Fourier analysis and medical image filtering / M'hamed Bentourkia.

Format:

Book

Author/Creator:

Bentourkia, M'hamed, author.

Language:

English

Subjects (All):

Imaging systems in medicine.

Physical Description:

1 online resource (652 pages)

Place of Publication:

Newcastle upon Tyne, England : Cambridge Scholars Publishing, [2022]

Summary:

Even after completing a course on Fourier transform, it is difficult for many students to mentally represent a function or an image in the frequency domain. Several technologies exclusively work in the frequency domain like television and magnetic resonance imaging (MRI), making an understanding of this issue essential. As such, this book depicts the transformation into the frequency domain in detail, covering topics from Fourier series to image filtering and enhancement. It also provides a progressive introduction to programming in Matlab, as well as detailed operations of Fourier series and Fourier transforms, convolution and filtering, with numerical applications on functions and images at each step of the data processing.

Contents:

Intro

Table of Contents

About the Author

Preface

Summary

Chapter I

I.1 Brief history of complex numbers

I.2 Definition

I.3 Geometric representation of complex numbers

I.4 Operations on complex numbers

I.5 The polar form of a complex number, or trigonometric representation

I.6 Euler's notation

I.7 The nth roots of a complex number

I.8 Usefulness of complex numbers

I.9 Exercises

Chapter II

II.1 Periodicity

II.2 Parity

II.3 Orthogonality

II.4 Exercises

Chapter III

III.1 Definition of Fourier series

III.2 The conditions of Dirichlet

III.3 The harmonics

III.4 The complex form of FS

III.5 The properties of least squares approximation

III.6 Parseval's theorem

III.7 Symmetry and Fourier series

III.8 Gibbs phenomenon

III.9 Exercises

Chapter IV

IV.1 From the Fourier series to the Fourier transforms

IV.2 Definitions

IV.3 Determination of frequencies

IV.4 Amplitude and phase spectra

IV.5 Fourier transforms and symmetry

IV.6 The Dirac function

IV.7 Properties of Fourier transforms

IV.8 Generalized Fourier transforms

IV.9 Two-dimensional Fourier transforms

IV.10 Properties of 2D Fourier transforms

IV.11 Exercises

Chapter V

V.1 From Fourier transforms to discrete Fourier transforms

V.2 Decomposition of the operations of discrete Fourier transforms

V.3 Symmetry in discrete Fourier transforms

V.4 Interpolation of discrete Fourier transforms

V.5 The periodicity

V.6 Frequency folding

V.7 Nyquist critical frequency

V.8 The two-dimensional DFT

V.9 Exercises

Chapter VI

VI.1 The elements of the fast Fourier transform (FFT)

VI.2 Inverse FFT (IFFT)

VI.3 Two-dimensional FFT

VI.4 Clarifications on the use of the FFT

VI.5 Exercises

Chapter VII

VII.1 Intuitive definitions of convolution.

VII.2 Definition and properties of the convolution

VII.3 Correlation

VII.4 Deconvolution

VII.5 Deconvolution by direct methods

VII.6 Deconvolution by the Wiener method

VII.7 Deconvolution with other approaches

VII.8 Examples of convolution with classical kernels

VII.9 Exercises

Chapter VIII

VIII.1 Sampling

VIII.2 The sampling theorem

VIII.3 The cardinal sine function

VIII.4 Theory of function reconstruction from samples

VIII.5 Aliasing and folding, again

VIII.6 Reconstruction of sampled functions

VIII.7 Exercises

Chapter IX

IX.1 Filters and frequency bands

IX.2 The transfer function

IX.3 Examples of current filters

IX.4 Utilization of filters

Bibliography

Matlab Functions

Index.

Notes:

Includes bibliographical references and index.

Description based on print version record.

Other Format:

Print version: Bentourkia, M'hamed Fourier Analysis and Medical Image Filtering

ISBN:

9781527589278