Applying an Ideal Lowpass Filter130.243.105.49/Research/Learning/courses/dip/2014/... · DIP'14 © A. J. Lilienthal (Dec 4, 2014) # 69. 2. Applying an Ideal Lowpass Filter . Filtering

# 68 DIP'14 © A. J. Lilienthal (Dec 4, 2014)

2.

Applying an Ideal Lowpass Filter

Filtering in the Frequency Domain

Input Image ⋅ (-1)x+y F(u,v)


2.



F(u,v)

ideal lowpass filter

⋅ H(u,v)


2.



F(u,v) G(u,v) = F(u,v) ⋅ H(u,v) ILPF


2.

G(u,v)



f(x,y) Re[f(x,y)] ⋅ (-1)x+y

inverse DFT


2.

Ideal Lowpass Filter – "Ringing"


ideal lowpass filter (D0 = 15 px-1)


2.



ideal lowpass filter (D0 = 80 px-1)


2.

Convolution Theorem o establishes a fundamental relationship

between spatial and frequency domain

Link between Filtering in the Spatial Domain and the Frequency Domain? o to explain "ringing" effects o to better understand spatial filtering



2.

∫+∞

∞−

−=∗ ααα dxhfxhxf )()()()( Definition

o1D, continuous

» flip one function h about the origin » shift h with respect to f » compute the sum of products for each displacement x

Convolution

flip one function and slide it past the other


2.

Definition o1D, continuous

o2D, continuous

o2D, discrete

Convolution

∫+∞

∞−

−=∗ ααα dxhfxhxf )()()()(

βαβαβα ddyxhfyxhyxf ∫ ∫+∞

∞−

−−=∗ ),(),(),(),(

∑∑−

=

−

=

−−=∗1

0

1

0),(),(),(),(

M

m

N

nnymxhnmfyxhyxf


2.

∫+∞

∞−

−=∗ ααα dxhfxhxf )()()()( Definition

o1D, continuous

» flip one function h about the origin » shift h with respect to f » compute the sum of products for each displacement x

Convolution


2.

Derivation of the Convolution Theorem

Convolution Theorem – Derivation

∫+∞

∞−

⋅== dueuFuFxf uxjπ2)()]([)( 1-uF

∫+∞

∞−

⋅== dueuHuHxh uxjπ2)()]([)( 1-uF


2.



( ) ( ) ( ') ( ') 'f x h x f x h x x dx+∞

−∞

∗ = −∫

∫+∞

∞−


∫+∞

∞−



2.



( ) ( ) ( ') ( ') 'f x h x f x h x x dx+∞

−∞

∗ = −∫

∫+∞

∞−


∫+∞

∞−


replace h(x-x’) with the inverse FT of its Fourier transform


2.



( ) ( ) ( ') ( ') 'f x h x f x h x x dx+∞

−∞

∗ = −∫

∫+∞

∞−


∫+∞

∞−


∫ ∫

∫∞+

∞−

∞+

∞−

−⋅

+∞

∞−

=

−=∗

')()'(

')'()'()()(

)'(2 dxdueuHxf

dxxxhxfxhxf

xxujπ


2.

Swap the order of integration ...


[ ])()()()(

')'()(

')()'(

')'()'()()(

12

2'2

)'(2

uHuFdueuFuH

duedxexfuH

dxdueuHxf

dxxxhxfxhxf

uxj

uxjuxj

xxuj

⋅==

=

=

−=∗

−∞+

∞−

⋅

∞+

∞−

⋅∞+

∞−

⋅−

∞+

∞−

∞+

∞−

−⋅

+∞

∞−

∫

∫ ∫

∫ ∫

∫

Fπ

ππ

π

replace h(x-x’) with its Fourier transform

identify F(u)

change integration order;

move the "x" part outside the dx'

integral


2.

Convolution Theorem

o convolution in the spatial domain equals

point-wise multiplication in the frequency domain and vice versa

owrap around problem needs to be considered ⇒ padding to the sum of the length of both functions f and h

Convolution Theorem

),(),(),(),( vuHvuFyxhyxf ⋅⇔∗

),(),(),(),( vuHvuFyxhyxf ∗⇔⋅


2.

Convolution Theorem – Wrap Around Problem o expected result?

Convolution Theorem

300 0

400

f(m)

200 0

400

h(m)

3

2


2.

Convolution Theorem – Wrap Around Problem o expected result?

Convolution Theorem

300 0

400

f(m)

-200 0

h(-m)

3

2


2.

Convolution Theorem – Wrap Around Problem o expected result

Convolution Theorem

200

0

400

f(m)

-200 0

3 h(-m)

0

f∗g

300

500 800

2


2.

Convolution Theorem – Wrap Around Problem o considering the implicit periodicity assumption

Convolution Theorem

200

0

400

f(m)

-200 0

h(-m)

0

f∗g

300

400

* range of Fourier transform computation

*


2.

Convolution Theorem – Wrap Around Problem o solution: padding

Convolution Theorem

300 0

400

f(m)

200 0

400

h(m)

3

2

800

800


2.

Convolution Theorem – Wrap Around Problem o expected result with padding

Convolution Theorem

300 0

400

f(m)

200 0 400

h(m) 3

2

800

800

f∗g * range of Fourier transform computation

*


2.

Convolution Theorem

onow we want to find a relation between h(x,y) and H(u,v) alone

Convolution Theorem


),(),(),(),( vuHvuFyxhyxf ∗⇔⋅


2.

Link between Filtering in the Spatial Domain and the Frequency Domain o impulse function δ

oparticularly

Relation Between Spatial and Frequency Filters

∑∑−

=

−

=

=−−1

0

1

00000 ),(),(),(

M

m

N

nyxfAyyxxAyxf δ

)0,0(),(),(1

0

1

0syxsyx

M

x

N

y∑∑

−

=

−

=

=δ


2.

Link Between Spatial and Frequency Domain o consider FT of the unit impulse function at (0,0)

» Fourier transform of an impulse at the origin


[ ]MN

eyxMN

yxM

x

N

y

MvyMuxj 1),(1),(1

0

1

0

)//(2∑∑−

=

−

=

+− == πδδF


2.

Link Between Spatial and Frequency Domain o consider FT of the unit impulse function at (0,0)

o convolution of the unit impulse function at (0,0)

» copies the convolution function h at x,y (up to the factor 1/M/N)


),(1

),(),(1),(),(1

0

1

0

yxhMN

nymxhnmMN

yxhyxM

m

N

n

=

−−=∗ ∑∑−

=

−

=

δδ

[ ]MN

eyxMN

yxM

x

N

y

MvyMuxj 1),(1),(1

0

1

0

)//(2∑∑−

=

−

=

+− == πδδF


2.

Link Between Spatial and Frequency Domain o from the last slide


[ ]MN

yx 1),( =δF ),(1),(),( yxhMN

yxhyx =∗δ


2.


ousing the convolution theorem



[ ]MN

yx 1),( =δF ),(1),(),( yxhMN

yxhyx =∗δ


2.




[ ] ),(),(),(),( vuHvuyxhyx ⋅⇔∗ δδ F),(),( yxyxf δ=

[ ]MN

yx 1),( =δF ),(1),(),( yxhMN

yxhyx =∗δ



2.




[ ]MN

yx 1),( =δF ),(1),(),( yxhMN

yxhyx =∗δ


[ ] ),(),(),(),( vuHvuyxhyx ⋅⇔∗ δδ F),(),( yxyxf δ=

),(1),(1 vuHMN

yxhMN

⇔


2.

Link Between Spatial and Frequency Domain

o filters in the spatial and the frequency domain constitute a Fourier pair

o from a filter in the frequency domain, an equivalent filter in the spatial domain can be obtained (F -1[H(u,v)] )

o all involved filters are linear o all involved filters are of the size M×N

⇒ not necessarily faster to use the spatial filter


),(),( vuHyxh ⇔


2.

Gaussian Filters


22 2/)( σueAuH −=


2.

Gaussian Filters

o Fourier transform of a real Gaussian is a real Gaussian oH(u) and h(x) behave reciprocal in terms of width onarrow frequency domain filter

⇒ attenuate high frequencies ⇒ blurring of an image (after inverse Fourier transform) ⇒ implies a larger mask


22222)( xeAxh σπσπ −=⇔ 22 2/)( σueAuH −=


2.

Gaussian Filters

o Fourier transform of a real Gaussian is a real Gaussian oH(u) and h(x) behave reciprocal in terms of width onarrow frequency domain filter

⇒ attenuate high frequencies ⇒ blurring of an image (after inverse Fourier transform) ⇒ implies a larger mask

owhat happens if σ→∞ ?


22222)( xeAxh σπσπ −=⇔ 22 2/)( σueAuH −=


2.

Basic Steps


Input Image ⋅ (-1)x+y F(u,v)

center the transform DFT

⋅ H(u,v)

apply filter

f(x,y)

inverse DFT

Re[f(x,y)]

Output Image

⋅ (-1)x+y


2.

Basic Steps


f = imread('bubbles.tif'); imshow(f,[]);[M N] = size(f);


2.

Basic Steps


f = imread('bubbles.tif'); imshow(f,[]);[M N] = size(f); F = fft2(f); figure; imshow(log(1 + abs(fftshift(F))),[]);


2.

Basic Steps


f = imread('bubbles.tif'); imshow(f,[]);[M N] = size(f); F = fft2(f); figure; imshow(log(1 + abs(fftshift(F))),[]); sig = 10; H = lpfilter('gaussian', M, N, sig); figure; imshow(fftshift(H),[]);


2.

Basic Steps


f = imread('bubbles.tif'); imshow(f,[]);[M N] = size(f); F = fft2(f); figure; imshow(log(1 + abs(fftshift(F))),[]); sig = 10; H = lpfilter('gaussian', M, N, sig); figure; imshow(fftshift(H),[]); G = H.*F; g = real(ifft2(G)); figure; imshow(g,[]);


2.

Basic Steps




2.

Basic Steps




2.

Basic Steps




2.

Ideal Lowpass Filter – "Ringing" (re-visited)


ideal lowpass filter


2.

Link Between Spatial and Frequency Domain

o filters in the spatial and the frequency domain constitute a Fourier pair

o from a filter in the frequency domain, an equivalent filter in the spatial domain can be obtained (F -1[H(u,v)] )


),(),( vuHyxh ⇔


2.



f = imread('bubbles.tif'); % only to get dimensions [M N] = size(f); D0 = 12; Hilp = lpfilter('ideal', M, N, D0); figure; imshow(fftshift(Hilp),[]);


2.



f = imread('bubbles.tif'); % only to get dimensions [M N] = size(f); D0 = 12; Hilp = lpfilter('ideal', M, N, D0); hilp = ifft2(Hilp); figure; imshow(ifftshift(abs(hilp)), []);


2.



f = imread('bubbles.tif'); % only to get dimensions [M N] = size(f); D0 = 12; Hilp = lpfilter('ideal', M, N, D0); hilp = ifft2(Hilp); figure; imshow(log(1 + ifftshift(abs(hilp))), []);


2.



),( vuH ILPF

→

),( yxhILPF

F -1


2.

Ideal Lowpass Filter – Avoid "Ringing"



2.

Gaussian Lowpass Filter

ono ringing


ideal lowpass filter Gaussian lowpass filter


2.

Highpass Filters o suppress low frequencies

⇒ edge enhancing o reverse of a lowpass filter

o ideal, Gaussian


),(1),( vuHvuH lphp −=


2.

Ideal Highpass Filter (IHPF)

o ideal highpass filter (also shows "ringing")


>⇔≤⇔=

0

0

),(1),(0),( DvuD

DvuDvuH


2.

Highpass Filters



2.

Highpass Filters – IHPF



2.

Highpass Filters – IHPF



2.

Gaussian Highpass Filter

o smoother and no "ringing"


20

2 2/),(1),( DvuDevuH −−=


2.

Derivatives and their Fourier Transform

Laplacian in the Fourier Domain


[ ])()()( xfujdx

xfd nn

n

FF =

)(),( 22 vuvuH Laplacian +−=⇒

),()()],([ 222 vuFvuyxf +−=∇F


2.

Derivatives and their Fourier Transform



[ ])()()( xfujdx

xfd nn

n

FF =

( ) ( ) ( )vuFNvMuyxf ,22),(222

−+−−⇔∇


2.




2.

Laplacian in the Spatial Domain


( ) ( ) yxvuvu ++− −−+− 1]1)([ 221F


2.

General Idea o select a filter in the frequency domain o transform this filter to the spatial domain o try to specify a small filter mask

that captures the "essence" of the filter function

Deriving Spatial Filter Masks


2.

Other Filters obandpass

» allows frequencies in a band between the two frequencies D0 and D1

obandstop: » stops frequencies in a band

between the two frequencies D0 and D1

onon-symmetric filters: allow different frequencies in the u and v direction



Recovering Intrinsic Images, Homomorphic Filtering

3


3.

Reminder: Image Formation Model o illumination i(x,y) from a source o reflectivity r(x,y) = reflection / absorption in the scene f(x,y) = r(x,y) i(x,y)

Image Formation


3.

Intrinsic Images o "midlevel description" of scenes

» proposed by Barrow and Tenebaum [H.G. Barrow and J.M. Tenenbaum. Recovering Intrinsic Scene Characteristics from Images". In Computer Vision Systems. Academic Press, 1978]

» not a full 3D description of the scene » viewpoint dependent » physical causes of changes in illumination are not made explicit

Intrinsic Images


3.

Intrinsic Images o "midlevel description" of scenes

» proposed by Barrow and Tenebaum [H.G. Barrow and J.M. Tenenbaum. Recovering Intrinsic Scene Characteristics from Images". In Computer Vision Systems. Academic Press, 1978]

o "The observed image is a product of two images: an illumination image and a reflectance image."

Intrinsic Images


3.

Intrinsic Images o "midlevel description" of scenes o (input) image is decomposed into two images …

Intrinsic Images

from "Deriving Intrinsic Images From Image Sequences", Yair Weiss , Proc. ICCV 2001


3.


» a reflectance image

Intrinsic Images

r(x,y)



3.


» a reflectance image and » an illumination image

Intrinsic Images

r(x,y) i(x,y)



3.

Intrinsic Images o "midlevel description" of scenes o "The observed image is a product of two images: an illumination

image and a reflectance image." » segmentation on the intrinsic reflectance should be much simpler than on

the original image » 3D information can be obtained from the illumination picture

Intrinsic Images

r(x,y) i(x,y)



3.


» a reflectance image and » an illumination image

o but: decomposition is an ill-posed problem » number of unknowns is twice as high as the number of equations

(for example: set i(x,y) = 1 r(x,y) = f(x,y))

Intrinsic Images

),(),(),( yxryxiyxf =


3.

Idea: Separate Illumination and Reflectance

onot separable directly …

o… but the logarithm is separable

Homomorphic Filtering


)],([)],([)],([ yxryxiyxf FFF ≠

[ ]),(ln),( yxfyxz ≡

[ ][ ] [ ][ ]),(),(),(),(ln),(ln)],([

vuFvuFvuZyxryxiyxz

ri +==+= FFF


3.

Frequency Domain Approximation to Homomorphic Filtering – Assumption o illumination component varies slowly o reflectance component

tends to vary abruptly ⇒ use filter that affects

low- and high-frequency components in a different way (decreases influence of illumination, increases influence of reflectance)



γL<1

γH >1


3.

Idea: Separate Illumination and Reflectance


[ ][ ] [ ][ ]),(ln),(ln),(),(),( yxryxivuFvuFvuZ ri FF +=+=

),(),(),(),(),(),(),( vuFvuHvuFvuHvuZvuHvuS ri +==

[ ] [ ] [ ]),('),('

),(),(),(),(),(),( 111

yxryxivuFvuHvuFvuHvuSyxs ri

+=+== −−− FFF

),('),('),(),( zxrzxiyxs eeeyxg ==


3.

Example o consider non-uniform illumination


disturbance pattern


3.

Example o histogram equalization does not perform well


histogram equalization


3.

Example o homomorphic filtering


homomorphic filtering


3.

Example o homomorphic filtering


homomorphic filtering, then histogram equalization


3.

Recovering Intrinsic Images from a Single Image ["Recovering Intrinsic Images from a Single Image", Marshall F. Tappen, William T Freeman, Edward H Adelson, MIT AI Memo, 2002, http://hdl.handle.net/1721.1/6703]

o use colour information

Remarks


3.


o use colour information » find chromaticity changes by classifying image derivatives by thresholding

scalar product of normalized RGB vector neighbours

Remarks


3.


o use colour information » find chromaticity changes by classifying image derivatives by thresholding

scalar product of normalized RGB vector neighbours

Remarks


3.


o use colour information o learn appearance models of shading patterns

» classify gray-scale image

Remarks


3.




Remarks


3.




Remarks


3.



» classify gray-scale image o propagate evidence (MRF model with learned parameters)

Remarks


3.


o use colour information + o learn appearance models of shading patterns + o propagate evidence (MRF model with learned parameters)

Remarks


3.



Remarks


3.



Remarks


3.


Remarks


3.


Remarks


3.


Remarks


Properties of the Fourier Transform

4


4.

Translation


a shift in f(x,y) does not affect |F(u,v)|

)//(200

00),(),( NyvMxujevuFyyxxf +−⇔−− π


4.

Translation o a shift in f(x,y)

does not affect the spectrum |F(u,v)|



4.

Distributive Over Addition

Not Distributive Over Multiplication

Scaling

Rotation


)],([)],([)],(),([ 2121 yxfyxfyxfyxf FFF +=+

)],([)],([)],(),([ 2121 yxfyxfyxfyxf FFF ≠

),(),( vuaFyxaf ⇔ )/,/(1),( bvauFab

byaxf ⇔

),(),( 00 θϕωθθ +⇔+ Frf


4.

Rotation


rotating f(x,y) rotates F(u,v) by the same angle

F(u,v)

f(x,y)


4.

),(),(),(),( NvMuFNvuFvMuFvuF ++=+=+=

),(),(),(),( NyMxfNyxfyMxfyxf ++=+=+=

),(),( * vuFvuF −−= ),(),( * vuFvuF −−=⇒

Periodicity

o the discrete Fourier transform is periodic o also the inverse of the discrete Fourier transform is periodic

Conjugate Symmetry

o the spectrum is symmetric about the origin



4.

Separability

o F(x,v) is the Fourier transform along one row o F(u,v) can be obtained by two successive applications of

the simple 1D Fourier transform instead of by one application of the more complex 2D Fourier transform


[ ][ ]),(),(1

),(11

),(1),(

1

0

/2

1

0

/21

0

/2

1

0

1

0

)//(2

yxfvxFeM

eyxfN

eM

eyxfMN

vuF

vu

M

x

Muxj

N

y

NvyjM

x

Muxj

M

x

N

y

NvyMuxj

FF==

==

==

∑

∑∑

∑∑

−

=

−

−

=

−−

=

−

−

=

−

=

+−

π

ππ

π


4.

Fourier Transform


from "Computer Vision – A Modern Approach", Forsyth and Ponce, Prentice Hall, 2002

( )1 ,box x ysin sinu v

u v( )( ),u vF f a b

ab

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Applying an Ideal Lowpass Filter130.243.105.49/Research/Learning/courses/dip/2014/... · DIP'14 © A. J. Lilienthal (Dec 4, 2014) # 69. 2. Applying an Ideal Lowpass Filter . Filtering