Chapter 9

Basic Signal Processing

Motivation Many aspects of computer imagery differ from

aspects of conventional imagery Computer representations are digital and discrete Natural representations are continuous

Reqires a basic understanding of signal processing

ReconstructionDisplay creates a continuous light image fr

om these discrete digital values.Ex) framebuffer

Sampling Make digital image from an analog

image.Ex)CCD camera

Discussion Discussion of reconstruction and sampling

leads to an interesting question: Is it possible to sample an image and th

en reconstuct it without any distortion?

Jaggies, AliasingFigure 9.3 the jagged edges along the edges of the

checkered pattern.

MoireSampling the equation sin(x2+y2).rather than a single set of rings ce

ntered at the origin, notice there are several sets of superimposed rings

UsefulnessSignal processing is very useful tool in computer

graphics and image processing.

1.images can be filtered to improve their appearance

2.Multiple signals can be cleverly combined into a single signal.

Mathematical fact Anyperiodic funtion can always be written as a sum of

sine and cosine waves.

More generally, a non-periodic function can also be represented as a sum of sin’s and cos ’s

Fourier transform

Example a square pulse

12

)12cos()1(

2

2

1)( 1

1

k

xkxs k

n

kn

Fourier Fourier Series

주기적인 함수 혹은 신호를 삼각함수 (sin, cos 함수 ) 들의 Linear Combination 의 형태로 전개함으로써 대상 신호를 해석

Fourier Transform

자연계에는 단순히 주기적인 신호만이 존재하는 것이 아닌 비주기 신호까지도 존재하므로 신호 해석의 범주안에서 비주기적인 신호에 대한 부분도 고려 .

비주기적인 신호를 주기가 무한대인 주기신호로 가정한다면 이 역시 Fourier Series 와 같이 삼각함수 들의 Linear Combination 으로 전개할 수 있다 .

dxexff xi

)()(

Frequency domain

FilteringModifying a signal or an image in

this way is called filtering.

H : the spectrum of the filtered functionF : spectrum of the original functionG : spectrum of the filter.

Symbol X indicates simple multiplication.

)()()( wgwfwh

filters

Low pass filtering

High pass filtering

Convolution In the space domain, filtering is achieve

d by a more complicated operation called convolution

* 입력신호 및 시스템의 임펄스응답 (impulse response) 이 주어졌을

경우에 선형시스템의 출력신호를 구하고자 할 때에는 입력함수 및 임펄스응답 함수에 대해 특별한 형태로 주어지는 적분

dxxygxfgfyh )()()(

Convolution in square pulse

One square pulse, the one corresponding to the input signal, is shown stationary and centered at the origin. The other square pulse, representing the filter, moves along the output axis from left to right.Each output value is the sum of the product of the filter and the input.

Results of convolving

Sampling

Reconstruction

Sampling theoremA signal can be reconstucted from its sma

ples without loss of information, if the original signal has no frequencies above 1/2the sampling frequency.

-Claude shannon (1949)

AliasingPre-aliasing

Due to undersampling(sampled at less than its nyquist frequency)

Post-aliasingDue to bad reconstuction(low-pass filter is not perfect : in general, rec

onstruction is a property of the hardware and media)

Undersampling

Poor reconstruction