Slide 1

CS 112 - Sampling and Aliasing

Slide 2

Analog vs Digital

n Nature is analog signal n We discretize to create digital signals

Slide 3

Analog signalsn Function dependent on single or multiple variablesn Defined at any value of the independent variables

t

Ampl

itude

1D: A = f ( t )

x

y

2D: I = f ( x, y )

x

y

z

3D: S = f ( x, y, z )

Slide 4

Digital Signals

n Defined at only few values of t

Sampling Correct Reconstructiont

Slide 5

Digital Signalsn Whether you can reconstruct correctly depends

on how you sample – sampling rate

Sampling Incorrect Reconstructiont

Slide 6

Nyquist Rate

n Consider only sine wavesn If you sample at least at twice the

frequency (2 samples per cycle), signal can be reconstructed correctlyn More the sampling rate, better the

reconstructionn If less than twice the frequency, cannot

reconstruct correct

Slide 7

Nyquist Rate Sampling

Sampling Correct Reconstructiont

Slide 8

Aliasingn Aliasing: Incorrect representation of some entity

A much lower frequency Zero frequency

Slide 9

How does sinusoids help?

n Any signal can be expressed as a sum of sinusoids of different frequenciesn Amplituden Phase

Slide 10

Spectral Analysis

Time Domain Frequency Domain

Slide 11

For 2D images

n Any signal can be expressed as a sum of sinusoids of different frequenciesn Amplituden Phasen Orientation

Slide 12

Extending it to 2D

Amplitude

Phase

Slide 13

Frequency Contentn Lower frequencies : Global Patternn Higher frequencies : Detailsn Required sampling rate lower for low frequency

image (lower number of pixels, lower resolution)

Slide 14

Amplituden Amplitude

n How much details?n Sharper details signify higher frequenciesn Will deal with this mostly

Slide 15

Phase

n Where are the details?

n Though we do not use it much, it is important, especially for perception

Slide 16

Reducing Frequency contentn Filtering: Applying mathematical function over a window

around every pixeln Simplest: Averaging pixels (Box Filter)

n Other sophisticated methods n Size of the window usedn Mathematical function used is more complicated

Slide 17

How does it help?

Input (256 x 256)

Subsampled(128 x 128) Subsampled from filtered image(128 x 128)

Insufficient sampling.Hence, aliasing.

Filtering reduces frequency content.

Hence, lower sampling is sufficient.

Filtered (256 x 256)ANTI-ALIASING

Slide 18

Aliasing in Scan Conversion

n Rasterized line segments and edges of polygons look jagged

Slide 19

Aliasing in Scan Conversionn 1-pixel wide ideal line span partial pixelsn Scan conversion method forces us to choose

exactly one pixel for every value of x

Slide 20

Aliasing in Scan Conversionn Supersampling and Filtering: Render a super-

sampled image and then filtern Area Averaging: Shade each pixel by gray value

= the percentage of the actual line crossing it at x

Slide 21

Aliasing in Scan Conversionn Very expensive – Usually not implemented for real-

time renderingn Only when you have lot of time to render each

frame – Like in animation movies

Slide 22

Aliasing during z-buffering

n A pixel shared by three primitivesn Z intersection – identified in an integer leveln Front-most gets drawn

n Same technique: Area weighted average

Slide 23

Temporal Aliasing

n Animationn Speed of the object too fastn Jittered Motion