CS148: Introduction to Computer Graphics and Imaging Basic Signal Processing: Sampling and Aliasing Key Concepts Frequency space Filters and convolution Sampling Aliasing Nyquist frequency Antialiasing CS148 Lecture 15 Pat Hanrahan, Fall 2009 Page 1 Frequency Space Spatial and Frequency Domain Spatial Domain Frequency Domain (Center is fx = fy = 0) Can convert from one to the other using the Fourier Transform CS148 Lecture 15 Pat Hanrahan, Fall 2009 Page 2 Filtering in Frequency Space Spatial Domain Frequency Domain Multiply filter times image in frequency space In this case, the filter removes high frequencies CS148 Lecture 15 Pat Hanrahan, Fall 2009 Filtering in Frequency Space Spatial Domain Frequency Domain Multiply filter times image in frequency space In this case, the filter removes low frequencies CS148 Lecture 15 Pat Hanrahan, Fall 2009 Page 3 Filtering in Frequency Space Spatial Domain Frequency Domain CS148 Lecture 15 Pat Hanrahan, Fall 2009 Filtering in Frequency Space Spatial Domain Frequency Domain CS148 Lecture 15 Pat Hanrahan, Fall 2009 Page 4 Filters = Convolution Convolution Signal / Image 1 3 0 4 2 1 Filter 1 2 CS148 Lecture 15 Pat Hanrahan, Fall 2009 Page 5 Convolution 1 3 0 4 2 1 1 2 1 * 1 + 3 * 2 = 7 7 CS148 Lecture 15 Pat Hanrahan, Fall 2009 Convolution 1 3 0 4 2 1 1 2 3 * 1 + 0 * 2 = 3 7 3 CS148 Lecture 15 Pat Hanrahan, Fall 2009 Page 6 Convolution 1 3 0 4 2 1 1 2 0 * 1 +4 * 2 = 8 7 3 8 CS148 Lecture 15 Pat Hanrahan, Fall 2009 Convolution Theorem A filter can be implemented in the spatial domain using convolution A filter can also be implemented in the frequency domain Convert image to frequency domain Convert filter to frequency domain Multiply filter times image in frequency domain Convert result to the spatial domain This works because of the convolution theorem Filter properties are easier to understand in the frequency domain CS148 Lecture 15 Pat Hanrahan, Fall 2009 Page 7 Box Filter 1 1 1 1 CS148 Lecture 15 Pat Hanrahan, Fall 2009 Box Filter = Low-Pass Filter Spatial Domain Frequency Domain CS148 Lecture 15 Pat Hanrahan, Fall 2009 Page 8 Wider Filters, Lower Frequencies Spatial Domain Frequency Domain CS148 Lecture 15 Pat Hanrahan, Fall 2009 Size of Filter As a filter is localized in space, it spreads out in frequency Conversely, as a filter is localized in frequency, it spreads out in space A box filter is very localized in space; it has infinite extent in frequency space A small box filter has more high frequencies than a larger box filter CS148 Lecture 15 Pat Hanrahan, Fall 2009 Page 9 Efficiency? When would it be faster to apply the filter in the spatial domain? When would it be faster to apply the filter in the frequency domain? CS148 Lecture 15 Pat Hanrahan, Fall 2009 Sampling Page 10 Image Generation = Sampling Rasterization is performed by testing whether a point is inside a triangle More generally, evaluate a function at a point !for( int x = 0; x < xmax; x++ )! !!for( int y = 0; y < ymax; y++ )! !! I[x][y] = f( float(x), float(y));! Take a continuous function f and convert to set of samples I, a discrete representation of the func. CS148 Lecture 15 Pat Hanrahan, Fall 2009 Aliasing Page 11 Zone Plate CS148 Lecture 15 Pat Hanrahan, Fall 2009 Zone Plate Aliases (0,0) CS148 Lecture 15 Pat Hanrahan, Fall 2009 Page 12 Wagon Wheel Effect http://www.michaelbach.de/ot/mot_wagonWheel / “Aliases” Two frequencies are indistinguishable CS148 Lecture 15 Pat Hanrahan, Fall 2009 Page 13 Nyquist Frequency Definition: Sampling period Ts Definition: The Nyquist frequency is ½ the sampling frequency = 1/(2 Ts) A periodic signal above the Nyquist frequency cannot be distinguished from a periodic signal below the Nyquist frequency Indistinguishable frequencies look alike Hence, they are called aliases CS148 Lecture 15 Pat Hanrahan, Fall 2009 Sampling in Computer Graphics All of computer graphics involve sampling Artifacts due to sampling - Aliasing Jaggies – sampling in space Flickering small objects – sampling in space Moire – sampling texture coordinates Sparkling highlights – sampling normals Temporal strobing – sampling in time Preventing these artifacts - Antialiasing CS148 Lecture 15 Pat Hanrahan, Fall 2009 Page 14 Jaggies Retort sequence by Don Mitchell Staircase pattern or jaggies CS148 Lecture 15 Pat Hanrahan, Fall 2009 Moire Pattern http://commons.wikimedia.org/wiki/File:Moire-%28quadrat%29-1.png CS148 Lecture 15 Pat Hanrahan, Fall 2009 Page 15 Zone Plate – Moire Patterns Aliases (0,0) Moire pattern (sum of signal and aliased signal) CS148 Lecture 15 Pat Hanrahan, Fall 2009 Antialiasing Page 16 Antialiasing Simple idea: Remove frequencies above the Nyquist frequency before sampling How? Filtering before sampling Filter during rasterization CS148 Lecture 15 Pat Hanrahan, Fall 2009 Prefiltering by Computing Coverage Pixel Area = Box Filter CS148 Lecture 15 Pat Hanrahan, Fall 2009 Page 17 Point- vs. Area-Sampled Point Area Checkerboard sequence by Tom Duff CS148 Lecture 15 Pat Hanrahan, Fall 2009 Supersampling Take a set of samples and average them together 4 x 4 supersampling CS148 Lecture 15 Pat Hanrahan, Fall 2009 Page 18 Point-sampling vs. Super-sampling Point 4x4 Super-sampled Checkerboard sequence by Tom Duff CS148 Lecture 15 Pat Hanrahan, Fall 2009 Area-Sampling vs. Super-sampling Exact Area 4x4 Super-sampled CS148 Lecture 15 Pat Hanrahan, Fall 2009 Page 19 Antialiasing Jaggies Prefilter CS148 Lecture 15 Pat Hanrahan, Fall 2009 Antialiasing vs. Blurred Aliases Blurred Jaggies Prefilter CS148 Lecture 15 Pat Hanrahan, Fall 2009 Page 20 Things to Remember Convolutions Frequency domain vs. spatial domain Filtering in the spatial domain vs. frequency domain Sampling converts continuous image to discrete image Image generation involves sampling May also sample geometry, motion, … Nyquist frequency is ½ the sampling rate Frequencies above the Nyquist frequency appear as other frequencies – aliasess Antialiasing – Filter before sampling CS148 Lecture 15 Pat Hanrahan, Fall 2009 Page 21 .