Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Part IIntroduction to image processing
- Part IIFundamental concepts of image processing
- 5Fundamentals of digital image processing
- 6Image enhancement in the spatial domain
- 7Image enhancement in the frequency domain
- 8Image restoration
- Part IIIImage analysis
- Part IVMedical applications and ongoing developments
- Appendix AThe Fourier series and Fourier transform
- Appendix BSet theory and probability
- Appendix CShape and texture
- Bibliography
- Index
- Plate Section
7 - Image enhancement in the frequency domain
from Part II - Fundamental concepts of image processing
Published online by Cambridge University Press: 12 September 2018
Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Part IIntroduction to image processing
- Part IIFundamental concepts of image processing
- 5Fundamentals of digital image processing
- 6Image enhancement in the spatial domain
- 7Image enhancement in the frequency domain
- 8Image restoration
- Part IIIImage analysis
- Part IVMedical applications and ongoing developments
- Appendix AThe Fourier series and Fourier transform
- Appendix BSet theory and probability
- Appendix CShape and texture
- Bibliography
- Index
- Plate Section
Summary
Overview
A number of mathematical transformations can be applied to images to obtain information that is not readily available in the raw image. The Fourier transform is the most popular although other transforms, such as wavelets and the Gabor transform, are being increasingly used. The Fourier transform converts the spatial domain representation of an image into an alternative representation in the Fourier domain, in terms of spatial frequencies. Convolution of the input data with the point spread function of an imaging system results in the formation of an image. The convolution operation in the spatial domain is equivalent to multiplication in the Fourier domain, which is a more efficient method of performing smoothing or sharpening of an image.
Learning objectives
After reading this chapter you will be able to:
• describe how periodic waveforms consist of a linear superposition of sinusoids;
• explain how the Fourier transform is derived from the Fourier series;
• illustrate the concept of the discrete Fourier transform in two dimensions, with its dependence on sample and hold;
• describe the phenomenon of aliasing and apply appropriate procedures to eliminate it;
• outline the properties of the Fourier transform;
• use cross-correlation to perform template matching;
• obtain the spatial resolution of an imaging system both from its point spread function (PSF) and from its modulation transfer function (MTF), and show that they are equivalent;
• use frequency domain filters to smooth or sharpen an image while avoiding ringing artifacts;
• explain the need for filters in filtered backprojection and summarize the filtered backprojection algorithm;
• outline the properties of the Radon transform;
• describe the process of direct Fourier reconstruction.
The Fourier domain
Although the convolution process provides a model for the formation of an image from input signals by a (linear shift-invariant) imaging system, there exists an alternative and equivalent way of modeling the process in terms of the spatial frequency content of the image. Spatial frequency is a measure of how frequently gray values change over distance.
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- Digital Image Processing for Medical Applications , pp. 194 - 245Publisher: Cambridge University PressPrint publication year: 2009
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