Reply to “Comments on ‘The Discrete Periodic Radon Transform’”

Article (PDF Available)inIEEE Transactions on Signal Processing 58(11):5963 - 5964 · December 2010with52 Reads
DOI: 10.1109/TSP.2010.2059021 · Source: IEEE Xplore
Abstract
This paper presents the reply for the comment made on "The Discrete Periodic Radon Transform" by A. M. Grigoryan. This comment presents a series of paper about tensor and paired transform as applied to the fast realization of two-dimensional discrete Fourier transform. The reply indicates that the claim made by A.M Grigoryan was incorrect. The contributions of T. Hsung and D. P. K. Lun are far more than that described by A.M Grigoryan.The DPRT is not only a forward transform for computing 2-D DFT. It is a complete discrete transformation method that includes efficient inverse transform.

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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 11, NOVEMBER 2010 5963
Each set of numbers , as shown, is the split-
ting signal , where .
The summation of values of the image in is performed
by points such that
. If we change by and
by , and denote the even number by , we obtain the fol-
lowing: .
The set of numbers , is the splitting-signal
written in the second line in the tensor
transform in (5). Thus, in the case, the DPRT repeated the
concept of the tensor transform.
Fast 2-D DFT. The splitting-signal carries the spectral information of
the image at frequency-points of the corresponding group , where
. In tensor representation, the following holds [2]:
(7)
The 2-D DFT of the image is split by 1-D DFTs of the splitting-signals.
In [1], formulas (6), (7), (11), and (12) of the discrete Fourier slice
theorem repeat the concept of splitting the 2-D DFT by the covering
consisting of and cyclic groups.
REFERENCES
[1] T. Hsung, D. P. K. Lun, and W.-C. Siu, “The discrete periodic
radon transform,” IEEE Trans. Signal Process., vol. 44, no. 10, pp.
2651–2657, 1996.
[2] A. M. Grigoryan, “An algorithm of the two-dimensional Fourier trans-
form,” (in Russian) Izvestiya VUZ SSSR, Radioelectronica, vol. 27, no.
10, pp. 52–57, 1984.
[3] A. M. Grigoryan, “An optimal algorithm for computing the two-dimen-
sional discrete Fourier transform,” (in Russian) Izvestiya VUZ SSSR,
Radioelectronica, vol. 29, no. 12, pp. 20–25, 1986.
[4] A. M. Grigoryan, “New algorithms for calculating discrete Fourier
transforms,” (in Russian) J. Vichislitelnoi Matematiki i Matematich-
eskoi Fiziki, vol. 26, pp. 1407–1412, 1986.
[5] A. M. Grigoryan and M. M. Grigoryan, “Two-dimensional Fourier
transform in the tensor presentation and new orthogonal functions,
Avtometria, AS USSR Siberian section, no. 1, pp. 21–27, 1986.
[6] A. M. Grigoryan and M. M. Grigoryan, “A new method of image recon-
struction from projections,” (in Russian) Electronnoe Modelirovanie,
vol. 8, no. 6, pp. 74–77, 1986.
[7] A. M. Grigoryan and M. M. Grigoryan, “Algorithm ofcomputerized to-
mography,” (in Russian) Electronoe Modelirovanie, vol. 12, pp. 96–98,
1990.
[8] A. M. Grigoryan, “An algorithm for computing the discrete Fourier
transform with arbitrary orders,” in J. Vichislitelnoi Matematiki i
Matematicheskoi Fiziki (in Russian), 1991, vol. 30, pp. 1576–1581.
Reply to “Comments on ‘The Discrete
Periodic Radon Transform’”
Tai-Chiu Hsung, Daniel P. K. Lun, and Wan-Chi Siu
We would like to thank the author of “Comments on “The Discrete
Periodic Radon Transform (DPRT)’” [1] for indicating the series of
paper about tensor and paired transforms [1]–[8] as applied to the fast
realization of two-dimensional (2-D) discrete Fourier transform. As the
authors of DPRT [9], we would firstly like to point out that it has been
clearly indicated in [9] that the forward DPRT is based on an earlier
work given in [10]. We acknowledge the similarity as indicated in (4)
and (5) in [1] that the work in [10] may be a subset of the Tensor trans-
form [5]. Since [5] was published earlier than [10], it could be more
accurate to indicate in [9] that the forward DPRT is based on the work
in [5] which is written in Russian.
However, we believe that a complete transformation method should
contain not only the forward transform. The existence of an efficient in-
verse transform is extremely important so as to enable the processing of
the transform coefficients to be meaningful. Besides, the transform co-
efficients should have special properties that can facilitate certain pro-
cessing objectives which cannot be achieved or will be less efficient to
be achieved in the original domain. The DPRT as proposed in [9] is to
achieve these objectives. The DPRT mimics the classical continuous
Radon transform for discrete sampled images that the line inte-
grals are replaced by summation over discrete periodic line segments.
More specifically, the DPRT of an image , where
, is defined as
(1)
where , and . The
operator refers to the positive residue of any integer modulo a
positive integer . The corresponding dimension is of size
. We proved in [9] that the inverse transform of DPRT exists and is
then given by
(2)
where
(3)
Manuscript received June 17, 2010; accepted July 02, 2010. Date of publi-
cation July 19, 2010; date of current version October 13, 2010. The associate
editor coordinating the review of this manuscript and approving it for publica-
tion was Prof. Olgica Milenkovic.
The authors are with the Centre for Signal Processing, Department of Elec-
tronic and Information Engineering, The Hong Kong Polytechnic University,
Hong Kong (e-mail: enpklun@polyu.edu.hk).
Digital Object Identifier 10.1109/TSP.2010.2059021
1053-587X/$26.00 © 2010 IEEE
5964 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 11, NOVEMBER 2010
The above formulations are indeed the major difference from the tensor
transform. In the original tensor transform [5, eqs. (2), (3)] proposed in
1986, the projections are generated by taking all possible discrete pe-
riodic line summation for
where
. This generates projections which are highly
redundant. The inverse transform based on these pro-
jections certainly will also be highly redundant. For ,itwas
proposed that the reconstruction from tensor projections can be imple-
mented as follows [5, eq. (27)]:
(4)
where
, for , and , for
. Direct implementation of the above formulation
requires additions, which is an extremely heavy
loading ever for modern computer systems. Here we have not counted
the computational cost for indexing. For DPRT, (3) implies that the
inverse transform requires only additions and
bit-wise shifting operations. Let us take as an
example. Equation (4) requires additions while (3) requires
only additions plus bit-wise shifting operations.
It is a saving of more than an order of magnitude. Interested readers
may find the orthogonal version of DPRT, its efficient realization and
applications in [11]–[13]. One may argue that (3) is a subset of (4).
However, it is never trivial, if possible, to derive (3) from (4). In fact,
due to the redundancy of (4), the author of [1] proposed another opti-
mized inverse paired transform in [14], which was however published
seven years after our work.
Besides the efficient inverse transform, another important contribu-
tion of [9] is the discovery of the convolution property of the DPRT.
Similar to the classical continuous Radon transform, we proved in [9]
that the DPRT possesses the convolution property such that a 2-D cyclic
convolution of two functions and of size can be
converted into 3 /2 1-D cyclic convolutions in the DPRT domain. By
performing an inverse DPRT on the 1-D cyclic convolution results, the
2-D convolution of and can be obtained. We have shown in
[13] and [15] that such property is extremely useful for the fast realiza-
tion of discrete convolution, deconvolution and image restoration.
To conclude, we would like to point out that the claim in [1] that the
DPRT repeated the concept of the tensor transform or the later paired
transform is incorrect. The contributions of [9] are far more than that
described in [1]. The DPRT is not only a forward transform for com-
puting 2-D DFT. It is a complete discrete transformation method that
includes efficient inverse transform. We have also shown in [9] some
very important properties of DPRT such as the convolution property
which has never been mentioned in any publication of tensor transform
and paired transform earlier than our work.
REFERENCES
[1] A. M. Grigoryan, “Comments on ‘The discrete periodic Radon trans-
form’,” IEEE Trans. Signal Process., vol. 58, no. 11, Nov. 2010.
[2] A. M. Grigoryan, “An algorithm of the two-dimensional Fourier trans-
form,” (in Russian) Izvestiya VUZ SSSR, Radioelectronica, vol. 27, no.
10, pp. 52–57, 1984.
[3] A. M. Grigoryan, “An optimal algorithm for computing the two-dimen-
sional discrete Fourier transform,” (in Russian) Izvestiya VUZ SSSR,
Radioelectronica, vol. 29, no. 12, pp. 20–25, 1986.
[4] A. M. Grigoryan, “New algorithms for calculating discrete Fourier
transforms,” (in Russian) J. Vichislitelnoi Matematiki i Matematich-
eskoi Fiziki, vol. 26, no. 9, pp. 1407–1412, 1986.
[5] A. M. Grigoryan and M. M. Grigoryan, “Two-dimensional Fourier
transform in the tensor presentation and new orthogonal functions,”
(in Russian) Avtometria AS USSR Siberian Section, no. 1, pp. 21–27,
1986.
[6] A. M. Grigoryan and M. M. Grigoryan, “A new method of image recon-
struction from projections,” (in Russian) Electronnoe Modelirovanie,
vol. 8, no. 6, pp. 74–77, 1986, AS USSR, Kiev.
[7] A. M. Grigoryan and M. M. Grigoryan, “Algorithm of computerized
tomography,” (in Russian) Electronoe Modelirovanie, vol. 12, no. 4,
pp. 96–98, 1990.
[8] A. M. Grigoryan, “An algorithm for computing the discrete Fourier
transform with arbitrary orders,” (in Russian) J. Vichislitelnoi Matem-
atiki i Matematicheskoi Fiziki, vol. 30, no. 10, pp. 1576–1581, 1991.
[9] T. Hsung, D. P. K. Lun, and W.-C. Siu, “The discrete periodic
Radon transform,” IEEE Trans. Signal Process., vol. 44, no. 10, pp.
2651–2657, 1996.
[10] I. Gertner, “A new efficient algorithm to compute the two-dimensional
discrete Fourier transform,” IEEE Trans. Acoust. Speech, Signal
Process., vol. 36, no. 7, pp. 1036–1050, Jul. 1988.
[11] T.-W. Shen, T.-C. Hsung, and D. P.-K. Lun, “Inversion algorithm
for the discrete periodic Radon transform and application on image
restoration,” in Proc. IEEE Int. Symp. Circuits Systems, 1997, vol. 4,
pp. 2665–2668.
[12] D. P. K. Lun, T.-C. Hsung, and T. W. Shen, “Orthogonal discrete peri-
odic radon transform Part I: Theory and realization,” Signal Process.,
vol. 83, no. 5, pp. 941–955, May 2003.
[13] D. P. K. Lun, T.-C. Hsung, and T. W. Shen, “Orthogonal discrete pe-
riodic radon transform Part II: Applications,” Signal Process., vol. 83,
no. 5, pp. 957–971, May 2003.
[14] A. M. Grigoryan, “Method of paired transforms for reconstruction
of images from projections: Discrete model,” IEEE Trans. Image
Process., vol. 12, no. 8, pp. 985–994, Sep. 2003.
[15] D. P. K. Lun, T. C. L. Chan, T.-C. Hsung, D. D. Feng, and Y.-H. Chan,
“Efficient blind image restoration using discrete periodic Radon trans-
form,” IEEE Trans. Image Process., vol. 13, no. 2, pp. 188–200, 2004.
  • [Show abstract] [Hide abstract] ABSTRACT: This paper discusses the decomposition of the image by direction images, which is based on the concept of the tensor representation and its advanced form, the paired representation. The 2-D image is considered of the size N × N , where N is prime, a power of two, and a power of odd primes. The tensor and paired representations in the frequency-and-time domain define the image as a set of 1-D signals, which we call splitting-signals. Each of such splitting-signals is calculated as the sum of the image along the parallel lines, and it defines the direction image as a component of the original image. The unique decomposition of the image by direction images is described, and formulas for the inverse tensor and paired transforms are given. These formulas can be used for image reconstruction from projections, when splitting-signals or their direction images are calculated directly from the projection data. The number of required projections is uniquely defined by the tensor representation of the image.
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