CS Sampling and Aliasing. Analog vs Digital

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1 CS Sampling and Aliasing Aditi Majumder, CS 112, Winter 2007 Slide 1 Analog vs Digital God has created the world analog Man has created digital world Aditi Majumder, CS 112, Winter 2007 Slide 2 1

y t x Aditi Majumder, CS 112, Winter 2007 Slide 3 z x Digital Signals Defined at only few

2 Amplitude Analog signals Function dependent on single or multiple variables Defined at any value of the dependent variable 3D: S = f ( x, y, z ) 1D: A = f ( t ) 2D: I = f ( x, y ) y y t x Aditi Majumder, CS 112, Winter 2007 Slide 3 z x Digital Signals Defined at only few values of t Sampling t Correct Reconstruction Aditi Majumder, CS 112, Winter 2007 Slide 4 2

3 Digital Signals Whether you can reconstruct correctly depends on how you sample sampling rate Sampling t Incorrect Reconstruction Aditi Majumder, CS 112, Winter 2007 Slide 5 Nyquist Rate Consider only sine waves If you sample at least at twice the frequency (2 samples per cycle), signal can be reconstructed correctly More the sampling rate, better the reconstruction If less than twice the frequency, cannot reconstruct correct Aditi Majumder, CS 112, Winter 2007 Slide 6 3

4 Nyquist Rate Sampling Sampling t Correct Reconstruction Aditi Majumder, CS 112, Winter 2007 Slide 7 Aliasing Aliasing: Incorrect representation of some entity A much lower frequency Zero frequency Aditi Majumder, CS 112, Winter 2007 Slide 8 4

5 How does sinusoids help? Any signal can be expressed as a sum of sinusoids of different frequencies Amplitude Phase Aditi Majumder, CS 112, Winter 2007 Slide 9 Spectral Analysis Time Domain Frequency Domain Aditi Majumder, CS 112, Winter 2007 Slide 10 5

6 For 2D images Any signal can be expressed as a sum of sinusoids of different frequencies Amplitude Phase Orientation Aditi Majumder, CS 112, Winter 2007 Slide 11 Extending it to 2D Phase Amplitude Aditi Majumder, CS 112, Winter 2007 Slide 12 6

Majumder, CS 112, Winter 2007 Slide 13 Amplitude Amplitude How much details?

7 Frequency Content Lower frequencies : Global Pattern Higher frequencies : Details Required sampling rate lower for low frequency image (lower number of pixels, lower resolution) Aditi Majumder, CS 112, Winter 2007 Slide 13 Amplitude Amplitude How much details? Sharper details signify higher frequencies Will deal with this mostly Aditi Majumder, CS 112, Winter 2007 Slide 14 7

8 Phase Where are the details? Though we do not use it much, it is important, especially for perception Aditi Majumder, CS 112, Winter 2007 Slide 15 Reducing Frequency content Filtering: Applying mathematical function over a window around every pixel Simplest: Averaging pixels (Box Filter) Other sophisticated methods Size of the window used Mathematical function used is more complicated Aditi Majumder, CS 112, Winter 2007 Slide 16 8

9 How does it help? Filtering reduces frequency content. Hence, lower sampling is sufficient. Input (256 x 256) Filtered (256 x 256) ANTI-ALIASING Insufficient sampling. Hence, aliasing. Subsampled(128 x 128) Subsampled from filtered image(128 x 128) Aditi Majumder, CS 112, Winter 2007 Slide 17 Aliasing in Scan Conversion Rasterized line segments and edges of polygons look jagged Aditi Majumder, CS 112, Winter 2007 Slide 18 9

Aliasing in Scan Conversion 1-pixel wide ideal line span partial pixels Scan conversion method forces us to choose exactly one pixel for every value of x Aditi Majumder, CS 112, Winter 2007 Slide 19

10 Aliasing in Scan Conversion 1-pixel wide ideal line span partial pixels Scan conversion method forces us to choose exactly one pixel for every value of x Aditi Majumder, CS 112, Winter 2007 Slide 19 Aliasing in Scan Conversion Supersampling and Filtering: Render a supersampled image and then filter Area Averaging: Shade each pixel by gray value = the percentage of the actual line crossing it at x Aditi Majumder, CS 112, Winter 2007 Slide 20 10

Aliasing in Scan Conversion Very expensive Usually not implemented for realtime rendering Only when you have lot of time of render each frame Like in animation movies Aditi Majumder, CS 112, Winter

11 Aliasing in Scan Conversion Very expensive Usually not implemented for realtime rendering Only when you have lot of time of render each frame Like in animation movies Aditi Majumder, CS 112, Winter 2007 Slide 21 Aliasing during z-buffering A pixel shared by three primitives Z intersection identified in an integer level Front-most gets drawn Same technique: Area weighted average Aditi Majumder, CS 112, Winter 2007 Slide 22 11

12 Temporal Aliasing Animation Speed of the object too fast Jittered Motion Aditi Majumder, CS 112, Winter 2007 Slide 23 12

Images have structure at various scales

Images have structure at various scales Frequency Frequency of a signal is how fast it changes Reflects scale of structure A combination of frequencies 0.1 X + 0.3 X + 0.5 X = Fourier transform Can we