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Gonzalez & Woods, and Efros)
Sampling and Aliasing Gilad Lerman Math 5467 (stealing slides from Gonzalez & Woods, and Efros)
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The Sampling Theorem Theorem: If f is in L1() & supported on [-B0, B0], then Recall Proof: We view as (2B0)-periodic function with coefficients: At last, find f using IFT and using FS of Shannon sampling theorem, and is also known as Nyquist–Shannon–Kotelnikov, Whittaker–Shannon–Kotelnikov, Whittaker–Nyquist–Kotelnikov–Shannon, WKS, etc.
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More on the Sampling Theorem
Frequency band: Time: Note: Theorem holds for B>B0. Indeed, then If B<B0, the above equation is not true for all
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Sampling Theorem (meaning)
Interpretation: If a function f(t) contains no frequencies higher than W cps, it is completely determined by giving its ordinates at a series of points spaced 1/(2W) seconds apart Remark: For L1 function a freq. = W is fine but for more general functions we need > W… "cps" for "cycles per second"
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Simple Example (not L1) Assume a cosine
(it is not L1() but will be instrumental) Freq: a (“& -a”), Freq Band: =[-a,a], Time: 1/(2a) Here one needs B>B0 (B=B0 doesn’t work) Example: for all 3 functions freq: 0.5, time: 1 The sampled function has different aliases…
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Aliasing If the sampling condition is not satisfied, frequencies will overlap (high freq → low freq) The reconstructed signal is said to be an alias of the original signal
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Example: Increased Frequency
Input signal: Related Image: x = 0:.05:5; imagesc(sin((2.^x).*x)) Matlab output: Picket fence receding Into the distance will produce aliasing…
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One more example at the Fourier domain
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Aliasing in Images (Fourier domain)
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Good and Bad Sampling Good sampling: Bad sampling: Sample often or,
Sample wisely Bad sampling: see aliasing in action!
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Texture makes its worse (high frequencies)
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Even worse for synthetic images
Slide by Steve Seitz
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Really bad in video Slide by Paul Heckbert
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Wheels of Wagons in Westerns
Can watch Ballad of the wagon war (1967), start watching around 1:40sec.
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Moiré pattern Definition: Interference pattern created, e.g., when two grids are overlaid at an angle, or when they have slightly different mesh sizes. In images produced e.g., when scanning a halftone picture or due to undersampling a fine regular pattern. See e.g. Though if I put it on a slide I could not view aliasing
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Moiré pattern due to undersampling
See e.g. Though if I put it on a slide I could not view aliasing Original image downsampled image
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Antialiasing What can be done? Raise sampling rate by oversampling
Sampling rate ≥ 2 * max frequency in the image Raise sampling rate by oversampling Sample at k times the resolution continuous signal: easy discrete signal: need to interpolate 2. Lower the max frequency by prefiltering Smooth the signal enough Works on discrete signals 3. Improve sampling quality with better sampling Nyquist is best case! Stratified sampling Importance sampling Relies on domain knowledge
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Gaussian pre-filtering
Solution: filter the image, then subsample Filter size should double for each ½ size reduction.
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Subsampling with Gaussian pre-filtering
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Compare with... 1/2 1/4 (2x zoom) 1/8 (4x zoom)
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Correcting some Moiré patterns
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