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CS 519 – Signal & Image Processing
Sampling
Sampling
t
f ( t )
t
g ( t )
Continuous
Discrete
Sampling: Spatial/Temporal Domain
Sampling a continuous function f at time/space interval ∆ t to produce a discrete function g g [ n ] = f ( n ∆ t ) is the same as multiplying it by a comb: g = f comb h where h = ∆ t
t
g ( t )
Sampling: Spatial/Temporal Domain
t
f ( t )
t
g ( t )
Continuous
Discrete
Sampling Comb
t
comb h ( t )
Sampling: Frequency Domain
Sampling in the spatial/temporal domain by multiplying with comb h g = f comb h is the same as convolution in the frequency domain with the transform of comb h : G = F * comb (^) 1/ h Convolution of a function and a comb causes a copy of the function to “stick” to each tooth of the comb, and all of them add together
Sampling: Frequency Domain
Spectrum
Spectrum of Discrete Signal
Comb’s Spectrum
s
comb1/ h ( s )
s
F ( s )
s
G ( s )
Reconstruction
In theory, we can reconstruct the original continuous function by removing all of the extraneous copies of its spectrum created by the sampling process:
F ( s ) = G ( s ) Π1/ h ( s )
In other words, keep everything in the frequency domain between and throw the rest away
h
s h 2
Reconstruction: Graphical Example
s
Reconstructed Signal Spectrum
Rectangular (Box) Filter
F ( s )
Spectrum of Discrete Signal s
G ( s )
s
Π1/ h ( s )
The Sampling Theorem
We can only do this reconstruction if the duplicated copies do not overlap
They do not overlap iff:
- The signal is band limited, and
- The highest frequency in the signal is less than
In other words, the sampling rate 1/ h must be twice the frequency of the highest frequency in the image
This is called the Nyquist rate
2 h
1
Aliasing
What if the duplicated copies in the frequency domain do overlap? High frequency parts of the signal (those higher than ) intrude into neighboring copies The higher the frequency, the lower the point of overlap in the adjacent copy These high frequencies masquerading as low frequencies is called aliasing False low-frequency patterns are called Moiré patterns
2 h
1
Sampling: Frequency Domain
Spectrum
Spectrum of Discrete Signal
Comb’s Spectrum
s
comb1/ h ( s )
s
F ( s )
s
G ( s )
Sampling: Above the Nyquist Rate
Imperfect Reconstruction
Correcting Imperfect Reconstruction:
- Sample well above the Nyquist rate
- Low-pass filter after reconstruction
Imperfect Reconstruction
Spectrum of Discrete Signal s
G ( s )
s
Π1/ h ( s )
Typical Processing Pipeline
- Low-pass filter to reduce aliasing
- Sample
- Do something with the digitized signal/image
- Reconstruct
- Low-pass filter to correct for imperfect reconstruction
The Discrete Frequency Domain
- If sampling in the time/spatial domain is multiplication by a comb, so is sampling (discretizing) the frequency domain
- Multiplication by a comb in one domain is convolution with a comb of inverse spacing in the other
- Discrete time/spatial samples Æ replicated copies of the signal’s spectrum appear in the frequency domain
- Discrete frequencies Æ replicated copies of the signal appear in the time/spatial domain (i.e., the signal is periodic)
The Discrete Frequency Domain
- If a signal is N time samples long, and we disccretize the frequency domain a 1/ N intervals (like the DFT), we reproduce the signal every N samples in the time domain
- The Discrete Fourier Transform of a truncated (finite- domain) signal is the Continuous Fourier Transform of the same periodic signal
Frequency Resolution
- An N -element signal is accurate in the frequency domain only to 1/ N
- To be more accurate in the spatial domain, sample more frequently
- To be more accurate in the frequency domain, sample longer
