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

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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Available from: Tai-Chiu Hsung, Dec 03, 2014IEEE 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.

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**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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