computer graphics1 antialiasing & filters. computer graphics2 introduction n as seen in class,...

Computer Graphics 1

Antialiasing & Filters

Computer Graphics 2

Introduction

As seen in class, we have aliasing when high As seen in class, we have aliasing when high frequencies pop up as low frequencies, resulting in frequencies pop up as low frequencies, resulting in undesired or even strange results.undesired or even strange results.

For example, in a movie, when there is a speeding For example, in a movie, when there is a speeding car, in a certain point the wheels will look as if car, in a certain point the wheels will look as if they are not moving, and when the car keeps they are not moving, and when the car keeps speeding up, the wheels will even appear to rotate speeding up, the wheels will even appear to rotate backwards...backwards...

Computer Graphics 3

Introduction - Cont.

This occurs since the camera filming the action This occurs since the camera filming the action can only ‘shoot’ the wheel 24 times per second. If can only ‘shoot’ the wheel 24 times per second. If the wheel of 24 rods is now rotating exactly 24 the wheel of 24 rods is now rotating exactly 24 rotations per second, it will be in exactly the same rotations per second, it will be in exactly the same position every time the camera snaps. We have position every time the camera snaps. We have here a frequency of 24Hz which was transformed here a frequency of 24Hz which was transformed to 0Hzto 0Hz..

We will now see different kinds of filtersWe will now see different kinds of filters

Computer Graphics 4

Box

Box trange

( )

1

0

t

else

Triangle

Triangle tt

( )

1

0

1- t < 0

else

Computer Graphics 5

quadratic (quadratic B-spline)

else 0

1.5t0.5 1.5-t5.0

0.5t0.5- t-0.75

-0.5t1.5- 1.5+t5.0

)(2

2

2

tQuadratic

cubic (cubic B-spline)

Cubic t

t

t t

t

2

6

4 6 3

61

2

60

3

2

3

- 2 t < -1

t

1 t < 2

else

Computer Graphics 6

Catrom

else 0

2<t1 5845.0

1<t0 3525.0

0<t1- 3525.0

-1<t2- 5845.0

)( 2

2

ttt

tt

tt

ttt

tCatrom

Computer Graphics 7

Mitchel First, for a chosen constants b,c create the First, for a chosen constants b,c create the

following:following:

p b

p b c

p b c

q b c

q b c

q b c

q b c

0

2

3

0

1

2

3

6 2 6

18 12 6 6

12 9 6 6

8 24 6

12 48 6

6 30 6

6 6

/

/

/

/

/

/

/

Computer Graphics 8

Mitchel - Cont.

Mitchel t

q t q t q tq

p t p tp

p t p tp

q t q t q tq

( )

0 1 2 3

02

2 3

02

2 3

0 1 2 3

0

- 2 t < -1

-1 t < 0

0 t < 1

1 t < 2

else

Computer Graphics 9

Gaussian (infinite)

Gaussian t e t 2 2 2

Sinc

Sinc t t

t

1 t = 0

elsesin

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Hanning

Hanning t t 0 5 1. cos

Ham g t tmin . . cos 0 54 0 46

Hamming

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Blackman

Blackman t t t 0 42 0 5 0 08 2. . cos . cos

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Kaiser First, for some small First, for some small we need to define Bessel_i0:we need to define Bessel_i0:

double Bessel_i0(double x)double Bessel_i0(double x)

{{

register int i;register int i;

double sum, y, t;double sum, y, t;

sum = 1.;sum = 1.;

y = x*x/4.; y = x*x/4.;

t = y;t = y;

for (i=2; t>for (i=2; t>; i++) {; i++) {

sum += t;sum += t;

t *= (double)y/(i*i); t *= (double)y/(i*i);

} }

return sum;return sum;

}}

First, for a chosen range d and constants b,c create the following:First, for a chosen range d and constants b,c create the following:

Computer Graphics 13

Kaiser - Cont.

Initialize the following (for chosen constant a):Initialize the following (for chosen constant a): i0a=1/Bessel_i0(a)i0a=1/Bessel_i0(a)

And now:And now:

Bessel t Bessel i a t i a _ 0 1 02