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8
Transforms
The usual mathematical representation of an image is a function of two spatial variables: f ( x, y ) . The value of the function at a particular location ( x, y ) represents the intensity of the image at that point. The term transform refers to an alternative mathematical representation of an image. This section defines several important transforms and shows examples of theirapplication to image processing. Topics covered include Terminology (p. 8-2) Fourier Transform (p. 8-3) Discrete Cosine Transform (p. 8-16) Provides definitions of image processing terms used in this section Defines the Fourier transform and some of its applications in image processing Describes the Discrete Cosine Transform (DCT) of an image and its application, particularly in image compressionDescribes how the Image Processing Toolbox radon function computes projections of an image matrix along specified directions.

Radon Transform (p. 8-20)

8

Transforms

Terminology
An understanding of the following terms will help you to use this chapter. Note that this table includes brief definitions of terms related to transforms; a detailed discussion of these terms and the theory behindtransforms is outside the scope of this User’s Guide.

Terms Discrete transform

Definitions

A transform whose input and output values are discrete samples, making it convenient for computer manipulation. Discrete transforms implemented by MATLAB and the Image Processing Toolbox include the discrete Fourier transform (DFT) and the discrete cosine transform (DCT). The domain in which animage is represented by a sum of periodic signals with varying frequency. An operation that when performed on a transformed image, produces the original image. The domain in which an image is represented by intensities at given points in space. This is the most common representation for image data. An alternative mathematical representation of an image. For example, the Fourier transform is arepresentation of an image as a sum of complex exponentials of varying magnitudes, frequencies, and phases. Transforms are useful for a wide range of purposes, including convolution, enhancement, feature detection, and compression.

Frequency domain

Inverse transform

Spatial domain

Transform

8-2

Fourier Transform

Fourier Transform
The Fourier transform is a representation of an imageas a sum of complex exponentials of varying magnitudes, frequencies, and phases. The Fourier transform plays a critical role in a broad range of image processing applications, including enhancement, analysis, restoration, and compression. This section includes the following subsections: • “Definition of Fourier Transform” • “Discrete Fourier Transform” on page 8-8, including a discussion of fastFourier transform • “Applications” on page 8-11 (sample applications using Fourier transforms)

Definition of Fourier Transform
If f ( m, n ) is a function of two discrete spatial variables m and n, then we define the two-dimensional Fourier transform of f ( m, n ) by the relationship
∞ ∞

F ( ω 1, ω 2 ) =





f ( m, n )e – jω 1 m e – jω 2 n

m = –∞ n = –∞

The variables ω1 andω2 are frequency variables; their units are radians per sample. F ( ω 1, ω 2 ) is often called the frequency-domain representation of f ( m, n ) . F ( ω 1, ω 2 ) is a complex-valued function that is periodic both in ω 1 and ω 2 , with period 2π . Because of the periodicity, usually only the range – π ≤ ω 1, ω 2 ≤ π is displayed. Note that F ( 0, 0 ) is the sum of all the values of f ( m, n ) . Forthis reason, F ( 0, 0 ) is often called the constant component or DC component of the Fourier transform. (DC stands for direct current; it is an electrical engineering term that refers to a constant-voltage power source, as opposed to a power source whose voltage varies sinusoidally.) The inverse two-dimensional Fourier transform is given by 1 f ( m, n ) = --------2 4π

∫ω

π

1

= –π...
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