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Image Processing:6. Linear Filters6. Linear Filters

Aleix M. Martinezaleix@ece.osu.edu

Convolution

•Represent these weights as an image, H.•H is usually called the kernel.•The operation is called a convolution:

•Notice order of indices–all examples can be put in this form–it’s a result of the derivation expressing any shift-

invariant linear operator as a convolution.

Rij H iu, jvFuvu ,v

Example: Smoothing byAveraging Smoothing with a Gaussian

• Smoothing with an averageactually doesn’t compare at all well with a defocussedlens–Most obvious difference

is that a single point oflight viewed in adefocussed lens lookslike a fuzzy blob; but theaveraging process wouldgive a little square.

•A Gaussian gives a goodmodel of a fuzzy blob

exp x2 y2

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An Isotropic Gaussian Smoothing with a Gaussian

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Differentiation and convolution

•Recall

•This is linear and shiftinvariant, so must bethe result of aconvolution.

•We could approximatethis as:

fx

lim0

f x , y

f x, y

f

x

f xn1,y f xn , y x

Finite differences

Finite differences responding tonoise

Increasing noise => (this is zero mean additive gaussian noise).

Gradients and edges

•Points of sharp change in an image are interesting:–change in reflectance,–change in object,–change in illumination,–and noise.

•Sometimes called edge points.

•General strategy–determine image gradient.–now mark points where gradient magnitude is

particularly large wrt neighbours (ideally, curves of suchpoints).

There are three major issues:1) The gradient magnitude at different scales is different; which should

we choose?2) The gradient magnitude is large along thick trail; how

do we identify the significant points?3) How do we link the relevant points up into curves?

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Let be a transform of size J x J where,

Fourier Transform:

)2

exp(1

),( klJ

iJ

lkJJ

k, l = 0,1,…..,J-1JJ

Then the discrete Fourier transform can be defined as:NNQ ,P MM

M M N NF f and

1,,1,0

2exp),(1

),(1

0

1

0

Mu

Nnv

Mmu

inmfMN

vuFM

m

N

n

1,,1,0 Nv

eq(1)

The properties of the discrete Fourier Transform are the sameas the continuous Fourier transform wrt linearity, shift,modulation, convolution, multiplication and correlationproperties.

The inverse transform matrix is

)2

exp(),(1 klJ

ilkJJ

1JJ

And thus the inverse Fourier transform is given by:

1,,1,0

2exp),(),(1

0

1

0

Mm

Nnv

Mmu

ivuFnmfM

u

N

v

1,,1,0 Nn

The kernel function of the discrete transform (seeeq 1) is given by:

Nnv

Mmu

i2exp

(image values can be viewed as weighted sum of periodicpatterns!)

For implementation of the discrete Fouriertransform, equation (1) can be modified to:

1,,1,0

2exp),(

2exp

11),(

1

0

1

0

Mu

Mimu

nmfN

invNM

vuFM

m

N

n

1,,1,0 Nv

The term in the square bracket is the 1D Fourier transformof the mth line, and can be calculated using a FFT. Thus,to perform the 2D Fourier transform each line of the imagecan be computed and replaced with it Fourier transform,and then the 1D FFT of the columns can be computed.

The frequency spectrum is defined as:

),(),(),( 22 vuIvuRvuF

The phase spectrum is given by:

),(),(

tan),( 1vuRvuI

vu

The power spectrum (power spectrum density) is givenby:

),(),(),(),( 222 vuIvuRvuFvuP

The Fourier transform of real function is a complex function, thatis,

F(u,v)=R(u,v) + iI(u,v)

where R(u,v) and I(u,v) are real and imaginary parts of F(u,v)

Power

Square

What happens when thesquare is shifted?

Shifted square

What happens when thesquare is scaled? Say,what if we make itsmaller… What will the FFT look like?

Explain why this makessense.

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Now, what if we take this tothe extreme, a single-pixelsquare? What do youpredict the DFT will looklike

A single pixel is “on” –roughlycorresponding to animpulse.

Does this make sense?How can we test ouridea that this is “white”?

OK, so now let’s spread thoseimpulses out. Whatshould happen?

And spread them out a bitmore? By now, it’s obvious…

Examples

We’ve placed the zero frequency at the center of the plot.

The “cross” pattern arises from two sources: the edges in the image and the edges of the image.

imshow(log(abs(fftshift(F))),[],'notruesize');colormap(jet); colorbar

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Another example The convolution of two images f and h of the same size M x Nresults in periodic image g given by:

1

0

1

0

),(),(1

),(M

m

N

n

nbmahnmfMN

bag

According to the convolution theorem f, g, and h and theirtransforms are related by the equation:

),(),(),( vuHvuFvuG Thus we get the following–which dramaticallyreduces the computational cost of convolving“large” masks:

1

0

1

0

2exp),(),(),(M

u

N

v Nbv

Mau

ivuHvuFbag

Using Fourier Transform to calculate image convolutions.

Other Transforms•There are a great many other tranforms, each

with different characteristics and differentuses.•These include:–Hadamard–DCT–Wavelets,–Etc.

The basic forms•Low pass: small values located far from the

co-ordinate regions. Suppresses edges,smoothes/blurs.•High-pass: small values near the origin.

Enhances sharp edges.•Band-pass: select frequencies in a range that

you wish to enhance, or pick directionalresponses that you wish to enhance.

The basic forms.lowpass highpass

bandpass

Example: LPF

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Example: Band-pass filter Bandstop Filter

Example: High Pass Filter