Sampling—50 Years After Shannon MICHAEL UNSER , FELLOW, IEEE

Sampling—50 Years After Shannon MICHAEL UNSER , FELLOW, IEEE
Sampling—50 Years After Shannon
MICHAEL UNSER, FELLOW, IEEE
This paper presents an account of the current state of sampling,
50 years after Shannon’s formulation of the sampling theorem. The
emphasis is on regular sampling, where the grid is uniform. This
topic has benefited from a strong research revival during the past
few years, thanks in part to the mathematical connections that were
made with wavelet theory. To introduce the reader to the modern,
Hilbert-space formulation, we reinterpret Shannon’s sampling procedure as an orthogonal projection onto the subspace of band-limited functions. We then extend the standard sampling paradigm for
a representation of functions in the more general class of “shift-invariant” functions spaces, including splines and wavelets. Practically, this allows for simpler—and possibly more realistic—interpolation models, which can be used in conjunction with a much
wider class of (anti-aliasing) prefilters that are not necessarily ideal
low-pass. We summarize and discuss the results available for the
determination of the approximation error and of the sampling rate
when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets,
multiwavelets, Papoulis generalized sampling, finite elements, and
frames. Irregular sampling and radial basis functions are briefly
mentioned.
Keywords—Band-limited functions, Hilbert spaces, interpolation, least squares approximation, projection operators, sampling,
sampling theorem, Shannon, splines, wavelets.
I. INTRODUCTION
In 1949, Shannon published the paper “Communication in
the Presence of Noise,” which set the foundation of information theory [105], [106]. This paper is a masterpiece; both in
terms of achievement and conciseness. It is undoubtedly one
of the theoretical works that has had the greatest impact on
modern electrical engineering [145]. In order to formulate his
rate/distortion theory, Shannon needed a general mechanism
for converting an analog signal into a sequence of numbers.
This led him to state the classical sampling theorem at the
very beginning of his paper in the following terms.
contains no freTheorem 1 [Shannon]: If a function
(in radians per second), it is comquencies higher than
pletely determined by giving its ordinates at a series of points
seconds apart.
spaced
Manuscript received September 17, 1999; revised January 4, 2000.
The author is with the Biomedical Imaging Group, Swiss Federal Institute
of Technology Lausanne CH-1015 Lausanne EPFL, Switzerland (e-mail:
[email protected]).
Publisher Item Identifier S 0018-9219(00)02874-7.
While Shannon must get full credit for formalizing this
result and for realizing its potential for communication
theory and for signal processing, he did not claim it as his
own. In fact, just below the theorem, he wrote: “this is a
fact which is common knowledge in the communication
art.” He was also well aware of equivalent forms of the
theorem that had appeared in the mathematical literature;
in particular, the work of Whittaker [144]. In the Russian
literature, this theorem was introduced to communication
theory by Kotel’nikov [67], [68].
The reconstruction formula that complements the sampling theorem is
(1)
may be interpreted
in which the equidistant samples of
as coefficients of some basis functions obtained by appropriate shifting and rescaling of the sinc-function: sinc
. Formula (1) is exact if
is bandlimited to
; this upper limit is the Nyquist frequency, a
term that was coined by Shannon in recognition of Nyquist’s
important contributions in communication theory [88]. In the
mathematical literature, (1) is known as the cardinal series
expansion; it is often attributed to Whittaker in 1915 [26],
[143] but has also been traced back much further [14], [58].
Shannon’s sampling theorem and its corresponding reconstruction formula are best understood in the frequency domain, as illustrated in Fig. 1. A short reminder of the key
sampling formulas is provided in Appendix A to make the
presentation self-contained.
Nowadays the sampling theorem plays a crucial role in
signal processing and communications: it tells us how to
convert an analog signal into a sequence of numbers, which
can then be processed digitally—or coded—on a computer.
While Shannon’s result is very elegant and has proven to
be extremely fruitful, there are several problems associated
with it. First, it is an idealization: real world signals or
images are never exactly bandlimited [108]. Second, there is
no such device as an ideal (anti-aliasing or reconstruction)
low-pass filter. Third, Shannon’s reconstruction formula is
rarely used in practice (especially with images) because of
the slow decay of the sinc function [91]. Instead, practitioners typically rely on much simpler techniques such as
0018–9219/00$10.00 © 2000 IEEE
PROCEEDINGS OF THE IEEE, VOL. 88, NO. 4, APRIL 2000
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Fig. 1. Frequency interpretation of the sampling theorem: (a)
Fourier transform of the analog input signal f (x), (b) the sampling
process results in a periodization of the Fourier transform, and (c)
the analog signal is reconstructed by ideal low-pass filtering; a
=T .
perfect recovery is possible provided that !
linear interpolation. Despite these apparent mismatches with
the physical world, we will show that a reconciliation is
possible and that Shannon’s sampling theory, in its modern
and extended versions, can perfectly handle such “nonideal”
situations.
Ten to 15 years ago, the subject of sampling had reached
what seemed to be a very mature state [26], [62]. The research in this area had become very mathematically oriented,
with less and less immediate relevance to signal processing
and communications. Recently, there has been strong revival
of the subject, which was motivated by the intense activity
taking place around wavelets (see [7], [35], [80], and [85]).
It soon became clear that the mathematics of wavelets were
also applicable to sampling, but with more freedom because
no multiresolution is required. This led researchers to reexamine some of the foundations of Shannon’s theory and develop more general formulations, many of which turn out to
be quite practical from the point of view of implementation.
Our goal in this paper is to give an up-to-date account of
the recent advances that have occurred in regular sampling.
Here, the term “regular” refers to the fact that the samples
are taken on a uniform grid—the situation most commonly
encountered in practice. While the paper is primarily conceived as a tutorial, it contains a fair amount of review material—mostly recent work: This should make it a useful complement to the excellent survey article of Jerri, which gives
a comprehensive overview of sampling up to the mid-1970’s
[62].
The outline of this paper is as follows. In Section II, we
will argue that the requirement of a perfect reconstruction
is an unnecessarily strong constraint. We will reinterpret the
standard sampling system, which includes an anti-aliasing
prefilter, as an orthogonal projection operator that computes
the minimum error band-limited approximation of a not-necessarily band-limited input signal. This is a crucial observation that changes our perspective: instead of insisting that
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the reconstruction be exact, we want it to be as close as possible to the original; the global system, however, remains unchanged, except that the input can now be arbitrary. (We can
obviously not force it to be bandlimited.)
In Section III, we will show that the concept extends nicely
to the whole class of spline-like (or wavelet-like) spaces generated from the integer shifts of a generating function. We
will describe several approximation algorithms, all based on
the standard three-step paradigm: prefiltering, sampling, and
postfiltering—the only difference being that the filters are
not necessarily ideal. Mathematically, these algorithms can
all be described as projectors. A direct consequence is that
they reconstruct all signals included within the reconstruction space perfectly—this is the more abstract formulation
of Shannon’s theorem.
In Section IV, we will investigate the issue of approximation error, which becomes relevant once we have given up
the goal of a perfect reconstruction. We will present recent
results in approximation theory, making them accessible to
an engineering audience. This will give us the tools to select
the appropriate sampling rate and to understand the effect of
different approximation or sampling procedures.
Last, in Section V, we will review additional extensions
and variations of sampling such as (multi)wavelets, finite elements, derivative and interlaced sampling, and frames. Irregular sampling will also be mentioned, but only briefly, because it is not the main focus of this paper. Our list of sampling topics is not exhaustive—for instance, we have completely left out the sampling of discrete sequences and of
stochastic processes—but we believe that the present paper
covers a good portion of the current state of research on
regular sampling. We apologize in advance to those authors
whose work was left out of the discussion.
II. SHANNON’S SAMPLING THEOREM REVISITED
Shannon’s sampling theory is applicable whenever the
input function is bandlimited. When this is not the case, the
standard signal-processing practice is to apply a low-pass
filter prior to sampling in order to suppress aliasing. The
, which
optimal choice is the ideal filter
suppresses aliasing completely without introducing any
distortion in the bandpass region. Its impulse response is
. The corresponding block diagram is shown
in Fig. 2. In this section, we provide a geometrical Hilbert
space interpretation of the standard sampling paradigm. For
and rescale the time
notational simplicity, we will set
dimension accordingly.
In 1941, the English mathematician Hardy, who was referring to the basis functions in Whittaker’s cardinal series
(1), wrote: “It is odd that, although these functions occur repeatedly in analysis, especially in the theory of interpolation,
it does not seem to have been remarked explicitly that they
form an orthogonal system” [55]. Orthonormality is a fundamental property of the sinc-function that has been revived
recently.
To understand the modern point of view, we have to consider the Hilbert space , which consists of all functions that
PROCEEDINGS OF THE IEEE, VOL. 88, NO. 4, APRIL 2000
where the inner product
represents the signal
—the apcontribution along the direction specified by
proximation problem is decoupled component-wise because
of the orthogonality of the basis functions. The projection
theorem (see [69]) ensures that this projection operation is
well defined and that it yields the minimum-error approximation of into .
Fig. 2. Schematic representation of the standard three-step
sampling paradigm with T = 1: 1) the analog input signal is
prefiltered with h(x) (anti-aliasing step), 2) the sampling process
yields the sampled representation c (x) =
c(k ) (x
k ),
and 3) the reconstructed output f~(x) =
c(k )'(x
k)
is obtained by analog filtering of c with '. In the traditional
approach, the pre- and postfilters are both ideal low-pass:
h(x) = '(x) = sinc(x). In the more modern schemes, the filters
can be selected more freely under the constraint that they remain
biorthogonal: '(x k ); '
~(x
l) = .
0
0
h
0
(7)
By a lucky coincidence, this inner product computation is
equivalent to first filtering the input function with the ideal
low-pass filter and sampling thereafter. More generally, we
observe that any combination of prefiltering and sampling
can be rewritten in terms of inner products
0 i
are squareintegrable in Lebesgue’s sense. The corresponding
-norm is
with
(8)
(2)
It is induced by the conventional
-inner product
(3)
We now assume that the input function that we want to
sample is in , a space that is considerably larger than the
usual subspace of band-limited functions, which we will
refer to as . This means that we will need to make an
approximation if we want to represent a non-band-limited
signal in the band-limited space . To make an analogy,
is to what (the real numbers) is to (the integers).
is apparent if we rewrite the
The countable nature of
as
normalized form of (1) with
with
(4)
’s are some coefficients, and where the ’s
where the
are the basis functions to which Hardy was referring. It is not
difficult to show that they are orthonormal
That is, the underlying analysis functions correspond to the
, the time-reversed impulse
integer shifts of
response of the prefilter (which can be arbitrary). In the
and
is the
present case,
.
ideal low-pass filtered version of
The conclusion of this section is that the traditional sampling paradigm with ideal prefiltering yields an approxima, which is the orthogonal projection of the
tion
input function onto (the space of band-limited functions).
In other words, is the approximation of in with minimum error. In light of this geometrical interpretation, it is
(since
is a projection
obvious that
operator), which is a more concise statement of Shannon’s
theorem.
III. SAMPLING IN SHIFT-INVARIANT (OR SPLINE-LIKE)
SPACES
Having reinterpreted the sampling theorem from the more
abstract perspective of Hilbert spaces and of projection operators, we can take the next logical step and generalize the
approach to other classes of functions.
A. Extending Shannon’s Model
(5)
where the autocorrelation function is evaluated as follows:
While it would be possible to consider arbitrary basis functions, we want a sampling scheme that is practical and retains
the basic, shift-invariant flavor of the classical theory. This is
by a more general temachieved by simply replacing
. Accordingly, we specify
plate: the generating function
our basic approximation space as
(9)
This orthonormality property greatly simplifies the implementation of the approximation process by which a function
is projected onto . Specifically, the orthogonal procan be written as
jection operator
(6)
UNSER: SAMPLING—50 YEARS AFTER SHANNON
, which is conThis means that any function
tinuously defined, is characterized by a sequence of coeffi; this is the discrete signal representation that will
cients
be used to do signal processing calculations or to perform
’s are not necessarily the samples
coding. Note that the
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of the signal, and that
can be something quite different
. Indeed, one of our motivations is to discover
from
functions that are simpler to handle numerically and have a
much faster decay.
For such a continuous/discrete model to make sense, we
need to put a few mathematical safeguards. First, the se.
quence of coefficients must be square-summable:
Second, the representation should be stable1 and unambiguously defined. In other words, the family of functions
should form a Riesz basis of
, which
is the next best thing after an orthogonal one [35]. The definition of a Riesz basis is that there must exist two strictly
and
such that
positive constants
(10)
is the squared -norm (or enwhere
. A direct consequence of the lower inequality
ergy) of
implies
. Thus, the basis
is that
functions are linearly independent, which also means that
is uniquely specified by its coevery signal
. The upper bound in (10) implies that the
efficients
-norm of the signal is finite, so that
is a valid subspace of . Note that the basis is orthonormal if and only if
, in which case we have a perfect norm equivalence between the continuous and the discrete domains (Parseval’s relation). Because of the translation-invariant structure of the construction, the Riesz basis requirement has an
equivalent expression in the Fourier domain [9]
. It is easy to verify that the corresponding
, which is consistent
Riesz bounds in (11) are
with the orthonormality property (5). We now show that the
sinc function satisfies the partition of unity: using Poisson’s
summation formula (cf. Appendix A), we derive an equivalent formulation of (13) in the Fourier domain2
(14)
, the
a relation that is obviously satisfied by
.
Fourier transform of
The sinc function is well localized in the frequency domain
but has very poor time decay. At the other extreme, we can
look for the simplest and shortest function that satisfies (13).
It is the box function (or B-spline of degree 0)
.
(15)
The corresponding basis functions are clearly orthogonal for
they do not overlap.
By convolving this function with itself repeatedly, we construct the B-splines of degree , which are defined recursively as
(16)
In practice, it is this last condition that puts the strongest
constraint of the selection on an admissible generating func.
tion
Let us now look at some examples. The first one, which has
already been discussed in great length, is the classical choice
These functions are known to generate the polynomial
splines with equally spaced knots [98], [99]. Specifically,
, then the signals defined by (9) are
if
(
polynomials of degree within each interval
odd), respectively,
(1/2)
(1/2) when is even,
with pieces that are patched together such as to guarantee
the continuity of the function and its derivatives up to order
(i.e.,
). The B-spline basis functions up
to degree 4 are shown in Fig. 3. They are symmetric and
. Yet,
well localized, but not orthogonal—except for
they all satisfy the Riesz basis condition and the partition
of unity. This last property is easily verified in the Fourier
domain using (14). The B-splines are frequently used in
practice (especially for image processing) because of their
short support and excellent approximation properties [126].
The B-spline of degree 1 is the tent function, and the
corresponding signal model is piecewise linear. This representation is quite relevant because linear interpolation is
one of the most commonly used algorithm for interpolating
signal values.
As additional examples of admissible generating functions, we may consider any scaling function (to be defined
below) that appears in the theory of the wavelet transform
[35], [79], [115], [139]. It is important to note, however,
that scaling functions, which are often also denoted by
, satisfy an additional two-scale relation, which is
not required here but not detrimental either. Specifically,
a scaling function is valid (in the sense defined by Mallat
1By stable, we mean that a small variation of the coefficients must result
in a small variation of the function. Here, the upper bound of the Riesz condition (10) ensures L -stability.
2The equality on the right hand side of (14) holds in the distributional
sense provided that '(x) 2 L , or by extension, when '(x) + '(x + 1) 2
L , which happens to be the case for '(x) = sinc(x)
(11)
is the Fourier transform of
where
. Note that the central term in (11) is the Fourier transform of the sampled autocorrelation function
(12)
It can therefore also be written
[see (A.7) in Appendix A].
The final requirement is that the model should have the
capability of approximating any input function as closely as
desired by selecting a sampling step that is sufficiently small
). As shown
(similar to the Nyquist criterion
in Appendix B, this is equivalent to the partition of unity
condition
(13)
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PROCEEDINGS OF THE IEEE, VOL. 88, NO. 4, APRIL 2000
where
is the autocorrelation sequence in (12). By
,
imposing the biorthogonality constraint
and by solving this equation in the Fourier domain [i.e.,
], we find
that
Fig. 3. The centered B-splines for n = 0 to 4. The B-splines of
degree n are supported in the interval [ ((n +1)=2); ((n +1)=2)];
as n increases, they flatten out and get more and more Gaussian-like.
0
[81]) if and only if 1) it is an admissible generating function
(Riesz basis condition partition of unity) and 2) it satisfies
the two-scale relation
(17)
is the so-called refinement filter. In other words,
where
, a propthe dilated version of must live in the space
erty that is much more constraining than the conditions imposed here.
(20)
Note that the Riesz basis condition (11) guarantees that this
solution is always well defined [i.e., the numerator on the
right-hand side of (20) is bounded and nonvanishing].
Similar to what has been said for the band-limited case [see
(4)], the algorithm described by (18) has a straightforward
signal-processing interpretation (see Fig. 2). The procedure
is exactly the same as the one dictated by Shannon’s theory
(with anti-aliasing filter), except that the filters are not necessarily ideal anymore. Note that the optimal analysis filter
is entirely specified by the choice of the generating function (reconstruction filter); its frequency response is given
by (20). If is orthonormal, then it is its own analysis func) and the prefilter is simply a flipped
tion (i.e.,
version of the reconstruction filter. For example, this implies
that the optimal prefilter for a piecewise constant signal approximation is a box function.
C. Consistent Sampling
B. Minimum Error Sampling
Having defined our signal space, the next natural question
’s in (9) such that the signal model
is how to obtain the
is a faithful approximation of some input function
. The optimal solution in the least squares sense is the
orthogonal projection, which can be specified as
We have just seen how to design an optimal sampling
system. In practice however, the analog prefilter is often
specified a priori (acquisition device), and not necessarily
optimal or ideal. We will assume that the measurements of
are obtained by sampling its prefiltered
a function
, which is equivalent to computing the series
version,
of inner products
(18)
.
where the ’s are the dual basis functions of
This is very similar to (6), except that the analysis and synthesis functions ( and , respectively) are not identical—in
general, the approximation problem is not decoupled. The
with
is unique and is deterdual basis
mined by the biorthogonality condition
(19)
It also inherits the translation-invariant structure of the basis
.
functions:
, it is a linear conbination of the ’s.
Since
Thus, we may represent it as
where
is a suitable sequence to be determined next. Let
us therefore evaluate the inner product
(21)
[see (8)]. Here, it is
with analysis function
such that the meaimportant to specify the input space
. In the
surements are square-summable:
), we will be able to
most usual cases (typically,
; otherwise ( is a Delta Dirac or a difconsider
ferential operator), we may simply switch to a slightly more
constrained Sobolev space.3 Now, given the measurements
in (21), we want to construct a meaningful approximation of
. The solution is
the form (9) with synthesis function
to apply a suitable digital correction filter , as shown in the
block diagram in Fig. 4.
Here, we consider a design based on the idea of consistency [127]. Specifically, one seeks a signal approximation
that is such that it would yield exactly the same measurements if it was reinjected into the system. This is a reasonable requirement, especially when we have no other way of
probing the input signal: if it “looks” the same, we may as
well say that it is the same for all practical purposes.
Sobolev space W specifies the class of functions that
times differentiable in the L -sense. Specifically, W
=
f : (1 + ! ) f^(!) d! < + where f^(!) is the Fourier transform
of f (x).
3The
are
f
UNSER: SAMPLING—50 YEARS AFTER SHANNON
r
j
j
1g
573
Fig. 4. Sampling for nonideal acquisition devices. The block
diagram is essentially the same as in Fig. 2, except for the addition
of the digital correction filter q .
Fig. 5. Principle of an oblique projection onto V perpendicular to
V in the simplified case of one-component signal spaces.
Before stating the main theorem, we define the cross-correlation sequence
where
is the spectral coherence function defined by
(22)
is the analysis function and where
is the generwhere
ating (or synthesis) function on the reconstruction side.
be an unknown input funcTheorem 2 [127]: Let
such that
tion. Provided there exists
, then there is a unique signal approximation in
that is consistent with in the sense that
(23)
This signal approximation is given by
(24)
where
(25)
into
and the underlying operator is a projector from
.
satisfies the Riesz
If, in addition, the analysis function
and the approximation
basis condition (11), then
operator has an interesting geometrical interpretation: it is
perpendicular to
[127]. A
the projection onto
simplified graphical illustration of the situation is given in
Fig. 5.
It is interesting to compare the performance of this oblique
projection algorithm to the least squares solution, which is
typically not realizable here. Specifically, we can derive the
following optimal error bound (see [127, Theorem 3]):
(26)
denotes the orthogonal projection of into
where
and where
is an abstract quantity that can be assimilated to the cosine of the maximum angle between the suband
. This latter quantity is given by (see
spaces
[127])
(27)
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(28)
Equation (26) indicates that the behavior of both methods
is qualitatively the same. The upper bound, which can be
quantified, corresponds to the worst possible scenario. The
present algorithm provides the least squares solution when
. In the case where
, the analysis
and synthesis spaces are identical and the oblique projection
is equivalent to the orthogonal one. Interestingly, the factor
also corresponds to the norm of the operator
1
(see [136, Theorem 4]); it is therefore also a good indicator
of the performance of the algorithm in the presence of noise.
Theorem 2 provides a generalization of Shannon’s result
for nonideal acquisition devices. The projection property implies that the method can reconstruct perfectly any signal that
,
is included in the reconstruction space; i.e.,
. This is true not only for band-limited functions but
also for functions living in the more general “shift-invariant”
.
subspaces
The stability of the reconstruction process will obviously
depend on the invertibility of the cross-correlation sequence
. Note that, in the special case
, one recovers
the classical inverse filtering solution to the deconvolution
problem. Approximate, regularized solutions are also conceivable to improve the robustness in the presence of noise
[124].
An attractive feature of the above formulation is the symmetric role assumed by the analysis and synthesis functions
and . It is therefore possible to model the distortions
that happen at either end of the chain. This allows for the
design of a digital filter that compensates for the nonideal
response ( ) of some specific ouput device; for example,
the sample-and-hold of a digital-to-analog converter or the
pixel shape of an image display.
D. Interpolation Revisited
The most basic form of sampling occurs when the signal
. To make sure
is specified in terms of its sample values
that these samples are in (see [19, Appendix III.A]), we
(Sobolev space of order
select the input space
one), which simply adds the constraint that the derivative
PROCEEDINGS OF THE IEEE, VOL. 88, NO. 4, APRIL 2000
of be in
as well. The question is now how to find the
intercoefficients in (9) such that the signal
polates the specified samples values exactly. We now show
that the solution can be derived directly from Theorem 2. To
model the sampling process, we take the analysis function to
(Dirac delta). In this way, we are able to
be
reformulate the interpolation condition as a consistency re.
quirement:
If we now substitute this choice together with
in Theorem 2, we get the correction filter
(29)
which provides an efficient digital filtering solution to our
problem. Note that the filter is the identity if and only if the
generating function has the interpolation property:
; this is for example the case for the sinc function and the
B-splines of degree 0 and 1. For higher order splines a nontrivial filter is necessary; this kind of inverse filter can be
implemented very efficiently using a simple combination of
causal and anticausal recursive filters [131], [132].
that results from this
The interpolating signal
process is
This solution can also be presented in a form closer to the
traditional one [see (1)]
(30)
where
is the interpolation function given by
(31)
is well defined, the interpolating function
Assuming that
is uniquely defined.
has the
within the space
remarkable property that it is one at the origin and zero at all
other integers. The interpolator for the space of cubic splines
is shown in Fig. 6. It is rather similar to the sinc function,
which is shown by the dotted line; for more details, refer to
[10].
A slight modification of the scheme allows for the representation of signals in terms of samples shifted by some fixed
amount [60]. Walter has developed a similar representation in the more constrained setting of the wavelet transform
[141].
E. Equivalent Basis Functions
So far, we have encountered three types of basis functions:
the generic ones ( ), the duals ( ), and the interpolating ones
). In fact, it is possible to construct many others using
(
equivalent generating functions of the form
(32)
UNSER: SAMPLING—50 YEARS AFTER SHANNON
Fig. 6. Comparison of the cubic spline (solid line) and sinc (dashed
line) interpolators.
where
is an appropriate sequence of weights. The
necessary and sufficient conditions for
to yield an equivalent Riesz basis of
are that there
and
such that
exist two strictly positive constants
, where
is the Fourier trans[9]. The most important types of basis functions
form of
are summarized in Table 1. Note that the orthogonalized
plays a special role in the theory of the
version
wavelet transform [81]; it is commonly represented by the
symbol .
Interestingly, it has been shown that the interpolator
and the orthonormal function
associated with many
(splines, Dubuc–Deslaufamilies of function spaces
rier [42], etc.) both converge to Shannon’s sinc interpolator
as the order (to be defined in Section IV-B) tends to infinity
[9], [10], [130]. The sinc function is rather special in the
sense that it is both interpolating and orthonormal, which also
means that it is its own dual.
A few remarks are in order concerning the usefulness of
the various representations; these are applicable not only for
single resolution schemes but also for wavelet-like representations, which offer the same kind of freedom [8]. If one is
only concerned with the signal values at the integers, then
the so-called cardinal representation is the most adequate;
here, the knowledge of the underlying basis function is only
necessary when one wishes to discretize some continuous
signal-processing operator (an example being the derivative)
[122]. If, on the other hand, one wishes to perform computation involving values in between samples, one has advantage
of using an expansion formula as in (9) where the basis functions have minimal support (e.g., B-splines). This is especially true in higher dimensions where the cost of prefiltering
is negligible in comparison to the computation of the expansion formula. More details on the computational issues can
be found in [120]. From the point of view of computation,
the Shannon model has a serious handicap because there are
no band-limited functions that are compactly supported—a
consequence of the Paley–Wiener theorem [117]. The use of
windowed or truncated sinc functions is not recommended
because these fail to satisfy the partition of unity; this has
the disturbing consequence that the reconstruction error will
575
where
is specified as in (25). By construction, this func(i.e.,
tion is biorthogonal to
). The oblique projection operator (24) may therefore be
written in a form similar to (18)
Table 1
Primary Types of Equivalent
Generating Functions with Their Specific Properties
Formally, this is equivalent to
where we have now identified the projection kernel
(37)
not vanish as the sampling step tends to zero (cf., Appendix
B).
Another interesting aspect is the time-frequency localization of the basis functions. B-splines are close to optimal
because they converge to Gaussians as the order increases;
(cubic splines), their time-frequency bandwidth
for
product (TFBP) is already within 1% of the limit specified
by the uncertainty principle [129]. Furthermore, by using this
type of basis function in a wavelet decomposition (see Section V-A), it is possible to trade one type of resolution for the
other [31], [129]. This is simply because the TFBP remains
a constant irrespective of the scale of the basis function.
Because we are dealing with a projection operator, we can
write the identity
(38)
which is the same as (34), except that the projection kernel
may be different from
. Note that it is the inclusion constraint (33) that specifies the reprodcuing kernel
in a unique fashion. The requirement
is equivalent to the condition
or
,
. Thus, the reprowhich implies that
corresponds to the orthogonal projecducing kernel
tion.
F. Sampling and Reproducing Kernel Hilbert Spaces
G. Extension to Higher Dimensions
We now establish the connection between what has been
presented so far and Yao and Nashed’s general formulation of
sampling using reproducing kernel Hilbert spaces (RKHS’s)
[87], [148].
Definition: A closed vector space is an RKHS with
if and only if
reproducing kernel
All results that have been presented carry over directly for
the representation of multidimensional signals (or images)
, provided that the sampling is performed on
. This generalization holds bethe cartesian grid
cause our basic mathematical tool, Poisson’s summation for, remains valid in
mula
dimensions. Thus, we can extend all formulas presented so
far by considering the multidimensional time (or space) and
and
frequency variables
, and by replacing simple summations
and integrals by multiple ones.
In practice, one often uses separable generating functions
of the form
(33)
and
(34)
is
The reproducing kernel for a given Hilbert space
. For the
unique; is is also symmetric
specified by (9), it is given by
shift-invariant space
(35)
is the dual of
, as defined in Section III-B.
where
The concept can be generalized slightly by introducing the
notion of a projection kernel. We now show how these can be
and
constructed with the help of Theorem 2. We set
such that the invertibility condition in Theorem 2
select a
is satisfied. We can define the equivalent analysis function
(36)
576
This greatly simplifies the implementation because all filtering operations are separable. Another advantage of separability is that the one-dimensional Riesz basis condition is
equivalent to the multidimensional one. The dual functions
remain separable as well.
IV. CONTROLLING THE APPROXIMATION ERROR
The projection interpretation of the sampling process that
has just been presented has one big advantage: it does not require the band-limited hypothesis and is applicable for any
. Of course, perfect reconstruction is generally
function
. It is therefore crucial to
not possible when
PROCEEDINGS OF THE IEEE, VOL. 88, NO. 4, APRIL 2000
get a good handle on the approximation error. In the classical scheme with ideal anti-aliasing filtering, the error is entirely due to the out-of-band portion of the signal; its magnitude can be estimated simply by integrating the portion of the
spectrum above the Nyquist frequency [22], [142]. For more
, the situation is more comgeneral spline-like spaces
plicated. One possibility is to turn to approximation theory
and to make use of the general error bounds that have been
derived for similar problems, especially in connection with
the finite element method [38]–[41], [73], [113]. Specialized
error bounds have also been worked out for quasi-interpolation, which is an approximate form of interpolation without
any prefilter [40], [74], [109], [110]. Unfortunately, these
results are mostly qualitative and not suitable for a precise
determination of the approximation error. This has led researchers in signal processing, who wanted a simple way to
determine the critical sampling step, to develop an accurate
error estimation technique which is entirely Fourier-based
[20], [21]. This recent method is easy to apply in practice and
yet powerful enough to recover most classical error bounds;
it will be described next.
The key parameter for controlling the approximation error
is the sampling step . We therefore consider the rescaled
signal space
(39)
where the basis functions are dilated by a factor of
and shifted by the same amount (sampling step). For a
, the interesting question is then
given input signal
’s in
to determine the approximation error when the
(39) are obtained using one of the above algorithms (in
Section III-B–D). In this way, we will be able to select a
such that the approximation
critical sampling step
error is below some acceptable threshold. The premise is
that the error should decrease and eventually vanish as the
sampling step gets smaller; as mentioned before, this is
possible only if satisfies the partition of unity condition.
A. Calculating the Error in the Frequency Domain
denote a linear approximation operator with samLet
pling step . The most general mathematical description of
is
a -shift-invariant4 approximation operator in
Our goal is now to determine the dependence of the apon the samproximation error
pling step . Since the initial starting point of the signal with
respect to the origin is arbitrary, we may as well consider an
averaged version of the error over all shifts of the input signal
, where is the displacement with respect to the sampling grid. A remarkable fact is that this error measure can
be computed exactly by simple integration in the frequency
domain (see [19, Theorem 2])
(41)
Here,
mate and
is the Fourier transform of the signal to approxiis the error kernel given by
(42)
and
are the Fourier transform of
and
where
, respectively. This result allows us to predict the general
error behavior of an algorithm by simple examination of the
.
function
In the least squares case (see Section III-B), the error
kernel reduces to
(43)
which is consistent with the fact that the orthogonal projection minimizes the approximation error in
. For the standard Shannon paradigm, which uses
ideal analysis and reconstruction filters, we find that
; this confirms the fact that
the approximation error is entirely due to the out-of-band
portion of the signal.
is an average measure of the error,
Even though
it turns out to be an excellent predictor of the true error
. This is a direct consequence of
the following approximation theorem.
Theorem 3 [19]: The -approximation error of the opdefined by can be written as
erator
(44)
(40)
where is a suitable analysis function (or distribution). This
corresponds to the general sampling system described in
Fig. 2 and includes all the algorithms described so far.
4Our
8 2
f
0
-shift-invariance is that f
L ; Q
f (x
. In other words, Q commutes with the shift
operator by integer multiples of T .
definition of
g=
kT )
Q
T
f g( 0
f
x
kT )
UNSER: SAMPLING—50 YEARS AFTER SHANNON
is a correction term negligible under most cirwhere
(Sobolev space of order
cumstances. Specifically, if
) with
, then
where is some
, provided that is banknown constant. Moreover,
(Nyquist frequency).
dlimited to
accounts for the dominant part
Thus, the estimate
, while
is merely a perturbaof the true error
tion. This latter correction, which may be positive or negative, is guaranteed to vanish provided that is bandlimited
577
or at least sufficiently smooth to have
derivatives
). In the latter case, the error
in the -sense (i.e.,
can be made arbitrarily small by selecting a sampling step
sufficiently small with respect to the smoothess scale of as
, the norm of its th derivative.
measured by
Thanks to (41) and Theorem 3, the approximation problem
has thus been recast in a framework that is reminiscent of
filter design and that should be familiar to a signal-processing
audience.
B. Approximation Order
In light of Theorem 3, the minimum requirement for the
is
, a condition that
error to vanish as
implies the partition of unity (see Appendix B). More generally, we can predict the rate of decay of the approxima(or from
tion error from the degree of flatness of
its Taylor series) near the origin. Specifically, if
as
(because of symmetry all
odd powers of are zero), then a simple asymptotic argument on (44) yields
as
(45)
so that
is fiwhere we are assuming that
nite. Under the same hypotheses, one can also derive upper
bounds of the error having the same form as the right-hand
side of (45) but with a larger leading constant, and which are
valid for all values of [21], [38], [135]. This implies that
. This rate of decay is
the error decays globally like
called the order of approximation; it plays a crucial role in
wavelet and approximation theory [73], [111], [112], [115].
Through the Strang–Fix conditions [113], this order property
is equivalent to the reproduction of polynomials of degree
. The order parameter determines the approximation power of a given function . In wavelet theory, it
corresponds to the number of vanishing moments of the analysis wavelet [35], [79], [115]; it also implies that the transfer
function of the refinement filter in (17) can be factorized as
. As an example, the B-splines of de;
gree [see (16)] have an order of approximation
they are also the shortest and smoothest scaling functions of
order [125].
Relations (41) and (45) provide alternatives to the Nyquist
frequency criterion for selecting the appropriate sampling
step . The error will not be zero in general, but it can be
.
made arbitrarily small without any restriction on
C. Comparison of Approximation Algorithms
The kernel in (42), or the asymptotic relation (45), can be
the basis for the comparison (or the design) of approximation/sampling procedures. It is especially interesting to predict the loss of performance when an approximation algorithm such as (30) and (24) is used instead of the optimal
least squares procedure (18). As an example, we concentrate
on the case of linear splines. Fig. 7 provides a comparison
of the error kernel for three standard algorithms: a) sampling
578
Fig. 7. Error kernels for three linear spline sampling methods: 1)
interpolation, 2) oblique projection, and 3) orthogonal projection.
without prefiltering, b) sampling with a suboptimal prefilter
(simple box function), and c) least squares sampling. The
first case corresponds to the standard piecewise linear in. The second alterpolation
gorithm uses the simplest possible analog prefilter—the box
function, or B-spline of degree 0—combined with the digital
filtering correction described by Theorem 2; its geometrical
interpretation is a projection onto the space of linear splines
perpendicular to the space of splines of degree 0. The third
optimal algorithm uses the optimal prefilter specified by (20).
These error kernels all follow a standard pattern: They
are close to zero within the Nyquist band and more or less
constant outside. As expected, the best algorithm is the
third one; in terms of performance, it is the closest to the
Shannon paradigm which uses (nonrealizable) ideal filters.
The oblique projection is only slightly suboptimal. Sampling
without analog prefiltering (interpolation) is by far the least
favorable approach. In particular, this method suffers from
aliasing; this explains why the signal components above the
Nyquist frequency contribute twice to the approximation
error: a first time because they cannot be reproduced in
, and a second time because of the spectral folding
induced by sampling (aliasing).
This comparison clearly emphasizes the importance of
prefiltering for the suppression of aliasing. Interestingly,
the prefiltering does not need to be optimal—a simple box
function, as in algorithm b), may do. In this particular case,
[136]. This
the bound constant in (26) is
is another indication that this oblique projector is essentially
equivalent to the least squares solution; both spline algorithms have been used successfully for image resizing with
arbitrary scale factors [72], [133]. The oblique solution is
generally simpler and faster to implement.
It is also interesting to look at the asymptotic performance
of these algorithms. Their order of approximation is
because they all reproduce linear polynomials [123], [135].
PROCEEDINGS OF THE IEEE, VOL. 88, NO. 4, APRIL 2000
The leading constant in (45) for linear spline interpolation
; this is more than a factor of two
is
above the projection algorithms b) and c), which both achieve
. More
the smallest possible constant
generally, it has been shown that the performances of th
order orthogonal and oblique projectors are asymptotically
equivalent, provided that the analysis and synthesis functions
are biorthogonal and that satisfies the partition of unity
[123]. Under those hypotheses, the leading constant in (45)
is given by
(46)
denotes the th derivative of the Fourier
where
transform of . A simpler and more direct formula in terms
of the refinement filter is available for wavelets [20].
D. Comparison of Approximation Spaces
The other interesting issue that we may address using the
above approximation results is the comparison of approximation subspaces. Indeed, it is desirable to have some quantitative criteria for selecting a generating function . In particular, we would like to identify functions that have good approximation properties and that are reasonably short to allow
rapid computation. To factor out the effect of the algorithm,
we will base the comparison on the least squares procedure
characterized by the kernel (43).
As a first example, we have plotted the error kernels for
, to in Fig. 8. This graph clearly
splines of degrees
shows that, for signals that are predominantly low-pass (i.e.,
with a frequency content within the Nyquist band), the error
tends to be smaller for higher order splines. Of course, the
price to pay for better performance is the larger support of
the basis functions, which may also induce Gibbs oscillations [63]. As increases, the spline approximation converges to Shannon’s band-limited solution [130]. Since the
convergence happens quite rapidly, there is usually not much
benefit beyond quintic splines.
Based on the knowledge of these kernels, it is easy, using
(41), to predict the behavior of the error as a function of for
. The corresponding log–log
some given input function
plot for the approximation of the Gaussian test function is
given in Fig. 9. We observe that a polynomial spline approx1
imation of degree provides an asymptotic decay of
20 dB per decade, which is consistent with (45).
The next graph in Fig. 10 provides a performance comparison of four piecewise cubic polynomial kernels of
: a) Keys’ interpolation function [66],
same support
which is frequently used for image interpolation, b) a cubic
Lagrange-like interpolation [97], c) the cubic B-spline [59],
[128], and d) the optimal O-Moms function recently derived
in [18]. The last three generating functions are of order
and are members of the so-called Moms (maximum
order minimum support) family; the cubic spline is the
smoothest member while the O-Moms is the one with the
smallest asymptotic constant (46), which explains its better
performance. The least favorable case is Keys’ interpolating
UNSER: SAMPLING—50 YEARS AFTER SHANNON
Fig. 8. Frequency plot of the error kernels for the least-squares
spline approximations of degree n = 0; 1; 2; 3. Below the Nyquist
frequency ! = , the kernels (and therefore the errors) tend to be
smaller for splines of higher degree. The dotted line corresponds to
the Shannon solution with ideal low-pass prefilter.
Fig. 9. Approximation error as a function of the sampling step T
for the least squares approximation of the function f (x) = e
with splines of degree n = 0 to 3.
function [with optimal parameter
], which happens to have one less order of approximation. This example
demonstrates that enforcing the interpolation constraint
[cases a) and b)] is detrimental to overall performance.
Comparisons of this nature are quite relevant because all
four kernels have the same computational complexity. It thus
appears to be more advantageous to use noninterpolating
functions such as the B-splines (if derivatives are required),
or the O-Moms; this is especially true for medical imaging
applications where quality is a key concern [89], [120].
It is only recently that researchers have realized there
may be a lot to gain from relaxing the usual interpolation
constraint. Keys’ short cubic convolution is still considered
579
The wavelet function
, which may be represented as
, is designed to generate a
; these
Riesz basis of the difference spaces
are also rescaled versions of each other but their pairwise
—in contrast with the ’s, which
intersections are all
are included hierarchically. Since the multiresolution is
(i.e., we can approximate any
-function as
dense in
go to zero),
closely as we wish by letting the scale
in
it is thus possible to represent any function
terms of the wavelet expansion
(47)
Fig. 10. Frequency plot of the error kernels for four piecewise
cubic generating functions of equal support: 1) Keys’, 2)
Langrange-like interpolator, 3) cubic B-spline, and 4) cubic
O-Moms.
the state-of-the art interpolation method in image processing
[91], [94], but the situation is likely to change. There is
still room for optimization and design work in order to find
the basis functions that give the best quality for a given
computational budget (or support).
where and are the position and scale indexes, respectively.
This is quite similar to the sampling expansion (18), except
that (47) includes an additional summation over the scale
index . The wavelet expansion works because the analysis
and synthesis wavelets and generate a biorthogonal basis
such that
[32].
of
While a detailed discussion of wavelets is beyond the
scope of this paper, we want to point out that the analysis
tools and mathematics are essentially the same as those used
in the modern formulations of sampling theory. In this sense,
wavelets have had a very positive feedback on sampling
and have contributed to a revival of the subject. The reader
who wishes to learn more about wavelets is referred to the
standard texts [35], [79], [115], [139].
B. Generalized (or Multichannel) Sampling
V. VARIATIONS AND EXTENSIONS OF SAMPLING THEORY
In this section, we briefly mention some related topics
(such as wavelets), which can be thought of as variations and
extensions of sampling theory. Our intention here is not to
be exhaustive but rather to bring to the attention of the reader
some interesting variations of Shannon’s theory, while providing pointers to the appropriate literature. We have also
tried to give an up-to-date account of the current research in
the field, which is greatly influenced by wavelets. What is essential to a subject like sampling is the communication taking
place between engineering and mathematical communities.
One of the places where this has been happening recently is
the International Workshop on Sampling Theory and Applications, held biannually since 1995 [1], [2].
For a more classical perspective on sampling, we refer to
Jerri’s excellent tutorial article, which gives a very comprehensive view of the subject up to the mid 1970’s [62]. Another useful source of information are the survey articles that
appeared in the mathematical literature [26], [27], [57].
A. Wavelets
In Section II, we have already encountered the scaling
function , which plays a crucial role in wavelet
,
theory. There, instead of a single space
one considers a whole ladder of rescaled subspaces
using the standard
. If
satisfies the
notation
two-scale relation (17), then these spaces are nested and
[80], [81], [85].
form a multiresolution analysis of
580
In 1977, Papoulis introduced a powerful extension of
Shannon’s sampling theory, showing that a band-limited
signal could be reconstructed exactly from the samples of
linear shift-invariant systems sampled
the response of
the reconstruction rate [90]. The main point is
at 1
that there are many different ways of extracting information from a signal—a reconstruction is generally possible
provided there are as many measurements as there are
degrees of freedom in the signal representation. If the
measurements are performed in a structured manner, then
the reconstruction process is simplified: for example, in the
Papoulis framework, it is achieved by multivariate filtering
[23], [83]. Typical instances of generalized sampling are
interlaced and derivative sampling [75], [149], both of
which are special cases of Papoulis’ formulation. While the
generalized sampling concept is relatively straightforward,
the reconstruction is not always feasible because of potential
instabilities [24], [30], [82].
More recently, Papoulis’ theory has been generalized
in several directions. While still remaining with band-limited functions, it was extended for multidimensional, as
well as multicomponent signals [25], [101]. Djokovic and
Vaidyanathan applied similar ideas for the reconstruction of
functions in certain wavelet spaces [44]; they concentrated
on the special cases of interlaced sampling, sampling of a
function and of its derivative, and reconstruction from local
averages. A further step was taken by Unser and Zerubia
who generalized the approach for a reconstruction in the
without any constraining hypothesis on the
space
PROCEEDINGS OF THE IEEE, VOL. 88, NO. 4, APRIL 2000
input signal [137]. Instead of an exact reconstruction, which
, they looked
is obviously not possible as soon as
for a solution that is consistent in the sense that it yields the
exact same measurements if it is reinjected into the system.
Their key result is a multivariate extension of the sampling
theorem described in Section III-C. The computational
solution takes the form of a multivariate filterbank and
is compatible with Papoulis’ theory in the special case
. These authors also looked at performance issues
and derived general formulas for the condition number of
the system, as well as error bounds for the comparison with
the least squares solution [136]. Janssen and Kalker carried
out an explicit stability analysis for the particular case of
interlaced sampling with a reconstruction in the space of
piecewise linear splines [61].
Recent applications of generalized sampling include
motion-compensated deinterlacing of televison images [11],
[121], and super-resolution [107], [138]. The latter is an
attempt to reconstruct a higher resolution image from a
series of lower resolution ones, which are shifted slightly
with respect to each other.
C. Finite Elements and Multiwavelets
An interesting generalization of (9) is to consider
generating functions
instead of a single
one; this corresponds to the finite element—or multiwavelet—framework. To obtain a signal representation that
has the same sampling density as before, the multifunctions
are translated by integer multiples of
(48)
In the finite-elements method, the basis functions are typically chosen as short as possible and with minimal overlap,
to facilitate the inversion of the matrices involved in the numerical solution of differential equations [114], [118]. In this
area of application, the approximation order of the representation is the key parameter and the ’s do not need to be
particularly smooth [113].
In the more recent multiwavelet constructions, the multiscaling functions satisfy a vector two-scale relation—similar
to (17)—that involves a matrix refinement filter instead of a
scalar one [4], [45], [51], [56], [116]. One of the primary motivation for this kind of extension is to enable the construction
of scaling functions and wavelets that are symmetric (or antisymmetric), orthonormal, and compactly supported. These
are properties that cannot be enforced simultaneously in the
conventional wavelet framework, with the notable exception
of the Haar basis [33].
For multiwavelet applications, one of the key issues is
’s in (48) for
the appropriate determination of the
the initial representation of the signal at the finest scale
available—the so-called initialization problem [140], [146].
The situation is very much the same as in the scalar case,
when the generating function is noninterpolating (see
Section III-D). Given the equidistant samples (or mea, the expansion coefficients are
surements) of a signal
UNSER: SAMPLING—50 YEARS AFTER SHANNON
usually obtained through an appropriate digital prefiltering
procedure (analysis filterbank) [54], [140], [146]. The
initialization step—or prefiltering—can be avoided for
the class of so-called balanced multiwavelets [71], [103].
Recently, Selesnick has designed multiscaling functions that
are interpolating, orthonormal, and compactly supported
[104]; these are the vector counterparts of the interpolating
function
in Section III-D. A simultaneous fulfillment of
all these properties is not possible in the scalar case, except
for the Haar scaling function, as proven by Xia [147].
Because of the importance of the finite elements in
engineering, the quality of this type of approximation has
been studied thoroughly by approximation theorists [64],
[73], [111]. In addition, most of the results presented in
Section IV are also available for the multifunction case [19].
D. Frames
The notion of frame, which generalizes that of a basis, was
introduced by Duffin and Schaffer [47]; it plays a crucial
role in nonuniform sampling [12]. A frame is essentially a
set of functions that span the whole space, but are not necessarily linearly independent [3]. To be more precise, a sewith
constitutes a frame of the
quence
function space if there exist two strictly positive constants
and (frame bounds) such that
(49)
(except zero)
This implies that there is no function
that is orthogonal to all frame components simultaneously,
which ensures a complete coverage of the space . The main
difference with the Riesz basis condition (10) is that the
frame definition allows for redundancy: there may be more
template functions than are strictly necessary. In terms of
sampling, this translates into situations where one has more
measurements (or samples) than the minimum requirement.
This is especially advantageous for reducing noise [119].
Practically, the signal reconstruction is obtained from the
solution of an overdetermined system of linear equations
[92]; see also [34], for an iterative algorithm when is close
to .
When dealing with frames, the important fact to realize
is that the signal representation in terms of the ’s is generally not unique. However, there is one representation (the
minimum norm solution) that is especially attractive because
it yields the same type of expansion as a Riesz basis:
where
is the so-called dual
is the inverse of the frame operator , defined
frame—
[3], [47]. In particular, when the
as
—the operator
is a simple
frame is tight—i.e.,
renormalization, and one gets
(50)
which is almost the same formula as (9), except for the normalization by . When all vectors have a unit norm, then the
factor also gives the redundancy of the frame [34].
581
E. Irregular Sampling
Irregular or nonuniform sampling constitutes another
whole area of research that we mention here only briefly to
make the connection with what has been presented. There
are essentially two strategies for finding a solution: 1) by
considering the same kind of “shift-invariant” spaces as in
the uniform case and by fitting the model to the measurements or 2) by defining new basis functions (or spaces) that
are better suited to the nonuniform structure of the problem.
The two approaches are not incompatible; for instance, one
may very well construct nonuniform bases (or even frames)
.
for any of the shift-invariant spaces
1) Irregular Sampling in Shift-Invariant Spaces: The
problem that has been studied most extensively is the
recovery of a band-limited function from its nonuniform
[12], [48], [52], [70], [96], [102].
samples
for which a stable reconstruction is possible for
A set
is called a set of sampling for
. The
all
stability requirement is important because there exist sets
of samples that uniquely determine a band-limited function
but for which the reconstruction is unstable—this happens,
for example, when the samples are all bunched together.
One of the deepest and most difficult results in this area is
the Beurling–Landau density theorem [17], [53], [70]. In its
simplifed version, the theorem states that all sets of sampling
must have
for the space of band-limited functions
—roughly speaking,
a (lower) sampling density
represents the (minimum) average number of samples
per unit length in . Conversely, if the Beurling density of
is
, then
is a set of sampling for
the set
, which means that that a perfect reconstruction of a
is
band-limited function from its nonuniform samples in
possible [17]. Efficient numerical methods for performing
such reconstructions are described in [49] and [50]. More
recently, researchers have extended these techniques to the
[5], [29],
more general wavelet and spline-like spaces
[76], [77]. Aldroubi and Gröchenig derived generalized
versions of the Beurling–Landau theorems based on an
[6].
appropriate definition of the sampling density
is
Specifically, they showed that the condition
necessary to have a set of sampling for the general spaces
. Conversely, they also proved that the condition
is sufficient to have a set of sampling for the
polynomial splines. The reconstruction can be achieved
using the iterative algorithm described in [5].
The first paper on irregular sampling in terms of frames
was [15]. In particular, Benedetto et al. analyzed the role of
the coefficients as sampled values [12], [15]. They also gave
frame sampling theorems for non-band-limited functions, allowing for a quantitative means of measuring aliasing.
2) Nonuniform Splines and Radial Basis Functions: Another fruitful approach to irregular sampling
is to use specially tailored basis functions such as the
nonuniform splines [37], [100]. The B-spline formalizm, in
particular, is well suited for practical applications: the underlying B-spline basis functions are compactly supported and
can be constructed systematically using de Boor’s recursion
582
[36], [37]. The expansion coefficients of the model are then
determined from the solution of a sparse (band-diagonal)
system of equations [37], [43]. One remarkable theoretical
result, which makes the connection with Shannon’s theory,
is that the method can be used to recover a band-limited
function from its nonuniform samples; the key theorem is
that the nonuniform polynomial spline interpolant converges
to the band-limited function as the order of the spline goes
to infinity [78].
In higher dimensions, the B-spline formalism is no longer
applicable unless the grid is separable. A more general
approach is to use radial basis functions [93], which are
closely related to splines as well [46]. The radial basis
function model has the form
(51)
are centered on the
where the basis functions
; they are radially symmetric besampling points
only. Typcause they depend on the distance
(membrane
ical examples of radial functions are
( even) or
(
spline) and
odd). This latter choice yields the interpolant whose Laplacian energy is minimum (thin plate splines) [46]; it is the
natural variational extension of the cubic splines in multiples dimensions. The expansion coefficients in (51) are determined by solving a linear system of equations which ex[28],
presses the interpolation constraint
[84]. Often, the solution includes an additional low-order
polynomial term that is constrained to be orthogonal to the
rest of the expansion. Micchelli has proven that the radial
basis function interpolation problem with an arbitrary set of
sampling points has a unique solution for a relatively wide
[86].
class of positive, increasing radial functions
At first sight, the representation (51) looks rather similar
to (9) for it also involves the shifts of a single template. However, the nature of the basis functions is fundamentally difis an -function that is well
ferent. In the first case,
localized (typically, compactly supported). Practically, this
gets translated into a sparse and well-conditioned interpois unbounded at
lation problem. In the second case,
. Thus, the structure of
infinity and is certainly not in
the interpolation equations is not so favorable, which makes
working with radial basis functions more delicate and much
less efficient computationally. This appears to be the price to
pay for their generality.
VI. CONCLUSION
Fifty years later, Shannon’s sampling theory is still alive
and well. It treats such a fundamental problem, with so many
practical repercussions, that it is simply unavoidable. The
sampling theorem is part of the basic knowledge of every
engineer involved with digital signals or images. The subject
is far from being closed and its importance is most likely to
grow in the future with the ever-increasing trend of replacing
analog systems by digital ones; typical application areas are
PROCEEDINGS OF THE IEEE, VOL. 88, NO. 4, APRIL 2000
communications including the Web, sound, television, photography, multimedia, medical imaging, etc.
Recently, sampling has experienced a strong research revival, especially after it was realized that some of mathematics developed for wavelets (Riesz bases, projection operators) were the right tools for this problem as well. This
has motivated many researchers with different background
in engineering (e.g., signal and image processing) and mathematics (harmonic analysis, mathematical physics, and approximation theory) to pick up the subject, and has resulted
in a substantial advancement of the field—especially its theoretical aspects. Many of the results reviewed in this paper
have a potential for being useful in practice because they
allow for a realistic modeling of the acquisition process and
offer much more flexibility than the traditional band-limited
framework. The newer formulations of sampling tend to give
better practical results because the solutions are designed to
be optimal in a well-defined sense (e.g., least squares).
Last, we believe that the general unifying view of sampling
that has emerged during the past decade is beneficial because
it offers a common framework for understanding—and hopefully improving—many techniques that have been traditionally studied by separate communities. Areas that may benefit from these developments are analog-to-digital conversion, signal and image processing, interpolation, computer
graphics, imaging, finite elements, wavelets, and approximation theory.
APPENDIX A
SAMPLING FORMULAS
We briefly review two alternative ways of understanding
the basic sampling formulas which are at the heart of
Shannon’s theory. To simplify the argument, we use a
.
normalized time scale with a sampling step
A. Sampling and Dirac Distributions
It is a common engineering practice to model the sampling process by a multiplication with a sampling sequence
of Dirac impulses
in Fig. 1(b). The reconstruction is achieved by convolving
the sampled signal
with the reconstruction function
(A.4)
In the Fourier transform domain, this gives
(A.5)
Thus, as illustrated in Fig. 1(c), we see that a perfect reconis an ideal low-pass filter [e.g.,
struction is possible if
] and
for
(Nyquist criterion).
If, on the other hand, is not bandlimited, then the periodization of its Fourier transform in (A.5) results in spectral
. This disoverlap that remains after postfiltering with
tortion, which is generally nonrecoverable, is called aliasing.
B. Sampling and Poisson’s Summation Formula
The standard form of Poisson’s summation formula is (see
[117])
(A.6)
denotes the Fourier transform of the continuous
where
. The reader is referred to [13], [16],
time function
or [65] for a rigorous mathematical treatment.
, the Fourier
Considering the function
(modulation proptransform of which is
erty), we get
(A.7)
This is precisely the discrete-time Fourier transform of the
with
as the frequency variable. The
sequence
central term of (A.7) is identical to (A.3), which means
and
are in fact
that the -periodic functions
equivalent, even though they have very different interpretations—the former is a discrete-time Fourier transform,
while the latter is a continuous-time one.
The last step in this formulation is to derive the Fourier
transform of the reconstructed signal
(A.1)
The corresponding sampled signal representation is
(A.2)
Exchanging the order of integration and making the change
, we get
of variable
In the Fourier domain, multiplication corresponds to a convolution, which yields
(A.8)
Together with (A.7), (A.8) is equivalent to (A.5).
(A.3)
where the underlying Fourier transforms are to be taken in
the sense of distributions. Thus, the sampling process results
in a periodization of the Fourier transform of , as illustrated
UNSER: SAMPLING—50 YEARS AFTER SHANNON
APPENDIX B
NECESSITY AND SUFFICIENCY OF THE PARTITION OF UNITY
Our goal is to find conditions on such that the approxfor all
(the set
imation error vanishes as
583
of -functions that are differentiable once). We make use
of Theorem 3 to get the asymptotic form of the error as the
sampling step gets sufficiently small
Note that
is bounded if
is bounded and
satisfies the Riesz condition (11). Thus, we can apply
Lebesgue’s dominated convergence theorem and move the
limit inside the integral
This final manipulation requires that
be continuous
. Consequently, under suitable technical conditions,
at
is
the requirement
. Using (42), we obtain
equivalent to
This expression is a sum of positive terms; it can obviously
vanish only if all are zero. In particular, we need to have
, which is possible only if both factors are
nonzero. We can therefore renormalize such that
. Thus, the conditions that need to be satisfied
are
and
(B.1)
The second part is the same as the right-hand side of
(14); it is equivalent to the partition of unity condition
(in the sense of distributions).
Note that a more restrictive form of the above equivalence
can be found in [95]. This result is also
for
closely related to the Strang–Fix conditions of order one
[113]—which give a strict equivalence between the partition
of unity (reproduction of the constant) and a first order of
, as
.
approximation; i.e.,
However, the converse part of the Strang–Fix equivalence
(see Section IV-B) requires some additional decay condition
on : compact support [113], or at least some inverse polynomial decay [19]. Here, we have considered the weakest
around
possible assumption—the continuity of
—which allows for slower rates of decay. Examples of
admissible generating functions, which are not covered by
the Strang–Fix theory, are the fractional B-splines of degree
; these satisfy the partition of unity but
[134].
have a fractional order of approximation
ACKNOWLEDGMENT
The author wishes to thank A. Aldroubi and T. Blu
for mathematical advice. He is also grateful to T. Blu, P.
Thévenaz, and four anonymous reviewers for their constructive comments on the manuscript.
584
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UNSER: SAMPLING—50 YEARS AFTER SHANNON
Michael Unser (M’89–SM’94–F’99) received the M.S. (summa cum laude)
and Ph.D. degrees in electrical engineering from the Swiss Federal Institute
of Technology in Lausanne (EPFL), Switzerland, in 1981 and 1984, respectively.
From 1985 to 1997, he was with the Biomedical Engineering and Instrumentation Program, National Institutes of Health, Bethesda, MD. He is now
Professor and Head of the Biomedical Imaging Group at EPFL. His main research area is biomedical image processing. He has a strong interest in sampling theories, multiresolution algorithms, wavelets, and the use of splines
for image processing. He is the author of more than 80 published journal
papers in these areas. He serves as regular Chair for the SPIE conference on
Wavelet Applications in Signal and Image Processing, held annually since
1993.
Dr. Unser is an Associate Editor for the IEEE TRANSACTIONS ON
MEDICAL IMAGING. He is on the editorial boards of several other journals,
including Signal Processing, IEEE TRANSACTIONS ON IMAGE PROCESSING
(1992–1995), and IEEE SIGNAL PROCESSING LETTERS (1994–1998). He
received the IEEE Signal Processing Society’s 1995 Best Paper Award.
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