Topic
Hartley transform
About: Hartley transform is a research topic. Over the lifetime, 2709 publications have been published within this topic receiving 79944 citations.
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TL;DR: This note makes a critical comparison of some matlab programs for the digital computation of the fractional Fourier transform that are freely available and describes the own implementation that lters the best out of the existing ones.
134 citations
TL;DR: The aim of this study is to show how the F-transform technique can be generalized from the cases of constant components to the case of polynomial components.
134 citations
Book•
31 Jan 2004
TL;DR: In this paper, two-dimensional Fourier Transform of the Shah Function was used for line detection in two dimensions, one dimension and the other dimension of the Hartley Transform of a Shah Function.
Abstract: 1 Introduction.- Summary of the Chapters.- Notation.- Teaching a Course from This Book.- The Problems.- Aspects of Imaging.- Computer Code.- Literature References.- Recommendation.- 2 The Image Plane.- Modes of Representation.- Some Properties of a Function of Two Variables.- Projection of Solid Objects.- Image Distortion.- Operations in the Image Plane.- Binary Images.- Operations on Digital Images.- Reflectance Distribution.- Data Compression.- Summary.- Appendix: A Contour Plot Programt.- Literature Cited.- Further Reading.- Problems.- 3 Two-Dimensional Impulse Functions.- The Two-Dimensional Point Impulse.- Rules for Interpreting Delta Notation.- Generalized Functions.- The Shah Functions iii and 2III.- Line Impulses.- Regular Impulse Patterns.- Interpretation of Rectangle Function of f(x).- Interpretation of Rectangle Function of f(x,y).- General Rule for Line Deltas.- The Ring Impulse.- Impulse Function of f(x,y).- Sifting Property.- Derivatives of Impulses.- Summary.- Literature Cited.- Problems.- 4 The Two-Dimensional Fourier Transform.- One Dimension.- The Fourier Component in Two Dimensions.- Three or More Dimensions.- Vector Form of Transform.- The Corrugation Viewpoint.- Examples of Transform Pairs.- Theorems for Two-Dimensional Fourier Transforms.- The Two-Dimensional Hartley Transform.- Theorems for the Hartley Transform.- Discrete Transforms.- Summary.- Literature Cited.- Further Reading.- Problems.- 5 Two-Dimensional Convolution.- Convolution Defined.- Cross-Correlation Defined.- Feature Detection by Matched Filtering.- Autocorrelation Defined.- Understanding Autocorrelation.- Cross-Correlation Islands and Dilation.- Lazy Pyramid and Chinese Hat Function.- Central Value and Volume of Autocorrelation.- The Convolution Sum.- Computing the Convolution.- Digital Smoothing.- Matrix Product Notation.- Summary.- Literature Cited.- Problems.- 6 The Two-Dimensional Convolution Theorem.- Convolution Theorem.- An Instrumental Caution.- Point Response and Transfer Function.- Autocorrelation Theorem.- Cross-Correlation Theorem.- Factorization and Separation.- Convolution with the Hartley Transform.- Summary.- Problems.- 7 Sampling and Interpolation in Two Dimensions.- What is a Sample?.- Sampling at a Point.- Sampling on a Point Pattern, and the Associated Transfer Function.- Sampling Along a Line.- Curvilinear Sampling.- The Shah Function.- Fourier Transform of the Shah Function.- Other Patterns of Sampling.- Factoring.- The Two-Dimensional Sampling Theorem.- Undersampling.- Aliasing.- Circular Cutoff.- Double-Rectangle Pass Band.- Discrete Aspect of Sampling.- Interpolating Between Samples.- Interlaced Sampling.- Appendix: The Two-Dimensional Fourier Transform of the Shah Function.- Literature Cited.- Problems.- 8 Digital Operations.- Smoothing.- Nonconvolutional Smoothing.- Trend Reduction.- Sharpening.- What is a Digital Filter?.- Guard Zone.- Transform Aspect of Smoothing Operator.- Finite Impulse Response (FIR).- Special Filters.- Densifying.- The Arbitrary Operator.- Derivatives.- The Laplacian Operator.- Projection as a Digital Operation.- Moire Patterns.- Functions of an Image.- Digital Representation of Objects.- Filling a Polygon.- Edge Detection and Segmentation.- Discrete Binary Objects.- Operations on Discrete Binary Objects.- Union and Intersection.- Pixel Morphology.- Dilation.- Coding a Binary Matrix.- Granulometry.- Conclusion.- Literature Cited.- Problems.- 9 Rotational Symmetry.- What Is a Bessel Function?.- The Hankel Transform.- The jinc Function.- The Struve Function.- The Abel Transform.- Spin Averaging.- Angular Variation and Chebyshev Polynomials.- Summary.- Table of the jinc Function.- Problems.- 10 Imaging by Convolution.- Mapping by Antenna Beam.- Scanning the Spherical Sky.- Photography.- Microdensitometry.- Video Recording.- Eclipsometry.- The Scanning Acoustic Microscope.- Focusing Underwater Sound.- Literature Cited.- Problems.- 11 Diffraction Theory of Sensors and Radiators.- The Concept of Aperture Distribution.- Source Pair and Wave Pair.- Two-Dimensional Apertures.- Rectangular Aperture.- Example of Circular Aperture.- Duality.- The Thin Lens.- What Happens at a Focus?.- Shadow of a Straight Edge.- Fresnel Diffraction in General.- Literature Cited.- Problems.- 12 Aperture Synthesis and Interferometry.- Image Extraction from a Field.- Incoherent Radiation Source.- Field of Incoherent Source.- Correlation in the Field of an Incoherent Source.- Visibility.- Measurement of Coherence.- Notation.- Interferometers.- Radio Interferometers.- Rationale Behind Two-Element Interferometer.- Aperture Synthesis (Indirect Imaging).- Literature Cited.- Problems.- 13 Restoration.- Restoration by Successive Substitutions.- Running Means.- Eddington's Formula.- Finite Differences.- Finite Difference Formula.- Chord Construction.- The Principal Solution.- Finite Differencing in Two Dimensions.- Restoration in the Presence of Errors.- The Additive Noise Signal.- Determination of the Real Restoring Function.- Determination of the Complex Restoring Function.- Some Practical Remarks.- Artificial Sharpening.- Antidiffusion.- Nonlinear Methods.- Restoring Binary Images.- CLEAN.- Maximum Entropy.- Literature Cited.- Problems.- 14 The Projection-Slice Theorem.- Circular Symmetry Reviewed.- The Abel-Fourier-Hankel Cycle.- The Projection-Slice Theorem.- Literature Cited.- Problems.- 15 Computed Tomography.- Workingfrom Projections.- An X-Ray Scanner.- Fourier Approach to Computed Tomography.- Back-Projection Methods.- The Radon Transform.- The Impulse Response of the Radon Transformation.- Some Radon Transforms.- The Eigenfunctions.- Theorems for the Radon Transform.- The Radon Boundary.- Applications.- Literature Cited.- Problems.- 16 Synthetic-Aperture Radar.- Doppler Radar.- Some History of Radiofrequency Doppler.- Range-Doppler Radar.- Radargrarnmetry.- Literature Cited.- Problems.- 17 Two-Dimensional Noise Images.- Some Types of Random Image.- Gaussian Noise.- The Spatial Spectrum of a Random Scatter.- Autocorrelation of a Random Scatter.- Pseudorandom Scatter.- Random Orientation.- Nonuniform Random Scatter.- Spatially Correlated Noise.- The Familiar Maze.- The Drunkard's Walk.- Fractal Polygons.- Conclusion.- Literature Cited.- Problems.- Appendix A Solutions to Problems.
133 citations
TL;DR: The sampling and reconstruction formulas are deduced, together with the construction methodology for the multiplicative filter in the time domain based on fast Fourier transform (FFT), which has much lower computational load than the construction method in the linear canonical domain.
Abstract: As generalization of the fractional Fourier transform (FRFT), the linear canonical transform (LCT) has been used in several areas, including optics and signal processing. Many properties for this transform are already known, but the convolution theorems, similar to the version of the Fourier transform, are still to be determined. In this paper, the authors derive the convolution theorems for the LCT, and explore the sampling theorem and multiplicative filter for the band limited signal in the linear canonical domain. Finally, the sampling and reconstruction formulas are deduced, together with the construction methodology for the above mentioned multiplicative filter in the time domain based on fast Fourier transform (FFT), which has much lower computational load than the construction method in the linear canonical domain.
133 citations
TL;DR: An algorithm for the in-place computation of the discrete Fourier transform on real data: a decimation-in-time split-radix algorithm, more compact than the previously published one and a new fast Hartley transform algorithm with a reduced number of operations.
Abstract: This paper highlights the possible tradeoffs between arithmetic and structural complexity when computing cyclic convolution of real data in the transform domain. Both Fourier and Hartley-based schemes are first explained in their usual form and then improved, either from the structural point of view or in the number of operations involved. Namely, we first present an algorithm for the in-place computation of the discrete Fourier transform on real data: a decimation-in-time split-radix algorithm, more compact than the previously published one. Second, we present a new fast Hartley transform algorithm with a reduced number of operations. A more regular convolution scheme based on FFT's is also proposed. Finally, we show that Hartley transforms belong to a larger class of algorithms characterized by their "generalized" convolution property.
131 citations