, which must satisfy certain conditions for stability. Examples are provided to demonstrate the application of the new approach. The approach reduces computational efforts compared to traditional methods and provides the necessary information for stability analysis in a concise manner.
History and Real Life Applications of Fourier AnalaysisSyed Ahmed Zaki
This document discusses the history and applications of Fourier analysis. It notes that Fourier analysis was invented by Jean Baptiste Joseph Fourier, a French mathematician and physicist born in the late 18th century. The document then lists some of the main applications of Fourier analysis, such as signal processing, image processing, heat distribution mapping, wave simplification, and light simplification. It provides examples of how Fourier analysis can be used to transform signals from the time domain to the frequency domain using Fourier series equations. Charts are shown demonstrating this transformation for simple sine waves. The document cautions that Fourier analysis works best for stationary waves and that more advanced techniques are needed for non-stationary waves like music or speech.
This document provides an overview of key concepts related to functions and their properties. It discusses function definition and notation, domain and range, continuity, increasing and decreasing functions, boundedness, local and absolute extrema, symmetry, asymptotes, and end behavior. Examples are provided to illustrate key concepts such as finding the domain and range of functions, identifying points of discontinuity, determining if a function is increasing or decreasing, and identifying local extrema. The document is intended to teach readers about important properties of functions.
This ppt covers following topics of Unit - 2 of B.Sc. 2 Mathematics Rolle's Theorem , Lagrange's mean value theorem , Mean value theorem & its example .
4.5 continuous functions and differentiable functionsmath265
The document discusses continuous and differentiable functions. It defines elementary functions as those constructed using basic operations like addition and multiplication. Continuous functions over a closed interval are bounded and have absolute maximum and minimum values. The Intermediate Value Theorem states that a continuous function takes on all values between its minimum and maximum. Differentiable functions are continuous. Rolle's Theorem says that if a differentiable function is equal at the endpoints of an interval, its derivative is zero somewhere in between.
The document defines the limit of a function and how to determine if the limit exists at a given point. It provides an intuitive definition, then a more precise epsilon-delta definition. Examples are worked through to show how to use the definition to prove limits, including finding appropriate delta values given an epsilon and showing a function satisfies the definition.
The document discusses building new functions from existing functions through algebraic combinations, composition, and implicit definitions. It provides examples of combining two functions using addition, subtraction, multiplication, and division. Composition of functions is defined as applying one function to the output of another. Implicitly defined functions relate sets of ordered pairs as relations that can define multiple functions. Examples are provided for composing, decomposing, and graphing implicitly defined functions.
American Journal of Multidisciplinary Research and Development is indexed, refereed and peer-reviewed journal, which is designed to publish research articles.
This document discusses the mean value theorem and continuity in calculus. It defines Rolle's theorem, which states that if a function is continuous on a closed interval and differentiable on the open interval, and if the function is equal at the endpoints, then its derivative must be equal to zero for at least one value between the endpoints. It then uses Rolle's theorem to prove the mean value theorem, which states that the rate of change of a function over an interval is equal to the derivative of the function at some value between the endpoints. Finally, it introduces the Cauchy mean value theorem, which relates the rates of change of two functions over an interval to their derivatives at some interior point.
1. Rolle's theorem states that if a function f(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), and f(a) = f(b), then there exists at least one number c in (a,b) where the derivative f'(c) = 0.
2. The mean value theorem states that if a function f(x) is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), there exists a number c in (a,b) such that f'(c) = (f(b) - f(a))/(b - a).
This document discusses key concepts related to rates of change and derivatives:
1) It defines average rate of change (ARC) as the slope of a secant line on a graph or using the slope formula algebraically, and instantaneous rate of change (IRC) as the slope of the tangent line.
2) It introduces the difference quotient as a way to define ARC and IRC algebraically without a graph by taking the limit as h approaches 0.
3) A derivative is defined as a function that gives the IRC, allowing it to be evaluated at any point without graphing by taking the limit of the difference quotient.
This document provides an overview of Fourier series and the Fourier transform. It defines the Fourier transform and its inverse, and how they allow transforming between the time domain and frequency domain. It discusses how Fourier series can be used to decompose functions into sums of sinusoids. It also gives examples of Fourier transforms, such as for rectangle, triangle, exponential and Gaussian functions. It notes properties like symmetry and conditions for the existence of Fourier transforms.
This document discusses representing signals as tempered distributions to provide a unified framework for signal processing theory. It introduces:
- Tempered distributions, which allow discrete-time signals to be expressed as distributions involving the Dirac delta function.
- Conditions for continuous- and discrete-time signals to be considered tempered distributions. Continuous signals must satisfy certain growth conditions, while discrete signals must be bounded by a polynomial.
- Existing definitions of multiplication and convolution of distributions have limitations for signal processing. The paper proposes new definitions of multiplication and convolution of distributions appropriate for unified signal processing theory.
The document discusses optical waveguide analysis using the Beam Propagation Method (BPM). BPM is a numerical technique to determine electromagnetic fields inside complicated waveguide structures like Y-couplers. It works by decomposing the optical mode into plane waves that are propagated through the structure and then recombined. The author implemented BPM in MATLAB to simulate double slit diffraction and a Gaussian beam. Code examples are provided for the Gaussian beam, BPM algorithm, and double slit simulation.
This document provides an overview of chapter six which discusses applications of the definite integral in geometry, science, and engineering. It introduces how definite integrals can be used to calculate volume, surface area, length of a plane curve, and work done by a force. It reviews key concepts like Riemann sums and finding the area between two curves. It then explains the specific applications of using integrals to find volume of solids obtained by rotating an area about an axis, surface area of revolution, and work done by a variable force. Examples are provided for each application.
Rolle's theorem states that if a function is continuous on a closed interval and differentiable on the open interval with equal values at the endpoints, then the derivative is 0 for at least one value in the interval. The mean value theorems - Lagrange's and Cauchy's - generalize this idea, relating the average rate of change over an interval to the instantaneous rate at a point within the interval. Examples are provided to illustrate the theorems and exceptions that can occur when their conditions are not fully met.
The document discusses fuzzy set theory and fuzzy logic. It introduces fuzzy sets and membership functions, which allow for gradual or "fuzzy" boundaries between values. Membership functions assign a value between 0 and 1 to indicate the degree to which an element belongs to a fuzzy set. Several common types of membership functions are presented, including triangular, trapezoidal, Gaussian, and generalized bell-shaped functions. The key set-theoretic operations of union, intersection, and complement are also defined for fuzzy sets using max-min or min-max composition rules.
The document discusses the Mean Value Theorem and its applications. It states that according to the Mean Value Theorem, the slope of the tangent line at some point in an interval must match the slope of the secant line between the endpoints. It provides examples of using the Mean Value Theorem to find a tangent line where the derivative is equal to the slope of the secant line, and to determine if a truck exceeded the speed limit based on its speeds passing two police cars.
This document discusses Rolle's Theorem from calculus. Rolle's Theorem states that if a function f is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), and if f(a) = f(b), then there exists at least one value c in the interval (a,b) where the derivative of f is equal to 0. The document provides an example of applying Rolle's Theorem to show that the derivative of the function f(x) = x^2 - 3x + 2 is equal to 0 at some point between the two x-intercepts of the function.
The document provides an overview of the Mean Value Theorem and Rolle's Theorem. It discusses that the Mean Value Theorem states that for any function continuous on a closed interval, there exists a point where the slope of the tangent line equals the slope of the secant line through the endpoints. Rolle's Theorem is a special case where if a function is continuous on a closed interval and differentiable on the open interval, if the function is equal at the endpoints, the derivative at some interior point is zero. Graphical interpretations are also provided to illustrate these theorems.
This document discusses continuity and the Intermediate Value Theorem (IVT) in mathematics. It defines continuity, examines examples of continuous and discontinuous functions, and establishes theorems about continuity. The IVT states that if a function is continuous on a closed interval and takes on intermediate values within its range, there exists at least one value in the domain where the function value is intermediate. An example proves the existence of the square root of two using the IVT and bisection method.
New approach to modified schur cohn criterion for stability analysis of a dis...Ritesh Keshri
This document describes a new approach to the modified Schur-Cohn criterion for stability analysis of discrete time invariant systems. The approach formulates the problem in the form of an array, similar to Chen-Chang array and Jury's criterion. The array provides values for the coefficients of the inverse polynomials at each step of iteration, as well as the ratios αi, which must satisfy certain conditions for stability. Examples are provided to demonstrate the application of the new approach. The approach reduces computational efforts compared to traditional methods and provides the necessary information for stability analysis in a concise manner.
History and Real Life Applications of Fourier AnalaysisSyed Ahmed Zaki
This document discusses the history and applications of Fourier analysis. It notes that Fourier analysis was invented by Jean Baptiste Joseph Fourier, a French mathematician and physicist born in the late 18th century. The document then lists some of the main applications of Fourier analysis, such as signal processing, image processing, heat distribution mapping, wave simplification, and light simplification. It provides examples of how Fourier analysis can be used to transform signals from the time domain to the frequency domain using Fourier series equations. Charts are shown demonstrating this transformation for simple sine waves. The document cautions that Fourier analysis works best for stationary waves and that more advanced techniques are needed for non-stationary waves like music or speech.
This document provides an overview of key concepts related to functions and their properties. It discusses function definition and notation, domain and range, continuity, increasing and decreasing functions, boundedness, local and absolute extrema, symmetry, asymptotes, and end behavior. Examples are provided to illustrate key concepts such as finding the domain and range of functions, identifying points of discontinuity, determining if a function is increasing or decreasing, and identifying local extrema. The document is intended to teach readers about important properties of functions.
This ppt covers following topics of Unit - 2 of B.Sc. 2 Mathematics Rolle's Theorem , Lagrange's mean value theorem , Mean value theorem & its example .
4.5 continuous functions and differentiable functionsmath265
The document discusses continuous and differentiable functions. It defines elementary functions as those constructed using basic operations like addition and multiplication. Continuous functions over a closed interval are bounded and have absolute maximum and minimum values. The Intermediate Value Theorem states that a continuous function takes on all values between its minimum and maximum. Differentiable functions are continuous. Rolle's Theorem says that if a differentiable function is equal at the endpoints of an interval, its derivative is zero somewhere in between.
The document defines the limit of a function and how to determine if the limit exists at a given point. It provides an intuitive definition, then a more precise epsilon-delta definition. Examples are worked through to show how to use the definition to prove limits, including finding appropriate delta values given an epsilon and showing a function satisfies the definition.
The document discusses building new functions from existing functions through algebraic combinations, composition, and implicit definitions. It provides examples of combining two functions using addition, subtraction, multiplication, and division. Composition of functions is defined as applying one function to the output of another. Implicitly defined functions relate sets of ordered pairs as relations that can define multiple functions. Examples are provided for composing, decomposing, and graphing implicitly defined functions.
American Journal of Multidisciplinary Research and Development is indexed, refereed and peer-reviewed journal, which is designed to publish research articles.
This document discusses the mean value theorem and continuity in calculus. It defines Rolle's theorem, which states that if a function is continuous on a closed interval and differentiable on the open interval, and if the function is equal at the endpoints, then its derivative must be equal to zero for at least one value between the endpoints. It then uses Rolle's theorem to prove the mean value theorem, which states that the rate of change of a function over an interval is equal to the derivative of the function at some value between the endpoints. Finally, it introduces the Cauchy mean value theorem, which relates the rates of change of two functions over an interval to their derivatives at some interior point.
1. Rolle's theorem states that if a function f(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), and f(a) = f(b), then there exists at least one number c in (a,b) where the derivative f'(c) = 0.
2. The mean value theorem states that if a function f(x) is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), there exists a number c in (a,b) such that f'(c) = (f(b) - f(a))/(b - a).
This document discusses key concepts related to rates of change and derivatives:
1) It defines average rate of change (ARC) as the slope of a secant line on a graph or using the slope formula algebraically, and instantaneous rate of change (IRC) as the slope of the tangent line.
2) It introduces the difference quotient as a way to define ARC and IRC algebraically without a graph by taking the limit as h approaches 0.
3) A derivative is defined as a function that gives the IRC, allowing it to be evaluated at any point without graphing by taking the limit of the difference quotient.
This document provides an overview of Fourier series and the Fourier transform. It defines the Fourier transform and its inverse, and how they allow transforming between the time domain and frequency domain. It discusses how Fourier series can be used to decompose functions into sums of sinusoids. It also gives examples of Fourier transforms, such as for rectangle, triangle, exponential and Gaussian functions. It notes properties like symmetry and conditions for the existence of Fourier transforms.
This document discusses representing signals as tempered distributions to provide a unified framework for signal processing theory. It introduces:
- Tempered distributions, which allow discrete-time signals to be expressed as distributions involving the Dirac delta function.
- Conditions for continuous- and discrete-time signals to be considered tempered distributions. Continuous signals must satisfy certain growth conditions, while discrete signals must be bounded by a polynomial.
- Existing definitions of multiplication and convolution of distributions have limitations for signal processing. The paper proposes new definitions of multiplication and convolution of distributions appropriate for unified signal processing theory.
The document discusses optical waveguide analysis using the Beam Propagation Method (BPM). BPM is a numerical technique to determine electromagnetic fields inside complicated waveguide structures like Y-couplers. It works by decomposing the optical mode into plane waves that are propagated through the structure and then recombined. The author implemented BPM in MATLAB to simulate double slit diffraction and a Gaussian beam. Code examples are provided for the Gaussian beam, BPM algorithm, and double slit simulation.
This document provides an overview of chapter six which discusses applications of the definite integral in geometry, science, and engineering. It introduces how definite integrals can be used to calculate volume, surface area, length of a plane curve, and work done by a force. It reviews key concepts like Riemann sums and finding the area between two curves. It then explains the specific applications of using integrals to find volume of solids obtained by rotating an area about an axis, surface area of revolution, and work done by a variable force. Examples are provided for each application.
IB Mathematics Extended Essay Decoding Dogs Barks With Fourier Analysis - G...Kimberly Williams
This document is a 3,990 word extended essay on using Fourier analysis to analyze dog bark recordings. The essay has two main parts: 1) an explanation of Fourier analysis including waveforms, Fourier series, and the transition to Fourier transforms; and 2) an application of Fourier transforms to analyze different types of bark recordings from a dog in various emotional states (distress, anger, excitement) to find correlations between the bark patterns and emotions. The goal is to determine if Fourier analysis can differentiate a dog's motivational changes by examining bark recordings.
Here are the answers to the exercises:
1. The 2007th digit after the period in the decimal expansion of 1/7 is 7, since the expansion repeats with a period of 7 digits (142857...).
2. a) and b) have finite decimal expansions, while c) does not.
3. A = [-1, 2], B = (-∞, -1] ∪ (2, ∞). C = (-∞, 1) ∪ (2, ∞). D = (-∞, 1) ∪ (3, ∞). E = [-1, 2].
The finite sets are A and E.
3. Functions
3
This document provides an overview and preface for online notes covering Calculus III topics taught by Paul Dawkins at Lamar University. It outlines the course contents, which include vectors, vector-valued functions, and multiple coordinate systems. It notes some differences between the material covered in class versus the online notes and warns students that the notes alone are not a substitute for attending class. The document is copyrighted and governed by the website's terms of use.
Motivated by presenting mathematics visually and interestingly to common people based on calculus and its extension, parametric curves are explored here to have two and three dimensional objects such that these objects can be used for demonstrating mathematics.
Epicycloid, hypocycloid are particular curves that are implemented in MATLAB programs and the motifs are presented here. The obtained curves are considered to be domains for complex mappings to have new variation of Figures and objects. Additionally Voronoi mapping is also implemented to some parametric curves and some resulting complex mappings.
Some obtained 3 dimensional objects are considered as flowers and animals inspiring to be mathematical ornaments of hypocycloid dance which is also illustrated here.
This document contains lecture notes for a first semester calculus course. It begins by discussing different types of numbers like integers, rational numbers, and real numbers which are represented by possibly infinite decimal expansions. It then introduces functions and their properties like inverse functions and implicit functions. The notes provide examples and exercises to accompany the explanations.
This document discusses various topics related to Fourier series and partial differential equations, including:
- Periodic functions and their properties.
- Fourier series representations of functions over intervals, including the calculation of Fourier coefficients.
- Using Fourier series to solve partial differential equations, including first and second order equations.
- Applications of Fourier series such as image compression using the discrete cosine transform.
The document discusses key concepts in Laplace transforms including:
1) The Laplace transform is defined as an integral transform that transforms a function of time into a function of a complex variable, simplifying analysis of differential equations.
2) Important properties include the Laplace transforms of derivatives and integrals, which allow transforming differential equations into algebraic equations.
3) The existence theorem guarantees a unique solution to initial value problems under certain conditions on the function.
The document discusses various techniques for edge detection and line detection in images, including:
- Canny edge detection, which uses thresholds to detect and link edges.
- Hough transforms, which detect shapes like lines and circles by counting points that agree with a shape model.
- RANSAC for line detection, which forms line hypotheses from random samples and counts supporting points.
- Techniques for thinning thick edges and detecting edge contours.
Wavelets for computer_graphics_stollnitzJuliocaramba
This document provides an introduction to wavelets for computer graphics applications. It begins with an overview of how wavelet transforms can hierarchically decompose functions. It then describes the Haar wavelet basis, including how one-dimensional and two-dimensional signals can be decomposed into lower resolution approximations and detail coefficients. The document focuses on explaining the mathematical foundations of wavelet transforms using the Haar basis as an example, covering topics like multiresolution analysis, scaling functions, wavelets, and orthogonal bases. It aims to give intuition for what wavelets are and the theory needed to understand and apply them.
This document summarizes a student paper about using the shortest path algorithm to interpolate contours in images. It discusses how the human visual system perceives 3D representations from 2D images and how extracting meaningful contours is challenging due to noise and discontinuities. The paper proposes using a modified Dijkstra's algorithm to find the shortest path in log-polar space, which maps circles in images to straight lines. This approach aims to identify simple, closed curves representing object contours while ignoring irrelevant edges.
This document discusses functions of several variables. It introduces notation for functions with multiple independent variables, defines the domain of such functions, and explains how to graph functions of two and three variables by sketching traces in coordinate planes and parallel planes. Level curves and contour maps are presented as ways to visualize functions of two variables in the xy-plane. Examples demonstrate finding domains, sketching graphs using traces and level curves, and interpreting contour maps.
1) Fourier analysis transforms images from the spatial domain to the frequency domain, allowing images to be manipulated in unexpected ways.
2) It represents any signal as a sum of sinusoids, encoding spatial frequency, magnitude, and phase information for each pixel.
3) This frequency domain representation can then be modified and transformed back, providing a means to filter images and extract geometric information.
The document discusses the natural logarithm function ln(x) and the natural exponential function exp(x). It begins by defining ln(x) as the area under the curve y=1/t from 1 to x, and noting that its derivative is 1/x. It then defines exp(x) as the inverse of ln(x). It is shown that for rational r, exp(r) = er, and this definition is extended to irrational r. The derivative of exp(x) is then shown to be exp(x) itself.
This document presents the results of a MATLAB workshop (taller) on Fourier analysis. It includes 4 sections: an introduction describing the objectives of Fourier analysis and the workshop; a development section showing the steps and code to solve 3 problems involving varying the amplitude, frequency, and phase of signals; solving a problem involving harmonic functions; and analyzing an audio signal. Graphs are presented of the signals analyzed. The conclusion emphasizes that practice with tools like MATLAB is important for engineering students to better understand signals and waves.
This document describes a model for measuring positional error when projecting harmonic points onto a circular screen. The model assumes projection of four collinear harmonic points onto a circle, representing a flat image and curved projection surface. Positional error is measured as the angle between projected and actual locations of points from the viewer's perspective. The goals are to determine optimal projector placement and viewer seating to minimize this error, especially for central points where viewers focus most.
The document discusses transformations in geometry. It defines a geometric transformation as a bijective mapping between two geometries that maps points to points and lines to lines. Reflections, rotations, and translations are provided as examples of geometric transformations in Euclidean plane geometry. It is shown that reflections, rotations, and translations are isometries that preserve distance, angle measure, and area. The composition of transformations is also discussed, and it is shown that the composition of isometries is again an isometry.
This document discusses different representations of curves and surfaces in computer graphics, including explicit, implicit, and parametric representations. Explicit representations express one variable as a function of others, but cannot represent all shapes. Implicit representations use equations like f(x,y)=0 but can be difficult to render points on. Parametric representations express each coordinate as a function of one or more parameters and have advantages over explicit and implicit forms like versatility and ease of rendering. The document focuses on using rational parametric polynomials, as they are simple yet powerful representations.
Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapte...University of Maribor
Slides from talk presenting:
Aleš Zamuda: Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapter and Networking.
Presentation at IcETRAN 2024 session:
"Inter-Society Networking Panel GRSS/MTT-S/CIS
Panel Session: Promoting Connection and Cooperation"
IEEE Slovenia GRSS
IEEE Serbia and Montenegro MTT-S
IEEE Slovenia CIS
11TH INTERNATIONAL CONFERENCE ON ELECTRICAL, ELECTRONIC AND COMPUTING ENGINEERING
3-6 June 2024, Niš, Serbia
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Literature Review Basics and Understanding Reference Management.pptx
Fourier analysis
1. Tutorial on Fourier Theory
Yerin Yoo
March 2001
1 Introduction: Why Fourier?
During the preparation of this tutorial, I found that almost all the textbooks on dig-
ital image processing have a section devoted to the Fourier Theory. Most of those
describe some formulas and algorithms, but one can easily be lost in seemingly
incomprehensible mathematics.
The basic idea behind all those horrible looking formulas is rather simple, even
fascinating: it is possible to form any function
¢¡¤£¦¥
as a summation of a series
of sine and cosine terms of increasing frequency. In other words, any space or
time varying data can be transformed into a different domain called the frequency
space. A fellow called Joseph Fourier first came up with the idea in the 19th
century, and it was proven to be useful in various applications, mainly in signal
processing.
1.1 Frequency Space
Let us talk about this frequency space before going any further into the details.
The term frequency comes up a lot in physics, as some variation in time, describ-
ing the characteristics of some periodic motion or behavior. The term frequency
that we talk about in computer vision usually is to do with variation in brightness
or color across the image, i.e. it is a function of spatial coordinates, rather than
time. Some books even call it spatial frequency.
For example, if an image represented in frequency space has high frequencies
then it means that the image has sharp edges or details. Let’s look at figure 1,
which shows frequency graphs of 4 different images. If you have trouble inter-
preting the frequency graphs on the top low; The low frequency terms are on the
1
2. Figure 1: Images in the spatial domain are in the middle row, and their frequency
space are shown on the top row. The bottom row shows the varying brightness of
the horizontal line through the center of an image.
(Taken from p.178 of [1].)
2
3. Figure 2: Images with perfectly sinusoidal variations in brightness: The first three
images are represented by two dots. You can easily see that the position and
orientation of those dots have something to do with what the original image looks
like. The 4th image is the sum of the first three.
(Taken from p.177 of [1].)
center of the square, and terms with higher magnitude are on the outer edges.
(Imagine an invisible axis with its origin at the center of the square.) Now, the
frequency space on the top left consists of higher frequencies as well as low ones,
so the original image has sharp edges. The second image from the left, however,
is much fuzzier, and of course the frequency graph for it only has lower frequency
terms.
Another thing to note is that if an image has perfectly sinusoidal variations in
brightness, then it can be represented by very few dots on the frequency image
as shown in figure 2. From those images, you can also see that regular images or
images of repeating pattern generate fewer dots on the frequency graph, compared
to images on figure 1 which don’t have any repeating pattern.
1.2 So, What’s the Point?
Frequency domain offers some attractive advantages for image processing. It
makes large filtering operations much faster, and it collects information together
in different ways that can sometimes separate signal from noise or allow measure-
ments that would be very difficult in spatial domain. Furthermore, the Fourier
3
5. Figure 3: (a) Our dirty looking photocopied image. (b) The representation of our
image in the frequency space, i.e. the star diagram. Look, you can see stars! (c)
Those stars, however, do no good to the image, so we rub them out. (d) Recon-
struct the image using (c) and those dirty spots on the original image are gone!
(Images taken from p.204 of [1].)
5
6. The analogy of complex numbers being coordinates on a plane lets us repre-
sent a complex number in a different way. In the above paragraph, we talked about
a complex number as a rectangular coordinate, but we can also write it as a polar
coordinate, i.e. in terms of its distance from the origin (magnitude), and the angle
that it makes with the positive real axis (angle):
2 ¡436587@9 ¨ 7BADCE9F¥
where 2
§HGI¨PQG
, and
9 PRQS CUTUVW¡QXY ¥
. The fact that
36587`9 Ya
and
7bAcCE9 Xa
makes it obvious that the two forms of representation denote the same complex
number.
Example:
¨P$6
can also be expressed as d
¡e3587@9 ¨f 7bAcCE9g¥
, where
R1S CE9
G
V
h$
which makes
9 pi8q8rcq dts degrees. The magnitude is d and the angle is
i8q8rcq dts degrees or
r
8u
radians.
2.2 Euler’s Formula
Euler’s formula is: vwyx 36587@9 ¨ 7BADC€9
(1)
where
v $8rcu
ƒ‚
$
‚…„Q„Q„ , and
9
is an angle which can be any real number. This is
proven to be true for any real number
9
.
This gives yet another representation of complex numbers to be:
2
v wyx
where r is the magnitude of a polar form of complex number, and
9
is the angle.
Example:
¨†$
can also be expressed as d
v wyx
, where
9 #i8q8rcq dts degrees or
r
8u
radians.
In some textbooks, a complex number is often expressed in the form of the
Euler’s formula without indicating so. (It is the case specially in the formulas as-
sociated with the Fourier transforms.) If you see anything in the form of 2
v wyx
, that
be sure that you know that is is just an ordinary complex number. Furthermore,
9
usually means the angle in radians if not indicated otherwise.
3 Fourier Transform
First, we briefly look at the Fourier transform in the purely mathematical point of
view, i.e. we will talk about “continuous” or “infinite” things. I will assume that
6
7. you know what mathematical symbols like ‡ , and
v
means, and you are famil-
iar with the complex numbers. Beware that the upcoming section have complex
mathematics in it, so if you suffer from “integral-o-phobia” then just skim through
to the next section and look at the discrete Fourier transform. Remember that the
Fourier transform of a function is a summation of sine and cosine terms of differ-
ent frequency. The summation can, in theory, consist of an infinite number of sine
and cosine terms.
3.1 Equations
Now, let
ˆ¡¤£¦¥
be a continuous function of a real variable
£
. The Fourier transform
of
ˆ¡¤£¦¥
is defined by the equation:
‰ ¡¤W¥ TU‘
‘
ˆ¡¤£¦¥ v T w G“’”–•8— £
(2)
where
˜
and
is often called the frequency variable. The summation of
sines and cosines might not be apparent just by looking at the above equation, but
applying Euler’s equation (see Eq. 1 in the previous section) gives
‰ ¡¤W¥ TU‘
‘
ˆ¡¤£¦¥1¡436587 $
‡
@£
7bAcC $
‡
@£¦¥ — £ r
(3)
Given
‰ ¡¤W¥
, we can go backwards and get
ˆ¡¤£¦¥
by using inverse Fourier trans-
form:
ˆ¡¤£¦¥ TU‘
‘
‰ ¡™W¥ vw G“’6”ƒ•g— r
(4)
Equations 2 and 4 are called Fourier transform pairs, and they exist if
ˆ¡¤£¦¥
is continuous and integrable, and
¢¡¤W¥
is integrable. These conditions are usually
satisfied in practice.
Note that the only difference between the forward and inverse Fourier trans-
form is the sign above
v
, which makes it easy to go back and forth between spatial
and frequency domains; it is one of the characteristics that make Fourier transform
useful.
Some of you might ask what
‰ ¡¤W¥
is.
‰ ¡¤W¥
’s are the data in the frequency
space that we talked about in the first section. Even if we start with a real function
ˆ¡™£¦¥
in spatial domain, we usually end up with complex values of
‰ ¡¤W¥
. It is
because a real number multiplied by a complex number gives a complex number,
7
8. Figure 4: A simple function and its Fourier spectrum.
(Taken from p.83 of [2].)
so
¢¡¤£¦¥ v w G“’”–•
is complex, thus the sum of these terms must also give a complex
number, i.e.
‰ ¡™d¥
. Therefore,
‰ ¡¤W¥ e§ ¡™W¥ ¨) ¡™W¥
where
§ ¡™d¥
is a real component (terms that don’t have
), and
¡¤W¥
is an imag-
inary component, (terms that involve
) when you expand the equation 3. (The
“
¡™W¥
” part is just there to remind you that the terms are the functions of
.) Just
like any other complex numbers we can also write it in the polar form, giving
‰ ¡¤W¥ 2 ¡47BADCE9 ¨f 3587@9F¥ 2
v wyx
. In most of the textbooks, this form of the
Fourier transform is written as
‰ ¡™W¥ hgi‰ ¡¤W¥ g v wyx–j ”Fk
(5)
but it’s essentially the same thing.
There are some words that we use frequently when talking about Fourier
transform. The magnitude
gl‰ ¡™d¥ g
from equation 5 is called the Fourier spec-
trum of
ˆ¡™£¦¥
and
9m¡™d¥
is phase angle. The square of the spectrum,
gl‰ ¡™W¥ gG
§ G ¡™W¥ ¨n G ¡™W¥
is often denoted as o
¡¤W¥
and is called the power spectrum of
ˆ¡¤£¦¥
.
The term spectral density is also commonly used to denote the power spectrum.
The Fourier spectrum is often plotted against values of
. The Fourier spectrum
is useful because it can be easily plotted against
on a piece of paper. (See figure
4 for an example of the Fourier spectrum.) Note that
‰ ¡™W¥
’s themselves are hard
to plot against
on the 2-D plane because they are complex numbers.
8
9. 3.2 Discrete Fourier Transform
Now that you know a thing or two about Fourier transform, we need to figure out a
way to use it in practice. Going back to the example where we transform an image
by taking brightness values from pixels, those pixel values are never continuous
to begin with. (Remember that the Fourier transform we talked about in previous
section was about a continuous function
¢¡¤£¦¥
.) Our mathematicians came up with
a good solution for this, namely the discrete Fourier transform.
Given p discrete samples of
ˆ¡¤£¦¥
, sampled in uniform steps,
‰ ¡¤W¥
p
q TUV
•srtt
ˆ¡™£¦¥ v T w G“’6”–•u q
(6)
for
vw
$81r1r1r–
pxy , and
ˆ¡™£¦¥
q TUV
”grtt
‰ ¡™W¥ vw G“’6”ƒ•u q
(7)
for
£ vw
$81r1r1r–
pxy .
Notice that the integral is replaced by the summation, which is a simple “for
loop” when programming. For those of you who are curious, the calculation inside
z
is multiplying
ˆ¡™£¦¥ {§f¨#1
with
v w G“’6”–•u q 3587d¡™|U¥
7bAcC!¡¤|U¥
where
|
$
‡
`£
:
ˆ¡¤£¦¥¦} vw G“’”–•u q ¡ §(¨P1 ¥¦}~¡e3587W¡™|U¥
7bAcC!¡¤|U¥B¥
§ 3587W¡™|U¥
§ 7bAcC!¡¤|U¥ ¨€1 3587W¡™|U¥
1 G 7bAcC!¡¤|U¥
§ 3587W¡™|U¥
§ 7bAcC!¡¤|U¥ ¨€1 3587W¡™|U¥ ¨P 7bAcC~¡™|U¥
¡ § 36587W¡¤|U¥ ¨€ 7BADC!¡¤|m¥b¥ ¨ ¡ 3587W¡™|U¥
§ 7bAcC¡™|U¥B¥
where
§01
are real numbers, and
~'
when
ˆ¡¤£¦¥
is a real number.
The implementation of this transform in C is included in the appendix, but in
the mean time, here is the pseudo-code in C style which I’m sure will make some
readers happy:
/* Data type for N set of complex numbers */
double fx[N][2];
double Fu[N][2];
/* Fourier transform to get F(0)...F(N-1) */
9
10. for (u=0; uN; u++) {
for (k=0; kN; k++) {
p = 2*PI*u*k/N;
/* real */
Fu[u][0] += fx[k][0]*cos(p) + fx[k][1]*sin(p);
/* imaginary */
Fu[u][1] += fx[k][1]*cos(p) - fx[k][0]*sin(p);
}
/* multiply the result by 1/N */
Fu[u][0]/=N;
Fu[u][1]/=N;
}
3.3 Fast Fourier Transform
The discrete Fourier transform allows us to calculate the Fourier transform on a
computer, but it is not so efficient. The number of complex multiplications and
additions required to implement Eq. 6 and 7 is proportional to p
G
. For every
‰ ¡¤W¥
that you calculate, you need to use all
ˆ¡ ¥ 1r1rr ˆ¡
pyv
¥
and there are N
‰ ¡™d¥
’s
to calculate.
It turns out proper decomposition of Eq. 6 can make the number of multi-
plication and addition operations proportional to p#‚
58ƒ G p . The decomposition
procedure is called the fast Fourier transform (FFT) algorithm. There are number
of different ways that this algorithm can be implemented, and we will not discuss
this further in this tutorial. For those of you who are interested, there is a section
devoted to this algorithm in “Numerical Recipes in C”.
3.4 Applications of Fourier Transform
There are many situations in Graphics and Vision, specially in image processing
and filtering, where Fourier transform is useful.
The Fourier method is often used on images from astronomy, microbiology,
images of repetitive structures such as crystals and so on. It is because the Fourier
transform is good for identifying a periodic component or lattice in an image.
Identifying regular patterns on an image has other advantages like removing reg-
ular dirty spots or noise from an image as illustrated in figure 3 .
The example in figure 5 also shows another application of the Fourier trans-
form; Some Fourier components of higher frequency can be removed to achieve
10
11. Figure 5: Top: Lines with zaggy edges, and the frequency components. Bottom:
Removing some frequency components results in smoother lines.
(Taken from p.180-190 of [1].)
11
12. anti-aliasing effect, i.e. removing ugly zaggy edges.
There are other techniques associated with Fourier transform, namely convolu-
tion theory, correlation, sampling, reconstruction, image compression, and more.
Since much of these topics were covered in Graphics course, I will include no
further. Instead, the rest of the tutorial will focus on a particular application of the
Fourier theory, namely the Fourier descriptors.
4 Fourier Descriptor
The Fourier descriptor is used to describe the boundary of a shape in 2 dimen-
sional space using the Fourier methods.
4.1 Parameterization of the Boundary
First, we take p point digital boundary of a shape on
£U„
-plane. We can choose
to take all the pixels occupied by the boundary, or we can take p samples from
them. This can be done by traveling the boundary anti-clockwise keeping the
constant speed for, say, p seconds, taking a coordinate every second. [3] suggests
a method for choosing an appropriate speed, which is out of the scope of this
tutorial, so I will leave it to my keen readers to look it up.
Now we have a complete set of coordinates describing the boundary. We can
call each coordinate
¡¤£`… „w…s¥
where
††ˆ‡Š‰
pˆh . You will have no trouble
imagining these coordinates plotted on the
£U„
-plane. Let us replace the labels
on each axis; name the horizontal axis
§
for “real”, and the vertical axis
for
“imaginary”. Now on the graph you have complex numbers that you know and
love. We can call those ‹
¡ ‡ ¥
’s,
‹
¡ ‡ ¥ £`… ¨ „8…
for
‡Œ8
$wr1r1rƒ
ppŽ . Although the interpretation of the sequence has been
recast, the nature of the boundary itself has not been changed. The advantage of
this representation is that it reduces a 2-D into a 1-D problem, i.e. you now have
p complex numbers instead of 2*p real numbers.
12
13. 4.2 Applying Fourier Transform
The discrete Fourier Transform of ‹
¡ ‡ ¥
gives
¡¤W¥
p
q TUV
… rtt ‹
¡ ‡ ¥ v T w G“’”–•u q
(8)
for
xw
$81r1r1r–
pp . The complex coefficients ¡¤W¥
are called the Fourier
descriptors of the boundary. Applying inverse Fourier Transform to ¡¤W¥
restores
‹
¡ ‡ ¥
.
‹
¡ ‡ ¥
q TUV
”Frtt
¡™d¥ v w G“’”–•u q
(9)
for
‡‘xw
$w1r1rrƒ
phŽ . The restored pixel values are exactly the same as the
ones that we started with.
However, we don’t have to take all p pixel values to reconstruct the original
image. We can “drop” the Fourier descriptors with higher frequencies because
their contribution to the image is very small. Expressing this as an equation,
’
‹
¡ ‡ ¥ €“
TUV
”Frtt
¡¤W¥ vw G“’”–•u q
(10)
where
‡ew
$81r1r1rƒ
phy . This is equivalent to setting ¡¤W¥ ”
for all terms
where
‡•'–
y .
The more descriptors you use to reconstruct the original image, i.e. the bigger
the
–
in the Eq. 10, the closer the result gets to the original image. (See figure 6.)
In practice, we can reconstruct an image reasonably well even though we didn’t
use all the descriptors.
4.3 Geometrical Centroid
The descriptor ¡ ¥
yields the geometrical centroid of the shape, with the
£
value
given by the real part, and
„
value by the imaginary part. Consider only having
the value of ¡ ¥
as your Fourier descriptor, (and ¡
¥
„Q„Q„ ¡
p—x
¥ h
)
we can still try to reconstruct the original shape from it. The result we get is
a circle, situated about the center of the original shape. This circle is called the
geometrical centroid. In fact, eliminating all but the first
$
Fourier descriptors will
always result in a circle. (Proof found in p456 of [4].)
The concept of the geometrical centroid relates perfectly well to the Fourier
theory in general. When a function
¢¡¤£¦¥
is represented in the frequency space as
13
14. ‰ ¡¤W¥
,
‰ ¡ ¥
is the lowest frequency term which is the sinusoid making the major
contribution to the image, i.e. the average brightness.
Talking more about the significance of this geometrical centroid, ¡ ¥
is the
only component in Fourier descriptors that is dependent on the actual location of
the shape. For example, say, you had a shape which was centered on the origin,
and you calculated Fourier descriptors of that image. Even if you translated the
same image to somewhere else on the plane, you don’t have to re-calculate the
Fourier descriptors. All you need to adjust is the centroid, ¡ ¥
.
Moreover, the Fourier descriptors should be insensitive to other geometrical
changes like rotation, scale and also the choice of the starting point. (The pixel
point on the boundary where we start taking coordinates, i.e.
¡¤£ t „ t ¥
.) It turns
out that the Fourier descriptors are not strictly insensitive to these geometrical
changes, but the changes can be related to simple transformations on the descrip-
tors. (Table 8.1 on p.501 of [2] lists some basic properties of Fourier descriptors
that might help you understand this point.)
4.4 Applications of Fourier Descriptors
Fourier descriptors are often used to smooth out fine details of a shape. As you
have seen in the previous section, using the portion of Fourier descriptors to re-
construct an image smooths out the the sharp edges and fine details found in the
original shape. Filtering an image with Fourier descriptors provides a simple tech-
nique of contour smoothing.
Fourier description of an edge is also used for template matching. Since all
the Fourier descriptors except the first ¡ ¥
do not depend on the location of the
edge within the plane, this provides a convenient method of classifying objects
using template matching of an object’s contour. A set of Fourier descriptors is
computed for a known object. Ignoring the first component of the descriptors,
the other Fourier descriptors are compared against the Fourier descriptors of un-
known objects. The known object, whose Fourier descriptors are the most similar
to the unknown object’s Fourier descriptors, is the object the unknown object is
classified to.
Fourier descriptors can also be used for calculation of region area, location
of centroid, and computation of second-order moments; Describing the specific
techniques involving Fourier descriptors is out of the scope of this tutorial, but
p.207 of [3] has various pointers to some journal articles for some further reading
for those who are keen. Some limitations of Fourier descriptors are also discussed
on the same page of [3], again giving various pointers.
14
15. Figure 6: Examples of reconstructions from Fourier descriptors. M is the number
of descriptors taken to reconstruct.
(Taken from p.500 of [2].)
15
16. Figure 7: Examples using Fourier descriptors: (a) the original edge image with
1024 edge pixels, (b) 3 Fourier coefficients, (c) 21 Fourier coefficients, (d) 61
Fourier coefficients, (e) 201 Fourier coefficients, and (f) 401 Fourier coefficients.
(Taken from p.457 of [4].)
16
17. 5 Summary and Comments
The Fourier theory is based on the idea that any function can be composed of sines
and cosines of different frequencies. In computer vision, images in the spatial do-
main can be transformed into the frequency domain by the Fourier transform.It is
a very useful technique in image processing, because some operations and mea-
surements are better be done in the frequency space than in the spatial domain.
The implementation of the Fourier transform is called discrete Fourier transform
(DFT), and other algorithms like fast Fourier transform (FFT) was also devel-
oped to reduce the complexity of the DFT. One of the techniques that involve
the Fourier transform is the Fourier descriptors. They describe the boundary of a
shape, and holds various properties that are useful in various applications.
The purpose of this tutorial is to give you the basic overview of the Fourier
theory as well as to show some techniques that uses the Fourier theory. Like I said
in the introduction, there are many textbooks that cover the Fourier theory, and I
hope that you’ll find those books less difficult to understand after going through
this tutorial. Each textbooks that I looked at discuss somewhat different aspects
of the Fourier theory, so here are some pointers to “where to look”:
[2] seems to be the book that is referenced the most. It, however, has a hu-
mongous chapter describing all sorts of aspects of the Fourier transform without
really explaining “why”. The chapter is also very mathematically inclined, and it
takes a while to see the relevance. I suggest that you look through the chapters
when you feel somewhat confident about the topic.
[1] has very good introductory chapter on the Fourier theory. It’s rather easy
to read, and it has many good examples and figures.
[4] has a good section on the Fourier descriptors with good examples and
figures. However, it is very basic, and the author never discusses the limitations
of the technique.
[3] talks about the Fourier descriptors in 2 pages, and yet it seems to contain
the most information. It is very hard to get the basic concepts, but it provides
plenty of pointers to journal articles.
6 Focus Questions
Let us try out a few questions to make sure that we learned something:
1 Why is the Fourier transform useful in computer vision and graphics?
17
18. 2 There are three ways of representing a complex number. Express the com-
plex number, whose value of the real part is 3 and the imaginary part 4, by
using those three representations.
3 You took 4 discrete samples of brightness values from a scan line of an
image. Those were
ˆ¡ ¥ ywr s ,
ˆ¡
¥ ywrcu s ,
ˆ¡ $ ¥
rc
and
¢¡
d
¥
rc$ s .
Work out the first two data values in frequency domain by using the Fourier
transform, i.e.
‰ ¡ ¥
and
‰ ¡
¥
. What are the Fourier spectrums of these
terms?
4 What are the properties of the Fourier descriptor?
5 What are the advantages of using the Fourier descriptors to describe a shape?
References
[1] The Image Processing Handbook, chapter 4. CRC Press, 1992.
[2] Digital Image Processing, chapter 3 and 8.2.3. Addison-Wesley Publishing
Company, 1993.
[3] Image Processing, Analysis and Machine Vision, chapter 6.2.3. Chapman and
Hall, 1993.
[4] Fundamentals of Electronic Image Processing, chapter 8.4. IEEE Press, 1996.
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