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Signal sinc‐interpolation: A fast computer algorithm

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An efficient algorithm for discrete signal sinc-interpolation that is suitable for use in image and signal processing is described. Being mathematically equivalent to the commonly used zero padding interpolation method, the algorithm surpasses it in terms of flexibility, computational complexity and usage of computer memory.
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Bioimaging 4(1996) 225–231. Printed in the UK
Signal sinc-interpolation: a fast
computer algorithm
L P Yaroslavsky
Interdisciplinary Department, Engineering faculty, Tel-Aviv University, Tel-Aviv,
69978, Israel
Submitted 24 April 1996, accepted 13 August 1996
Abstract. An efficient algorithm for discrete signal sinc-interpolation that is suitable
for use in image and signal processing is described. Being mathematically equivalent to
the commonly used zero padding interpolation method, the algorithm surpasses it in
terms of flexibility, computational complexity and usage of computer memory.
Keywords: image processing, interpolation
1. Introduction
Discrete signal interpolation is a very common operation
in digital signal and image processing. It is required
when one needs resolution in the signal domain higher
than that corresponding to the sampling rate. Typical
examples are signal positioning and object location with
subpixel accuracy, image zooming and other geometrical
transformations. The most commonly used signal
interpolation methods are nearest neighbor and linear and
bilinear (in 2-D) interpolation methods. They are popular
due to their computational simplicity. Unfortunately, these
simple approaches have low interpolation accuracy and
produce considerable aliasing artifacts.
The nearest neighbor and bilinear interpolation methods
correspond to zero and first order spline models,
respectively. General polynomial spline interpolation
is asymptotically equivalent to sinc-interpolation [1],
the most accurate method for representing signals with
monotonically decreasing spectra. When properly used,
this method entirely avoids aliasing errors. In sinc-
interpolation, a continuous signal a(x)is restored from its
samples {an}taken with a discretization interval 1x in the
following way:
a(x) =
n=∞
X
n=−∞
ansinc(π(x /1x n)) (1)
where
sinc(x) =sin x
x(2)
On leave from the Institute of Information Transmission Problems,
Russian Academy of Sciences, Moscow, Bolshoi Karetny 19, Russia.
E-mail address: yaro@eng.tau.ac.il
is the interpolation sinc-function. In digital signal
processing, the exact sinc-interpolation is replaced by the
signal interpolation from its finite number of Nsamples
a(x) =
N1
X
k=0
ak
sin(π M(x/1x k)/N )
Nsin(π(x/1x k)/N)(3)
with a function
sincd(M;N;x) =sin(πM x/N )
Nsinx/N ) (4)
that is a discrete analog of the sinc-function (2) and
approximates it to the accuracy of boundary effects. M
is a parameter equal to N1,N or N+1 depending on
the algorithmic implementation of the interpolation formula
(3). The well known and commonly used digital signal
processing method for discrete sinc-interpolation is ‘zero
padding’. It is implemented by padding the signal discrete
Fourier transform (DFT) spectrum with an appropriate
number of zeros and performing the inverse transformation
of the padded spectrum. Since the number of signal samples
has (usually) to be a power of two, and therefore an even
number, to permit the fast Fourier transform to be used,
three methods of zero padding are possible.
Let {αr}be DFT coefficients of a discrete signal
{an,n=0,1,...,N 1}:
α
r=1
N
N1
X
n=0
a
nexpi2πnr
N.(5)
In the first method the signal spectral coefficient αN/2
is discarded in the padded spectrum, and this results in
0966-9051/96/040225+07$19.50 c
1996 IOP Publishing Ltd 225
L P Yaroslavsky
Figure 1. Interpolation functions for different versions of discrete sinc-interpolation.
interpolation by equation (3) with M=N1, while in
the second method this coefficient is repeated twice which
results in the interpolation by equation (3) with M=N+1.
The third method is a combination of the first two methods
when the coefficient αN/2is halved and then repeated
twice as in the second method. This results in a signal
interpolation with a function
sincd(±1;N;x) =(sincd(N 1;N;x)
+sincd(N +1;N;x))/2 (6)
that converges to zero faster than the functions (4) with
M=N±1 (see figure 1, where the interpolation functions
(4), for M=N1 and M=N+1, and (6) are plotted
from top to bottom, respectively) and therefore produces
fewer boundary effects.
The computational complexity of this method is
O(N L logNL), where L(the expansion factor) is the
number of interpolated signal samples per initial one. One
can reduce this complexity to O(N L logN) with the use
of so-called ‘pruned’ FFT algorithms [2–4]. They exploit
the fact that among NL samples of the zero padded
spectrum only Nsamples are nonzero. However, the ‘zero
padding’ method has some important disadvantages. It is
very inefficient storage-wise because it requires a buffer
memory for NL signal samples while actually working
with the sequences of Nsamples; the use of most widely
226
Signal sinc-interpolation: a fast computer algorithm
Figure 2. An algorithm for discrete sinc-interpolation.
used FFT subroutines requires Lto be a power of two
which does not allow signal expansion by arbitrary factors;
implementation of the ‘pruned FFT’ algorithms requires
cumbersome programming.
In the following, we present an alternative method of
sinc-interpolation which eliminates these restrictions. The
method is based on the so-called shifted DFTs [4–6].
2. Discrete sinc-interpolation by shifted DFTs
Shifted DFTs (SDFTs) take into account the possibility of
an arbitrary shift of signal discretization sample points with
relation to the signal coordinate system and are defined as
αu,v
r=1
N
N1
X
n=0nanexpi2πnv
Noexp i2π(n +u)r
N(7)
for the direct SDFT and
au,v
n=1
N
N1
X
r=0nαrexpi2πru
No
×expi2πn(r +v)
N(8)
for the inverse DFT and arbitrary shift parameters
(u, v). These parameters describe respective shifts (in
fractions of the corresponding discretization intervals)
of the signal and its spectrum sampling points with
relation to the corresponding coordinate systems. These
definitions are obtained by eliminating irrelevant phase
factors exp(±i2πuv/N) from the formulas
au,v
nu,v
r)=1
N
N1
X
r(n)=0
αu,v
r(an)
exp(+)i2π(n +u)(r +v)
N(9)
that are obvious generalizations of the conventional DFTs
(5) that account for shifts (u, v) of samples in signal and
spectral domain.
A possibility of signal interpolation by SDFT follows
from the fact that one can perform direct and inverse SDFTs
with different shifts in the signal domain. With respective
shifts (u, v) and (p, q) for the direct and inverse DFTs, this
will result in a signal
˜au/p,v /q
n=1
N
M1
X
r=0nαu,v
rexpi2πrp
No
×expi2πn(r +q)
N
=
N1
X
k=0
akexpi2πkv+M1
2/N
×sincd(M;N;kn+up)
×expi2πnq+M1
2
×expi2πM1
N(u p)(10)
which, with an appropriate choice of the parameters,
coincides with the sincd-interpolated signal (3) for x/1x =
kp+u. For instance, when u=0 and q=v=
(M 1)/2,
a0/p,v/q
n=(N1
X
k=0
aksincd(M;N;kn+up))
×expiπM1
Np.(11)
This expression suggests an algorithm for discrete sinc-
interpolation as shown in figure 2. The algorithm involves
direct and inverse FFTs and modulations of the input signal,
its spectrum, and the output signal by corresponding phase
multipliers that provide the required signal shift. When
M=N1, interpolation is equivalent to the first version
of the zero padding algorithm. When M=N+1,
interpolation is equivalent to the second version of the
zero padding algorithm. When M=N, the algorithm
implements interpolation by a function
sincd(N;N;x) =sin(πx )
Nsin(πx/N)(12)
that approximates continuous sinc-interpolation most
closely and that cannot be implemented by zero padding
methods for even N. In terms of the speed of convergence
227
L P Yaroslavsky
Figure 3. A modified algorithm for discrete sinc-interpolation.
to zero and boundary effects in signal interpolation, this
function behaves as the functions (4) with M=N±1as
shown in figures 1(a) and (b).
One can eliminate modulations of input and output
signals required by the algorithm of figure 2 if the signal
spectrum modulation coefficients {µr=exp(i2πrp/N)}
are made pair-wise complex conjugate {µr=µ
Nr}to
guarantee that the interpolation function, being DFT of the
set of coefficients {µr}, is a real valued function. The
algorithm is shown, modified in this way, in figure 3. One
can show that selections of a parameter A=0 and A=2
in the multiplier µN/2in this modified algorithm correspond
to signal interpolation with function (4) where M=N1
and M=N+1, respectively (that is, to the above first
and the second methods of spectrum zero padding) while
the selection A=1 corresponds to signal interpolation by
function (6). One can regard the selection A=1 as a kind
of spectrum shaping with a window function
Wr=(1/2 for r=N/2
1 elsewhere. (13)
With this shaping, a sharp spectrum limitation (WN/2=0or
1) is substituted by a softer one. This explains why function
(6) converges to zero faster than function (5). Naturally,
one can include in this algorithm arbitrary spectrum shaping
windows without any increase of the computational cost.
3. Applications and computational complexity of
SDFT-based sinc-interpolation
We will describe three basic applications of the
above SDFT-based sinc-interpolation algorithms: signal
translation, signal interpolation in the vicinity of an
individual signal sample and signal zooming.
For signal translation by a fraction pof the sampling
interval, one needs to perform one pass of the above
algorithms with shift parameter p. This requires
O(N logN) operations. A typical application example of
signal translation is image rotation by a 3 pass algorithm
[7–9]. The algorithm decomposes image rotation into three
successive image shearings, in the x, y and then again in
the xdirections, that can be performed by corresponding
row- and column-wise translations. This is illustrated in
figure 4 for an image rotation by 30. The advantages
of the SDFT-based interpolation algorithm over the zero
padding algorithm in this application are: (i) arbitrary
shift versus a quantized shift by (1/L) with Lequal to
a power of two and (ii) reduced computational complexity
(O(N logN) operations versus O(N L logNL) operations
for the zero padding algorithm and conventional FFT, or
O(N L logN) operations for the zero padding algorithm and
‘pruned’ FFT).
Signal interpolation in the vicinity of an individual
signal sample is required, for instance, when one
needs to determine the position of a signal maximum
with subpixel accuracy. With the use of the above
algorithms this will require performing Ltimes, for
only single output signal sample, the inverse DFT with
shift parameters {pk,k=1,...,L
}, that correspond to
the required positions of Ladditional signal samples
(for uniform subpixel spacing pk=k/(L +1)). The
computational complexity of this process is only O(NL)
versus O(N L logNL) or O(N L logN) operations for zero
padding (plus no intermediate buffer for zero padded signal
spectrum of NL samples is needed).
In signal and image L-fold expansion (magnification),
one needs to perform (L 1)consecutive signal shifts by
1/L. The computational complexity of this procedure is
O(N(L 1)logN), as for the zero padding algorithms
implemented by pruned FFTs, although the SDFT-based
algorithms remain advantageous when compared to zero
padding. They are much simpler in programming, and more
flexible in terms of the expansion factor. In addition, they
are more storage-wise efficient. This advantage may be of
importance in hardware implementation. Note also that,
with the described SDFT based interpolation algorithms,
signal expansion is also possible when the expansion factor
228
Signal sinc-interpolation: a fast computer algorithm
Initial image First pass
Second pass Third pass: rotated image
Figure 4. Image rotation by a three-pass algorithm of successive image shearings (fragments of black background seen in figures
represent circular shift effects).
is an arbitrary rational number rather than an integer
number. If, for instance, the expansion factor is m/n,
expansion has to be performed in two steps. First, m-fold
signal expansion is performed by the above algorithms and
then every nth sample has to be taken from the interpolated
signal. Examples of 2-, 5/2- and 3-fold expansion of an
image fragment are illustrated in figure 5.
Two-dimensional (multi-dimensional) signal discrete
interpolation is more involved than the one-dimensional
case, just as two-dimensional (multi-dimensional) sampling
is more involved than one-dimensional sampling [4]. The
most common and simple implementation is sampling
in a rectangular grid. Such sampling corresponds to
separable 2-D (or, respectively, multi-dimensional) sinc-
interpolation when the interpolation function is a product of
single-dimensional sinc-functions (2) of the corresponding
coordinates. Its discrete implementation is, obviously, a
separable discrete sinc-interpolation consecutively applied
to each coordinate. In order to estimate the computational
complexity of 2-D signal expansion (zooming) let Lxand
Lybe expansion coefficients in two dimensions of a 2-D
signal of Nx·Nysamples. Separable interpolation requires
in this case
ONyNxlogNx+Ny(Nx(Lx1)log Nx)
=O(NyNxLxlogNx)
229
L P Yaroslavsky
Initial image
Fragment zoom 2.5 Fragment zoom 3
Fragment zoom 2
Figure 5. Examples of 2-, 5/2- and 3-fold expansion of an image region.
operations for the interpolation in x-direction plus
ONxLxNylogNy+Ny(Ly1)log Ny
=O(NxNyLxLylogNy)
operations for the interpolation in the y-direction. The total
number of operations is therefore
Qsep
op =O(NxNy{LxlogNx+LxLylog Ny}). (14)
Note that this formula is not symmetric with regard to
the interpolation directions. One can see that if
NLx(Ly1)
y<N
L
y
(Lx1)
x(15)
interpolation first in xdirection and then in ydirection
requires fewer operations than interpolation in the reverse
order. In particular, when Nx=Ny, interpolation ‘first
along x, then along y’ is less time consuming if Ly<L
x
.
2-D inseparable discrete interpolation is also possible,
and in different versions, depending on the way of
treating the modulating multipliers µNx/2,0,µ0,Ny/2,
and µNx/2,Ny/2. The computational complexity of the
inseparable interpolation:
Qinsep
op =ONxNy(LxLy+1)logNxNy(16)
is, however, higher than that of separable algorithms Qsep
op
which makes inseparable interpolation less attractive than
the separable case.
230
Signal sinc-interpolation: a fast computer algorithm
4. Conclusion
Computational aspects of discrete signal sinc interpolation
have been discussed. Two modifications of SDFT-based
interpolation algorithms have been described and shown
to be superior to commonly used spectrum zero padding
algorithms, both in terms of computational expense and
flexibility. The same approach can be applied also
for signal correlation by FFT and spectrum analysis
with subpixel resolution. In signal correlation with
subpixel resolution, one needs to perform an inverse
DFT with appropriate shifts as described above for signal
interpolation. In signal spectral analysis with subpixel
resolution, repeated direct SDFTs with appropriate shift
parameters in the spectral domain have to be performed to
obtain interpolated spectral samples. In general, the above
algorithms are well suited to any type of signal processing
in the spectral domain.
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[2] Smith T, Smith M S and Nichols S T 1990 Efficient sinc
function interpolation technique for center padded data
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[3] Markel J D 1971 FFT pruning IEEE Trans. Audio Electron.
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[4] Yaroslavsky L P 1985 Digital Picture Processing. An
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[5] Yaroslavsky L P 1979 Shifted discrete Fourier transforms
Problems Informat. Transmission 15 324–6
[6] Yaroslavsky L P 1980 Shifted discrete Fourier transforms
Digital Signal Processing ed V Cappellini and
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[7] Kiesewetter H and Graf A 1985 Rotation of digital grids
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Akademie der Wissenshaften der DDR
[8] Paeth A W 1986 A fast algorithm for general raster
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231
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Image up-scaling employs various polynomial interpolation schemes for their reduced computational complexity and suitability for various real-time applications. However, they give blurring artifacts in up-scaled images due to the loss of high frequency (HF) information. Likewise, most of the other edge directed and transform domain interpolation schemes available in the literature though produce lesser blurring as compared to polynomial interpolation schemes but are computationally more complex. To overcome these problems, an iterative spatial domain 2-D signal decomposition technique is proposed. It is meant for extracting the very high frequency (VHF) information from a low resolution (LR) image. The VHF information is obtained by performing the signal decomposition for an estimated number of iterations. Subsequently, the superimposition of this VHF extract with the low resolution image prior to image up-scaling reduces the blurring in its up-scaled counterpart. Since the degradation of higher order sub-band information such as HF and VHF is more than the low and medium frequency information during an up-scaling process, restoration of the most degraded VHF sub-band information would produce much lesser blurring. Simulation results reveal that the proposed scheme gives better performance than many of the existing schemes in terms of objective and subjective measures.
Chapter
This chapter covers extensively the methods used to determine the flow velocity starting from the recordings of particle images. After an introduction to the concept of spatial correlation and Fourier methods, an overview of the different PIV evaluation methods is given. Ample discussions devoted to explain the details of the discrete spatial correlation operator in use for PIV interrogation. The main features associated to the FFT implementation (aliasing, displacement range limit and bias error) are discussed. Methods that enhance the correlation signal either in terms of robustness or of accuracy are surveyed. The discussion of ensemble correlation techniques and the use of single-pixel correlation in micro-PIV and macroscopic experiments is a novel addition to the present edition. A detailed description is given of the standard image interrogation based on multigrid image deformation, where the advantages in the treatment of complex flows are discussed as well as the issues in terms of resolution and numerical stability. Another new feature introduced in this chapter is the discussion of the recent developments of algorithms in use for PIV time series as obtained by high-speed PIV systems. Namely, the algorithms to perform Multi frame-PIV, Pyramid Correlation and Fluid Trajectory Correlation and Ensemble Evaluation are treated. Furthermore, a new section that discusses the methods used for individual particle tracking is introduced. The discussion describes the working principles of PTV for planar PIV. The potential of the latter techniques in terms of spatial resolution as well as their limits of applicability in terms of image density are presented.
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Linear Spatial Transformations Nonlinear Transformations Registration Quality Metrics Interpolation Methods for Image Registration Biomedical Examples References
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In this paper, a highly nonlinear, fuzzy logic based, composite scheme is proposed by combining a preprocessing and a post-processing operation to efficiently restore high frequency (HF) and very high frequency (VHF) details in an up-scared image. The blurring in case of an up-sampled image is caused by the degradation of HF and VHF image details that correspond to fine details and edge regions during the up sampling process. The degradation of HF and VHF image details is more significant than that of the flat and slowly varying regions. In order to resolve this problem effectively, a fuzzy composite scheme is developed which is based on the inverse modeling approach of HF degradation. During the preprocessing operation, the VHF components of an image are boosted up using recursive Laplacian of Laplacian (LOL) operator prior to image up-scaling. Subsequent to the image up-scaling, a fuzzy local adaptive Laplacian post-processing scheme is used which enhances the HF image details more than the low frequency image details based on local statistics in the up-scaled image. The HF restoration performance of the fuzzy based composite scheme is enhanced by improving its nonlinearity through the variations of different parameters of the fuzzy inference system (FIS) such as slope, width and the number of input-, and output membership functions. The effective fusion of pre-processing and post-processing operations makes the proposed scheme much effective to tackle the non-uniform blurring than the standalone pre-processing and post-processing techniques. Experimental results reveal that the proposed composite scheme gives much less blurring in comparison to the standalone schemes and performs better than most of the widely used interpolation schemes in terms of objective and subjective measures.
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One of the first signs of cell differentiation in the Drosophila melanogaster embryo occurs 3 h after fertilization, when discrete groups of cells enter their fourteenth mitosis in a spatially and temporally patterned manner creating mitotic domains (Foe, V. E. and G. M. Odell, 1989, Am. Zool. 29:617-652). To determine whether cell residency in a mitotic domain is determined solely by cell position in this early embryo, or whether cell lineage also has a role, we have developed a technique for directly analyzing the behavior of nuclei in living embryos. By microinjecting fluorescently labeled histones into the syncytial embryo, the movements and divisions of each nucleus were recorded without perturbing development by using a microscope equipped with a high resolution, charge-coupled device. Two types of developmental maps were generated from three-dimensional time-lapse recordings: one traced the lineage history of each nucleus from nuclear cycle 11 through nuclear cycle 14 in a small region of the embryo; the other recorded nuclear fate according to the timing and pattern of the 14th nuclear division. By comparing these lineage and fate maps for two embryos, we conclude that, at least for the examined area, the pattern of mitotic domain formation in Drosophila is determined by the position of each cell, with no effect of cell lineage.
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The least-squares polynomial spline approximation of a signal g ( t ) ∈ L <sub>2</sub>( R ) is obtained by projecting g ( t ) on S <sup>n</sup>( R ) (the space of polynomial splines of order n ). It is shown that this process can be linked to the classical problem of cardinal spline interpolation by first convolving g ( t ) with a B-spline of order n . More specifically, the coefficients of the B-spline interpolation of order 2 n +1 of the sampled filtered sequence are identical to the coefficients of the least-squares approximation of g ( t ) of order n . It is shown that this approximation can be obtained from a succession of three basic operations: prefiltering, sampling, and postfiltering, which confirms the parallel with the classical sampling/reconstruction procedure for bandlimited signals. The frequency responses of these filters are determined for three equivalent spline representations using alternative sets of shift-invariant basis functions of S <sup>n</sup>( R ): the standard expansion in terms of B-spline coefficients, a representation in terms of sampled signal values, and a representation using orthogonal basis functions
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The Monte Carlo approach to testing a simple null hypothesis is reviewed briefly and several examples of its application to problems involving spatial distributions are presented. These include spatial point pattern, pattern similarity, space-time interaction and scales of pattern. The aim is not to present specific “recommended tests” but rather to illustrate the value of the general approach, particularly at a preliminary stage of analysis.
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A major activity of many sciences is to search for patterned behavior within complex phenomena. The fields of Biology and Psychology are just two examples, in which the discovery of patterns is an impetus for building explanatory models that could account for the patterns. This paper reports the invention of a powerful machine-oriented heuristic for finding complex patterned behavior in empirical data. The heuristic was developed by retrospecting on our own human reasoning during “field work” in experimental developmental biology, in which we detected a novel dynamic pattern in the mitoses of the early embryo. The new heuristic is broadly applicable: we also apply it to psychological data on memory in chess, with interesting results.
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Time-lapse microscopy of biological systems has provided new and exciting information about the dynamics of cellular and developmental events. However, these events are often complex and difficult to analyze. This paper describes a study in which computation was indispensable for formulating and evaluating a cellular/developmental hypothesis directly from observations of time-lapse fluorescence images. Previous analyses of time-lapse microscopy sequences of Drosophila melanogaster embryonic syncytial nuclear cycles 10-13, when the nuclei form an evenly spaced monolayer at the surface of the embryo, have failed to identify any pattern in these divisions. However, computational analysis of the data has provided evidence that the direction of syncytial nuclear mitosis is not random, but is clearly influenced by the relative positions of neighboring nuclei. An approximate law governing mitotic direction that is based on a scheme that compromises among "votes" made by neighboring nuclei is introduced.
An efficient zooming FFT (fast Fourier transform) algorithm that allows center padding sinc function interpolation of 2-D images is presented. This algorithm avoids the phase shifts that would be introduced if the efficient Skinner interpolation method is used. Output pruning is incorporated to allow efficient determination of a zoomed subimage. Time savings of more than 50% can be achieved. Example images illustrating the use of the algorithm in conjunction with zooming and ARMA (autoregressive moving average) modeling of data are given
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