FOURIER TRANSFORM

16/10/2008 A Java Advanced Imaging Implementation

This article presents the implementation of the Fourier

transform using the Java API JAI. This API is very powerful in

the construction of image processing applications.

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Fourier Transform I N T R O D U C T I O N

The French mathematician Baptiste Joseph Fourier (1768-1830) developed the known Fourier

transform when studying the heat propagation theory (Fourier, 2003). He considered that, the heat,

as well as gravity, infiltrate in each universe substance, and its primary cause is subject to constant

and simple laws. Starting with this reckoning, he developed the mathematical laws of heat

diffusion, obeyed by the elements. One of his main concerns was the terrestrial temperature, so

much so it is awarded to him the paternity of the term “greenhouse effect”, effet de serre (Prestini,

2003).

The goal of this document is the Java implementation of the discrete Fourier transform and its

inverse, with the Java Advanced Imaging (JAI) application program interface (API), for mapping

between the spatial and frequency domains in digital images. The Fourier transform utility, among

other things, is its function as a filter, operating as low and high pass.

Generally, “Fourier analysis” means the use of Fourier “series” and “transforms”, being useful in

several fields of mathematics, physics, biology etc. In vector calculus, for instance, a vector v is an

entity with the length and direction properties. Usually, we do not do vector calculus directly with

these properties, but using the vector’s components. For example, in 2D space, they would be (v1,

v2). The length, or modulus, of v is calculated as:

These components are the coefficients of the only representation of v as a linear combination of

vectors in the standard base [i, j]:

v = v1i + v2j

As each vector can be represented this way and as [i, j] is a simple basis (their elements are

orthogonal and of unitary length), vector manipulation can be simplified to direct calculations with

each vectors components. Similarly, when Fourier was describing mathematically the heat

conduction process, he considered working with a linear combination of several elements. For the

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heat conduction in a uniform, finite length rod, he represented the temperature, u, in the x position

in the rod, u(x), in determined time, subject to certain initial and boundary conditions as:

(1)

Where c is π divided by the rod’s length, and the ak’s and bk’s are constants to be determined after

conditioning this representation to u in the equations modeling the heat flow. Fourier’s approach

was correct1, and his idea of representing a function as sine and cosines lead to the development of

several mathematical applications (Howell, 2001).

F O U R I E R S E R I E S A N D T R A N S F O R M S

If the initial temperature distribution, f(t, θ) = f(0, θ), is given, this problem is knows as initial

condition problem, for the heat equation (2). Fourier proposed a linear solution to the problem,

expanding the initial distribution of temperature as a trigonometric series.

In the lecture that Fourier presented in the French Academy, December 1807, he stated that any

arbitrary function, defined in a finite interval, could be expressed as a summation of sine and cosine

functions. The validity of the theorem was steadily extended and accepted with the works from

Dirichlet, 1829 (Johann Peter Gustav Lejeune Dirichlet, 1805-1859), Riemann, 1854 (Georg

Friedrich Bernhard Riemann, 1826-1866), Lipót Fejér, 1904 (Lipót Fejér, 1880–1959), and

Lebesgue, 1905 (Henri Léon Lebesgue, 1875-1941), compiled by O’ Connor & Robertson (2006).

The validity of Fourier series was conditioned only by the integrability of the function f(θ)

(Kanasewich, 1981).

Consider a tiny, circular ring. To simplify, consider its length as 2π. Let θ be the variable that

determines the position in the ring (angle in radians), such that θ and θ + 2π give the same position

in the ring. The temperature in time t and the position θ is defined by f(t, θ). The heat flow in the

ring is given by the following differential partial equation (Jayakumar, 1998):

(2)

1According to Duddeck (2002), Fourier submitted his paper On the propagation of heat in solid bodies to Paris Institute in December, 21st, 1807, being criticized by the scientific committee (Lagrange, Laplace, Monge and Lacroix). Lagrange and Laplace objection concerned the function’s expansion as trigonometric series, knows as Fourier series. His work became known from 1822, with the Théorie Analytique de la Chaleur publication.

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If the initial temperature distribution, f(t, θ) = f(0, θ), is given, this problem is knows as initial

condition problem, for the heat equation (2). Fourier proposed a linear solution to the problem,

expanding the initial distribution of temperature as a trigonometric series.

In the lecture that Fourier presented in the French Academy, December 1807, he stated that any

arbitrary function, defined in a finite interval, could be expressed as a summation of sine and cosine

functions. The validity of the theorem was steadily extended and accepted with the works from

Dirichlet, 1829 (Johann Peter Gustav Lejeune Dirichlet, 1805-1859), Riemann, 1854 (Georg

Friedrich Bernhard Riemann, 1826-1866), Lipót Fejér, 1904 (Lipót Fejér, 1880–1959), and

Lebesgue, 1905 (Henri Léon Lebesgue, 1875-1941), compiled by O’ Connor & Robertson (2006).

The validity of Fourier series was conditioned only by the integrability of the function f(θ)

(Kanasewich, 1981).

Fourier developed the idea to find the solution of the heat equation, as differential partial equations

with initial and boundary conditions (2). The Fourier series of a function f(θ), defined in the

interval (-I, I), is given by Debnath (2002)2:

(3)

And the Fourier coefficients, already known by Euler in 1777 (Leonhard Euler, 1707-1783), as the

following integrals formulas:

(4)

Where . The Fourier series represents the resolution of a given function in an infinity set,

but discrete, of harmonic components. There is also the continuous form of the Fourier series, not

showed here. The Fourier series and integrals, and its applications, were the main issues of his

treatise published in 1822. Carleson solved the main issue about the convergence of the Fourier

series in 1966 (Lennart Axel Edvard Carleson, 1928-), which proofed that these series of a

continuous function almost always are convergent (Debnath, 2002).

2 Considering sine and cosine: , as of:

, e

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The Fourier transform descended from the theorem of the Fourier integral, sustained in his treatise

Théorie analytique de la chaleur. The Fourier series and transforms are acknowledged as one of the

most important mathematic discoveries, with a wide application front in several scientific fields.

Much of these applications use the Fourier transform in the time and frequency domains. The

Fourier transform of a signal, or function f(t), is defined as (Debnath, 2002):

(5)

Where is the Fourier transform with variable ω or a frequency function ω, and is the

inner product in a Hilbert space. The transformation of a signal divides it in a sine wave with

different frequencies and phases, commonly knows as Fourier spectrum (Debnath, 2002).

Materially, the Fourier integral measures the f oscillation in ω frequency, and is called

frequency spectrum of the signal or wave f(t). One can call f(t) as a wave form in the time domain

and as the wave form in the frequency domain. This terminology describes the duality and

the equivalence of the waves forms representation.

The success of the Fourier transform was due to the fact that, under certain conditions, the signal

f(t) can be reconstructed with the inverse Fourier transform formula:

(6)

According to Champeney (1987), Fourier didn’t give, precisely, a set of conditions under what the

inversion theorem could happen. His legacy is based upon the discovery and demonstration of how

the theorem could be useful in applications related to physics problems, being burdened to

subsequent mathematicians to add rigor in Fourier ideas. Actually, the “Fourier theorem” consists

not only of a sole theorem, but in several theorems, all about a common theme.

To Champeney (1987), a lot of things depend on how one interprets the integral symbol in these

theorems. It is necessary to decide between Lebesgue or Riemann integration. Each interpretation

comes up with its own theorems. This way, it is mandatory to recognize that the own definition of

what one could understand by the Fourier transform, becomes uncertain. The initial developments

of the Fourier transform used the Riemann integration theory. An alternative approach for this

integration, developed by Lebesgue, is more powerful and simple to use, respect to the Fourier

theorem, and virtually all the modern mathematical approach uses the Lebesgue theory.

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Fundamentally, the difference between the integration processes of Lebesgue versus Riemann is:

the Lebesgue integral considers infinity integration limits and certain infinity singularities all in one

simple limit process, whilst the Riemann integral is not able to deal with infinity limits of

integration or with infinities functions singularities. Functions with singularity are either infinity or

do not have finite derivatives in all intervals. Singularity is a point where an equation, surface or

function degenerates (a limit issue in which a class of objects changes its nature, then belonging to

other class of objects, normally, simpler). Singularities are also known as singular points.

As shown in (5) and (6), the Fourier transform (FT) and its inverse (IFT), need two modifications

for use in digital image processing: (1) use the two-dimensional plane or spatial domain; and (2) use

a discrete version. The discrete version of the Fourier transform (DFT) is given as (Myler & Weeks,

1993):

(7)

Where M is the number of pixel in the x direction, N is the number of pixel in the y direction, u = 0,

1, 2,…, M – 1; and v = 0, 1, 2,…, N – 1. The inverse Fourier transform is given as (Myler & Weeks,

1993):

(8)

With x = 0, 1, 2,…, M – 1, and y = 0, 1, 2,…, N – 1.

The equations (7) and (8) show that the transform of a image M × N yields a matrix M × N of

coefficients. As the inverse transform rebuilds the original image using these coefficients, they must

build a complete representation of the information contained in the image. The Fourier transform

manages the image in the frequency domain and the inverse maps the image back to the spatial

domain. When working with digital images, a variant of the above equations, named fast Fourier

transform (FFT), is used, based upon the separability property (Gonzalez & Wintz, 1977; Lyon,

1999).

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J A V A A D V A N C E D I M A G I N G ( J A I )

JAI classes, or API, are not found in the standard J2SDK package. It is necessary to download them

from the web site:

http://java.sun.com/products/java-media/jai

Besides the JAI package, the above site has the API documentation, JAI developers’ community

help addresses, and a tutorial.

JAI has two main operators to work with the Fourier frequency domain: DFT and IDFT. The discrete

Fourier transform (DFT) operator creates an image in the frequency domain with complex data,

where the even bands represent the real domain, and the odd bands represent the imaginary domain

of the Fourier transform.

This means that each pixel in the frequency domain has a magnitude and a phase angle, so as to say

that the generated band quantity in this domain is twice the number in the spatial domain. The

mapping from the frequency to the spatial domain is accomplished using the inverse discrete

Fourier transform (IDFT). In addition to these operators, there are some that work the complex side

of the image: Magnitude, MagnitudeSquared, Phase and PeriodicShift

The Magnitude operator computes each pixel’s magnitude or amplitude spectrum from the complex

image. The source image must have an even pair of bands (2, 4 etc.), representing the real domain

and an odd pair of bands (1, 3 etc.), representing the imaginary domain of each complex pixel. The

MagnitudeSquared operator powers by two each magnitude pixel, yielding the potency spectrum.

These operators’ implementations are:

public PlanarImage imagemMagnitude(PlanarImage dft) {

System.out.println("Creating magnitude image...");

ParameterBlock pb = new ParameterBlock();

pb.addSource(dft);

return JAI.create("magnitude", pb);

}

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// Compute the potency spectrum

public PlanarImage imagemQuadradoMagnitude(PlanarImage dft) {

System.out.println("Creating squared magnitude...");

ParameterBlock pb = new ParameterBlock();

pb.addSource(dft);

return JAI.create("magnitudeSquared", pb);

}

The Phase operator computes the phase angle of each complex image pixel. It uses the even and

odd bands set, a DFT image. The implementation code is:

public PlanarImage imagemFase(PlanarImage dft) {

System.out.println("Creating phase image...");

ParameterBlock pb = new ParameterBlock();

pb.addSource(dft);

return JAI.create("phase", pb);

}

Another JAI frequency operator is PeriodicShift, that computes the periodic image translation in

the x and/or y direction. This operator is used to make a translation in the amplitude spectrum

origin, transferring, e.g., the zero frequency to the center of image, an usual procedure when

showing this spectrum:

public PlanarImage imagemDeslocaXY(PlanarImage src, int shiftX,

int shiftY) {

System.out.println("Calculating periodic translation...");

ParameterBlock pb = new ParameterBlock();

pb.addSource(src);

pb.add(shiftX);

pb.add(shiftY);

return JAI.create("periodicShift", pb);

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}

After obtaining the separated images with the amplitude spectrum and phase, one can apply a low

or high pass filter, or another operation, individually. To show the result of procedures done in the

frequency domain, it is necessary to aggregate again the real and imaginary components, permitting

the use of the IDFT operator. The PolarToComplex operator does this, creating a complex image

from the phase and magnitude images:

// Compute complex image

public PlanarImage imagemPolar2Complex(PlanarImage mag,

PlanarImage phase) {

System.out.println("Creating complex image...");

ParameterBlock pb = new ParameterBlock();

pb.addSource(mag);

pb.addSource(phase);

return JAI.create("polarTocomplex", pb);

}

The resulting PlanarImage is sent to IDFT, whose function is to rebuild the original image. Fig. 1

shows the process. JAI requires a chain process for the Fourier operators. The picture in Fig. 1

shows the implementation of the program JAIDFT.java. Its purpose is to show the importance of

the information contained in the phase component of the original image. If one erases it, as making

it zero in the example, the reconstruction of the image is damaged, as one can see running the

program.

Figure 1 JAIDFT.java flowchart

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As shown in the picture, for one to find the Fourier transform of an image, just feed the image name

to DFT. This operator generates two images, representing the magnitude and the phase. These

images synthesize the complex image, created by DFT, in the polar coordinates space, (r, θ), being

the magnitude the radial coordinate, r, and the phase, the angular coordinate, θ. In order to map the

image from the frequency domain back to the spatial domain, the image must be of complex type.

The PolarToComplex operator does this, creating a complex image, from the magnitude and phase

images. From this point on, the IDFT operator is used to generate the result in the spatial domain.

C A S E S T U D Y

Fig. 2 shows the result from the program JAIDFT.java. When the phase information from the

image is turned to zero in the frequency domain, the original image cannot be recovered, showing

that the image decomposition from the spatial domain to the frequency domain store the same

information, although in diverse planes.

Figure 2 JAIDFT.java result

W H E R E T O G E T T H E S O U R C E C O D E

Source code is available, under the General Public License GNU, from the site:

http://repositorio.agrolivre.gov.br/projects/pid.

Please note: if you intend to use this material in any publication, please give credits to:

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MIRANDA, J. I.; CAMARGO NETO, J. Implementação da transformada de Fourier com o Java

Advanced Imaging. Campinas: Embrapa Informática Agropecuária, 2006. Páginas: 5 (Embrapa

Informática Agropecuária. Comunicado Técnico, 74). Available in:

<http://www.cnptia.embrapa.br/files/ct74.pdf>.

FOR ANY additional information, please contact:

jose.ig.miranda@gmail.com

B I B L I O G R A P H Y

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DUDDECK, F. M. E. Fourier BEM: generalization of boundary element methods by Fourier

transform. Berlin: Springer-Verlag, 2002. 183p.

FOURIER, J. B. J. The analytical theory of heat. Mineola, NY: Courier Dover Publications, 2003.

496p.

GONZALEZ, R. C.; WINTZ, P. Digital image processing. Reading, MA: Addison-Wesley

Publishing Co., 1977. 431p.

HOWELL, K. B. Principles of Fourier analysis. Boca Raton, FL: CRC Press, 2001. 792p.

JAYAKUMAR, R. Method of applied Fourier analysis. Boston: Birkhäuser, 1998. 385p.

KANASEWICH, E. R. Time sequence analysis in geophysics. Alberta: University of Alberta,

1981. 480p.

LYON, D. Image processing in Java. Upper Saddle River, NJ: Prentice Hall, 1999. 532p.

MYLER, H. R.; WEEKS, A. R. Computer imaging in C. Englewood Cliffs: Prentice Hall, 1993.

284p.

PRESTINI, E. The evolution of applied harmonic analysis: models of the real world. Boston:

Birkhäuser, 2003. 368p.