J. Comp. & Math. Sci. Vol.4 (1), 1-3 (2013)
Fourier Transform and Applications in Image Processing PRAVEEN SHRIVASTAVA1, ARVIND GUPTA2 and S. K. VIJAY3 1
Sadhu Vaswani College Bhopal, INDIA. Govt. M.V.M. College Bhopal, INDIA. 3 Govt. Gitanjali College Bhopal, INDIA. 2
(Received on: December 28, 2012) ABSTRACT In this paper we propose a computationally efficient methods of Fourier imaging analysis using Fourier transform Fourier Analysis is a powerful tool even when periodicity is not directly a part of the problem being solved. Discrete Fourier Transforms (DFT) are well-suited for computation by computer, especially when using Fast Fourier Transform (FFT) algorithms. Keywords: Image processing, discrete fourier transform.
HISTORY AND INTRODUCTION
TRANSFORM
Fourier transform (FT) is named in the honor of Joseph Fourier (1768-1830), one of greatest names in the history of mathematics and physics1,13. 1822, Jean Baptiste Joseph Fourier, The Analytic Theory of Heat and periodic function can be expressed as sum of sines and cosines of different frequencies (Fourier series) if it has some mild mathematical conditions2,14. Functions which are not periodic but have finite area under the curve, can be expressed as integral of sines and cosines multiplied by a weighting function (Fourier transform); inverse transform without lost of information15.
Expansion of continuous function into weighted sum of sines and cosines3,4. ∞
x(t ) = a0 + ∑ 2 k =1
[ a ⋅ cos (k ω t ) + b ⋅ sin (k ω t )] k
0
k
0
(1) x(t ) ⋅ cos(k ω t ) dt T ∫T 2 b = ∫T x(t ) ⋅ sin(k ω t ) dt T
a
k
=
2
0
0
0
0
k
0
0
2π
ω =T 0
= 2π
f
0
0
If x(t) is even, i.e., x(-t) = x(t) like cosine, then bk = 0.
Journal of Computer and Mathematical Sciences Vol. 4, Issue 1, 28 February, 2013 Pages (1-79)
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Praveen Shrivastava, et al., J. Comp. & Math. Sci. Vol.4 (1), 1-3 (2013)
If x(t) is odd, i.e., x(-t) = -x(t) like sine, then ak = 0. Can be thought of as a substitution Example of a substitution
X (ω ) = F{x(t )} =
Original equation: x + 4x² – 8 = 0 Familiar form: ax² + bx + c = 0 Let: y = x² Solve for y x = ±√y Take inverse Transform: y = L¯¹(y)
Complex Exponential Fourier Series Expansion of continuous function into weighted sum of complex exponentials5,6. ∞
ω ⋅t
i⋅k ⋅
∑c e
0
(2)
k
k = −∞
converts a function from the time
ck =
T ∫T 0
ω
0
=
2π
T
ω ⋅t
− i ⋅k ⋅
1
x(t ) ⋅ e
0
dt
0
= 2π
f
∞
− iω t
∫ x(t ) ⋅ e
dt
(3)
−∞
Property of transforms
x(t ) =
continuous-to-continuous mapping. Fourier transform of x(t) is X(ω)7: (converts from time space to frequency space)8,9.
0
0
If x(t) is real, c-k = ck*. For k = 0, ck = average value of x(t) over one period. a0/2 = c0; ak = ck + c-k; bk = i · (ck - c-k) Discrete Fourier Transform In practice, we often deal with discrete functions (digital signals, for example) Discrete version of the Fourier Transform is much more useful in computer science: Maps one function to another:
Fourier inverse transform of X(ω) recovers x(t): (converts from frequency space to time space) ∞ iω t 1 x (t ) = F-1{ X (ω )} = X (ω ) ⋅ e dω ∫ 2π −∞ x(t) and X(ω) form a Fourier transform pair: x(t) ↔ X(ω) The Fourier Transform is a special case of the Laplace Transform, s = i ·ω Fast Fourier Transform X [ k ] = DFT { x [ n ]} =
N −1
∑
n=0
x[ n ] ⋅ e
− i ( 2 π kn / N )
k = 0, 1, …, N-1 • The FFT is a computationally efficient algorithm to compute the Discrete Fourier Transform and its inverse. • Evaluating the sum above directly would take O(N2) arithmetic operations. • The FFT algorithm reduces the computational burden to O(N log N) arithmetic operations. • FFT requires the number of data points to be a power of 2 (usually 0 padding is used to make this true) • FFT requires evenly-spaced time series
REFERENCES 1. H. J. Nussbaumer,. Fast Fourier Transform and Convolution Algorithms, 2nd ed. New York : Springer-Verlag (1982).
Journal of Computer and Mathematical Sciences Vol. 4, Issue 1, 28 February, 2013 Pages (1-79)
Praveen Shrivastava, J. Comp. & Math. Sci. Vol.4 (1), 1-3 (2013)
2. P. Robert, Performance of the Discrete Fourier Transform Satellite Imagery Classification Technique. Systems and Applied Sciences Corp Riverdale MD (1980). 3. D. B. Rowe,. Modeling both magnitude and phase of complex-valued fMRI data. NeuroImage, 25, 1310-1324 (2005). 4. D. B. Rowe and Logan, B.R.,A complex way to computefMRI activation. Neuro Image, 24, pp. 1078-1092 (2004). 5. D.B. Rowe, Nencka, A.S. and Hoffman, R.G., Signal and noise of Fourier reconstructed fMRI data, Journal of Neuroscience Methods, 159, pp. 361369 (2007). 6. Ramirez, R. W.,The FFT: Fundamentals and Concepts. Englewood Cliffs, NJ: Prentice-Hall (1985). 7. C. Shu, and Jain, R., Efficient Fourier image analysis algorithm for aligned rectangular and trapezoidal wafer structures, Proc. SPIE, Volume 1661, pp. 345-356 (1992). 8. C. D. Sogge., Fourier Integrals in Classical Analysis. New York: Cambridge University Press (1993). 9. X. Tang and Stewart W.K., Optical and Sonar Image Classification: Wavelet
10.
11.
12.
13.
14.
15.
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Packet Transform vs Fourier Transform. Computer Vision and Image Understanding, Volume 79, Number 1, pp. 25-46(22) (2000). M.Trott, The Mathematica GuideBook for Programming. New York: SpringerVerlag. (2004). J. S. Walker. Fast Fourier Transform, 2nd ed. Boca Raton, FL: CRC Press (1996). H.S Wu,., Barba, J. and Gil, J., An iterative algorithm for cell segmentation using short-time Fourier transform. J Microsc. 184(Pt 2):127-32 (1996). C. Xu and Prince, J., Snakes, Shapes, and Gradient Vector Flow, IEEE Transactions on Image Processing, 7(3), pp. 359-369 (1998). W Zou,. and Wang, D., Texture identification and image segmentation via Fourier transform. Proc. SPIE, Vol. 4550, pp. 34-39, Image Extraction, Segmentation, and Recognition, Tianxu Zhang; Bir Bhanu; Ning Shu; Eds (2001). E. Zwicker and Fastl H., Psychoacoustics: Facts and Models, Berlin: Springer Verlag, second updated edition (1999).
Journal of Computer and Mathematical Sciences Vol. 4, Issue 1, 28 February, 2013 Pages (1-79)