vision artificial parte2

8/11/2019 Vision Artificial Parte2

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Visin Artificial

Dr. Fernando Merchn

Facultad de Ingeniera Elctrica

Universidad Tecnolgica de Panam

2010


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Filtros en el Dominio de la Frecuencia


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Transformada de Fourier


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Ejemplo 1

ANLISIS ESPECTRALEste ejemplo muestra el uso de la funcin FFT para el anlisis espectral, para encontrar los componentes

de la frecuencia de una seal enterrada en una seal ruidosa con dominio de tiempo.Se va ha crear datos, muestreados en 1000 Hz

t = 0:.001:.25; Se forma un eje de tiempo desde t=0 hasta t=25 con pasosde 1 milisegundo.

x = sin(2*pi*50*t) + sin(2*pi*120*t); Formamos una seal x, con ondas seno a 50 Hz y 120 Hz

y = x + 0.02*rand(size(t));subplot(3,1,1), plot(t, y)title (Seal con ruidosa')

Adicionamos ruido aleatorio con STD = 2Se genera la seal con ruido y, es dficil identificar loscomponentes de la frecuencia mirando esta seal

Y = fft(y,256); Se aplica la transformada discreta de Fourier de la sealruidosa usando la FFT.

Pyy = Y.*conj(Y)/256; Calcular la densidad espectral de la energa, una medida dela energa a varias frecuencias, usando la compleja

conjugada (CONJ).f = 1000/256*(0:127); Formar un eje de frecuencia para los primeros 127 puntos y

usarlo para diagramar el resultado. (Los restantes 256 puntosson asimtricos)

subplot(3,1,2), plot(f, Pyy(1:128))title('Densidad espectral de la energa')xlabel('Frecuencia(Hz)')

subplot(3,1,3), plot(f, Pyy(129:end))

Presentar los resultados


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Ejemplo 1

0 0.05 0.1 0.15 0.2 0.25-5

0

5

Tiempo de Dominio de la Seal Ruidosa

0 50 100 150 200 250 300 350 400 450 5000

50

100

0 50 100 150 200 250 300 350 400 450 5000

50

100Densidad espectral de la energa

Frecuencia(Hz)


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Ejemplo 2

Haciendo uso de la funcin genwave generar y sumar 4

ondas de frecuencias: 10, 25, 35 y 60, con amplitudes: 10,5, 10 y 5 respectivamente, 512 Hz de frecuencia demuestreo, fase = 0, la seal debe durar 0.25 segundos.

Aplicar la FFT para identificar las frecuencias de muestreo.

0 0.05 0.1 0.15 0.2 0.25

-20

-10

0

10

20


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function ejercicio03

clcamp = 10; % amplitud

fr1 = 10; % frecuencia 1




sm = 512; % muestras en 1 segundo

% (frecuencia de muestreo)

tiempo= 0.25; % duracin de la onda

% en segundosph = 0; % fase

% eje de tiempo

t = 0:1/sm:tiempo;

Y1 = genwave(amp, fr1, sm, tiempo, ph);

Y2 = genwave(amp/2, fr2, sm, tiempo, ph);

Y3 = genwave(amp, fr3, sm, tiempo, ph);

Y4 = genwave(amp/2, fr4, sm, tiempo, ph);

% sumas las seales generadas

y = Y1 + Y2 + Y3 + Y4;

LEN = min(length(t), length(y));

subplot(3,1,1), plot(t(1:LEN),y(1:LEN));

axis([0 tiempo min(y) max(y)])

% calcula la FFT

Y = fft(y);

% calcula la densidad espectral de/la energaPyy = Y.*conj(Y)/LEN;

%considera la primera mitad para la frecuencia

f = sm/LEN*(0:LEN/2-1);

subplot(3,1,2), plot(f,

Pyy(1:LEN/2)); title('DEE');

xlabel('Frecuencia(Hz)')

subplot(3,1,3), plot(f(1:LEN/4),

Pyy(1:LEN/4)); title('DEE');

xlabel('Frecuencia(Hz)')

Ejemplo 2


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Ejemplo 2

0 0.05 0.1 0.15 0.2 0.25

-20

0

20

0 50 100 150 200 250 3000

1000

2000

3000DEE

Frecuencia(Hz)

0 20 40 60 80 100 120 1400

1000

2000

3000

DEE

Frecuencia(Hz)


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Transformada de Fourier en 2D


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The 2D Fourier Transform of a Digital Image

,,,21

0

1

0

Cuc

R

vriR

u

C

v

euvcrI

1

0

21

0

1 ),(C

c

C

uc

R

vriR

rRC

ecrIv,u

LetI(r,c) be a single-band (intensity) digital image withR rows and C columns.Then,I(r,c) has Fourier representation

where

are theRx CFourier coefficients.

these complexexponentials are 2Dsinusoids.


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What are 2D sinusoids?

Contd. on next page.

,

sinsin22

2 crNiucvr

Nii

eee Cuc

Rvr

. 122 tanand,,cos,sin uvuvuv

where

To simplify the situation assumeR= C=N. Then

,

N

Write

.cossinsincossincos 22cossin2 1

cricre cri

Then by Eulers relation,

Note: since images are indexedby row & col with r down and cto the right, , is positive in

the counterclockwise direction.


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What are 2D sinusoids? (contd.)

cossincosRe 2cossin2 1

cre cri

cossinsinIm 2cossin2 1

cre cri

Both the real part of this,

and the imaginary part,

are sinusoidal gratings of unit amplitude, period and direction .

ThenN

2is the radian frequency, and the frequency, of the wavefront

N

and

N is the wavelength in pixels in the wavefront direction.


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13

2D Sinusoids:

orientation

... are plane waves with

grayscale amplitudes, periods in

terms of lengths, ...

1cossin

2cos2,

cr

AcrI

A

= phase shift


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2D Sinusoids:

... specific orientations,and phase shifts.


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Points on the Fourier Plane

This point represents this particular sinusoidal grating

x

y


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is a complexnumber with areal part and animaginary part.

The Value of a Fourier Coefficient

If you representthat number as amagnitude,A, anda phase, ,

..these represent the amplitudeand offset of thesinusoid withfrequency and direction .


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The Value of a Fourier Coefficient

The magnitude and phaserepresentation makesmore sense physically

since the Fourier magni-tude, A (,), at point (,)represents the amplitudeof the sinusoid

and the phase, (,),represents the offset of thesinusoid relative to origin.


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The Fourier Coefficient at (u,v)

So, the point (u,v) on theFourier plane

represents a sinusoidal

grating of frequency and orientation .

represents the ampli-tude,A, and the phaseoffset, , of the sinusoid.

The complex value,F(u,v),of the FT at point (u,v)


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The Sinusoid from the Fourier Coeff. at (u,v)


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FT of an Image (Magnitude + Phase)

I log{|F{I}|2+1} [F{I}]


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FT of an Image (Real +

Imaginary)

I Re[F{I}] Im[F{I}]


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The Power SpectrumThe power spectrum of a signal is the square of the magnitude of its

Fourier Transform.

.ImRe

ImReImRe

22

2

vuvu

vuivuvuivuvuvuvu

,,

,,,,,,,

*

For display,

thelogofthe powerspectrum isoften used.

At each location (u,v) it indicates the squared intensity of the frequency component

with period and orientation22/1 vu

./tan 1 uv For display in Matlab:PS = fftshift(2*log(abs(fft2(I))+1));


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The power spectrum (PS) is defined by .

We take the base-e logarithm of the PS in order to view it. Otherwise its dynamic range could be

too large to see everything at once. We add 1 to it first so that the minimum value of the result is

0 rather thaninfinity, which it would be if there were any zeros in the PS. Recall that

log(f2) = 2log(f ).

Multiplying by 2 is not necessary if you are generating a PS for viewing, since you'll probablyhave to scale it into the range 0-255 anyway. It is much easier to see the structures in a Fourier

plane if the origin is in the center. Therefore we usually perform an fftshift on the PS before it is

displayed.

>> PS = fftshift(log(abs(fft2(I))+1));

>> M = max(PS(:));

>> image(uint8(255*(PS/M)));

If the PS is being calculated for later computational use -- for example the autocorrelation of a

function is the inverse FT of the PS of the function -- it should be calculated by

>> PS = abs(fft2(I)).^2;

On the Computation of the Power Spectrum 2,IIPS vu


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The Uncertainty Relation

FT

space frequency

FT

space frequency

A small object in spacehas a large frequencyextent and vice-versa.

216

1

vuyx

vu

yx

then,frequency

inextentitsisif

andspaceinobjectthe

ofextenttheisIf


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The Uncertainty Relation

Recall: a symmetricpair of impulses in thefrequency domain

becomes a sinusoid inthe spatial domain.

A symmetric pair of

lines in the frequencydomain becomes asinusoidal line in thespatial domain.

IFT

spacefrequency

l

argeexte

nt

small extent

sm

allextent

large extent

IFT

spacefrequency

small extent

sm

allextent

largeexten

t

large extent


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The Fourier Transform of an Edge

edge Power Spectrum Phase Spectrum


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The Fourier Transform of a Bar

bar Power Spectrum Phase Spectrum


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Coordinate Origin of the FFT

Center=

(floor(R/2)+1, floor(C/2)+1)

Even EvenOdd Odd

Image Origin Weight Matrix OriginImage Origin Weight Matrix Origin

After FFT shift After IFFT shiftAfter FFT shift After IFFT shift


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Matlabs fftshiftand ifftshift

J = fftshift(I):

I (1,1) J ( R/2+1, C/2+1)

I = ifftshift(J):

J ( R/2+1, C/2+1) I (1,1)

where x= floor(x) = the largest integer smaller thanx.

from FFT2or ifftshift

after fftshift

origin origin


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Inverse FFTs of Impulses

highest-possible-frequency horizontal sinusoid (Cis even)

horizontal is thewavefront direction.

fftshifted


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highest-possible-frequency vertical sinusoid (Ris even)

vertical is thewavefront direction.

fftshifted


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highest-possible-freq horizontal+vertical sinusoid (R&Ceven)

a checker-boardpattern.

fftshifted


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fftshifted

lowest-possible-frequency horizontal sinusoid

horizontal is thewavefront direction.


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fftshifted

lowest-possible-frequency vertical sinusoid

vertical is thewavefront direction.


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fftshifted

lowest-possible-frequency negative diagonal sinusoid

negative diagonal isthe wavefront direction.

I FFT f I l


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fftshifted

lowest-possible-frequency positive diagonal sinusoid

positive diagonal isthe wavefront direction.


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Frequencies and Wavelengths in the Fourier

Plane

+vdirection

+udirection

512 columns

384rows

frequencies: (u,v) = (4,3); wavelengths: (u, v) = (128,128)

Note this and this.


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Plane

frequencies: (u,v) = (1,0); wavelength: u= 512

u = # of complete cycles

in the horizontal direction

u= C/ u

512 columns

384rows


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frequencies: (u,v) = (0,1); wavelength: v= 384


Plane

v = # of complete

cycles in thevertical direction

v=R/ v

512 columns

384rows


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Frequencies and Wavelengths in the Fourier Plane




u= C/ u

512 columns

384rows


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v = # of complete


v=R/ v

512 columns

384rows


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u= C/ u

512 columns

384rows


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v = # of complete


v=R/ v

512 columns

384rows


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512 columns

384rows


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512 columns

384rows


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Power Spectrum of anImage

Transformada de Fourier de una imagen (en Matlab)


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Transformada de Fourier parauna imagen rectangular.

b. F = fft2(f);

S = abs(F);

imshow(S, [ ])

Para mover el origen de laTransformada al centro del

rectngulo de frecuencia:

c. Fc = fftshift(F);

imshow(abs(Fc), [ ])

Para hacer visible el rango

dinmico:d. S2 = log(1 + abs(Fc));

imshow(S2, [ ])

Transformada de Fourier de una imagen (en Matlab)


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Es muy dificil hacer asociaciones

directas entre componentes de

una imagen y su transformada.

Amedida que uno se aleja del

origen de la transformada, lasbajas frecuencias corresponden a

las pequeas variaciones en la

imagen.

El espectro de Fourier muestra

componentes prominentes a 45,correspondientes a los bordes de

la imagen a 45, y una

componente vertical, inclinada,

correspondiente a la incrustacin.


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FILTRADO EN EL DOMINIO

DE LA FRECUENCIA

Filtrado en el dominio de la frecuencia


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Los componente de alta frecuencia en una imagen

corresponden a las transiciones bruscas en niveles de gris.

El suavizado permite eliminar las componentes de altafrecuencia

Filtro en el dominio de la frecuencia

donde:

F(u,v):transformada de Fourier de la imagen originalH(u,v):Filtro atenuador de frecuencias



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Pasos para filtrar una imagen en el dominio de la

frecuencia.


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Filtrado en el dominio de la frecuencia.

1. Obtener los parmetros de relleno

con la funcin paddedsize.

2. Obtener la transformada de Fourier:

F = fft2(f, PQ(1), PQ(2));

3. Generar la funcin del filtro.

4. Multiplicar la transformada por el

filtro: G = H.*F;

5. Obtener la parte real de la

transformada inversa de Fourier:

g = real(ifft2(G));

6. Recortar el rectngulo al tamao

original:

g = g(1:size(f, 1), 1:size(f, 2));

Filtros pasabajo en el dominio de la frecuencia


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Filtros Pasabajo en el dominio de la Frecuencia.

1. Filtro ideal (ILPF).

D(u,v) es la distancia desde un punto (u,v)

al centro del filtro. D0 es un num. no-neg.

2. Filtro Butterworth (BLPF). n es el orden del

filtro. D0 establece la frec. de corte.

3. Filtro Gausiano (GLPF). es la desviacin

estandar.

0

0

),(0

),(1),(

DvuDsi

DvuDsivuH

nDvuDvuH

2

0 ]/),([1

1),(

22 2/),(),( vuDevuH

Filtros pasabajo en el dominio de la frecuencia

Filtro pasabajo ideal


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0

0

),(0

),(1

),( DvuDsi

DvuDsi

vuH

Filtro pasabajo ideal

Filtro pasabajo Butterworth


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nDvuDvuH

2

0 ]/),([1

1),(

Filtro pasabajo Butterworth

Filtro pasabajo gausiano


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Filtro pasabajo gausiano

22 2/),(),( vuDevuH

Resultado de filtrado ideal, Butterworth y Gausiano.


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Resultado de filtrado ideal, Butterworth y Gausiano.

Aplicacin de un filtro pasabajo Gaussiano a una imagen


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Filtros Pasabajo.Aplicacin de un filtro pasabajo

Gaussiano a una imagen.

PQ = paddedsize(size(f));

[U, V] = dftuv(PQ(1), PQ(2));

D0 = 0.05*PQ(2);F = fft2(f, PQ(1), PQ(2));

H = exp(-(U.^2 + V.^2) /

(2*D0^2)));

g = dftfilt(f, H);

b. figure, imshow(fftshift(H), [ ])c. figure, imshow(log(1 +

abs(fftshift(F))), [ ])

d. figure, imshow(g, [ ])

dftuv calcula la matriz de frecuencias meshgrid

D0 = (la desviacin estandar)




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H = lpfilter(tipo, M,N,D0,n)




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Visin 3D de filtros


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Visin 3D de filtros.

a. H = fftshift(lpfilter(gaussian,

500, 500, 50));

mesh(H(1:10:500, 1:10:500))axis([0 50 0 50 0 1])

b. colormap([0 0 0])

axis off

grid off

c. View(-25, 30)

d. View(-25, 0)


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Funcin view.

Los valores por defecto son az = -37.5 y el = 30

Para determinar los valores corrientes: [az, el] =view;

Para utilizar los valores por defectos: view(3)


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Uso de la funcin Surf.

a. H = fftshift(lpfilter(gaussian, 500, 500, 50));

surf(H(1:10:500, 1:10:500))

axis([0 50 0 50 0 1])colormap(gray)

grid off; axis off

b. Shading interp

Filtros pasaalto


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Filtros pasaalto

0

0

),(0

),(1),(

DvuDsi

DvuDsivuH

nDvuDvuH

2

0 ]/),([1

1),(

22 2/),(),( vuDevuH

Filtro PA Ideal

Filtro PA Butterworth

Filtro PA Gausiano

D(u,v) es la distancia de un punto (u,v) al centro del filtro.

Filtros pasaalto


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Filtros pasa-alto.Utilizados para aumentar el contraste (agudizar) de

una imagen.

H = fftshift(hpfilter(ideal, 500, 500, 50));

mesh(H(1: 10: 500, 1: 10: 500));

axis([0 50 0 50 0 1])

colormap([0 0 0])

axis off; grid off

d. Figure, imshow(H, [ ])

),(1),( vuHvuH lphp

D(u,v) es la distancia de un punto (u,v) al centro del filtro.

Filtros pasaalto

Filtrado pasabajo y pasaalto


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Filtrado pasabajo y pasaalto

Filtro pasaalto ideal


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Filtro pasaalto ideal

Filtro pasaalto Butterworth


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o pasaa o u e o

Filtro pasaalto Gausiano


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p


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Filtro Gaussiano Pasa-alto.

b. PQ = paddedsize(size(f));

D0 = 0.05*PQ(1);

H = hpfilter(gaussian, PQ(1), PQ(2), D0);

g = dftfilt(f, H);

figure, imshow(g, [ ])

Se observa un aumento en la intensidad de

bordes y transiciones

Filtro de alta frecuencia de Enfasis.


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Filtro de alta frecuencia de

Enfasis.

PQ = paddedsize(size(f));

D0 = 0.05*PQ(1);

HBW = hpfilter(btw, PQ(1),

PQ(2), D0, 2);

H = 0.5 + 2*HBW;

gbw = dftfilt(f, HBW);gbw = gscale(gbw);

ghf = dftfilt(f, H);

ghf = gscale(ghf);

ghe = histeq(ghf, 256);

),(),( vubHavuH hphfe

El filtro pasa-alto atenua los trminos dc (constantes) de

la imagen, como los fondos. Un desplazamiento y una

amplificacin mejoran los resultados.

Filtro de alta frecuencia de Enfasis.


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Tcnicas de Restauracin de imagen


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Tipos de Ruido


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Tipos de Ruido

Tipos de Ruido


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Histogramas de modelos de ruido.

a. r = imnoise2(gaussian, 100000, 1, 0, 1);

%genera el vector columna r, con 100000

elementos, siendo cada uno un nmeroaleatorio de la distribucin gaussiana, con

mediana 0 y desviacin estndar 1. El

histograma se obtiene con

p = hist(r, 50)

Tipos de Ruido

Tipos de Ruido


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Tipos de Ruido

Tipos de Ruido


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Tipos de Ruido

Tipos de Ruido


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Tipos de Ruido

Tipos de Ruido


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Tipos de Ruido

Ruido Peridico


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Ruido Peridico

Ruido Peridico


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Generacin de ruido peridico.

uo y vo son las frecuencias con respecto

a x y.Bx y By son desplazamientos de fase con

respecto al origen.

La transformada de Fourier de esta

funcin es:

con un par de impulsos complejos

conjugados localizados en: (u+uo, v+vo)

y (u-u0, v-vo).

]/)(2/)(2[),( 00 NByvMBxusenAyxr yx

)],()(),()[(2

),(00

/2

00

/2 00 vvuuevvuueA

jvuR NBvjMBuj yx

Ruido Peridico


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Parmetros del Ruido.

Se determina analizando el

espectro de Fourier. Un ruido

peridico produce picos defrecuencia.

Para esto se utiliza el histograma,

donde se puede obtener la media

y la varianza utilizando momentos

estadsticos. La forma del

histograma est dado por sumomento central o momento

alrededor de la media:

1

0

)()(L

i

i

n

in zpmzn es el orden del momento , m es la media y 2 es la varianza.

1

0

)(L

i

ii zpzm

1

0

2

2 )()(L

i

ii zpmz 0

1

1

0


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Filtros espaciales implementados en la funcin spfilt(g, type, m, n, parameter).

Restauracin en presencia de ruido


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Restauracin en presencia de ruido.

a. [M, N] = size(f);

R = imnoise2(salt & pepper, M, N, 0.1, 0);

c = find(R==0);

gp = f;

gp(c) = 0;

b. R = imnoise2(salt & pepper, M, N, 0,0.1);

c = find(R==1);

gs = f;

gs(c) = 255;

c. fp = spfilt(gp, chmean, 3, 3, 1.5);

d. fs = spfilt(gs, chmean, 3, 3, -1.5);

e. fpmax = spfilt(gp, max, 3, 3);

f. fsmin = spfilt(gs, min, 3, 3);

),(),(),( yxyxfyxg


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Noise-Free Image and Uncorrelated NoiseField


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Gaussian noise fieldimage

Field

Spectra of Noise-Free Image and Uncorr. Noise Field


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noise field center row log power spectrumimage center row log power spectrum

IID noisespectrumis flat.

Imagespectrumfalls off.

Sum of Noise-Free Image and Uncorrelated Noise Field


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image + noise field image + noise field center row log PS

Noise energyexceeds imageenergy beyonda certain freq.

22 ,, uvuv 22 ,, uvuv 22 ,, uvuv

Power Spectra of Noise-Free Image and NoiseField


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Field

noise imageoriginal image


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blue indicates noise > imageoriginal image

Power Spectra of Sum of Image and Noise Field


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red indicates image > noisenoise image



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image & noisenoisy image


Additive Noise: Another Example


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original image noise image image+noise



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image PS noise PS image+noise PS

displayed:

1Ilog 2F



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image PS image PS > noise PSimage+noise PS

displayed:

1Ilog 2F


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red indicates image > noise image PS > noise PS

Additive Noise: Reduce Through

Blurring?

f0f0

At some frequency,f

0, there are more

components wherethe noise power isgreater than theimage power.


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pass

reject

pass

reject

red indicates image > noise image PS > noise PS

Additive Noise: Reduce Through

Blurring?

Thus, it makessense to apply a LPFwith cutoff f0, (ablurring filter) tothe images and seewhat happens.


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PS of Gaussian blurred image Gaussian Blurred Image

Additive Noise: Reduction Through Blurring.

The resultis actually

no better.Theres lessnoise butthe blurringlooks worse.


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PS of Gaussian blurred image Gaussian Blurred Image

Additive Noise: Reduction Through Blurring.

The resultis actually

no better.Theres lessnoise butthe blurringlooks worse.


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103

red: image > noise

blue: image < noisepower spec. of noisy image

Noise Masking

If the freq. comps.

for which noise-power > image-power are known1

1

of course they almost never are.


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Noise Masking

mask those out

of the FT andinvert the resultto get

image < noise masked outpower spec. of noisy image


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noisy image noise-masked mage

Noise Masking

this:


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blurred noisy image noise-masked mage

Noise MaskingAlthough the noise-masked image looks better than the blurred one, it is still

noisy. Moreover, this example is unrealistic because we know the exact noise

power spectrum. In any real case we will at most know its statistics.

Filtrado de ruido peridico


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Filtrado de ruido peridico


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Ejemplo


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Image Degradation Model

.,N,H,I,J crcrcrcr

undegraded

image

additive

noise

degradedimage

pointspreadfunction

So far, we have considered

only additive noise. Before

going further it will be

useful to consider a moregeneral model of image

degradation, one that

includes convolution with a

pointspread1function, H, as

well as additive noise.

1H is also referred to as the optical transfer function.


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Pointspread OperatorsA pointspread operator is a linear model of the distortion acquired during the

imaging process. Since it is a linear model, it is a convolution operator. One

example of this is aperture distortion, an unavoidable consequence of making

an image with a camera that has an opening larger than a point.


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Pointspread Operators

pinhole camera aperture camera

A pinhole camera maps one

object point to one image

point; it is one-to-one.

An aperture camera maps one

object point to many image

points; it spreads the points.

P i t d O t d


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Pointspread Operators and

Convolution J , I , H ,r c r c r c

H ,r c

I ,r c

Recall how a convolution works through multiply, shift, and add . That is precisely the

effect of imaging through an aperture. It results in a blurry image.


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LensesA properly designed lens will focus the light emanating from a point and thereby

reduce the blurring. But no lens can do this perfectly. In fact, the lens adds its

own distortion. The result is an optical transfer function, H(r,c), that is

convolved with the image.


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.,N,H,I,J crcrcrcr

undegraded

image

additive

noise

degradedimage

pointspreadfunction

Note: The termpointspread operatorrefers to convolution by the pointspread function.


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cr,I

cr,J

cr,N

crcr ,H,I

cr,H J , I , H , N ,r c r c r c r c

I D d ti M d l (F


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Image Degradation Model (FrequencyDomain)

uv,

uv,

uv,

uvuv ,,

uv,Images shown are log magnitude.

.,,,, uvuvuvuv


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Modelado de la Funcin de

Degradacin PSF.

a. f = checkerboard(8);

b. PSF = fspecial(motion, 7, 45);%7 pixeles a 45 grados.

gb = imfilter(f, PSF, circular);

c. noise = imnoise(zeros(size(f)),

gaussian, 0, 0.001);

d. g = gb + noise;

imshow(pixeldup(f, 8), [ ])

Estimacin de la funcin de degradacin


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Filtro Wiener


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Se utiliza cuando se tiene un amplio

conocimiento de las propiedades de la seal

de ruido.

H(u,v) es la func. de degradac.

G(u,v) es la transf. de Fourier de la imagendegradada.

Sn(u,v) es el espectro de potencia del ruido.

Sf(u,v) es el espectro de potencia de la

imagen sin degradar.

H*(u,v) es el complejo conjugado de H(u,v)

),(),(/),(),(),(

),(),(

2

2^

vuGvuSvuSvuHvuH

vuHvuF

f

),(),(*),( 2

vuHvuHvuH

Fr = deconvwnr(g, PSF)

% La relacin ruido a seal es 0.

Fr = deconvwnr(g, PSF, NSPR)

% La relacin ruido a seal es conocida.

Fr = deconvwnr(g, PSF, NACORR, FACORR)

% Las func. de autocorrelacin de ruido yde seal son conocidas.

22),(),(,),(),( vuFvuSvuNvuS f

Filtro Wiener


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b. fr1 = deconvwnr(g, PSF);

Sn = abs(fft2(noise)).^2;

nA =

sum(Sn(:))/prod(size(noise));Sf = abs(fft2(f)).^2;

fA = sum(Sf(:))/prod(size(f));

R = nA/fA;

c. fr2 = deconvwnr(g, PSF, R);

NCORR = fftshift(real(ifft2(Sn)));

ICORR = fftshift(real(ifft2(Sf)));

d. fr3 = deconvwnr(g, PSF, NCORR, ICORR);


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a. f = checkerboard(8);

imshow(pixeldup(f, 8);

PSF = fspecial(gaussian, 7, 10);SD = 0.01; %desviacin estandar.

b. imnoise(imfilter(f, PSF), gaussian, 0,

SD^2); %desv. del umbral de la imag

DAMPAR = 10*SD; %resultante.

LIM = ceil(size(PSF, 1)/2);%lmite

WEIGHT = zeros(size(g)); %calidadWEIGHT (LIM+1:end-LIM, LIM+1:end-

LIM)=1; %borde de 4 ceros

),(*),(

),(*),(),(),( ^

^

1

^

yxfyxh

yxgyxhyxfyxf

k

kk

Algoritmo no lineal iterativo Lucy-Richardson.

vision artificial parte2

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