
Sampling and Reconstruction The sampling and reconstruction process

tarix  07.11.2018  ölçüsü  2,43 Mb.   #78508 

The sampling and reconstruction process  Real world: continuous
 Digital world: discrete
Basic signal processing  Fourier transforms
 The convolution theorem
 The sampling theorem
Aliasing and antialiasing  Uniform supersampling
 Nonuniform supersampling
 Sensor response
 Lens
 Shutter
 Scene radiance
Imagers = Signal Sampling All imagers convert a continuous image to a discrete sampled image by integrating over the active “area” of a sensor. Examples:  Retina: photoreceptors
 CCD array
Virtual CG cameras do not integrate, they simply sample radiance along rays …
Displays = Signal Reconstruction All physical displays recreate a continuous image from a discrete sampled image by using a finite sized source of light for each pixel. Examples:  DACs: sample and hold
 Cathode ray tube: phosphor spot and grid
Artifacts due to sampling  Aliasing  Jaggies
 Moire
 Flickering small objects
 Sparkling highlights
 Temporal strobing
Preventing these artifacts  Antialiasing
Jaggies
Fourier Transforms Spectral representation treats the function as a weighted sum of sines and cosines Each function has two representations The Fourier transform converts between the spatial and frequency domain
Spatial and Frequency Domain
Convolution Definition
Convolution Theorem: Multiplication in the frequency domain is equivalent to convolution in the space domain.
Symmetric Theorem: Multiplication in the space domain is equivalent to convolution in the frequency domain.
Sampling: Spatial Domain
Sampling: Frequency Domain
Reconstruction: Frequency Domain
Reconstruction: Spatial Domain
Sampling and Reconstruction
Sampling Theorem This result if known as the Sampling Theorem and is due to Claude Shannon who first discovered it in 1949  A signal can be reconstructed from its samples
 without loss of information, if the original
 signal has no frequencies above 1/2 the
 Sampling frequency
For a given bandlimited function, the rate at which it must be sampled is called the Nyquist Frequency
Undersampling: Aliasing
Sampling a “Zone Plate”
Ideal Reconstruction Ideally, use a perfect lowpass filter  the sinc function  to bandlimit the sampled signal and thus remove all copies of the spectra introduced by sampling Unfortunately,  The sinc has infinite extent and we must use simpler filters with finite extents. Physical processes in particular do not reconstruct with sincs
 The sinc may introduce ringing which are perceptually objectionable
Sampling a “Zone Plate”
Mitchell Cubic Filter
Antialiasing by Prefiltering
Antialiasing Antialiasing = Preventing aliasing Analytically prefilter the signal  Solvable for points, lines and polygons
 Not solvable in general
 e.g. procedurally defined images
Uniform supersampling and resample Nonuniform or stochastic sampling
Uniform Supersampling Increasing the sampling rate moves each copy of the spectra further apart, potentially reducing the overlap and thus aliasing Resulting samples must be resampled (filtered) to image sampling rate
Point vs. Supersampled
Analytic vs. Supersampled
Nonuniform Sampling Intuition Uniform sampling  The spectrum of uniformly spaced samples is also a set of uniformly spaced spikes
 Multiplying the signal by the sampling pattern corresponds to placing a copy of the spectrum at each spike (in freq. space)
 Aliases are coherent, and very noticable
Nonuniform sampling  Samples at nonuniform locations have a different spectrum; a single spike plus noise
 Sampling a signal in this way converts aliases into broadband noise
 Noise is incoherent, and much less objectionable
Jittered Sampling
Jittered vs. Uniform Supersampling
Analysis of Jitter
Poisson Disk Sampling
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