Abstract

Algorithms for digitally processing sampled spatiotemporal signals typically ignore the fact that those samples were obtained time-sequentially, rather than instantaneously, at the beginning of each frame period The effect of displacing time-sequentially obtained samples to a non-time-sequential lattice is analyzed for both deterministic and nondeterministic signals Sample displacement is shown to cause unavoidable frequency-domain aliasing The resulting artifacts are compared in test signals that have been sampled with a lexicographic and bit-reversed ordering of the sample points

© 1983 Optical Society of America

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References

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  1. S Deutsch, “Pseudo-random dot scan television systems,” IEEE Trans Broadcast BC-11, 11–21 (1965)
    [Crossref]
  2. S Deutsch, “Visual displays using pseudo-random dot scan,” IEEE Trans Commun Technol COM-21, 65–76 (1973)
    [Crossref]
  3. J P Allebach, “Analysis of sampling-pattern dependence in time-sequential sampling of spatiotemporal signals,” J Opt Soc Am 71, 99–105 (1981)
    [Crossref]
  4. D E Pearson, Transmission and Display of Pictorial Information (Wiley, New York, 1975), pp 1–30
  5. J W Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp 4–25
  6. A Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), pp 336–384
  7. A J Jerri, “The Shannon sampling theorem—its various extensions and applications a tutorial review,” Proc IEEE 65, 1565–1596 (1977)
    [Crossref]
  8. T Thong, Digital Image Processing Test PatternsARG Tech Rep FP014 (Tektronix Laboratories, Beaverton, Ore, 1982)

1981 (1)

J P Allebach, “Analysis of sampling-pattern dependence in time-sequential sampling of spatiotemporal signals,” J Opt Soc Am 71, 99–105 (1981)
[Crossref]

1977 (1)

A J Jerri, “The Shannon sampling theorem—its various extensions and applications a tutorial review,” Proc IEEE 65, 1565–1596 (1977)
[Crossref]

1973 (1)

S Deutsch, “Visual displays using pseudo-random dot scan,” IEEE Trans Commun Technol COM-21, 65–76 (1973)
[Crossref]

1965 (1)

S Deutsch, “Pseudo-random dot scan television systems,” IEEE Trans Broadcast BC-11, 11–21 (1965)
[Crossref]

Allebach, J P

J P Allebach, “Analysis of sampling-pattern dependence in time-sequential sampling of spatiotemporal signals,” J Opt Soc Am 71, 99–105 (1981)
[Crossref]

Deutsch, S

S Deutsch, “Visual displays using pseudo-random dot scan,” IEEE Trans Commun Technol COM-21, 65–76 (1973)
[Crossref]

S Deutsch, “Pseudo-random dot scan television systems,” IEEE Trans Broadcast BC-11, 11–21 (1965)
[Crossref]

Goodman, J W

J W Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp 4–25

Jerri, A J

A J Jerri, “The Shannon sampling theorem—its various extensions and applications a tutorial review,” Proc IEEE 65, 1565–1596 (1977)
[Crossref]

Papoulis, A

A Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), pp 336–384

Pearson, D E

D E Pearson, Transmission and Display of Pictorial Information (Wiley, New York, 1975), pp 1–30

Thong, T

T Thong, Digital Image Processing Test PatternsARG Tech Rep FP014 (Tektronix Laboratories, Beaverton, Ore, 1982)

IEEE Trans Broadcast (1)

S Deutsch, “Pseudo-random dot scan television systems,” IEEE Trans Broadcast BC-11, 11–21 (1965)
[Crossref]

IEEE Trans Commun Technol (1)

S Deutsch, “Visual displays using pseudo-random dot scan,” IEEE Trans Commun Technol COM-21, 65–76 (1973)
[Crossref]

J Opt Soc Am (1)

J P Allebach, “Analysis of sampling-pattern dependence in time-sequential sampling of spatiotemporal signals,” J Opt Soc Am 71, 99–105 (1981)
[Crossref]

Proc IEEE (1)

A J Jerri, “The Shannon sampling theorem—its various extensions and applications a tutorial review,” Proc IEEE 65, 1565–1596 (1977)
[Crossref]

Other (4)

T Thong, Digital Image Processing Test PatternsARG Tech Rep FP014 (Tektronix Laboratories, Beaverton, Ore, 1982)

D E Pearson, Transmission and Display of Pictorial Information (Wiley, New York, 1975), pp 1–30

J W Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp 4–25

A Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), pp 336–384

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Figures (10)

Fig 1
Fig 1

Time-sequential sampling and sample displacement with a lexicographic pattern One sample is taken at a dot every T sec After B sec, the spatial region is uniformly sampled at interval X, and the pattern repeats Ignoring the times at which the samples were obtained during the frame interval effectively displaces them to the locations indicated by squares

Fig. 2
Fig. 2

Signal-to-noise ratio that is due to abasing for lexicographic sampling with and without sample displacement The curves were calculated for M = 16 but are approximately identical for other values of M when M/AU is constant

Fig 3
Fig 3

Signal-to-noise ratio that is due to aliasing for bit-reversed sampling with and without sample displacement The curves were calculated for M = 16 but are approximately identical for other values of M when M/AU is constant

Fig 4
Fig 4

Vertical chirped test signal sampled instantaneously at eight points during its temporal cycle The numbers indicate the sampling times as a fraction of the period. Note that, in all the figures that follow, these numbers indicate temporal frequencies rather than time points.

Fig 5
Fig 5

Vertical chirped test signal sampled instantaneously (INS) and time-sequentially with a lexicographic pattern (LEX) The signal is shown at four temporal frequencies, which are given in units of cycles/frame period

Fig 6
Fig 6

Vertical chirped test signal sampled instantaneously (INS) and time-sequentially with a bit-reversed pattern (BRV) The signal is shown at four temporal frequencies, which are given in units of cycles/frame period

Fig 7
Fig 7

Horizontal chirped test signal sampled instantaneously (INS) and time-sequentially with a lexicographic pattern (LEX) The signal is shown at two temporal frequencies, which are given in units of cycles/frame period

Fig 8
Fig 8

Horizontal chirped test signal sampled instantaneously (INS) and time-sequentially with a bit-reversed pattern (BRV) The signal is shown at two temporal frequencies, which are given in units of cycles/frame period

Fig 9
Fig 9

Vertical chirped test signal sampled instantaneously (INS) and time-sequentially with a bit-reversed pattern (BRV) The signal is shown at four additional temporal frequencies to demonstrate how contrast varies with this parameter The units are cycles/frame period.

Fig 10
Fig 10

Circular chirped test signal sampled time-sequentially with a bit-reversed pattern (BRV) The signal is shown at four temporal frequencies, which are given in units of cycles/frame period Each circular moiré pattern indicates the location in the spatial-frequency domain of the spectral replication that is responsible for the aliasing

Equations (34)

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h ( x , y , t ) = X 2 B l g ( α l X , β l X , l T ) × δ ( x α l X , y β l X , t l T )
α l = [ l / M ] , β l = l mod M ,
H ( u , υ , f ) = m n p Q m n p G ( u m / A , υ n / A , f p / B ) ,
Q m n p = 1 M 2 l = 0 M 2 1 exp [ ι 2 π ( m α l / M + n β l / M + p l / M 2 ) ]
Q 000 = 1 , Q m n 0 = δ m mod M δ n mod M , Q 00 p = δ p mod M 2 , Q m + a M , n + b M , p + c M 2 = Q m n p ,
U 1 / ( 2 X ) , W 1 / ( 2 B )
ĥ ( x , y , t ) = X 2 B l g ( α l X , β l X , l T ) × δ ( x α l X , y β l X , t l / M 2 B )
c ( x , y , t ) = δ ( x ) δ ( y ) rect { [ t + ( B T ) / 2 ] / B } ,
d ( x , y , t ) = 1 × 1 × l δ ( t l B )
ĥ ( x , y , t ) = d ( x , y , t ) [ c ( x , y , t ) h ( x , y , t ) ] ,
Ĥ ( u , υ , f ) = k sinc [ B ( f k / B ) ] × exp [ ι π ( B T ) ( f k / B ) ] H ( u , υ , f k / B )
Ĥ ( u , υ , f ) = m n p Q m n ( f p / B ) G ( u m / A , υ n / A , f p / B ) ,
Q m n ( f ) = k Q m n k sinc [ B ( f + k / B ) ] × exp [ ι π ( B T ) ( f + k / B ) ] .
Q 00 ( f ) sinc [ B f ] exp [ ι π ( B T ) f / B ]
S h h ( u , υ , f ) = m n p | Q m n p | 2 × S g g ( u m / A , υ n A , f p / B )
S ĥ ĥ ( u , υ , f ) = k sinc 2 [ B ( f k / B ) ] S h h ( u , υ , f k / B )
S ĥ ĥ ( u , υ , f ) = m n p R m n ( f p / B ) × S g g ( u m / A , υ n / A , f p / B )
R m n ( f ) = k | Q m n k | 2 sinc 2 [ B ( f + k / B ) ]
ê 0 ( x , y , t ) = ĥ ( x , y , t ) g ( x , y , t + ( B T ) / 2 ) .
σ ê 2 = Ω S ê 0 ê 0 ( u , υ , f ) d u d υ d f
S ê 0 ê 0 ( u , υ , f ) = S ĥ ĥ ( u , υ , f ) + S g g ( u , υ , f ) 2 Re { S ĥ g ( u , υ , f ) exp [ ι π ( B T ) f ] } ,
S ĥ g ( u , υ , f ) = sinc ( B f ) exp [ ι π ( B T ) f ] S g g ( u , υ , f )
σ ê 2 = Ω { R 00 ( f ) 2 sinc ( B f ) + 1 } S g g ( u , υ , f ) d u d υ d f + m n p Ω R m n ( f p / B ) S g g ( u m / A , υ n / A , f p / B ) d u d υ d f , ( m , n , p ) ( 0 , 0 , 0 ) .
S g g ( u , υ , f ) = { 1 / ( 8 U 2 W ) , ( u , υ , f ) Ω 0 , else , Ω = { ( u , υ , f ) : max [ | u | / U , | υ | / U , | f | / W ] < 1 } ,
g ( x , y , t ) = 1 2 ( 1 + cos { 2 π [ x 2 / ( 8 A X ) f 0 t ] } ) .
g ( r , t ) = 1 2 ( 1 + cos { 2 π [ r 2 / ( 2 A X ) f 0 t ] } ) ,
[ σ ê 2 ] a l = m n p Ω R m n ( f p / B ) S g g ( u m / A , υ n / A , f p / B ) d u d υ d f , ( m , n , p ) ( 0 , 0 , 0 ) .
[ σ ê 2 ] a l = 1 8 m n p = 0 m n p { [ 2 | m | / A U ) ] × [ 2 | n | / A U ) ] 1 1 | p | / ( B W ) R m n ( W f ) d f } + ,
m n p = { 0 , ( m , n , p ) = ( 0 , 0 , 0 ) , 1 , ( m , n ) ( 0 , 0 ) p = 0 , 2 , p 1 ,
R m n ( f ) = sinc 2 ( B f ) rep 1 T [ l = 0 M 2 1 | Q m n l | 2 δ ( f + l / B ) ] ,
r m n ( t ) = 1 M 2 k = M 2 + 1 M 2 1 r m n k δ ( t + k T ) ,
r m n k = [ 1 | k | / M 2 ] l = 0 M 2 1 | Q m n l | 2 exp [ ι 2 π k l / M 2 ]
R m n ( f ) = 1 M 2 + 2 M 2 k = 1 M 2 1 | r m n k | cos ( 2 π k B f / M 2 + | r m n k _ )
1 1 | p | / ( B W ) R m n ( W f ) d f = 1 M 2 [ 2 | p | / ( B W ) ] + 1 π B W k = 1 M 2 1 | r m n k | k ( sin { 2 π k B W [ 1 | p | / ( B W ) ] / M 2 + | r m n k _ } sin [ 2 π k B W / M 2 + | r m n k _ ] )