First color film Captured by Edward Raymond Turner Predates Prokudin-Gorskii collection Like Prokudin-Gorskii, it was an additive 3-color system. Never successfully commercialized

Recap of Friday Sampling and Reconstruction Aliasing and how to prevent it (blur / low pass filters) Linear filtering and convolution

Slides from Alexei Efros, Fei Fei Li, Steve Marschner, and others Linear Filtering (See Szeliski 3.2) CS 129: Computational Photography James Hays, Brown, Spring 2011

Slide credit Fei Fei Li

© 2006 Steve Marschner 33 Continuous convolution: warm-up Can apply sliding-window average to a continuous function just as well –output is continuous –integration replaces summation

© 2006 Steve Marschner 34 Continuous convolution Sliding average expressed mathematically: –note difference in normalization (only for box) Convolution just adds weights –weighting is now by a function –weighted integral is like weighted average –again bounds are set by support of f(x)

© 2006 Steve Marschner 35 One more convolution Continuous–discrete convolution –used for reconstruction and resampling

© 2006 Steve Marschner 36 Reconstruction

© 2006 Steve Marschner 37 Resampling Changing the sample rate –in images, this is enlarging and reducing Creating more samples: –increasing the sample rate –“upsampling” –“enlarging” Ending up with fewer samples: –decreasing the sample rate –“downsampling” –“reducing”

© 2006 Steve Marschner 38 Resampling Reconstruction creates a continuous function –forget its origins, go ahead and sample it

© 2006 Steve Marschner 39 Cont.–disc. convolution in 2D same convolution—just two variables now –loop over nearby pixels, average using filter weight –looks like discrete filter, but offsets are not integers and filter is continuous –remember placement of filter relative to grid is variable

© 2006 Steve Marschner 40 A gallery of filters Box filter –Simple and cheap Tent filter –Linear interpolation Gaussian filter –Very smooth antialiasing filter B-spline cubic –Very smooth

© 2006 Steve Marschner 41 Box filter

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© 2006 Steve Marschner 45 Effects of reconstruction filters For some filters, the reconstruction process winds up implementing a simple algorithm Box filter (radius 0.5): nearest neighbor sampling –box always catches exactly one input point –it is the input point nearest the output point –so output[i, j] = input[round(x(i)), round(y(j))] x(i) computes the position of the output coordinate i on the input grid Tent filter (radius 1): linear interpolation –tent catches exactly 2 input points –weights are a and (1 – a) –result is straight-line interpolation from one point to the next

© 2006 Steve Marschner 46 Properties of filters Degree of continuity Impulse response Interpolating or no Ringing, or overshoot interpolating filter used for reconstruction

© 2006 Steve Marschner 47 Ringing, overshoot, ripples Overshoot –caused by negative filter values Ripples –constant in, non-const. out –ripple free when:

© 2006 Steve Marschner 48 Yucky details What about near the edge? –the filter window falls off the edge of the image –need to extrapolate –methods: clip filter (black) wrap around copy edge reflect across edge vary filter near edge

© 2006 Steve Marschner 49 Median filters A Median Filter operates over a window by selecting the median intensity in the window. What advantage does a median filter have over a mean filter? Is a median filter a kind of convolution? Slide by Steve Seitz

© 2006 Steve Marschner 50 Comparison: salt and pepper noise Slide by Steve Seitz