Compact Encoding of 3D Voxel Surface Based on Pattern...
Transcript of Compact Encoding of 3D Voxel Surface Based on Pattern...
Sampling
Chang-Su Kim
Some figures have been excerpted from
1. The lecture notes of Dr. Benoit Boulet in McGill University
(http://www.cim.mcgill.ca/~boulet/304-304A/304-304A.htm)
2. Signals and Systems by M. J. Roberts
3. http://www.ics.uci.edu/~majumder/CG/classes/sampling_1nov.pdf
Sampling
Sampling is a procedure
to extract a DT signal
from a CT signals
(b), (c), (d) are obtained
by sampling (a)
Is (b) enough to
represent (a)?
What is the adequate
sampling rate to
represent a given CT
signal without information
loss?
In general, DT signal cannot represent CT signal
perfectly
Are these sample enough to reconstruct the original blue curve?
Sampling Theorem
Under certain conditions, a CT signal can be completely
represented by and recoverable from samples
A lowpass signal can be reconstructed from samples, if the
sampling rate is high enough. Because it is a lowpass signal,
the change between two close samples is constrained (or
expected).
Let x(t) be a band-limited signal with X(jw) = 0 for
|w| > wM. Then, x(t) is uniquely determined by its
samples x(nT), if
ws > 2 wM
where the sampling rate ws is defined as
ws = 2p/T
Information Equivalence
[ ] ( )x n x nT
( ) ( ) ( )p
n
x t x nT t nT
Given [ ], we can generate ( ).
Given ( ), we can generate [ ].
Therefore, [ ] and ( ) have
the same information.
p
p
p
x n x t
x t x n
x n x t
Let x(t) be a band-limited signal with X(jw) = 0 for
|w| > wM. Then, x(t) is uniquely determined by the
modulated impulse train
if
ws > 2 wM
where the sampling rate ws is defined as
ws = 2p/T
( ) ( ) ( )p
n
x t x nT t nT
Restated Sampling Theorem
2 wM: Nyquist rate
Frequency Domain Interpretation
Show that
( ) ( ) ( )
1( ) ( ( ))
p
n
p s
k
x t x nT t nT
X jw X j w kwT
Reconstruction: ideal lowpass filter
( ) 2
0 otherwise
sin sin2( ) sinc( )
s
s
wT w
H jw
wt t
tTh t Tt T
tT
p
pp
2
sw
2
sw
T
swsw
Reconstruction of Signal from Its Samples
1( ) ( ( ))p s
k
X jw X j w kwT
( )X jw
w w
( ) ( )* ( )
sinc( / )* ( ) ( )
( )sinc
r p
n
n
x t h t x t
t T x nT t nT
t nTx nT
T
Reconstruction of Signal from Its Samples
This is the way to reconstruct
or interpolate xr(t) from
samples x(nT)’s
Note that xr(t) = x(t), if the
sampling rate is higher than
the Nyquist rate
Practical Interpolation Filters
Ideal interpolation filters –sinc function
Infinite duration
Not implementable
Practical interpolation filtersZero order holding (repetition)
First order holding (linear)
Undersampling Causes Aliasing
Rotating disk
1 rotation/second
To avoid aliasing, it should be
motion-pictured with at least
2 frames/s.
0 1/12 2/12 3/12 4/12 5/12 6/12 7/12 8/12 9/12 10/12 11/12
Undersampling Causes Aliasing
12 frames/s
0 1/12 2/12 3/12 4/12 5/12 6/12 7/12 8/12 9/12 10/12 11/12
Undersampling Causes Aliasing
12/11 = 1.09 frames/s
0 1/12 2/12 3/12 4/12 5/12 6/12 7/12 8/12 9/12 10/12 11/12
DT Processing of CT Signals
We assume that the sampling rate is higher than Nyquist rate
So yc(t) = xc(t)
Relation between modulated impulse train and DT signal
[ ] [ ]d dx n y n
( ) ( / ) or ( ) ( ),
( ) ( / ) or ( ) ( )
j jwT
d p p d
j jwT
d p p d
X e X j T X jw X e
Y e Y j T Y jw Y e
DT Processing of CT Signals
CT system Hc(jw)
( ) ( ) ( )
or ( ) ( )
jwT
c c d
jwT
c d
Y jw X jw H e
H jw H e
This is true if xc(t) is band-limited and T satisfies the Nyquist condition