G52IIP, School of Computer Science, University of Nottingham 1 Image Transforms Fourier Transform Basic idea

G52IIP, School of Computer Science, University of Nottingham 2 Image Transforms Fourier transform theory Let f(x) be a continuous function of a real variable x. The Fourier transform of f(x) is Given F(u), f(x) can be obtained by using the inverse Fourier transform

G52IIP, School of Computer Science, University of Nottingham 3 Image Transforms Fourier transform theory The Fourier transform F(u) is in general complex It is often convenient to write it in the form

G52IIP, School of Computer Science, University of Nottingham 4 Image Transforms Fourier transform theory Magnitude and Phase Fourier Spectrum of f(x) Phase angle Power Spectrum (spectrum density function) of f(x)

G52IIP, School of Computer Science, University of Nottingham 5 Image Transforms Fourier transform theory Frequency Euler’s formula u is called the frequency variable

G52IIP, School of Computer Science, University of Nottingham 6 Image Transforms Fourier transform theory Intuitive interpretation An infinite sum of sine and cosine terms, each u determines the frequency of its corresponding sine cosine pair

G52IIP, School of Computer Science, University of Nottingham 7 Image Transforms Fourier transform

G52IIP, School of Computer Science, University of Nottingham 8 Image Transforms Fourier transform When W become smaller, what will happen to the spectrum?

G52IIP, School of Computer Science, University of Nottingham 9 Image Transforms Discrete Fourier transform Continuous function f(x) is discretized into a sequence by taking N samples x units apart

G52IIP, School of Computer Science, University of Nottingham 10 Image Transforms Discrete Fourier transform pair of the sampled function

G52IIP, School of Computer Science, University of Nottingham 11 Image Transforms Fourier transform of unit impulse function and 0 t

G52IIP, School of Computer Science, University of Nottingham 12 Image Transforms Fourier transform of unit impulse function 0 x (x)(x) 0 u 1 F(ju) F

G52IIP, School of Computer Science, University of Nottingham 13 Image Transforms Fourier transform of unit impulse train Here t = x and  = u

G52IIP, School of Computer Science, University of Nottingham 14 Convolution The convolution of two functions f(x) and g(x), denote f(x)*g(x)

G52IIP, School of Computer Science, University of Nottingham 15 Convolution An example

G52IIP, School of Computer Science, University of Nottingham 16 Convolution Convolution and Spatial Filtering f(x,y) w(x,y) f(x,y)*w(x,y)

G52IIP, School of Computer Science, University of Nottingham 17 Convolution Convolution theorem

G52IIP, School of Computer Science, University of Nottingham 18 Sampling FT tt -ww  t  t f(t)f(t) F(u)F(u) s(t)s(t) S(u)S(u) s(t)f(t)s(t)f(t) S(u)*F(u)

G52IIP, School of Computer Science, University of Nottingham 19 Sampling FT  t tt -ww G(u)G(u) G(u)[S(u)*F(u)]= F(u)] f(t)f(t) -ww

G52IIP, School of Computer Science, University of Nottingham 20 Sampling Theorem Bandwidth, Sample Rate, and Nyquist Theorem The sampling rate (Nyquist rate) must be at least two times the bandwidth of a bandlimited signal  t -ww G(u)G(u) G(u)[S(u)*F(u)]= F(u)] -ww

G52IIP, School of Computer Science, University of Nottingham 21 Aliasing Over- and under-sampling Anti-aliasing filtering

G52IIP, School of Computer Science, University of Nottingham 22 Aliasing Consider an image with 512 alternating vertical black and white stripes. (You may not even be able to see the alternating stripes because of poor screen resolution. But take my word for it, they are there.) Source:

G52IIP, School of Computer Science, University of Nottingham 23 Aliasing The image is created by sampling an image with 512 alternating values of black (gray = 0) and white (gray = 255). Starting in row 0, 512 samples of the image are taken. For each successive row, 1 fewer sample is taken from row 0, (i.e. for row 1, take 511 samples, for row 2, take 510 samples,... for row 511, take 1 sample). The whole row is then reconstructed from the samples by pixel replication. The result is a colossal aliasing pattern. Source:

G52IIP, School of Computer Science, University of Nottingham 24 Aliasing More examples

G52IIP, School of Computer Science, University of Nottingham 25 Aliasing More examples

G52IIP, School of Computer Science, University of Nottingham 26 Aliasing More examples

G52IIP, School of Computer Science, University of Nottingham 27 Image Transforms 2D Fourier Transform (Fourier Transform of Images)

G52IIP, School of Computer Science, University of Nottingham 28 Image Transforms 2D Fourier Transform (Fourier Transform of Images) Fourier Spectrum of f(x) Phase angle Power Spectrum (spectrum density function) of f(x)

G52IIP, School of Computer Science, University of Nottingham 29 Image Transforms 2D Discrete Fourier Transform (Fourier Transform of Digital Images)

G52IIP, School of Computer Science, University of Nottingham 30 Frequency Domain Processing What does frequency mean in an image?

G52IIP, School of Computer Science, University of Nottingham 31 Frequency Domain Processing What does frequency mean in an image?

G52IIP, School of Computer Science, University of Nottingham 32 Frequency Domain Processing What does frequency mean in an image?

G52IIP, School of Computer Science, University of Nottingham 33 Frequency Domain Processing What does frequency mean in an image? High frequency components – fast changing/sharp features Low frequency components – slow changing/smooth features

G52IIP, School of Computer Science, University of Nottingham 34 Frequency Domain Processing The foundation of frequency domain techniques is the convolution theorem

G52IIP, School of Computer Science, University of Nottingham 35 Frequency Domain Processing H(u, v) is called the transfer function

G52IIP, School of Computer Science, University of Nottingham 36 Frequency Domain Processing Typical lowpass filters and their transfer functions

G52IIP, School of Computer Science, University of Nottingham 37 Frequency Domain Processing Typical lowpass filters and their transfer functions

G52IIP, School of Computer Science, University of Nottingham 38 Frequency Domain Processing Example

G52IIP, School of Computer Science, University of Nottingham 39 Frequency Domain Processing Example

G52IIP, School of Computer Science, University of Nottingham 40 Frequency Domain Processing Typical lowpass filters and their transfer functions

G52IIP, School of Computer Science, University of Nottingham 41 Frequency Domain Processing Example

G52IIP, School of Computer Science, University of Nottingham 42 Frequency Domain Processing Typical lowpass filters and their transfer functions

G52IIP, School of Computer Science, University of Nottingham 43 Frequency Domain Processing Example

G52IIP, School of Computer Science, University of Nottingham 44 Frequency Domain Processing Example

G52IIP, School of Computer Science, University of Nottingham 45 Frequency Domain Processing Example

G52IIP, School of Computer Science, University of Nottingham 46 Frequency Domain Processing Typical highpass filters and their transfer functions

G52IIP, School of Computer Science, University of Nottingham 47 Frequency Domain Processing Typical highpass filters and their transfer functions

G52IIP, School of Computer Science, University of Nottingham 48 Frequency Domain Processing Typical highpass filters and their transfer functions

G52IIP, School of Computer Science, University of Nottingham 49 Frequency Domain Processing Examples

G52IIP, School of Computer Science, University of Nottingham 50 Frequency Domain Processing Examples

G52IIP, School of Computer Science, University of Nottingham 51 Frequency Domain Processing Examples

G52IIP, School of Computer Science, University of Nottingham 52 Frequency Domain Processing More examples

G52IIP, School of Computer Science, University of Nottingham 53 Frequency Domain Processing Examples

G52IIP, School of Computer Science, University of Nottingham 54 Frequency Domain Processing Examples

G52IIP, School of Computer Science, University of Nottingham 55 Frequency Domain Processing Spatial vs frequency domain

G52IIP, School of Computer Science, University of Nottingham 56 Frequency Domain Processing Spatial vs frequency domain

G52IIP, School of Computer Science, University of Nottingham 57 Frequency Domain Processing Examples