Sampling and Reconstruction The sampling and reconstruction process

Sampling and Reconstruction

• 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

Camera Simulation

• 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

Sampling in Computer Graphics

• 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

• Spatial domain - normal representation
• Frequency domain - spectral representation

• 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 low-pass 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

Distribution of Extrafoveal Cones

Non-uniform 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

• Non-uniform sampling

• Samples at non-uniform 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