IMAGE PROCESSING (RRY025)

THE CONTINUOUS 2D FOURIER TRANSFORM

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INTRODUCTION

• A vital tool in image processing.

Also a prototype of other image transforms, cosine, Wavelet

etc.

• Applications

Image Filtering - Smooth or sharpen images.

Image Restoration - Remove distortions, such as blurring,

image motion

Image Classification - Distinguish different types of images

Image Compression - Can be used but much better trans-

forms available.

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1-D CONTINUOUS FOURIER TRANSFORMS

• Definition

F (ν) =∫ ∞−∞ f(t)exp(−2πiνt)dt

f(t) =∫ ∞−∞ F (ν)exp(+2πiνt)dν

Note forward and inverse transforms not the same, sign dif-

ference in integral.

• Other Definitions

F (ω) =∫ ∞−∞ f(t)exp(−iωt)dt

f(t) =1

2π

∫ ∞−∞ F (ω)exp(+iωt)dω

or

F (ω) =1√2π

∫ ∞−∞ f(t)exp(−iωt)dt

f(t) =1√2π

∫ ∞−∞ F (ω)exp(+iωt)dω

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• FT has real and imaginary parts

F (ν) =∫ ∞−∞ f(t)exp(−2πνit)dt

Real[F (ν)] =∫ ∞−∞ f(t)cos(−2πνt)dt

Imag[F (ν)] =∫ ∞−∞ f(t)sin(−2πνt)dt

• If f(t) is even (f(t) = f(−t)) the imaginary part of trans-form Imag[F (ν)] = 0 because sin(−2πνt) is an odd (anti-symmetric) function.

• Likewise if f(t) is antisymmetric so (f(t) = −f(−t)) the realpart of the transform Real[F (ν)] = 0 because cos(−2πνt)is an even (symmetric) function.

• Define Fourier amplitude =√

Real[F (ν)]2 + Imag[F (ν)]2

and define Fourier Phase = tan−1(Imag[F (ν)]/Real[F (ν)])

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2-D CONTINUOUS FOURIER TRANSFORMS

F (u, v) =∫ ∞−∞

∫ ∞−∞ f(x, y)exp(−2πi(ux + vy))dxdy

f(x, y) =∫ ∞−∞

∫ ∞−∞ F (u, v)exp(+2πi(ux + vy))dudv

• where u,v are spatial frequencies.

• If the image has linear dimensions, e.g. centimetres. the u,vare in units of cycles/cm

• If image has angular dimensions (e.g degrees) then u,v are incycles/degree.

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• From the definition to calculate 2D FT component at particu-lar u,v multiply image f(x,y) by the complex kernel function

exp(−2πi(ux + vy)) and integrate.

• the kernel has constant value on lines in the the u,v domainsuch that ux + vy = constant.

• The Kernel function is therefore like a ’corrugated roof’ withan orientation which depends on u,v and a wavelength ∝1/√

u2 + v2.

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• 2D Fourier Transform has a separable kernel

exp(−2πi(ux + vy)) = exp(−2πiux)exp(−2πivy)

Therefore

F (u, v) =∫ ∞−∞

∫ ∞−∞ f(x, y)exp(−2πi(ux + vy))dxdy

=∫ ∞−∞

[∫ ∞−∞ f(x, y)exp(−2πiux)dx

]

exp(−2πivy)dy

• This property means we can do the FT of any object in twostages. First do a 1D transform of each row

f(x, y) ⇒ f ′(u, y).

Then do a 1D transform of all the columns of f ′(u, y) so

f ′(u, y) ⇒ g(u, v).

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• Some input images are also separable into functions of x andy i.e

f(x, y) = fx(x)fy(y)

.

• In this special case only.

F (u, v) =[∫ ∞−∞ fx(x)exp(−2πiux)dx

] [∫ ∞−∞ fy(y)exp(−2πivy)dy

]

i.e. the FT is the product of the 1D FTs of fx(x) and fy(y)

respectively.

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FOURIER TRANSFORM PROPERTIES

• If f(x, y) is real (as images always are) then

F (u, v) is Hermitian

i.e. F (−u,−v) = F ∗(u, v)where ∗ indicates complex conjugate

• If f(x, y) is also symmetric (i.e. f(−x,−y) = f(x, y) ) then

F (u, v) is entirely real, i,e, the phase of F(u,v) is either 0◦ or180◦.

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FOURIER TRANSFORMS PROPERTIES(cont)

• Inner Product

∫ ∞−∞

∫ ∞−∞ f(x, y)g

∗(x, y)dxdy =∫ ∞−∞

∫ ∞−∞ F (u, v)G

∗(u, v)dudv

if f(x, y) = g(x, y) and F (u, v) = G(u, v) then we obtain

Parsevals’ thereoem.

• Scaling

FT [f(ax, by)] =F (u/a, v/b)

|ab|

• Shifting

FT [f(x − xo, y − yo)] = F (u, v)exp(−2πi(uxo + vyo))

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2D Delta Function Definition

The delta function situated at x = xo, y = yo, called

δ(x − xo, y − yo)is defined in the limit as dx, dy go to zero below

height = 1/ (dx dy)

x

y

dy

FunctionValue

xo

yo

(x-xo, y- yo)δ

Is above function as dx and dy go to zero

it therefore has value infinity at x=xo,y=yoand value zero elsewhere.

The area under the function is 1.

dx

Figure 1:

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SOME IMPORTANT 2D FOURIER TRANSFORMS

Delta Function

FT (δ(x − xo, y − yo) = exp(−2πi(uxo + vyo))

−50

5

−5

0

50

0.5

1

x

Delta function at x=1,y=−2

y

−500

50

−50

0

50−1

0

1

u

Real Part of FT

v −500

50

−50

0

50−1

0

1

u

Imaginary Part of FT

v

Figure 2:

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Two Delta Functions

FT (0.5δ(x−xo, y−yo)+0.5δ(x+xo, y+yo)) = cos(−2π(uxo+vyo))

−50

5

−5

0

50

0.5

1

x

Two Delta functions at x=+/−2,y=0

y

−500

50

−50

0

50−1

0

1

u

Real Part of FT

v −500

50

−50

0

50−1

0

1

uv

Imaginary Part of FT

Figure 3:

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Sampling Function

Defined as

III∆x,∆y(x, y) =m=∞

∑

m=−∞

n=∞∑

n=−∞δ(x − m∆x, y − n∆y)

FT (III∆x,∆y(x, y)) = III∆u,∆u(u, v)

where ∆u = 1/∆x and ∆v = 1/∆y

−20

2

−20

2

0

0.5

1

1.5

x

Sampling Function (Infinite Delta grid sep=1)

y

−20

2

−20

2

0

0.5

1

u

Real Part of FT (Infinite Delta grid Sep =1)

v −20

2

−20

2

0

0.5

1

uv

Img Part of FT

Figure 4:

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Gaussian

FT (exp(−π(x2 + y2)) = exp(−π(u2 + v2))

−20

2

−2

0

20

0.5

1

x

Gaussian Function

y

−20

2

−2

0

20

0.5

1

u

Real Part of FT

v −20

2

−2

0

2−1

0

1

uv

Imaginary Part of FT

Figure 5:

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Circular Top Hat

Consider a function of x,y which is 1 inside r =√

(x2 + y2)

and 0 outside. Its FT is circulary symmetric and depends oly

on the radius in the u,v plane ru,v =√

(u2 + v2) . The FT is

related to a Bessel function of the first kind divided by ru,v.

−50

5

−5

0

50

0.5

1

x

Circular Top Hat Function

y

−50

5

−5

0

5−2

0

2

4

u

Real Part of FT

v −50

5

−5

0

5−1

0

1

uv

Imaginary Part of FT

Figure 6:

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CONVOLUTION

DEFINITION If

g(x, y) = h(x, y) ∗ f(x, y)where ∗ indicates convolution then mathematically g(x, y)

=∫ ∞−∞

∫ ∞−∞ h(x

′, y′)f(x − x′, y − y′)dx′dy′

=∫ ∞−∞

∫ ∞−∞ h(x − x

′, y − y′)f(x′, y′)dx′dy′

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VISUALIZING 2D CONVOLUTION

����

����

����

����

����

����

x’

y’

x’

y’

x’

y’

x’

y’

x

y

For this value of x,y shift of rotated For this value of x,y shift of rotated

second function, result of multiplicationof two functions and integration is 1

second function, result of multiplicatioof two functions and integration is 0

x

y

f(x’,y’) two delta functions

h(x’,y’) 1 inside triangle, 0 outside

f(x’,y’)h(x−x’,y−y’) f(x’,y’)h(x−x’,y−y’)

Figure 7:

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VISUALIZING 2D CONVOLUTION(cont)

����

���� x’

y’

x’

y’

f(x’,y’) two delta functions

h(x’,y’) 1 inside triangle, 0 outside

*

= x

yg(x,y)

Do the operation on the previous sheet for every possible x,y and then build uppoint by point the g(x,y) function. Effect of convolution with delta functions is to make

not rotateda copy of the triangle function centred on each of the deltas. Note in output triangles

Figure 8:

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CONVOLUTION THEOREM

The convolution theorem will be used many times. If g,h and

f functions are related by a convolution (see above) then their

fourier transforms G(u, v), F (u, v) and H(u, v) are related,

viz if

g(x, y) = h(x, y) ∗ f(x, y)

Then

G(u, v) = H(u, v) × F (u, v)

It also follows that if in the spatial domain

g(x, y) = h(x, y) × f(x, y)then in the Fourier domain

G(u, v) = H(u, v) ∗ F (u, v)

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CORRELATION

Exactly like convolution but with NO 180 degree rotation of

second function, before shift multiplication and summation.

DEFINITION If

g1(x, y) = h(x, y) ∗ ∗f(x, y)where ∗∗ indicates cross-correlation then mathematically g1(x, y)

=∫ ∞−∞

∫ ∞−∞ h(x

′, y′)f(x′ − x, y′ − y)dx′dy′

unlike convolution the order matters,

g2(x, y) = f(x, y) ∗ ∗h(x, y)is not the same output function where g2(x, y)

=∫ ∞−∞

∫ ∞−∞ f(x

′, y′)h(x′ − x, y′ − y)dx′dy′

In previous lecture have seen how correlation with various

filters can be used to smooth, sharpen and find gradients of

images. Can also be used to ’recognise’ objects in an image.

Note get ’Autocorrelation’ if wto imput images are teh same.

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CORRELATION THEOREM

Similar to convolution if we Fourier transform an image which

is the correlation of two input images, there is a simple re-

lationship between this FT and the FT of the two original

images.

If

g1(x, y) = h(x, y) ∗ ∗f(x, y)

and we take the FT then

G1(u, v) = H(u, v) × F ∗(u, v)where the star indicates complex conjugate. In the case of

autocorrelation, f =h amd F=H hence the FT of the auto-

correlation of h(x,y) is everywhere real and positive and is

called the ’power spectrum’ of h(x,y).

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