Table Of ContentTIME-VARYING IMAGE PROCESSING
AND MOVING OBJECT RECOGNITION, 3
Proceedings of the
4th International Workshop
Florence, Italy, June 10-11,1993
Edited by
V. CAPPELLINI
Department of Electronic Engineering
University of Florence
Florence, Italy
1994
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PREFACE
The area of Digital Image Processing is of high actual importance in terms of research
and applications. Through the interaction and cooperation with the near areas of Pattern
Recognition and Artificial Intelligence, the specific area of "Time-Varying Image
Processing and Moving Object Recognition" has become of increasing interest. This new
area is indeed contributing to impressive advances in several fields, such as
communications, radar-sonar systems, remote sensing, biomedicine, moving vehicle
recognition-tracking, traffic monitoring and control, automatic inspection and robotics.
This book represents the Proceedings of the Fourth International Workshop on Time-
Varying Image Processing and Moving Object Recognition, held in Florence, June 10-11,
1993. Extended papers reported here provide an authorative and permanent record of the
scientific and technical lectures, presented by selected speakers from 15 Nations. Some
papers are more theoretical or of review nature, while others contain new
implementations and applications. They are conveniently grouped into the following
fields:
A. Digital Processing Methods and Techniques
B. Pattern Recognition
C. Image Restoration
D. Implementation Techniques
E. Computer Vision
F. Image Coding and Transmission
G. Remote Sensing Image Processing
R Digital Processing of Biomedical Images
I. Motion Estimation
J. Recognition and Tracking of Moving Objects
K. Traffic Monitoring
New digital image processing and recognition methods, implementation techniques and
advanced applications (television, remote sensing, biomedicine, traffic, inspection,
robotics, etc.) are presented. New approaches (i.e. digital transforms, neural networks, ...)
for solving 2-D and 3-D problems are described. Many papers are concentrated on the
motion estimation and recognition - tracking of moving objects. In overall the book
presents - for the new outlined area - the state of the art (theory, implementation,
applications) with the next-future trends.
This work will be of interest not only to researchers, professors and students in
university departments of engineering, communications, computers and automatic
control, but also to engineers and managers of industries concerned with computer
vision, manufacturing, automation, robotics and quality control.
V. Cappellini
VI
WORKSHOP CHAIRMAN
V. CAPPELLINI, University of Florence, Florence, Italy
STEERING COMMITTEE
J.K. AGGARWAL, University of Texas, Austin, U.S.A.
M. BELLANGER, Conservatoire National des Arts et Metiers, Paris, France
A.G. CONSTANTINIDES, Imperial College, London, England
T.S. DURRANI, University of Strathclyde, Glasgow, England
G. GALATI, // University of Rome, Italy
G.H. GRANLUND, University of Linkoping, Sweden
T.S. HUANG, University of Illinois at Urbana-Champaign, U.S.A.
M. KUNT, Ecole Polytechnique Fedérale de Lausanne, Switzerland
D. LEBEDEV, Institute for Information Transmission Problems, Moscow, C.S.I.
S. LEVIALDI, University of Rome, Italy
A.R. MEG, Polytechnic of Turin, Italy
F. ROCCA, Polytechnic of Milan, Italy
J.L.S. SANZ, IBM Almadén Research Center, San José, U.S.A.
V. TAGLIASCO, University of Genoa, Italy
A.N. VENETSANOPOULOS, University of Toronto, Canada
G. VERNAZZA, University of Genoa, Italy
VI
WORKSHOP CHAIRMAN
V. CAPPELLINI, University of Florence, Florence, Italy
STEERING COMMITTEE
J.K. AGGARWAL, University of Texas, Austin, U.S.A.
M. BELLANGER, Conservatoire National des Arts et Metiers, Paris, France
A.G. CONSTANTINIDES, Imperial College, London, England
T.S. DURRANI, University of Strathclyde, Glasgow, England
G. GALATI, // University of Rome, Italy
G.H. GRANLUND, University of Linkoping, Sweden
T.S. HUANG, University of Illinois at Urbana-Champaign, U.S.A.
M. KUNT, Ecole Polytechnique Fedérale de Lausanne, Switzerland
D. LEBEDEV, Institute for Information Transmission Problems, Moscow, C.S.I.
S. LEVIALDI, University of Rome, Italy
A.R. MEG, Polytechnic of Turin, Italy
F. ROCCA, Polytechnic of Milan, Italy
J.L.S. SANZ, IBM Almadén Research Center, San José, U.S.A.
V. TAGLIASCO, University of Genoa, Italy
A.N. VENETSANOPOULOS, University of Toronto, Canada
G. VERNAZZA, University of Genoa, Italy
vil
SPONSORED BY:
European Association for Signal Processing (EURASIP)
IEEE Middle & South Italy Section
International Center for Signal & Image Processing (ICESP), Florence
Centro d'Eccellenza Optronica (CEO)
Dipartimento di Ingegneria Elettronica, University of Florence
Istituto di Ricerca sulle Onde Elettromagnetiche (IROE) - C.N.R., Florence
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Associazione Italiana di Telerilevamento (A.I.T.)
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C.N.R.
Ordine degli Ingegneri della Provincia di Firenze
Sezione di Firenze delVA.E.I.
CO-SPONSORED BY:
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AXIS
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Cassa di Risparmio di Firenze
Time-Varying Image Processing and
Moving Object Recognition, 3 -V. Cappellini (Ed.)
© 1994 Elsevier Science B.V. All rights reserved.
Fast Approximate Calculation of the Two-Dimensional Discrete
Fourier Transform
Sanjit K. Mitra^ and Ogjnan V. Shentov^
^ Department of Electrical & Computer Engineering
University of California, Santa Barbara, CA 93106, U.SA.
^Pennie & Edmonds, Inc.
1155 Avenue of the Americas, New York, NY 10036, U.S.A.
The well-known decimation-in-time or decimation-in-frequency FFT algorithm can be
viewed as computing the DFT of a longer length sequence as an weighted sum,^of the DFTs of
smaller length sub-sequences generated by a polyphase decomposition of the original sequence
followed by down-sampling. A similar approach based on a structural subband decomposition,
a generalization of the polyphase decomposition, has resulted in a new FFT algorithm. In the
case of two-dimensional (2-D) DFT computation, the algorithm is divided into two distinct
stages: a pre-processing 2-D analysis filter bank with down-sampled output and a recombination
network. The preprocessing stage generates a set of smaller length 2-D sub-sequences with
approximate spectral separations in the 2-D frequency domain from the original 2-D sequence.
The overall DFT is then computed by a weighted sum of smaller length DFTs with the weights
determined by the recombination network. The shorter length subsequences occupy essentially
different portions of the input frequency range. This spectral separation property of the sub
sequences permits elimination of the subsequences having negligible energy contribution
resulting in a fast approximate DFT computation method. An example is included illustrating
the effectiveness of the proposed method in computing approximately the DFT of digital
images.
1. INTRODUCTION
In many practical applications the 2-D signals of interest, such as images, occupy a rela
tively small portion of their transform domains. This spectral separation property has been ex
ploited very effectively, in a number of image processing applications, like subband image cod
ing [1], [2] etc. However, the property has so far not been exploited in other image processing
applications such as discrete transform computation. In this paper, we propose a new 2-D FFT-
type algorithm that can be used either for the computation of the full discrete Fourier transform
(DFT) or selected DFT samples by making use of the above property. The conventional 2-D
FFT algorithm is based on the computation of shorter size 2-D DFTs of subsequences obtained
by a polyphase decomposition followed by down-sampling. The proposed 2-D FFT algorithm
is based on a similar approach where the polyphase decomposition is replaced by a generaliza
tion called the structural subband decomposition. In Section 2 we review the polyphase decom
position and the structural subband decomposition. The new FFT algorithm is derived in
Section 3. Section 4 illustrates an image processing application of the new 2-D FFT algorithm.
Section 5 contains the concluding remarks.
2. POLYPHASE DECOMPOSITION OF 2-D SEQUENCES AND ITS
GENERALIZATION
2.1 The Polyphase Decomposition
Consider a 2-D sequence x(m,n) with a z-transform X(zj,Z2):
oo oo
X(Zl,Z2)= £ Xx(m,n)zf'"z2"". (1)
m=-oo n=-oo
The conventional polyphase decomposition of the transform X(zi,Z2) is given by
M-l N-1
(2)
r=0 s=0
where
Xr,s(^l'^2)= Σ Xx(niM + r,nN + s)zf%-",
m=-oo n=-oo
r = 0, 1, ...,M-l;s = 0, 1,. . . , N-1; (3)
where functions Xj.g(zj,Z2) are called ihtpolyphase components oíX{z^,Z2) [3].
The relation between the sub-sequences g(m,n) given by the inverse transform of
Xj. s(zi,Z2), and the original sequence x(m,n) is given by
Xj. g(m.n) = x(Mm+r, Nn + s); r = 0, 1, ...,M-l;s = 0, 1,. . . , N-1, (4)
or in other words, the sub-sequence x^ s(m,n) is simply obtained by down-sampling x(m,n) by
a factor of (M,N) with (r,s) indicating the phase of the sub-sampling process as indicated in
Figure 1 for Μ = 3 and Ν = 2.
The polyphase decomposition of Eq. (2) can be written in matrix form as:
X(zi,Z2) = 1 ζ -1 -(M-l) (5)
-(N-1)
where
Μ _Nx
^0,0 ^2^^ ^Ο,ΐ(^Γ'^Γ)
^Ο,Ν-1^^1 '^2 ^
X,l(z,^,Z3N) Χ ί ζ ^ ζ ^^
AJQVZJ ,Z2 )
X = (6)
x(m,n) I 3, 2—•χο,οί^,η)
-1
I 3,2 I—•XQjim.n)
i 3.2
' xj o(m,n)
-1
I 3, 2 I •xjjim,!!)
i 3,2 • X2Q(m,n)
I 3, 2 •X2j(m,n)
Figure 1 - Conventional polyphase decomposition of a sequence.
2.2 The Structural Subband Decomposition
A generalization of the polyphase decomposition of Eq. (5) is given by
1
-1
X(zi,Z2) = 1 z"^ z-(^-^> AVB (7)
1 Zj ... Zj
-(N-1)
L^2
where A = [a^j] is an MxM nonsingular matrix, Β = [b^j] is an NxN nonsingular matrix, and
^0,N-l(^l '^2 )
M,N-l^^l '^2 >
V = (8)
Μ Ν .Μ „N,
The expression of Eq. (7) is called the 2-D structural subband decomposition [4]. The relation
between the polyphase components Xj. 5(m,n) and the generalized polyphase components
Vj g(m,n), the inverse transform of Vj.^(zj,Z2) can be established easily from Eqs. (5) and (7),
and is given by:
ν=Α-1χΒ-ΐ. (9)
One possible implementation of the above generalized 2-D polyphase decomposition is shown
in Figure 2.
x(m,n) i 3.2 ^VQQ(m,n)
Z2-1
-1 13,2 .VQi(m,n)
.-1
l3,2 > ViQ(m,n)
-1 (B-»)Tl
-1 "TI 13.2 • vii(m,n)
13.2 ^V2o(m,n)
|3,2 —•V2i(m,n)
Figure 2 - Generalized polyphase decomposition of a 2-D sequence.
The subband decomposition matrices A and Β can be chosen judiciously to simplify the
digital filtering algorithms of interest. Particularly suitable for this purpose is the Hadamard
matrix [4]. A Kx Κ Hadamard matrix Rj^^, with Κ = 2^, is given by
(10)
Rj. =R2®^2®*"®^2
L-times
1 1
where R2 = and Θ denotes the Kronecker product. It can be shown that Rj^^ ^ =
1 -1
Rj^/K. Note from Eq. (10) that Κ must be a power-of-two, i.e., Κ = 2^. In this case, Eq. (9)
reduces to:
(11)
3. EFFICIENT APPROXIMATE 2-0 DFT COMPUTATION
The network to the left of the down-samplers in Figure 2 can be considered as a 2-D anal
ysis filter bank. By choosing appropriately the subband decomposition matrices, one can derive
at analysis filters with reasonable frequency separation. This property of the analysis filters is
exploited here in the fast approximate computation of the 2-D DFT. Recall that the (P,Q)-point
DFT X(k,/) of a 2-D array x(m,n) of size PxQ is simply given by the equally-spaced
ft-equency samples of its z-transform X(zj,Z2) on the unit bi-disk, i.e.
X(k,^) = X(zi,Z2)| j2^/p j2^./Q, k = 0,l,...,P-l;/ = 0,l,...,Q-l. (13)
