R.B., J.M. Daida, J.F. Veseclr, G.
Meadows,
and C. Wolf
"Reconstructing Incomplete Signals Using Nonlinear
Interpolation and Genetic Algorithms"
In Genetic Programming 1998: Proceedings of the Third
Annual Conference, July 22-25, 1998, University of
Wisconsin, Madison, Wisconsin. J.R. Koza, W. Banzhaf, K.
Chellapilla, K. Deb, M. Dorigo, D.B. Fogel, M.H. Garzon,
D.E. Goldberg, H. Iba, and R. Piolo (eds.). San Francisco:
Morgan Kaufmann. 8
PP.
Accepted.
TCHHIScHE LJIVERSrfl
Laboratofitim voor Scheepehydromechanica Archi of Mekeiweg 2,2628 CD Oeft TL 014 - Fz 015 181ß
Reconstructing Incomplete Signals Using
i'
Nonlinear Interpolation and Genetic Algorithms
Robert R. Bertram', Jason M. Daida', John F. Vesecky1, Guy A. Meadows2, and Christian Wolf3
1The University of Michigan, Artificial Intelligence Laboratory and Space Physics Research Laboratoy.2455 Hayward Avenue, Ann Arbor. Michigan 48109-2143
(734)647-4581 FAX (734)764-5137. E-MAIL: daida@eccs.umich.edu, WWW httpu/www.sprLumichedu/acers University of Michigan. Dept NAME. 2600 Draper Road, Ann Arbor, Michigan 48109-2145
3Thc University of Heidelberg. Institute of Environmental Physics, Germany
Abstract
This paper describes a general,
non-analytical method for deriving Fourier
series coefficients using a genetic
al-gorithm. Nonanalytical methods are
often needed in problems where lost
portions of a complex signal require
restoration. We discuss some of the
difficulties involved in working with
the associated trigonometric
polynomi-als and propose an alternative solution
for adapting genetic algorithms for this
class of problems. We demonstrate the
efficacy of our approach with a case
study. Our particular case study
fea-tures the processing of data that has
been collected by a novel optical
wave-slope instrument, which measures the
topography of water surfaces.
I. INTRODUCTION
The study of orthogonal functions has long existed in the
genetic algonthm (GA) literature. Bethke was among the
earliest to study GAs with orthogonal functions in his
clisser-tation [1]. DeJong referenced Bethke's work at the First
In-ternational Conference on Genetic Algorithms [6] and Gold-berg also mentioned that work in [7]. The particular
orthog-onal functions studied by BethkeWalsh functionsstill
remain of interest, as evidenced by [13].
Part of the reason for this sustained interest has been in the tse of Walsh functions to formulate epistatic, deceptive zest problems [13]. The degree to which any problem is
epistatic (e.g., [10]) or deceptive has served as an indication
of how difficult that problem would be to a GA (see [16]).
That Walsh functions have been chosen over other
orthogo-nal functions has not been surprising. Walsh functions take on values of±l, which makes for test functions that are ens-ily tractable for a GA binary representation.
However, as useful as Walsh functions have been, a great
number of problems in signal processing have relied on other orthogonal functions as a basis for decomposition. One of the most well established of these involve the basis functions of sine and cosine in trigonometric
polynomials-i.e., in Fourier series analysis and transforms.
As with Walsh functions, sine and cosine functions can
be used as a basis by which to express large classes of func-tions. As it turns out, the resulting trigonometric
polynomi-als polynomi-also apparently exhibit much of the behavior that make for GA-hard problems. Nevertheless, the literature on
us-ing GM to solve for problems featurus-ing Fourier series
anal-ysis is fairly sparse, in spite of a number of signal process-ing problems that could benefit, and in spite of the theoret-ical advances that could come in examining an alternative
basis. Our subsequent investigations have centered on
com-binations of Fourier and GA methods. Our motivation has not been in deriving methods that would supplant already well established analytical techniques (e.g., [23]). Rather, our motivation has been rooted in applying GA methods to those situations when analytical methods fail. The partic-ular situation of interest involves signal reconstruction and restoration.
The purpose of this paper is to demonstrate how a GA would work with a Fourier/GA problem in signal
recon-struction. We show in this paper how a standard GA search in the solution space of a trigonometric polynomial can
cre-ate problems in optimization. We propose an alternative method and demonstrate its efficacy. We have chosen to illustrate our method by drawing from a current situation
faced by the oceanographers who are authors on this paper.
This paper is organized as follows: Section II briefly
de-scribes and places into context our problem statement.
Sec-tion ifi describes the theory behind our alternative method and details our resulting algorithm. Section IV outlines our case study experiment and summarizes our results. Section
'cusses
our results. Section VI highlights our conclu-sions.II. PROBLEM STATEMENT
This paper focuses on the reconstruction of slope image data in one-dimensional slices (i.e., by row vectors). We consider the case of reconstructing a single scan line sam-pled at N points where k of these points have been dropped out. We assume that the function I(a) that describes the waveform of this scan line has a continuous first derivative (and converges uniformly to its Fourier decomposition). We assume that we do not have access to an interpolating func-tion and that we do not know anything about the frequen-cies of the function. Reconstruction is based solely on the remaining
N - k
points.A. Featured Case Study
The featured case study involves the reconstruction of
sig-nal daza from a novel in-situ wave-slope instrument that has been built at The University of Michigan (U-M). (See [25].) This instrument, and another more advanced high-resolution version of it, has been designed to measure the surface to-pographies of water using shape-from-refraction principles (see [11J,[12]).
Measuring surface topography using shape-fromrefraction
is indirectthe image data obtained from these instruments depict slopes, and not surfaces. Under ideal conditions, slope data is numerically integrated to derive for height. However, ideal conditions are the exception and not the rule when examining water under turbulent conditions [25]. These instruments have a tendency of dropping measure-ments when wave-slopes become too steep (e.g., the sec-ond U-M instrument has a slope measurement range of ±30°). Furthermore, data can be dropped out for a variety of other reasons, including air entrainment (i.e., bubbles), occlusion of the coded light beam by auxiliary instruments, and optical vignetting. (For more information, please refer httjvi/www.sprl.umich.edu/acers/.)
.1
s
s
C-FIg. 1. Density plot of ¡Jope data breaking wav
FIgure 1 illustrates a sample of slope data (gradient along tl long direction only). This figure is a density plot sampled
at 500 x 240 points; dark areas represent regions where data was dropped out. Visible structures where data drops Out are breaking waves and bubbles (underneath the water surface).
In [4], we introduced a preliminary hybrid technique for
estimating surface topography from one-dimensional slope
data (consisting of multiple scan lines) by estimating the
Fourier coefficients for each scan line. Then, in [5], we ex-tended this algorithm to reconstruct entire images. Although functional, this solution represents what would typically be
done in applying a GA to a trigonometric polynomial. l'his "typical" approach used a significant amount of computa-tional resources (i.e., IBM SP-2). This paper describes a
considerable refinement over that method.
Previous Work
The problem of signal reconstruction is not new and has been addressed by several researchers. However, we note that the context of our problem differs from previous work in the sense that not much information about a signal to be
reconstructed is known in advance. For example, one class
of reconstruction algorithms utilizes an iterative approxi-mation and refinement technique (e.g., [17], [3], and [22]). These require an interpolation function or some frequency
information, which are not available from our initial data. If a higher degree of precision were required, our
reconstruc-tion technique, once applied, can be used as an
interpola-tion funcinterpola-tion to facilitate a higher level of refinement using Park's algorithm in [17], for example.
Another common approach to signal reconstruction ad-dresses decomposition into frequencies (e.g.. [26], [18], [21], [24], [15], and [14]). l'his reconstruction technique is attractive because it often generates the original signal ex-actly; however, it requires advance knowledge of a func-tion's frequency information, which we do not have. De-spite this apparent lack of information, our algorithm has
been able to reconstruct the missing data points.
A significant amount of work has been put into recon-structing signals by linear methods (e.g., [8], [201). One
drawback to these types of methods is that they often expe-rience difficulties in solving nonlinear problems.
Problemin Context of GA
Changing a single coefficient in a Fourier series can
pro-duce significant changes in the modeled function. Figure 2
illustrates this behavior. Figure 2a portrays one scan line of
data taken from the wave-slope instrument. Each scan line
example described in this paper consists of 128 data points. For illustrative purposes only, the portrayal of this scan line
is actually of the scan line's corresponding trigonometric polynomial the coefficients to this polynomial have been
analytically derived since no data points were missing.
Figures 2b and 2c depict curves that have had just one coefficient perturbed by 0.05 (which approximately
seats the peak amplitude for any one frequency compo-nent). (Note: Just as there are temporal frequencies for
time-varying signals, there are spatial frequencies for
space-varying "signals"topography---that can be treated as a separable two-dimensional problem. See [2] for the original
extension in frequency analysis from time- to space-varying signals.) In Figure 2b, the frequency that was perturbed was
the lowest even (cosine) frequency; for Figure 2c, the high-est odd (sine) frequency.
As Figures 2b and 2c show, identically valued
perturba-tions result in significantly different waveforms, which does
suggest epistatic behavior. (We have left for future work a formal assessment of the degree of epistasis.) The wave-forms are also quantitatively differentthe squared error in Figure 2c is roughly twice that of Figure 2b.
A typical approach to reconstructing lost signal data is to reconstruct every frequency component up to the Nyquist frequency (e.g., [261). However, as this paper shows, this search space is too large to obtain convergence in adequate
Magnitude
Fig. 2. Example úgws
time (or to obtain convergence at all). Our aIternativ
been to constrain the search space selectively. ¡IL THEORY
The reconstruction methods we present in this paper use Fourier coefficient approximation. This section focuses on
the mathematical background and analysis.
A. Conventions
A surface topography vector can be modeled using a single-variable continuous function, F(z). Assuming that this continuous function can be expressed in terms of ¡ts Fourier series (and convergence is uniform), the following
equality holds,
a0
- (z) = -- +
(a, cos(nz) + b, sin(nz)).
(1)fl=1
Using this decomposition is desirable for the following
reasons:
The function F(z) can be represented as a linear
com-bination of infinitely integrable and infinitely differentiable functions.
Uniform convergence ensures that nonsensical,
discon-tinuous, or pointwisc convergent approximations will be ig-nored.
Spectral information is preserved.
The derivative F'(z) can be expressed using the same co-efficients with the following formula:
d
=
n (b cos(nz) - a sin(nz)).
(2)Since we work with a signal that represents the slope of a surface, desirable approximation solutions are those for the
derivative of a function or for functions that are integrable.
B. Standard-Search Method
The standard-search method, as presented in [Il], uses the
surface vector model as presented in Equation I. Digitized sampling (sampled at N points) admits the following fol-lowing property for the surface vector F(z):
F(x) =
+
(a, cos(nz) + b,, sin(nz)). (3)n= i
And the derivative can be expressed as
=
n (b,, cos(nz) - a 8in(nz)).
(4) The problem of reconstruction is now redud to finding thecoefficientsa,, and b,, for{F(z)}(orforF(z)the
coefficients are the same). Each point z, {F(z)} is rep-resented as a sum of i + N terms. Hence, all N points on
{F(z)} can be represented in N(i + N) terms. C. Alternative Method
Our alternative method is based on Fourier decomposition
with the following modifications:
d - M
{F(z)} =
+>2
(a cos(nz) +b sin(nz)) (5)Here, M is a moderat& upper bound and is indepen-dent of N. Complexity analysis shows that all N points on can be represented in N * M terms using this
method. A solution for the slope does not directly generate a
solution for the surface, however. It must go through an ad-ditional integration algorithm. Note this method is general and would likely apply to most continuous signals.
Note, too, that the standard-search method for solving the Fourier coefficients for {F(z) } in Equation 5 involves
si-multaneously solving for all coefficients, with exception of
the constant of integration (which can be calculated with
ini-tiai conditions).
For the remainder of this paper, we refer to our alternative
method as CF-search (constrained frequency search).
The major differences between the two methods is that the standard-search method emphasizes high frequency
compo-nests, while the CF-search method only uses the first M initial frequencies (with an equal weight on each frequency component). The motivation for the CF-search method has
been driven largely by the need for increasing computational efficiency while reducing coefficient search space. While N
depends on the signal length, M does not. Because of this, the CF-search method is an order N algorithm, 0(N), and the standard-search method is order N2 algorithm, 0(N2). In theory, the standard-search approach can produce an exact reconstruction of the slope and integrated
informa-tion, while the CF-search method may yield only an
approx-imation. The following section subsequently discusses the trade-offs of using CF-search. Once an approximation is found, it is natural to discuss how well the approximation models F(z). The next section focuses on this topic. D Metrics
Given a function, f(z), defined on a finite domain of dis-acte points, D, we want to find a function, g(z), defined on
aninterval, I, which contains D such that g(z) best
approx-imates 1(z). This can be summarized mathematically with the following sum
SquareError
= E
[f(z) g(z)]2. (6)zED
'MC
Here, a reasonable approximation minimizes the square
error.2 Using this metric, suppose thatf(z)and g(z) are
pe-riodic functions given by the following Fourier
decomposi-tions (with M < N) where both f'(z) and g'(z) are
contin-uous.
f(z)=
+
>2(a cos(nz) + b,, sin(nz)). (7)M
g(z)
+
>2 (c,,cos(nz) + 4 sin(nz)).
(8)n=i
We assert3 that g(z) best approximates f(z) only when
a,,
=
c,, and b,, = d,,. Hence, the square error for this ap-proximation isSquareError
=IN
2>2 LE
cos(nz) + b,, sin(nz)) tED =M+1IN
2 (IanI+ Ib,,I) tED M+iFurthermore, we know that the coefficients for f(z) go to zero asymptotically with
(i.e., 3K, L > O : Vn> L
Ja,, J, J&,,J < ). This produces the following inequality for the square error:f N K
Kl2
SqtiareError
<>2 I
>2 (
+ 3)]
ZED ln=M$-i tED>2{K12
M3J :
M6 IDIK2 (12) When M is moderately large (e.g., M = 32), the errorfrom approximation is very small (e.g., less than 0.01 when
IDI
=
1024, K = 100). Optimal values of M (and K) de-pend on the specific problem at hand; however, M 32(and K < 100) appears to be a reasonable starting point
for modeling turbulent water surfaces as they appear in our slope instruments.
The inequalities used here to compute the upper bound
for the square error can be strengthened to produce an even
smaller bound. However, this first approximation provides an adequate picture of the type of error expected from this
type of method.
L
AlgorithmThe algorithms for both CF-search and standard-search methods require a nonlinear optimization algorithm. The
following is the list of steps that each of the methods per-.
foi-m. Note that the Fourier coefficient approximation model
that each method uses, however, is different. Step I: Choose a set of Fourier coefficients.
Step 2:
Using these coefficients, construct an approximation using
the
con-strained method or the standard-searchapproach and compare this to the
provided signal by computing the
SquareError.
Based on the SqitareError, choose Step 3: another set of Fourier coefficients from
the search space and go to Step 2.
IV. EXPERIMENT
A. Setup
The optimization algorithm was a standard genetic algo-rithm (i.e., GENESIS 5.0 [9]) to search for Fourier coeffi-cienta. The problem specific portion was written in C. The
code was compiled and executed on a Pentium-1 66 running
the Linux operating system.
The data used in this series of tests came from the
wave-slope instrument developed by Wolf [5]. Small-scale waves
were measured in the context of steady-spilling breaking waves. We focused on one scan line from these measure-ments.
Two cases were investigated. We conducted a control and
experiment test for each case. The first case used the CF-search method with M = 31 as was developed and pre-sented in this paper. The second case used the standard-search method.
16 32 48 64 80 96 112 128 X
Fig. 3. Signal daza used in zecoesthiction: Qipped Signal Daza
For each test, the same signal (see Figure 3) sampled at
128 points4 was processed by each reconstruction algorithm.
4flic high-solution surface-slope instrument peoduces images of
di-..inu25 X 25cmandapixelsuof2 x 2 msn.whcrecachscanbne
s length 128 mm. The signal length used in this analysis (which features
For both experiment (CF-search) and control (standa
search) tests, only 68% of the original signal (see Fig. 2) was provided (i.e., 32% of image information has been clipped to
simulate instrument drop out). Dropped out portions of the
signal were flagged with an arbitrary value (in our case this was -2.0 radians, a value that is out of the instrument's nor-mal range).
0 ¡6 32 48 64 80 96 112 128
X
FIg. 4. Signal data used in consmctiou: Otiginal Signal
Since the search space for the coefficients is extremely large5, each run of the genetic algorithm is given the same
initial population of approximate solutions generated by an adjacent row of the image.
We ran the genetic algorithm 20 times on each of the
con-trol and experiment tests for both experiment (CF-search) and control (standard-search). Each run had a population
of 50 with approximately 2000 generations (100,000 trials).
Mutation and crossover rates were set to 0.001 and 0.6, re-spectively. Elitism and gray coding were used, while
gran-ularity for each coefficient was set to 1048576.
B. Results
Execution times for each run of CF-search were approxi-mately 3.0 minutes per run; 5.8 minutes for standard-search.
Both cases performed 100,000 evaluations using the
con-strained and standard-search methods, respectively. For each case, the mean of 20 runs are averaged and
plot-ted with error bars in Figures 4-5. A thick line depicts the area where the signal is available to each search method.
The constrained method is shown in Figure 5; the standard-search method, in Figures 6.
Figures 7 and 9 illustrate the best solutions from experi-ment and control runs, respectively and the residuals
asso-data from our low-resolution instrument) has been fruncated to contspond directly to that of our high-resolution insirumeni
5For -search, the scaseh space contains 2320 > 10192 possibilities.
These ait 2'° > 10 possibilities in the swvh space for standard-search. In comparison, sonic of the previous work in signal reconstruction
would be more on the order of 2> iO'5, which is a few hundred
oof-gnitudesmal}ec Sec. e.g., (g.
0.5
. o
0.5
I
-0.5-with each approximation.6 Note that the centerlines corresponding to the plots of the best solutions for either test or control have gaps: the locations of those gaps corre-spond to regions of data drop out. The dotted line in a plot for a best solution corresponds to the original signal shown in Figure 4.
Figure 7 represents the best approximation7 from the
con-strained method with only 68% of the data. For this result, the SquareError is 0.95. For reference, a spline interpola-tion, with SquareError> 100, is plotted in Figure 8. Here we note thai the spline algorithm8 can only approximate in-terior points. The large portion on the right section of the spline interpolation fails to produce any new information,9
in part because there exists no endpoint information. The approximation and error (SquareErrors = 20.37) from the standard-search method are depicted in Figure 9.
Frequency graphs in Figures 10-12 correspond to the sig-nais illustrated in Figures 7-9. Note that the high-frequency
6Residual is calculated as Approximation - Original. 'Measured by smallest SqisareError.
Here, we use a cubic spline.
91he error for this section goes beyond the bounds of this graph and are
noi shown
0.5
-0.5
0 16 32 48 64 80 96 112 128
X
Fig. 5. Statistical Summary of Experimental Data: Reconstruction of (Ipped Signal Using CF-Search Method
I
Mean Standard Search
Original Clipped 0 ¡6 32 48 64 80 96 112 128 X 0.5 .0.5
FIg 6. Sistical Summary of Control Data: Reconstruction of OEpped
Signal Using Standard-Search Method
1.0 0.5 10.0 -1.0 1.0 - 0.5 0.0 -0.5 -1.0
FIg. 7. Summary of best performance: constrained method 1.0 0.5 s
0
0.0 -0.5 -1.0 1.0 - 0.5Hg. * Summary of best performance: Spline interpolation 1.0 . 0.5 0.0 -1.0 1.0 -. 0.5 s 0.0 a s -0.5 -1.0 X A. r -j r i
i_
X 20 40 60 80 100 120 0 20 40 60 80 100 120 XFig. 9. Summary of best performance: standard-search method
o 20 40 60 80 100 120
0.0
-0.5 -1.0
components are absent in Figure 10. This happened because
the CF-search method focuses only on the lower
frequen-cies. Alternatively, note that standard-search method in
Fig-ure 12 retains, even enhances the high frequency compo-nents.
DiscussioN
Starting with Figure 5 and 6, we note that the average
re-construction from the CF-search method approximated the
original signal with far less error than approximations by the
standard-search method. Looking at the areas of
approxi-¡nation where signal information is not available, we further
note that the standard-search method grossly overestimated the signal. To get a bettei picture of the approximation, we turn to the residuals.
The residual calculations in Figures 7-9 help demonstrate how well each method performed the reconstructions within the given number of evaluations. Comparing the CF-method (Figure 7) to the standard-search method (Figure 9), we find that the error in estimation was much larger for the latter (in
fact, the SquareErrors differ by an order of magnitude). Preserving and reconstructing spectral components are also desirable factors. Looking at the spectral analysis
graphs in Figure 10-12, we can verify some assertions about the CF-search and standard-search methods. First, our con-strained method (Figure 10) had no high frequency
compo-nents (i.e., the compocompo-nents are zero). Second, the
standard-search method (Figure 12) emphasized the higher
frequen-cies. As we showed in Section ffl.D, the high frequency components contributed little to the overall shape (energy) of the signal being reconstructed.
Each method also exhibited different phase shifting
prop-erties of the first, second, and third sidelobes in Figures
10-12. The CF-search method (Figure 10) has closely approxi-mated the spectral locations of these peaks with only a slight phase shift. The spline interpolation missed the first sidelobe by an order of magnitude and was off by a significant phase
shift on the other two. The standard-search method, on the other hand, missed them all.
One point of interest is the computational effort expended
each method. As noted in Section ifi, the algorithm us-ing the CF-search method took about 3 minutes per run, and the standard-search method took 5.8 minutes per run. That is, the CF-search method took 52% as much time as the standard-search method and yet produced a closer ap-proximation. This speedup occurred because the CF-search method operates on an 0(N) algorithm, while the standard-search method performs at 0(N2).
CoNcLusioN
i5
10
Fig. IO. Summaiy of best perfocman(spectrai analysis): constrained method
25-Fig. II. Summa,yofbestperformance(spectralanalysis): Spline interpo-lation 25 20 - Standard-Search Spectra - Original Spectra oIO - CF-Search Spectra Original Spectra 2 4 6x103 Frequency (cycles per cm)
- Spline Spectra
Otiginal Spectra
0 2 4 6x1113
Frequency (cycles per cm)
0 2 4 6xlOE
Frequency (cycles pet cm)
This paper has described a method (CF-search) for rCCOfl Fig. 12.methodSummasy of best performan(spectrai analysis): standard-search
structing a signal from partial data and trigonometric poly-nomials. 11 general theoy and mathematics behind the
Vlarch method was described. Although the
mathemat-ical forms for both methods were similar, their correspond-¡ng outcomes were significantly different. The key
distinc-tion between CF-search and standard-search methods is how
each weights frequencies. Although this difference seems fairly minor, the consequences have been shown to be pro-found. We have demonstrated that although the standard-search approach can yield a perfect solution in theory (if, say, it were if given an infinite search time); in practice, its outcomes were significantly deficient. We have shown that the GA using CF-search search outperformed the GA using standard search in the case study tests, which featured data from a novel wave-slope instrument. As an added bonus, execution time for CF-seArch is approximately half that of standard-search.
For more information about this paper, please refer our group's web site at httpJ/www.sprl.umich.edu/acers/.
VII. FUTURE WORK
The material presented in this paper demonstrates a method to reconstruct one-dimensional slices of an im-age. However, this is just one part of reconstructing two-dimensional shape-from-refraction images. The next step is to scale this method to reconstruction problems in two di-mensions. Future studies include a two-dimensional imple-mentation that would use -search GA.
ACKNOWLEDGMENTS
This research has received support from the Office of Naval Research and NASA. We extend our appreciation to the following people: David Lyzenga, David Walker, Mes-son Gbah, Tom StoReR, Sandra Gregerman, Donald Lund, Stephen Stan hope, Catherine Grasso, Tommaso Bersano-Begey, John Polito, Jesse Berwald, and Siddharth Gandhi. We also thank the anonymous reviewers for their valuable comments.
[9] Grefenstetze, Ji., L Davis. D. Ceiys, GERESIS and OOGA: Two Ge-nencAigorithm Systems, Melrose: The Software Partnership. 1991 ( 101 Holland, 1H., Adaptation in NaniraiandArtsficial Systems: An
Mtro-ductos' AsLr with Applications in 8io1og) ControL and Artificial Intelligence, Cambridge: MIT Piess. 1992
[Il] Jaime, B.. J. Klinkt. & S. Waas, "Imaging ofshort oceanwaves: a critical theoretical review." J Opt Soc ofAm. vol I I . 2 197-2209, 1994 [I 2) Keller, W. & BL Gotwols, Two-dimensional optical measurements
ofwave slope," Appi Optics. vol 22, 3476-3478, 1983
Khuii, S., Transfonn Methods," Handbook ofEvolutionary Compu-talion, T. Baeck, D.B. Fog!, and Z. Michalewicz (cils), New York: Oxford University Press, 82.6:1-10, 1997
Li, i., T.Q. Nguyen, S. Tantaratana, "A Simple Design Method for Near-Perfect-Reconstruction Nonuniform Filter Banks," IEEE Trans on Signal Processing. vol 45. no 8:2105-9, Aug 1997
Murakarni, H., 'Implementarion and Perfect Reconstniction of a Maximally Decimated FIR Filter Bank Using Parallel Module Decom-position," iEEE Trans on Signal Processing. vol 45. no 2:328-32, Feb
1997
(161 Naudts. B.. D. Suya. A. rschon. "Epistasis as a Basic Concept in Formal Landscape Analysis," Proc Seventh ICGA, T. Baeck (ed.). San Francisco: Morgan Kaufmann Publishers. 65-72. 1997
[17) Park, Y.. M. Soumekh, "Reconstn,ction from Unevenly Spaced Sam-pled Data Using the Iterative Methods," IEEE Trans onSignal Process-ing. vol 43, no 1:303-8, J50 1995
Pawlak, M.. E Rafajlowicz, "On Restoring Band-Limited Signals." IEEE Trans on information Theory. vol 40, no 5:1490-503, Sept 1994
Radcliffe, NJ., "Schema Processing," Handbook of Evolutionary Computation, T. Baeck, DB. Fogel, and Z. Michalewicz (eds), New York: Oxford University Press, B23:1-lO, 1997
Reeves. Si, LP. Heck, "Selection of Observations in Signal Recon-stnsctioo," IEEE Trans on Signal Processing, vol 43, no 3:788-91 Mar
1995
(21) Sandbesg, I.W., "The Reconstruction of Band-Limited Signals from Nonuniformly Spaced Samples," IEEE Trans on Circuits and Systems-I: Fundamental Theory andApplicahons. vol 41, no 1:64-6, April 1994
[22) Sauer, K.D., LP. A&bach, Iterative ReconstnictionofBand-Lirnited Images from Nonunifonnly Spaced Samples," IEEE Trans on Circuits and Systems, vol CAS-34, no 12, Dec 1987
Tolstoy. GE, Fourier Series, RA. Silverman (trans), Englewood Cliffs: Prentice-HaIl, 1962
Wingham, Di., 'hie Reconstruction of a Band-Limited Function and its Fourier Transform From a Finite Number of Samples at Arbitrary Locations by Singular Value Decomposition," IEEE Trans on Signal Processing. vol 40, no 3:559-70. Mar 1992
Wolf, C., Surface Properties ofBreakiog Waves. Diploma thesis, Uni-versity of Heidelberg. Germany, 1994
Xiao, C., Reconstruction of Bandlimited Signal with Lost Samples at Its Nyquù: Rate - The Solution w a Nonuniform Sampling Problem, IEEE Trans on Signal Processing. vol 43. no4: 1008-9, April 1995
REFERENCES
III Bcthke, AD.. Generic Algorithms as Funcison Opumizers. Doctoral Dissenation, The University of Michigan. 1981
12] Bracewdll, R.N., "Two Dimensional Aerial Smoothing in Radio
As-u000my,"AusrfPhys.vol9,297-314, 1956
131 Cetin. A.E, "An Iterative Algorithm foi Signal Reconstniction from Bispcctrum," IEEE Trans on Signal Processing. vol 39. no 12:2621-8. [lee 1991
141 Daida, 1M., et al.. "Measuring Topography of Small-Scale Water Sur-face Waves," Proc IGARSS '95, 1881-83, 1995
(s] Daida, 1M., et al., "Measuring Small-Scale Water Surface Waves: Nonlinear Interpolation & Integration Techniques for Slope-Image Daxa." Proc JGARSS '96,2219-21,1996
Dc long. K. "Genetic Algosithma: A Ten Year Perspective," Proc First ICGA. J. Grefenstette (ed), Hillsdalc: Lawren Eribaum Asso-elates, 1985
Goldbesg. D.E., Genetic Algorithms in Searr& Optimization and Ma-chine Learning, Reading: Mdison-)ksley Publishing, 1989 ($] Gorodnitsky, LE, B.D. Rao, "Sparse Signal Reconsssuctioo from
Lim-ed Data Using FOCUSS: A Re-weightLim-ed Minimum Norm Algo-rithm, IEEE 11-ans on Si,sal Processing. vol 45, no 3:600-616, Mar
