orthogonal discrete radon transform over images

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Signal Processing 86 (2006) 2040–2050

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Orthogonal discrete Radon transform over pn � pn images

Andrew Kingston�

Center for X-ray Physics and Imaging, School of Physics and Materials Engineering, Monash University, Vic. 3800, Australia

Received 7 November 2003; received in revised form 3 September 2004; accepted 28 September 2005

Available online 5 December 2005

Abstract

This paper presents a discrete Radon transform based on arrays of size pn � pn where p is prime ðpX2Þ and n 2 N. The

finite Radon transform presented in [F. Matus, J. Flusser, Image representation via a finite Radon transform, IEEE Trans.

Pattern Anal. Mach. Intell. 15 (10) (1993) 996–1006] and the discrete periodic Radon transform (DPRT) presented in [T.

Hsung, D. Lun, W. Siu, The discrete periodic Radon transform, IEEE Trans. Signal Process. 44 (10) (1996) 2651–2657] are

subsets of this more general transform with n restricted to 1 and p restricted to 2, respectively. This transform exactly and

invertibly maps the image as pn þ pn�1 projections of length pn wrapped under modulo pn arithmetic. Since pn has factors if

n41, this mapping is partly redundant and a version utilising orthogonal bases is required. An invertible form of the DPRT

with orthogonal bases, the orthogonal DPRT (ODPRT), was developed in [D. Lun, T. Hsung, T. Shen, Orthogonal discrete

periodic Radon transform. Part I: theory and realization, Signal Processing 83 (5) (2003) 941–955]. An orthogonal version for

the generalised pn case is developed here with applications analogous to those of the ODPRT presented in [D. Lun, T. Hsung,

T. Shen, Orthogonal discrete periodic Radon transform, Part II: applications, Signal Processing 83 (5) (2003) 957-971].

r 2005 Elsevier B.V. All rights reserved.

Keywords: Discrete Radon transform; Discrete Fourier slice theorem; Convolution property

1. Introduction

The discrete Radon transform (DRT) mapsdiscrete 2-D arrays, Iðx; yÞ, to a set of discrete 1-Dprojections. A projection at angle tan�1ðmÞ, RmðtÞ, isdefined along the straight line x ¼ myþ t for arange of intercepts, t, where m and t are integers.This class of transforms was defined by Beylkin in[1]. This paper develops a subset of this class wherethe projections are based on congruent mathematicsand the array sizes, N �N are restricted to N ¼ pn

where p is prime ðpX2Þ and n 2 N.

e front matter r 2005 Elsevier B.V. All rights reserved

pro.2005.09.024

9905 3653; fax: +61 3 9905 3637.

ess: [email protected].

The properties of the continuous Radon trans-form (RT) are preserved in a discrete form for theseperiodic DRT projections. A discrete form of theFourier slice theorem is demonstrated for the DRTand orthogonal DRT (ODRT) over pn. Thus, theconvolution property also holds for both the DRTand ODRT, making these transforms powerfultools for image representation.

The DRTs based on congruent mathematicspresented to date have been restricted to apply tosquare arrays, N �N where N is prime [2] or 2n

[3,4]. If the DRT is required for an arbitrary sizeM �N, the array must be padded out with zeroes tothe nearest size p� p or 2n � 2n with p and2nX supfM ;Ng. Padded transforms are redundant

and hence inefficient data representations.

.

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Fig. 1. Sample pattern generated by the DRT on a 33 � 33 array

by discrete line sums: (a) R4ð8Þ the line x � 4yþ 8 (mod 27), (b)

R?2 ð19Þ the line y � 6xþ 19 (mod27).

A. Kingston / Signal Processing 86 (2006) 2040–2050 2041

This work was prompted by the search for a DRTbased on congruent mathematics that applies toarrays of arbitrary size M �N. It was discoveredthat the simplest set of wrapped projections with acommon origin that completely tile Fourier spacearises when the array size is based on a uniqueprime, i.e., pn. While this does not remove thenecessity for square arrays, it does increase thedensity of possible array sizes over which the DRTcan be performed relative to the choices of onlyprimes and powers of 2. The DRT for arbitrary sizeM �N is the subject of ongoing work.

The development of a DRT based on modulo pn

arithmetic is presented in Section 2. A discrete formof the Fourier slice theorem is demonstrated inSection 3, the convolution property is shown inSection 4 and the inversion process for projections ispresented in Section 5. This DRT is furtherdeveloped to have orthogonal bases. A multi-resolutional ODRT based on modulo pn arithmeticis proposed in Section 6. The convolution propertyis shown in Section 7 and the inversion process isdeveloped in Section 8. The work presented hereextends and generalises the work by Lun et al., whodeveloped the DPRT [3] and ODPRT [4].

2. DRT over pn

The definition of the DRT over N ¼ pn needs tobe broken up into two sets of projections, essentiallyhorizontal line sums R and essentially vertical linesums R?. These sets of projections have a uniqueorigin, however, the intercept axes, which alsodetermines the direction of the straight lines overwhich the sums are performed, are perpendicular.Intercepts, t are along the x-axis for R and along they-axis for R?. An element RmðtÞ is defined as thesum of all pixels centred on the line x � myþ t

(modN). This approximates continuous line inte-grals as discrete line sums with gradient m withrespect to the x-axis. The corresponding angles,ym ¼ tan�1ð1=mÞ of these lines lie in the range0oympp=4 (except for R0ðtÞ). In a similar fashion,R?s ðtÞ is defined as the sum of all pixels centred ondiscrete lines with gradient sp with respect to the y-axis, i.e., on the line y � spxþ t (modN). The angle,ys ¼ tan�1ðspÞ with respect to the x-axis of thesediscrete lines lie in the range p=4oysop=2 (exceptfor R?0 ðtÞ).

Let l denote the gradient of an essentially verticaldiscrete line y � lx (modN). For each loN that isco-prime with N, there exists some moN also co-

prime with N such that ml � 1 (modN). Since bothl and m are co-prime with N, the discrete line x �

my (modN) produces the equivalent lattice (orsampling pattern) to the discrete line y � lx

(modN). Therefore, all l co-prime with N areentirely redundant and since N ¼ pn, any l not co-prime with N must be a multiple of p, i.e., l ¼ sp for0psoN=p.

Let R ðR?Þ denote a projection with a horizontal(vertical) intercept axis with modulo arithmeticapplied to the horizontal (vertical) ordinate. TheDRT applied to square arrays of size N �N whereN ¼ pn is then defined for intercepts 0ptoN as

RmðtÞ ¼XN�1x¼0

XN�1y¼0

Iðx; yÞdhx�my� tiN

¼XN�1y¼0

Iðhmyþ tiN ; yÞ; 0pmoN,

R?s ðtÞ ¼XN�1x¼0

XN�1y¼0

Iðx; yÞdhy� spx� tiN

¼XN�1x¼0

Iðx; hspxþ tiN Þ; 0psoN

p, ð1Þ

where hZix denotes Z (mod x) and dðZÞ is theKronecker delta function, dðZÞ ¼ 1 for Z ¼ 0 other-wise dðZÞ ¼ 0. An example of these discrete linesums for N ¼ 27 is depicted in Fig. 1.

It can be seen that for n ¼ 1 this is equivalent tothe finite RT [2] and for p ¼ 2 it is equivalent to theDPRT [3]. The N ¼ pn DRT contains N2 elementsin Rm and N2=p elements in R?s . Fig. 2b is adepiction of the generalised DRT of the Lena imageof size 243 ¼ 35 shown in Fig. 2a. It can be seen thatthis transform is redundant (the case where n ¼ 1can be represented such that it is non-redundant).The properties and effects of redundant projections

ARTICLE IN PRESS

Fig. 2. (a) 35 � 35 image, Iðx; yÞ (b) DRT of Iðx; yÞ, RmðtÞ ½35 �

35� and R?s ðtÞ ½35 � 34� .

A. Kingston / Signal Processing 86 (2006) 2040–20502042

for composite arrays was investigated by Svalbe in[5]. Here, there are enough projections to exactlyreconstruct the image from Rm, however, since pn

has factors other than 1 and itself 8n41, there is adegree of redundancy, which is accounted for byR?s . The case n ¼ 1, which corresponds to the finiteRT [2], p1 has no factors, so only Rm and R?0 arerequired. As each 1-D projection samples all pixelsin the 2-D image, the sum of all elements from oneprojection, gives the sum of the entire image, I sum.Therefore, one element of each projection can beignored, provided I sum is included once. For n ¼ 1,there are p1 þ 1 projections with p1 � 1 intercepts +I sum, which gives ðp1Þ

2 elements, so the DRT over p1

can be represented in a non-redundant form. Anorthogonal non-redundant version of the DRT overpn8n 2 N is presented in Section 6.

3. Discrete Fourier slice theorem for DRT

A well-known property of the continuous RT isthe Fourier slice theorem which states that the 1-DFourier transform (FT) of a continuous projectionat angle y is equivalent to a central radial slicethrough the 2-D FT of the original object/functionat the angle y? ¼ yþ p=2 [6]. A similar property,the discrete Fourier slice theorem, can be demon-strated for the DRT over pn. Denote RmðuÞ as the 1-D DFT of RmðtÞ. Then RmðuÞ can be found as

RmðuÞ ¼XN�1t¼0

RmðtÞe�i2put=N

¼XN�1t¼0

XN�1x¼0

XN�1y¼0

Iðx; yÞe�i2put=Ndhmyþ t� xiN

¼XN�1x¼0

XN�1y¼0

Iðx; yÞe�i2pðux�uhmyiN Þ=N

¼ Iðu; h�muiN Þ, ð2Þ

where Iðu; vÞ is the 2-D discrete FT (DFT) of Iðx; yÞ.Similarly R?s ðvÞ ¼ Iðh�spviN ; vÞ. The discrete Four-ier slice theorem states that the 1-D DFT of adiscrete modulo projection at angle y ¼ tan�1ðmÞ isequivalent to a central wrapped discrete linethrough the 2-D DFT of the original object/function at the angle y? ¼ tan�1ð�1=mÞ.

4. Convolution property of DRT

The convolution property of the continuous RT isconserved in the DRT. The 2-D convolution of twofunctions can be simplified to a set of 1-Dconvolutions on the DRT of the functions. Supposeit is desired to obtain F ðx; yÞ, where

F ðx; yÞ ¼XN�1a¼1

XN�1b¼0

Gða; bÞHðhx� aiN ; hy� biN Þ.

(3)

Let the DRT of Zðx; yÞ be RZmðtÞ and RZ?

s ðtÞ. Thevalue of F ðx; yÞ can be found through RF

mðtÞ andRF?

s ðtÞ using RGmðtÞ, RG?

s ðtÞ, RHm ðtÞ and RH?

s ðtÞ as

RFmðtÞ ¼

XN�1k¼0

RGmðht� kiN ÞR

Hm ðkÞ,

RF?s ðtÞ ¼

XN�1k¼0

RG?s ðht� kiN ÞR

H?s ðkÞ. ð4Þ

This shows the 2-D convolution of arrays can beperformed in DRT space as a set of 1-D convolu-tions of each discrete projection. This providesconsiderable efficiency, for example, in filtering andmatching a test object to candidate solutions, bycomparing the DRT projections rather than usingreal space features.

5. DRT inversion

The frequencies in the 2-D DFT of Iðx; yÞ, Iðu; vÞ,can be represented by RmðuÞ and R?s ðvÞ, which canbe found from the above discrete Fourier slicetheorem. Since the transform is redundant, somefrequencies in Iðu; vÞ are represented more than onceby several projections. When the arrays are re-stricted to size pn, a distinct pattern emerges; thenumber of times each frequency is representedby the DRT, n, can be found as nðu; vÞ ¼

ARTICLE IN PRESSA. Kingston / Signal Processing 86 (2006) 2040–2050 2043

gcdðgcdðu; vÞ; pnÞ. An example of this pattern forN ¼ 27 is shown in Fig. 3c. This figure demon-strates that the function nðu; vÞ can be viewed as anover-representation of the frequencies at n resolu-tions, where u and v are a multiple of p; p2; . . . up topn, i.e., nðpu; pvÞ, nðp2u; p2vÞ; . . . ; nðpnu; pnvÞ. Thispattern is utilised in the inversion process. Fig. 3aand b shows nðu; vÞ for RmðtÞ and R?s ðtÞ, respectively.It can be seen that by combining the 1-D DFT of allprojections from Rm and R?s , all spatial frequenciesof the image are represented without omission (butwith some redundancy).

The inversion process to obtain Iðx; yÞ from the 2-D DFT, Iðu; vÞ is

Iðx; yÞ ¼1

N2

XN�1u¼0

XN�1v¼0

Iðu; vÞei2pðuxþvyÞ=N . (5)

Using the multi-resolutional structure of nðu; vÞfrom the 1-D DFT of all DRT projections as

Fig. 3. A portion of nðu; vÞ for N ¼ 33, the number of times each spa

R?s ðvÞ, ½nsðu; vÞ�, and (c) fRmðuÞ; R?s ðvÞg, ½nðu; vÞ�.

described above, the inversion process (5) can bedecomposed as

Iðx; yÞ ¼1

N2

XN�1m¼0

XN�1u¼0

Iðu; h�muiNÞei2pðux�muyÞ=N

þ1

N2

XNp�1s¼0

XN�1v¼0

Iðh�spviN ; vÞei2pð�spvxþvyÞ=N

�p� 1

N2

XNp�1u¼0

XNp�1v¼0

Iðpu; pvÞei2pðpuxþpvyÞ=N

�ðp� 1Þp

N2

XNp2�1u¼0

XNp2�1v¼0

Iðp2u; p2vÞei2pðp2uxþp2vyÞ=N

..

.

�1

NIð0; 0Þ. ð6Þ

tial frequency in Iðu; vÞ is represented by (a) RmðuÞ, ½nmðu; vÞ�, (b)

ARTICLE IN PRESSA. Kingston / Signal Processing 86 (2006) 2040–20502044

By the discrete Fourier slice theorem, the first termin (6) becomes

1

N2

XN�1m¼0

XN�1u¼0

XN�1t¼0

RmðtÞe�i2put=Nei2pðux�muyÞ=N

¼1

N2

XN�1m¼0

XN�1u¼0

XN�1t¼0

RmðtÞei2puðx�my�tÞ=N

¼1

N2

XN�1m¼0

NXN�1t¼0

RmðtÞdhmyþ t� xiN

¼1

N

XN�1m¼0

Rmðhx�myiN Þ ð7Þ

and similarly the second term in (6) becomesð1=NÞ

PN=p�1s¼0 R?s ðhy� spxiNÞ. The remaining terms

in (6) can be further decomposed in the samemanner as (5) was to (6). This is done below for thethird term of (6):

�p� 1

N2

XNp�1m¼0

XNp�1u¼0

Iðpu; h�pmuiNÞei2pðpux�pmuyÞ=N

�p� 1

N2

XNp2�1s¼0

XNp�1v¼0

Iðh�sp2viN ; pvÞei2pð�sp2vxþpvyÞ=N

þðp� 1Þ2

N2

XNp2�1u¼0

XNp2�1v¼0


þðp� 1Þ2p

N2

XNp3�1u¼0

XNp3�1v¼0


..

.

þðp� 1Þpn�1

N2Ið0; 0Þ. ð8Þ

Fig. 4. Depiction of the correction made by each resolution of the reco

x0ðx; yÞ �p

N2

PN�1t¼0 RmðtÞ, (b) also including x1ðx; yÞ, (c) also including x

required corrections and exactly reproduces the original image.

By the discrete Fourier slice theorem

Iðpu; h�pmuiNÞ

¼XN�1t¼0

RmðtÞe�i2pupt=N

¼XNp�1t¼0

Xp�1j¼0

Rm tþ jN

p

� �" #e�i2pupt=N , ð9Þ

so the first term in (8) becomes

�p� 1

pN

XNp�1m¼0

Xp�1j¼0

Rm hx�myiNpþ j

N

p

� �" #. (10)

Similarly the second term in (8) is found as

�p� 1

pN

XNp2�1s¼0

Xp�1j¼0

R?s hy� spxiNpþ j

N

p

� �" #.

The above process corrects the reconstruction atresolution Iðpu; pvÞ. The same process (7)–(10) isemployed to correct the reconstruction at resolu-tions Iðp2u; p2vÞ (the fourth term in (6)), through toIðpnu; pnvÞ ¼ Ið0; 0Þ (the last term in (6)). The effectof each of these resolution corrections, representedby xi in (11), is depicted in Fig. 4. The resultinginverse equation is therefore

Iðx; yÞ ¼ x0ðx; yÞ �Xn�1i¼1

xiðx; yÞ �p

N2

XN�1t¼0

RmðtÞ,

(11)

nstruction of 35 � 35 generalised DPRT of Fig. 2a. (a) using only

2ðx; yÞ, (d) also including x3ðx; yÞ. Including x4ðx; yÞ completes all


where

x0 ¼1

N

XN�1m¼0

Rmðhx�myiN Þ þXNp�1s¼0

R?s ðhy� psxiN Þ

24

35,

xi ¼p� 1

piN

PNpi�1

m¼0

Ppi�1

j¼0

Rm hx�myiNpiþ j N

pi

� �

þPNpiþ1�1

s¼0

Ppi�1

j¼0

R?s hy� psxiNpiþ j N

pi

� �

266666664

377777775.

The transform and inversion can be performed inOðN2ð1þ 1=pÞ log NÞ by utilising the discrete Four-ier slice theorem. The 2-D DFT of the image can beperformed in OðN2 log NÞ [7] to produce Iðu; vÞ,each of the Nð1þ 1=pÞ projections can be obtainedin OðN log NÞ as the inverse 1-D DFT of N

elements of I according to (2). For the case of theinverse transform, this equation can be used togenerate the 2-D DFT of the image, Iðu; vÞ from theinverse 1-D DFT of each projection. Some spatialfrequencies will be over-represented as describedearlier in this section. These must be divided bynðu; vÞ ¼ gcdðgcdðu; vÞ;NÞ. The correctly normalisedimage can then be obtained from the inverse 2-DDFT of I .

6. ODRT over pn

The DRT for N ¼ pn, through the discreteFourier slice theorem and convolution property,reduces the computational complexity of 2-Dproblems by transforming them into a set of 1-Dproblems. However, the 2-D computation over N �

Fig. 5. The pattern in Fourier space for pn ¼ 32, of spatial frequ

N elements is reduced to Nð1þ 1=pÞ 1-D computa-tions over N elements. A degree of redundancy isintroduced by the non-orthogonal bases of theDRT. To further reduce the computational com-plexity, the definition of the DRT shall next berefined to remove any redundancy in the transform,such that the discrete Fourier slice theorem and thusconvolution property still hold. It was seen inSection 5 that the DRT for pn � pn over-representsFourier space. To produce bases which are ortho-gonal, hence non-redundant, it is required to sampleeach of the N �N frequencies in 2-D Fourier spaceonly once. To produce these orthogonal bases, werestrict the projections of the DRT to tile Fourierspace uniquely and minimally and then take theinverse 1-D DFT of these wrapped discrete Fourierslices to find the form of the projections in Radonspace.

Taking note of the structure of the function thatgives the degree of oversampling of spatial frequen-cies nmðu; vÞ for Rm, (shown in Fig. 3a), it can beseen that nmðu; vÞ is only greater than 1 where u � 0(mod p). Let wm

q ðuÞ contain all elements of Iðu; vÞwhere uc0 (mod p). i.e., wm

q ðuÞ ¼ Iðpuþ

q; h�mðpuþ qÞiN Þ for 0pmoN, 0oqop. This isshown in Fig. 5a for N ¼ 32. Note from thestructure of spatial frequency representation forR?s demonstrated in Fig. 3b for N ¼ 33, nsðu; vÞ isalways 0 except where u � 0 (mod p) and is always 1in these columns except where v � 0 (mod p). Letw?s

q ðvÞ contain all elements of Iðu; vÞ where vc0(mod p). i.e., w?s

q ðvÞ ¼ Iðh�spðpvþ qÞiN ; pvþ qÞ for0psoN=p, 0oqop. This is shown in Fig. 5b forN ¼ 32. Let w1ðu; vÞ represent the remaining ele-ments of Iðu; vÞ. This is shown in Fig. 5c for N ¼ 32

and is of the form w1ðu; vÞ ¼ Iðpu; pvÞ for u and v in½0;N=pÞ.

encies represented by (a) wmq ðuÞ, (b) w

?sq ðvÞ and (c) w1ðu; vÞ.


These projections (which are derived in Section6.1) are defined as follows for m and t in ½0;NÞ andfor s, x and y in ½0;N=pÞ and 0oqop:

wmq ðtÞ ¼ e�i2ptq=N

Xp�1j¼0

Rm tþ jN

p

� �e�i2pjq=p,

w?sq ðtÞ ¼ e�i2ptq=N

Xp�1j¼0

R?s tþ jN

p

� �e�i2pjq=p,

w1ðx; yÞ ¼Xp�1j¼0

Xp�1k¼0

I xþ jN

p; yþ k

N

p

� �. ð12Þ

The e�i2ptq=N term represents a frequency shiftof 2pq=N in Fourier space. An example of this shiftis depicted in Fig. 6b. Define X to be the pro-jections w with this term removed. This givesX m

q ðtÞ ¼ wmq ðtÞe

i2ptq=N , X?sq ðtÞ ¼ w?s

q ðtÞei2ptq=N and

X 1ðx; yÞ ¼ w1ðx; yÞ. For the ODPRT case, wherep ¼ 2, these projections become equivalent to(10)–(11) in [4]

X mq ðtÞ ¼ RmðtÞ � Rm tþ

N

2

� �,

X?sq ðtÞ ¼ R?s ðtÞ � R?s tþ

N

2

� �,

X 1ðx; yÞ

¼ Iðx; yÞ þ I x; yþN

2

� �

þ I xþN

2; y

� �þ I xþ

N

2; yþ

N

2

� �. ð13Þ

The frequency shift of 2pq=N must be included forthe discrete Fourier slice theorem to apply directly,however, its inclusion is unnecessary for inversionof the ODRT and the convolution property holds

Fig. 6. (a) An example of a discrete line sum in the ODRT for N ¼ 27

from the DRT that are separated by N=p ¼ 9 from t multiplied by a pha

Fourier transform of the ODRT projections. In this case, N ¼ 27 and

regardless. Therefore, the phase factor is viewed asbeing superfluous and is omitted in all equationsfollowing the derivation of w and thus X in Section6.1. The subsequent equations and proofs have beenperformed on X, however, they still hold for w.

An example depicting the discrete line sums for X

is presented in Fig. 6a. They are essentially thesummation of all parallel DRT line sums R, withintercepts separated by N=p. The phase factorcomes from a frequency shift of multiples of 2p=p

to align N frequencies to N=p locations in Fourierspace.

6.1. Projection derivation

w1ðu; vÞ represents the 2-D DFT of a ‘‘collapsedimage’’, w1ðx; yÞ, termed in [4] as the ‘‘spatial alias’’.Substitute w1 into the inverse of (5) to obtain

w1ðu; vÞ ¼ Iðpu; pvÞ

¼XN�1x¼0

XN�1y¼0

Iðx; yÞe�i2pðpuxþpvyÞ=N

¼XNp�1x¼0

XNp�1y¼0

Xp�1j¼0

Xp�1k¼0

I xþ jN

p; yþ k

N

p

� �" #

�e�i2pðuxþvyÞ=Np

¼XNp�1x¼0

XNp�1y¼0

w1ðx; yÞe�i2pðuxþvyÞ=Np , ð14Þ

so

w1ðx; yÞ ¼Xp�1j¼0

Xp�1k¼0

I xþ jN

p; yþ k

N

p

� �

for x and y in ½0;N=pÞ.

;m ¼ 1; t ¼ 0; q ¼ 1, i.e., X 11ð0Þ. The sum of all discrete line sums

se factor. (b) An example of the frequency shift, q, required for the

projection w262 ðuÞ has a shift of q ¼ 2 from the origin.

ARTICLE IN PRESS

Fig. 7. (a) Diagram demonstrating the multi-resolutional decomposition performed by the ODRT. Re and Im denote real and imaginary

components. Note only q ¼ 1 is required for p ¼ 3. (b) 35 � 35 ODRT of Iðx; yÞ from Fig. 2a.

A. Kingston / Signal Processing 86 (2006) 2040–2050 2047

The remaining subsets of Fourier space can be pro-duced from the DRT projections. Utilising the discreteFourier slice theorem, substitute wm

q into (2) to obtain

wmq ðuÞ ¼ Iðpuþ q; h�mðpuþ qÞiN Þ

¼XN�1t¼0

RmðtÞe�i2pðpuþqÞt=N

¼XNp�1t¼0

e�i2pqt=NXp�1j¼0

Rm tþ jN

p

� �e�i2pjq=p

" #

�e�i2put=Np ¼

XNp�1t¼0

wmq ðtÞe

�i2put=Np , ð15Þ

where wmq ðtÞ ¼ e�i2pqt=N

Pp�1j¼0 Rmðtþ j N

pÞe�i2pjq=p. Si-

milarly w?sq ðvÞ ¼

PN=p�1t¼0 w?s

q ðtÞe�i2pðpvþqÞt=N where

w?sq ðtÞ ¼ e�i2pqt=N

Pp�1j¼0 R?s ðtþ j N

pÞe�i2pjq=p.

From this point onwards, the frequency shift of2pq=N will be omitted from the projections, givingX m

q ðtÞ ¼ wmq ðtÞe

i2ptq=N , X?sq ðtÞ ¼ w?s

q ðtÞei2ptq=N and

X 1ðx; yÞ ¼ w1ðx; yÞ. The orthogonality of these basisfunctions is shown in Appendix A.

The transform is then applied to the spatial aliasX 1ðx; yÞ to produce a further collapsed image,X 2ðx; yÞ, and complex projections, ðX 1Þ

?sq ðtÞ and

ðX 1Þmq ðtÞ. This process is repeated down to size

p� p, producing a multi-resolutional projectivetransform. An example for N ¼ 243 is depicted inFig. 7. The multi-resolutional nature may be utilisedfor scale-based image processing applications.

The problem of redundancy seems to remain,since the image with N2 real values has beenmapped to projection space with N2 complexvalues. Note, however, that X m

p�qðtÞ ¼ X m�q ðtÞ and

similarly X?sp�qðtÞ ¼ X?s�

q ðtÞ where z� is the complex

conjugate of z. Since there are p� 1 possible valuesfor q and p� 1 is even 8p42, only ðp� 1Þ=2, X m

q ðtÞ

and X?sq ðtÞ are required, the remaining elements for

q4ðp� 1Þ=2 can be found as the conjugate of thevalue for p� q. For the case of p ¼ 2, q ¼ p� q,which implies that there is no imaginary componentof the DRT.

7. Convolution property of ODRT

For the convolution given in (3), let the ODRT ofZðx; yÞ be ðX ZÞ

mq ðtÞ and ðX ZÞ

?sq ðtÞ. The value of

ðX F Þmq ðtÞ using the result of (4) can be found as

ðX F Þmq ðtÞ ¼

Xp�1j¼0

RFm tþ j

N

p

� �e�i2pjq=p

¼Xp�1j¼0

XN�1k¼0

RGmðkÞR

Hm

� tþ jN

p� k

� �N

� �e�i2pjq=p

¼XNp�1k¼0

Xp�1a¼0

Xp�1j¼0

RGm k þ a

N

p

� �

�RHm t� k þ ðj � aÞ

N

p

� �N

� �e�i2pjq=p

¼XNp�1k¼0

Xp�1a¼0

Xp�1b¼0

RGm k þ a

N

p

� �

�RHm t� k þ b

N

p

� �N

� �e�i2pðaþbÞq=p

¼XNp�1k¼0

ðX GÞmq ðkÞðX

HÞmq ðht� kiN Þ. ð16Þ


Similarly ðX F Þ?sq ðtÞ ¼

PNp�1

k¼0 ðXGÞ?sq ðkÞðX

H Þ?sq ðht�

kiN Þ. As with the DRT, the 2-D convolution of arrayscan be performed in ODRT space as a set of 1-Dconvolutions of each discrete projection. Thus, theODRT has applications analogous to those of theODPRT, such as the 2-D circular convolution andblind image restoration presented in [8] but for a moreflexible choice of array sizes as pn rather than 2n.

8. ODRT inversion

The inversion process for the orthogonal basis setis again developed in Fourier space. The inverse 2-DDFT, (5) can be written as follows:

Iðx; yÞ ¼1

N2

XNp�1u¼0

XNp�1v¼0

Iðpu; pvÞei2pðpuxþpvyÞ=N

24

þXp�1q¼1

XNp�1u¼0

XN�1v¼0

Iðpuþ q; vÞei2pððpuþqÞxþvyÞ=N

þXp�1q¼1

XNp�1u¼0

XNp�1v¼0

Iðpu; pvþ qÞ

�ei2pðpuxþðpvþqÞyÞ=N

35, ð17Þ

where the first term in (17) becomes ð1=p2ÞX 1ðx; yÞ,as from the inverse DFT of (14),

p2

N2

XNp�1u¼0

XNp�1v¼0

Iðpu; pvÞei2pðuxþvyÞ=Np ¼ X 1ðx; yÞ. (18)

The second term in (17) can be rewritten as

1

N2

Xp�1q¼1

XNp�1u¼0

XN�1m¼0

Iðpuþ q; h�mðpuþ qÞiNÞ

�ei2pððpuþqÞx�mðpuþqÞyÞ=N

¼1

N2

Xp�1q¼1

XNp�1v¼0

XN�1m¼0

XNp�1t¼0

X mq ðtÞe

�i2pðpuþqÞt=N

�ei2pððpuþqÞx�mðpuþqÞyÞ=N

¼1

N2

Xp�1q¼1

XNp�1v¼0

XN�1m¼0

XNp�1t¼0

X mq ðtÞe

i2pðpuþqÞðx�my�tÞ=N

¼1

N2

Xp�1q¼1

N

p

XN�1m¼0

X mq ðhy�mxiN ÞdNðtþmy� xÞ

¼1

pN

Xp�1q¼1

XN�1m¼0

X mq ðhy�mxiN Þ. ð19Þ

Similarly, the third term in (17) becomesð1=pNÞ

Pp�1q¼1

PN=p�1s¼0 X?s

q ðhx� psyiN Þ. Therefore,Iðx; yÞ is found from the spatial alias and allprojections as follows:

Iðx; yÞ ¼1

p2X 1ðhxiN

p; hyiN

pÞ

þ1

pN

Xp�1q¼1

XN�1m¼0

X mq ðhy�mxiNÞ

24

þXNp�1s¼0

X?sq ðhx� psyiNÞ

35. ð20Þ

This must be applied to each resolution of theimage from X nðx; yÞ and projections ðX nÞ

mq ðtÞ and

ðX nÞ?sq ðtÞ up to X 1ðx; yÞ and projections X m

q ðtÞ andX?s

q ðtÞ. i.e.,

X kðx; yÞ ¼1

p2ðkþ1ÞX kþ1ðhxi N

pkþ1; hyi N

pkþ1Þ

þ1

pkþ1N

Xp�1q¼1

XNpk�1

m¼0

ðX kÞmq ðhy�mxiN

pkÞ

264

þXNpkþ1�1

s¼0

ðX kÞ?sq ðhx� psyiN

pkÞ

375, ð21Þ

for k ¼ n� 1 down to k ¼ 0, giving X 0 ¼ I .

The orthogonal transform and inversion can beobtained in OðN2ð1þ 1=pÞ log NÞ by utilising thediscrete Fourier slice theorem to obtain projectionsw in a method analogous to that for the DRT overpn described in Section 5. w can then be converted toX by multiplying by the spatial frequency shiftei2ptq=N outlined in Section 6.

9. Conclusions

A generalised DRT based on modulo arithmetichas been introduced which applies to arrays of sizepn � pn where p is prime ðpX2Þ and n 2 N. Thisencompasses both the finite RT, proposed in [2],applied to prime sized arrays and the DPRT,developed in [3], based on arrays of size 2n. The


computational complexity for the forward andinverse transforms over pn presented is of the sameorder as the above transforms, with DRT(p)oDRTð2nÞÞpDRTðpnÞ�OðN3Þ operations. How-ever, as a discrete form of the Fourier slice theoremapplies to all of these transforms, they can beperformed in OðN2ð1þ 1=pÞ log NÞ operations. Amulti-resolutional transform with orthogonal basesdeveloped from this DRT over pn is developed thatis analogous to that for the DPRT and ODPRT.This may prove useful for scale-based imageanalysis and processing. A discrete form of theFourier slice theorem and the convolution propertywas shown to hold for both the DRT and ODRT,this enables reduction of computational complexityof 2-D image processing problems by breaking themdown into a set of 1-D problems.

Acknowledgements

Andrew Kingston is a Monash University post-graduate student in receipt of an AustralianPostgraduate Award from the Australian Govern-ment. Thanks to Graham Farr for assistance withcongruent mathematics and to supervisor ImantsSvalbe for his direction, inspiration and thoughtprovoking discussions.

Appendix A. Orthogonality of the ODRT

The inner product for basis functions P and Q

mapping ðx; yÞ image space to ðt;m; qÞ Radon space,i.e., Pðx; y; t;m; qÞ and Qðx; y; t;m; qÞ is defined as

ðPt;m;qjQt0;m0;q0 Þ ¼XN�1x¼0

XN�1y¼0

Pðx; y; t;m; qÞ

�Q�ðx; y; t0;m0; q0Þ. ðA:1Þ

The requirement for bases P and Q to be orthogonalis ðPjQÞ ¼ Kdðt� t0Þdðm�m0Þdðq� q0Þ for some K.

The basis functions for the ODRT are

Bðx; y; t;m; qÞ ¼Xp�1j¼0

d x�my� t� jN

p

� �N

e�i2pjq=p

such that X mq ðtÞ

¼XN�1x¼0

XN�1y¼0

Iðx; yÞBðx; y; t;m; qÞ, ðA:2Þ

B?ðx; y; t; s; qÞ ¼Xp�1j¼0

d y� spx� t� jN

p

� �N

e�i2pjq=p

such that X?sq ðtÞ

¼XN�1x¼0

XN�1y¼0

Iðx; yÞB?ðx; y; t; s; qÞ, ðA:3Þ

B1ðx; y; a; bÞ ¼ dhx� aiNpdhy� biN

p

such that X 1ða; bÞ

¼XN�1x¼0

XN�1y¼0

Iðx; yÞB1ðx; y; a; bÞ. ðA:4Þ

A.1. Self orthogonality of B

ðBt;m;qjBt0;m0;q0 Þ

¼XN�1x¼0

XN�1y¼0

Bðx; y; t;m; qÞB�ðx; y; t0;m0; q0Þ

¼XN�1x¼0

XN�1y¼0

Xp�1j¼0

d x�my� t� jN

p

� �N

e�i2pjq=p

�Xp�1k¼0

d x�m0y� t0 � kN

p

� �N

ei2pkq0=p

¼XN�1x¼0

XN�1y¼0

Xp�1j¼0

Xp�1k¼0

d Dmyþ Dt

�

þðk � jÞN

p

�N

ei2pðkq0�jqÞ=p ðA:5Þ

where DZ ¼ ðZ0 � ZÞ. It can be seen that A.5 is pN2 ifDm ¼ Dt ¼ Dq ¼ 0. From here there are two casesdepending on the value of Dm (mod p). If Dmc0(mod p) then gcdðDm;N=pÞ ¼ 1 and there exists Dm

such that DmDm � 1 (modN=p). Therefore, d inA.5 is 1 only when y � �DmDt (modN=p).

NpXp�1j¼0

Xp�1k¼0

Xp�1l¼0

d Dm DmDtþ lN

p

� ��

þDtþ ðk � jÞN

p

�N

ei2pðkq0�jqÞ=p

¼ NpXp�1j¼0

Xp�1k¼0

Xp�1l¼0

dhDml þ k � jipei2pðkq0�jqÞ=p

¼ NpXp�1k¼0

Xp�1l¼0

ei2pðkq0�ðkþDmlÞqÞ=p

¼ NpXp�1k¼0

ei2pkq0=p ¼ 0. ðA:6Þ


If Dm � 0 (mod p) then Dm ¼ mp and (A.5) becomes

NpXN�1y¼0

Xp�1j¼0

Xp�1k¼0

d mpyþ Dtþ ðk � jÞN

p

� �N

ei2pðkq0�jqÞ=p,

ðA:7Þ

if Dtc0 (mod p) then (A.7) is 0 since the term in dcannot be � 0modN. If Dt � 0 (mod p) then Dt ¼

tp and (A.7) becomes

NpXN�1y¼0

Xp�1j¼0

Xp�1k¼0

d myþ tþ ðk � jÞN

p2

� �N=p

ei2pðkq0�jqÞ=p,

ðA:8Þ

which is of the same form as (A.5) and the proof iscircular. Hence ðBt;m;qjBt0;m0;q0 Þ ¼ pN2dðt0 � tÞdðm0 �mÞdðq0 � qÞ and Bðx; y; t;m; qÞ is self-orthogonal.Similarly, it can be shown that B? is self-ortho-gonal.

A.2. Orthogonality of B and B?

ðBt;m;qjB?t0;s;q0 Þ

¼XN�1x¼0

XN�1y¼0

Bðx; y; t;m; qÞB?�ðx; y; t0; s; q0Þ

¼XN�1x¼0

XN�1y¼0

Xp�1j¼0

d x�my� t� jN

p

� �N

e�i2pjq=p

�Xp�1k¼0

d y� spx� t0 � kN

p

� �N

ei2pkq0=p

¼ NXN�1y¼0

Xp�1j¼0

Xp�1k¼0

d yð1� spmÞ � t0�

�spt� ðspj þ kÞN

p

�N

ei2pðkq0�jqÞ=p, ðA:9Þ

since gcdð1� spm; pÞ ¼ 1 there exists 1� spm suchthat 1� spmð1� spmÞ � 1 (modN=p). Therefore, din (A.10) is 1 only when y � 1� spmðt0 þ sptÞ

(modN=p).

NXp�1j¼0

Xp�1k¼0

Xp�1l¼0

d ðð1� spmÞl � spj � kÞN

p

� �N

� ei2pðkq0�jqÞ=p

¼ NXp�1j¼0

Xp�1k¼0

Xp�1l¼0

dhl � spj � kipei2pðkq0�jqÞ=p

¼ pNXp�1j¼0

Xp�1k¼0

dhspj þ kipei2pðkq0�jqÞ=p

¼ pNXp�1j¼0

e�i2pjq=p ¼ 0. ðA:10Þ

A.3. Orthogonality of B and B1

ðBt;m;qjB1a;bÞ

¼XN�1x¼0

XN�1y¼0

Bðx; y; t;m; qÞB1�ðx; y; a; bÞ

¼XN�1x¼0

XN�1y¼0

Xp�1j¼0

d x�my� t� jN

p

� �N

�e�i2pjq=pdhx� aiN=pdhy� biN=p

¼ p2Xp�1j¼0

dha�mb� tiN=pe�i2pjq=p ¼ 0.

ðA:11Þ

Similarly, it can be shown that B? and B1 areorthogonal.

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