generalised finite radon transform for n×n images

www.elsevier.com/locate/imavis

Image and Vision Computing 25 (2007) 1620–1630

Generalised finite radon transform for N·N images q

Andrew Kingston *, Imants Svalbe

School of Physics, Monash University, Clayton, Melbourne, Vic. 3800, Australia

Received 30 November 2005; accepted 9 March 2006

Abstract

This paper extends the domain of the finite radon transform (FRT) to apply to square arrays of arbitrary size. The FRT is a discreteformalism of the Radon transform that assumes the image is periodic over the finite array Z2

N and requires only arithmetic operations forboth forward and exact inverse transformation. The FRT is useful in image processing applications such as tomographic reconstruction[I. Svalbe, D. van der Spek, Reconstruction of tomographic images using analog projections and the digital Radon transform, LinearAlgebra and Its Applications 339 (15) (2001) 125–145.], image representation [M. Do, M. Vetterli, Finite ridgelet transform for imagerepresentation, IEEE Transactions on Image Processing 12(1).] image convolution [D. Lun, T. Chan, T. Hsung, D. Feng, Y. Chan, Effi-cient blind image restoration using discrete periodic Radon transform, IEEE Transactions on Image Processing 13(2) (2004) 188–200.],image watermarking and encryption [A. Kingston, I. Svalbe, Projective transforms on periodic discrete image arrays, to appear inP. Hawkes (Ed), Advances in Imaging and Electron Physics (2006).], and robust data transmission [A. Kingston, I. Svalbe, Geometricshape effects in redundant keys used to encrypt data transformed by finite discrete Radon projections, In: Proc. 8th Int. Conf. on DigitalImage Computing: Techniques and Applications (2005).].

The original definition by Matus and Flusser in 1993 [F. Matus, J. Flusser, Image representation via a finite Radon transform, IEEETransactions on Pattern Analysis and Machine Intelligence 15(10) (1993) 996–1006.] was restricted to apply only to square arrays ofprime size, p·p. Hsung, Lun and Siu developed an FRT that also applied to dyadic square arrays, 2n·2n, called the discrete periodicradon transform (DPRT) [T. Hsung, D. Lun, W. Siu, The discrete periodic Radon transform, IEEE Transactions on Signal Processing44(10) (1996) 2651–2657.]. Kingston further extended this to define an FRT that applies to prime-adic arrays, pn·pn.

This paper defines a generalised FRT that applies to square arrays of arbitrary size, N·N for N 2 N The Fourier slice theorem andconvolution property (two important properties of the classical Radon transform) are established for this FRT. The original image canbe reconstructed exactly from the FRT projections using Fourier inversion and back-projection. New methods are identified to correctfor the over-representation of pixels due to the compositeness of N. A remarkable result is established that enables 2D sampling patternsto be corrected by an angle invariant 1D filter prior to back-projection.� 2006 Elsevier B.V. All rights reserved.

Keywords: Discrete radon transform; Discrete projection; Fourier slice theorem; Image representation

1. Introduction

The classical radon transform (RT) projects a continu-ous 2D function f to a set of 1D functions

0262-8856/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.imavis.2006.03.002

q Part of this work was originally presented at DGCI-05 in Poitiers [1].Here the proofs are more rigorous and the section on reconstruction fromprojections has been extended, in particular to include the result obtainedusing filtered back projection with a digital form of the Ram Lak filterproviding exact reconstruction applicable to discrete N·N arrays.

* Corresponding author. Fax: +33 2 4068 3232.E-mail addresses: [email protected] (A. Kingston),

[email protected] (I. Svalbe).

r = {rh|0 � h < p} called projections. An element of theprojection, rh(q), is found as the integral of f over the liney sin h + x cos h = q. A projection, rh, is the set of all par-allel line integrals rh ¼ frhðqÞjq 2 Rg. Johan Radon firstdemonstrated in 1917 that the original function could berecovered from this projection mapping [9]. The existenceof the inverse transform is one of the most important prop-erties of the RT, enabling nondestructive tomographicimaging of an object from its projected images. This prop-erty is utilised in areas ranging from astronomy and seis-mology to medical imaging [9].

mailto:[email protected]

mailto:[email protected]

x

y

0

0(a) (b)

1 2 3 4 5 6 0 1 2 3 4 5 6

1

2

3

4

5

6

(t)

x

y

0

1

2

3

4

5

6(t)

ig. 1. Examples of discrete lines on p·p arrays. Shaded points lie on theentre of the wrapped lines, the values of these pixels are summed to givehe value of one element in the transform. (a) Pixels on x”2y + 3 (mod 7)um to give R2(3). Since, the array size is prime, only one element on eachow and column is sampled by a discrete wrapped line. (b) Pixels on”0x + 3 (mod 7) sum to give R?0 ð3Þ.

A. Kingston, I. Svalbe / Image and Vision Computing 25 (2007) 1620–1630 1621

Other important properties of the RT include the Fou-rier slice theorem and the convolution property. The Fou-rier slice theorem relates the RT to the Fourier transform(FT), as the 1D FT of a projection, rh, represents the spatialfrequencies along a slice at angle h through the 2D FT ofthe original function, f^

, i.e. rhðxÞ ¼ f^ðx cos h; x sin hÞ[10]. The convolution property as a consequence enablesthe 2D convolution of functions to be performed on theRT as a set of 1D convolutions over the projections.

The classic RT described above maps a point in theimage, (j,k), to a sinusoid in the RT as q ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffij2 þ k2

psinðtan�1ðj=kÞ þ hÞ, giving rise to the name sinogram forthis representation of the RT. Another mapping used byseismologists (known as the s�p transform) and describedby Edholm and Herman in 1987 [11], maps the image to alinogram where line integrals are based on gradient, m andintercept t, according to x + my = t. In this representationthe point (j, k) is mapped to a line—hence the term lino-gram—and back-projection is also performed using thesame projection operator. It is common to perform the lin-ogram mapping in two stages, essentially horizontal lines:sx + y = t for�1 � s < 1 and essentially vertical lines:x + my = t for�1 � m < 1, to ensure |tan h| � 1.

To reconstruct exactly a continuous function from pro-jections, continuous projection data is required at all anglesin [0, p). In practice however, only a finite set of projectionsare obtained, each with a finite resolution set by the detec-tor aperture, thus only a discrete approximation to the ori-ginal function can be reconstructed. As well astomographic reconstruction, the RT is useful for manyimage processing applications such as locating linear fea-tures, detecting and isolating motion, denoising, blurdeconvolution and feature representation. Since, all theseapplications of the RT are in essence discrete, a discreteformalism of the RT that minimises the need for interpola-tion in projection and reconstruction would be ideal.

The RT has no 1D analogue, therefore the definition ofa discrete RT is nontrivial and many formalisms have beenproposed [12–14]. The finite radon transform (FRT) [7] is avery simple but powerful definition requiring only arithme-tic operations for the transform and exact inversion withno interpolation. It is almost orthogonal with a minimumangle between basis functions of cos�1(1/p) approachingp/2 with increasing p [15]. The FRT applies to squareimage data, p·p where p is prime, and assumes the imageis periodic with p in both the x and y directions (as doesthe 2D discrete Fourier transform (DFT)) by defining thep·p array as the finite group Z2

p under addition. Both theFourier slice theorem and convolution property hold forthe FRT [7]. Since, p is prime there are only p + 1 distinctline directions (thus p + 1 projections) corresponding to thep + 1 unique subgroups. Every discrete line samples p

points in the image, has p translates, and intersects anyother distinct line only once (or never if parallel). All theseattributes match those of lines in continuous space.

The FRT is a discrete form of a linogram with linesmapped according to gradient and intercept (i.e. points in

the image are mapped to discrete lines in the FRT) andthe back-projection operator is a symmetry of the projec-tion operator. Matus and Flusser showed that p of the pro-jections, Rm ¼ fRmðtÞjt 2 Zpg for m 2 Zp can be defined asthe sum of pixels centred on lines x”my + t (mod p) andthe remaining projection, R?0 ¼ fR?0 ðtÞjt 2 Zpg, can befound as a perpendicular line sum over y”0x + t (mod p),i.e.

RmðtÞ ¼Xp�1

y¼0

Iðhmyþ tip;yÞ

for 0 � m < p;R?0 ðtÞ ¼Xp�1

x¼0

Iðx;tÞ; ð1Þ

where Æxæg denotes x (mod g). An example of these discretelines for p = 7 is presented in Fig. 1a and b.

The FRT has found application in tomographic recon-struction [2,16,17], image compression [7,15], image denois-ing [18], image representation [3], blind blur deconvolution[4], watermarking [19,5], encrypted and robust data trans-mission [5,6]. An adaptive FRT for a sliding and zoomingwindow was developed in [20] to facilitate the location ofsmall linear features and for object tracking. The majorrestriction of the FRT in these applications is that itrequires square prime sized arrays.

The FRT formalism was extended by Hsung, et al., in1996 to apply to dyadic square arrays, 2n·2n for any posi-tive integer n [8]. This version, termed the discrete periodicRadon transform (DPRT), makes the transform more con-ducive to image processing, which commonly utilisessquare dyadic images for computational efficiency. TheDPRT has essentially horizontal projections Rm(t), asdefined above, for 0 � m < N = 2n. However, since thearray size is not prime, the image size N now has factorsother than N”0 (mod N). Additional perpendicular (oressentially vertical) projections are required as R?s (t) for0 � s < N/2, which are discrete line sums along y”2sx + t

(mod N). An example of these discrete lines for N = 8 is

Fctsry

1622 A. Kingston, I. Svalbe / Image and Vision Computing 25 (2007) 1620–1630

presented in Fig. 2a and b. Including all these discrete pro-jections represents I(x, y) exactly in the transform space.The DPRT is then defined as

RmðtÞ ¼XN�1

y¼0

Iðhmyþ tiN ;yÞ for 0 � m < N ;

R?s ðtÞ ¼XN�1

x¼0

Iðx;h2sxþ tiN Þ for 0 � s <N2: ð2Þ

The DPRT is a redundant representation of I(x, y) as ithas N(1 + 1/2) projections of length N. A non-redundantform of the DPRT with orthogonal bases was presentedby Lun et al. in 2003 [21], it is termed the orthogonalDPRT (ODPRT).

The formalism for the FRT applied to arrays of sizepn·pn is a natural extension of the DPRT and was devel-oped in [22]. Projections are defined as for the DPRT, withp replacing two in (2). This is a redundant transform butrequires only N(1 + 1/p) projections of length N. A non-redundant form of this FRT, with orthogonal bases, wasalso presented in [22].

The above FRT definitions are all restricted to apply tosquare arrays with dimensions based on a single prime, i.e.prime, dyadic or prime-adic. Should the FRT be requiredof an array of arbitrary size M·N, the image must be paddedwith zeroes to a square array, N·N, with a minimum sizebeing the smallest N = pn � sup{M, N}. Padding arraysadds redundant information and unwanted computationalcomplexity. The motivation for this work is to remove theserestrictions and this paper extends the above FRT definitionsto apply to square arrays of any composite sizeN ¼ pn1

1 pn22 pn3

3 . . ., where pi is prime and ni is any positive inte-ger giving the prime decomposition of N. This definition isreferred to in this paper as the generalised FRT. An FRTfor non-square arrays is the subject of ongoing research.

Section 2 establishes the minimal set of projections thatare required for an exact and invertible FRT for N·N

x

y

0

0(a) (b)1 2 3 4 5 6 7 0 1 2 3 4 5 6 7

1

2

3

4

5

6

7

(t)x

y

0

1

2

3

4

5

6

7(t)

Fig. 2. Examples of discrete lines on 2n·2n arrays. Shaded points lie on thecentre of the wrapped lines, the values or these pixels are summed to givethe value of one element in the transform. (a) Pixels on the essentially

horizontal line x”2y + 3 (mod 8) sum to give R2(3). (b) Pixels on theessentially vertical line y”1·2x + 3 (mod 8) sum to give R?1 ð3Þ, i.e. s = 3.Since the array size is composite and the slope m or 2s may have acommon factor with 2n, each row and column is no longer necessarilyuniquely sampled.

arrays and provides a method to obtain that set. This leadsto the definition for the generalised FRT in Section 3. Adiscrete form of the Fourier slice theorem that applies tothis FRT and the related convolution property is alsoderived in this section, along with a method to perform afast FRT. A means to determine the level of redundancyin the sampling patterns that result on composite arraysis presented in Section 4. Knowledge of this over-samplingpattern is required for the inverse transform; three distinctinversion processes for exact image reconstruction fromFRT projections are then presented (i)-Fourier inversion(Section 4.2), (ii)-multi-resolutional corrected back-projec-tion (Section 4.3) and (iii)-filtered back-projection (Section4.4).

2. Projection sets for invertible mappings

This section begins by reviewing the properties of theprojection set required for the FRT and shows how thisset is extended to apply to arrays of powers of two in theDPRT (or dyadic FRT) and prime powers in the DRT overpn·pn images (or prime-adic FRT). A method is then for-mulated to determine the required basis that matches theseconditions for an FRT over arrays of composite size.

2.1. Projection set for Z2p

The set of pixels sampled by each discrete line used forintegration in the FRT produces a 2D lattice with basisvectors (m, 1) and (p, 0) for lines in Rm, denotedLfðm; 1Þ; ðp;0Þg, and (1, 0) and (0, p) for lines in R?0 ,denoted Lfð1;0Þ; ð0;pÞg These lattices can be generated by(0, p) and/or (p, 0), along with any other vector in the lat-tice (m, 1), (2m, 2),. . .((p�1)m, p�1), for lines in Rm and (1,0), (2, 0),. . .(p�1, 0), for lines in R?0 , reduced modulo p.These vectors are the same for all of the parallel lines,translated by t, that make up a projection. Therefore eachvector specifies the orientation for an entire projection. LetHp be the set of these vectors that define the projection setfor a p·p array. Hp contains (m, 1) for 0 � m < p and (1, 0)(or any set of basis vectors that generate the equivalent lat-tices). To demonstrate this principle, the lattice generatedby the discrete line x”5y + 3 (mod 7) in Fig. 3a has beendepicted in Fig. 3b.

It is useful to store the FRT basis vectors as the shortestvector in each lattice Lfðm;1Þ; ðp;0Þg as this gives the direc-tion of the lines with maximum point density. The reducedbasis vectors for the example in Fig. 3 are shown as theblack vectors in Fig. 3c, these are the shortest vectors whichgenerate the equivalent lattice to Lfð5;1Þ; ð7;0Þg shown asgrey vectors. Note that to determine the reduced basis vec-tors for the lattice, the infinite lattice Lfðm;1Þ; ðp;0Þg isrequired, as opposed to the periodic lattice Lpfðm;1Þg.For the example given in Fig. 3, the shortest vector in thelattice is (�2, 1), which becomes (5, 1) in L7 and is nolonger the shortest vector (in L7 the reduced basis vectorsare {(1, 3), (3, 2)}).

x

y

00

(b) (a) (b)1 2 3 4 5 6

123456

(t)x

y

(t)

(–2,1)

(1,3)

(5,1)(7,0)

Fig. 3. An FRT discrete line sum x”my + t (mod p) with p = 7, m = 5 and t = 3. (a) Shaded points are those summed in the periodic discrete line. (b)fLð5;1Þ; ð7;0Þg generates the sampling pattern of this periodic discrete line. (c) The lattice basis {(5, 1), (7, 0)} shaded grey can be reduced to vectors ofminimum length (shown in black) which still generate the same lattice ({(�2, 1), (1, 3)} in this case).


The points of each of the p + 1 distinct lattices corre-spond to all the unique cyclic subgroups of order p in thegroup Z2

p under addition. Each subgroup contains the iden-tity (0, 0) and, since p is prime, each subgroup contains p�1unique elements. Let !(N) represent the number of discretelines in the projection set for the FRT of an N·N array, i.e.!(p) is the cardinality of Hp. It can be seen� ðpÞ ¼ pþ 1 ¼ pð1þ 1=pÞ where p is prime. !(N) alsogives the number of unique cyclic subgroups of order N

in the group Z2N under addition.

2.2. Projection set for Z22n and Z2

pn

For the dyadic FRT over Z22n , the array size has factors

of 2j. Any two distinct lines will intersect at any of 2n�1,2n�2,. . .,2, 1 locations or not at all if parallel. Thereforethe set of elements included in lines generated by the pro-jection set defined for the FRT does not include all ofZ2

2n and additional perpendicular projections based on dis-crete lines y”2sx + t (mod 2n) are necessary. Here H2n con-tains (m, 1) for 0 � m < 2n and (1, 2s) for 0 � s < 2n�1.There are N + N/2 cyclic subgroups of order N in thegroup Z2

N under addition when N = 2n. There are also cyc-lic subgroups of order less than N, however it is unneces-sary to include discrete lines resulting from these, as eachone is entirely contained within a cyclic subgroup of orderN. Therefore � ð2nÞ ¼ 2n þ 2n�1 ¼ 2nð1þ 1=2Þ. Similarly,for the prime-adic FRT over Z2

pn , Hpn contains (m, 1) for0 � m < pn and (l, ps) for 0 � s < pn�1 with� ðpnÞ ¼ pn þ pn�1 ¼ pnð1þ 1=pÞ.

2.3. Projection set for Z2N

Define the discrete lines for the FRT over square arraysof composite size as ax”by + t (mod N). HN then containsa number of vectors (b, a). How many of these are there toensure the transform is invertible and how are they found?

A function f(p) is said to be multiplicative if gcd(p,q) = 1 implies that f(pq) = f(p)f(q) [23]. It can be shownthat !(N) is multiplicative and is therefore found as

� ðNÞ ¼ NYpjNð1þ 1=pÞ: ð3Þ

where p|N denotes some prime, p, that divides N.

Suppose that gcd(p, q) = 1 and denote Hp to be the com-plete projection set for Z2

p;fðbjp;a

jpÞjj 2 ½0;� ðpÞ � 1�g: Also

denote Hq to be the complete projection set forZ2

q; fðbkq; a

kqÞjk 2 ½0; � ðqÞ � 1�g. Any point ðxp;ypÞ 2 Z2

p canbe represented as gpðbj

p;ajpÞ for some gq 2 Zp and any point

ðxq;yqÞ 2 Z2q can be represented as gqðbk

q;akqÞ

for some gq 2 Zp.The set fqðxp;ypÞ þ pðxq;yqÞ ðmod pqÞjðxp; ypÞ 2

Z2p;ðxq;yqÞ 2 Z2

qg gives Z2pq. There are p2q2 points generated

in this set. It is required to establish that each point in thisset is unique. Assume the converse is true, that the point

ðxpq;ypqÞ 2 Z2pq generated as ðxpq;ypqÞ � qðx1

p;y1pÞþ

pðx1q;y

1qÞ ðmod pqÞ can also be generated as

qðxp2;y

2pÞ þ pðx2

q;y2qÞ ðmod pqÞ; then

qðx1p; y

1pÞ þ pðx1

q; y1qÞ � qðx2

p; y2pÞ þ pðx2

q; y2qÞ ðmod pqÞ:

Using the property, jk � jlðmod qÞ ) k � lðmod q=gcdðj;qÞÞ, this becomes the congruence

qðx1p;y

1pÞ þ pðx1

q;y1qÞ � qðx2

p;y2pÞ þ pðx2

q;y2qÞ ðmod

pqgcdðq;pqÞÞ;

and so

ðx1p;y

1pÞ � ðx2

p;y2pÞ ðmod pÞ:

All points contributing to (xpq, ypq) from Z2p are unique.

Similarly ðx1q;y

1qÞ � ðx2

q;y2qÞ ðmod qÞ, so all the points gen-

erated are incongruent (mod pq) and together form thecomplete set Z2

pq.The projection set Hpq ¼ fðbl

pq;alpqÞjl 2 ½0;� ðpqÞÞg is

required such that any point in Z2pq can be represented as

gpqðblpq;a

lpqÞ for some gpq 2 Zpq and ðbl

pq; alpqÞ 2 Hpq From

the above theorem,

ðxpq;ypqÞ �qðxp;ypÞ þ pðxq;yqÞ

qðgpðbjp;a

jpÞÞ þ pðgqðbk

q;akqÞÞ

( )ðmod pqÞðmod pqÞ

: ð4Þ

It is possible to find some gpq2Zpq, which satisfiesqgpq”qgp (mod pq) and also satisfies pgpq”pgq (mod pq)giving

gpq � ðqgp � pgqÞq� p ðmod pqÞ; ð5Þ

where x�ðmod gÞ denotes the multiplicative inverse of x

modulo g. Substitute this into (4) to obtain


ðxpq;ypqÞ �gpqðqðbj

p;ajpÞ þ pðbk

q; akqÞÞ

gpqðblpq;a

lpqÞ

8<:

9=;ðmod pqÞ

ðmod pqÞ: ð6Þ

Here ðblpq;a

lpqÞ � qðbj

p;ajpÞ þ pðbk

q;akqÞ ðmod pqÞ. This shows

the complete projection set for Z2pq can be generated from

those of its co-prime factors as Hpq � fqHpþpHqg ðmod pqÞ; it has been proven above that all thesevectors are unique. This set contains !(pq) = !(p)!(q) vec-tors, showing the function !(N) is indeed multiplicative.Below is an example demonstrating how to obtain the pro-jection set for N = pq = 6 from the sets for the two primes,p = 2 and q = 3:

GivenH2¼fð1;0Þ; ð0;1Þ; ð1;1Þg:

H3¼fð1;0Þ; ð0;1Þ; ð1;1Þ; ð2;1Þg:

(

H6¼f2H3þ3H2g¼

þ ð1;0Þ ð0;1Þ ð1;1Þ ð2;1Þ �2

ð1;0Þ ð5;0Þ ð3;2Þ ð5;2Þ ð1;2Þ

ð0;1Þ ð2;3Þ ð0;5Þ ð2;5Þ ð4;5Þ

ð1;1Þ ð5;3Þ ð3;5Þ ð5;5Þ ð1;5Þ

�3

This property is the foundation for generating the basisvectors, which specify the projections included in the FRTof an image defined on a square array of arbitrary compos-ite size, N·N.

3. Generalised FRT formalism

3.1. Definition of the generalised FRT

Denote each element of the FRT as Rb,a(t), defined asthe sum of all pixels in I(x, y) such that ax”by + t (modN). For the prime, dyadic and prime-adic FRT definitionsRm(t) = Rm,1(t) and R?s ðtÞ ¼ R1;2pðtÞ. The generalised FRTover square arrays of composite size, N·N, can be definedas

Rb;aðtÞ ¼XN�1

x¼0

XN�1

y¼0

Iðx;yÞdhax� by� tiN for ðb;aÞ 2 HN :

ð7Þ

where dÆxæg is 1 when x”0 (mod g) and 0 otherwise and HN

is the projection set required for N·N arrays as defined inSection 2.

3.2. Properties of the generalised FRT

A discrete form of the Fourier slice theorem can be dem-onstrated for the generalised FRT. Denote R^b;aðuÞ as the1D Discrete Fourier Transform (DFT) of Rb,a(t) for (b,a) 2 HN, then

R^b;aðuÞ ¼XN�1

t¼0

Rb;aðtÞe�i2put=N

¼XN�1

t¼0

XN�1

x¼0

XN�1

y¼0

Iðx; yÞe�i2put=Ndhax� by� tiN

¼XN�1

x¼0

XN�1

y¼0

Iðx;yÞe�i2pðaxu�byuÞ=N ¼ I^ðau;� buÞ; ð8Þ

where I^ðu;vÞ is the 2D DFT of I(x, y). The DFT of the dis-crete projection with basis (b, a), which is based on discreteline sums over ax”bx + t for some t2N, is a wrapped dis-crete line through the origin of Fourier space, �bu”av

(mod N), perpendicular to the discrete line of projection(as the product of these gradients is �1).

The FRT of an image, I(x, y), can be obtained inO(N!(N) log N) operations by utilising this property.I^ðu;vÞ can be obtained in O(N2 log N) operations. Each ofthe !(N) projections of length N can be obtained as theinverse 1D DFT of the corresponding discrete slices inO(N log N) operations.

A discrete form of the convolution property is also con-served in the generalised FRT as a result of the discreteFourier slice theorem (8). Assume the function F(x, y) onan N·N array is to be determined from the N·N functionG(x, y) and L·M function H(x, y), 0 < L, M � N, by the2D cyclic convolution

F ðx;yÞ ¼XL�1

a¼0

XM�1

b¼0

Gðhx� aiN ;hy� biN ÞHða;bÞ: ð9Þ

The same result can be obtained by convolution opera-tions on each 1D projection in the FRT. Denote the gener-alised FRT of g(x, y) as Rn

b;aðtÞ. The value of F(x, y) can befound through the DRT of G(x, y) and H(x, y) as

RFb;aðtÞ¼

XN�1

y¼0

F ðx;yÞdhax�by� tiN

¼XN�1

x¼0

XN�1

y¼0

XL�1

a¼0

XM�1

b¼0

Gðhx�aiN ;hy�biN ÞHða;bÞdhax�by� tiN

¼XL�1

a¼0

XM�1

b¼0

XN�1

q¼0

XN�1

r¼0

Gðq;rÞdhaq�brþaa�bb� tiN Hða;bÞ

¼XL�1

a¼0

XM�1

b¼0

RGb;aðht�aaþbbiN ÞHða;bÞ

¼XN�1

k¼0

RGb;aðkÞ

XL�1

a¼0

XM�1

b¼0

Hða;bÞdhaa�bb� tþ kiN

¼XN�1

k¼0

RGb;aðht� kiN ÞRH

b;aðkÞ: ð10Þ

The convolution of 2D images G and H can be per-formed in FRT space as a set of 1D circular convolutionsof discrete projections, RG

b;a and RHb;a. This reduces the com-

putational complexity for 2D applications in filtering or


matching, as outlined for the prime and dyadic FRT (orDPRT) [8]. It is expected that this property will extend to3D and nD and this remains a topic for further research.

4. Generalised FRT inversion

The inverse transform (or image reconstruction fromprojections) for the FRT can be achieved either via Fourierinversion or back-projection. To recover the value of theoriginal function at pixel (x, y), the discrete line sums ineach of the !(N) projections that contain this pixel aresummed. This incorporates the value of the pixel (x, y)!(N) times and all other pixels at least once (dependingon the degree of compositeness of N). It is first requiredto determine the degree with which some pixels are over-represented; this is the subject of the next section and thefollowing section establishes three distinct methods to cor-rect for that over-representation, allowing the exact valueof I for each pixel (x, y) to be recovered.

4.1. Sampling function, vN(x, y)

Let vN(x, y) represent the number of times a pixel (x, y)is sampled by all discrete lines or subgroups generated byvectors in HN. This over-sampling may be found by consid-ering the set of all discrete lines that pass through the imageorigin, i.e. Rb,a(0). The function, vN(x, y), shows the over-representation each pixel will have after back-projectionwhen reconstructing the origin. From the discrete Fourierslice theorem (8), vN defined about the origin of the imagealso gives the over-representation of spatial frequencies inI(u,v), due to the symmetry: vN(v,�u) = vN(u,v). This over-representation must be accounted for when reconstructingthe original function. Since the array is congruent (mod N),the over-representation of each pixel (x, y) after back-pro-jection to reconstruct pixel (j, k) can be found as vN(Æx�iæN,Æy�jæN), so it is sufficient to investigate reconstructing theorigin only.

For the FRT case, where N = p, a prime, vp(x, y) = 1 forall pixels except the origin where vp(0, 0) = !(N) = p + 1.An example of this (for p = 7) is shown in Fig. 4a. Thiscan be written as

Fig. 4. Examples of the sampling function (with origin in the top left corner) fFRT, v9(x, y).

vpðx;yÞ ¼ dYp=d

ð1þ 1=pÞ where d ¼ gcdðx;y;pÞ: ð11Þ

For the dyadic FRT case, where N = 2n, it was shown in[8] that v2nðx;yÞ ¼ gcdðx;y;2nÞ for all pixels, except the ori-gin, where v2nð0;0Þ ¼ � ðNÞ ¼ 2nð1þ 1=2Þ. An example ofthis for N = 23 is shown in Fig. 4b. This can be written as

v2nðx;yÞ ¼ dYpjd

1þ 1=2; if d ¼ 2n;

1 otherwise;

�

where d ¼ gcdðx;y;2nÞ: ð12Þ

Similarly, for the prime-adic FRT case, where N = pn, itwas shown in [22] that

vpnðx; yÞ ¼ dYpjd

1þ 1=p; if d ¼ pn;

1 otherwise;

�

where d ¼ gcdðx; y; pnÞ: ð13Þ

An example of this for N = 32 is shown in Fig. 4c.Suppose that gcd(m, n) = 1 and fðaj

m;bjmÞjj 2 ½0;vmðx;yÞÞg

is the set of solutions ajmx� bj

my � 0 ðmod mÞ andfðak

n;bknÞjk 2 ½0;vnðx;yÞÞg is the set of solutions

aknx� bk

ny � 0 ðmod nÞ. Then fðblpq;a

lpqÞ ¼ ðnðaj

m;bj

m þ mðbkn;a

knÞjj 2 ½0;vmðx;yÞÞ;k 2 ½0;vnðx;yÞÞ;l ¼ jkg gives

the solution set (mod pq) and there are vm(x, y)vn(x, y) dis-crete lines. The proof that each new discrete line is uniqueis identical to that given for !(N) in Section 2. All of thenew solutions so defined are incongruent (mod pq) andform the complete set of !(N)N solutions. Therefore, wecan say vN(x, y) is multiplicative and define

vN ðx;yÞ ¼ dYpjd

1þ 1=p; if gcdðp;N=dÞ ¼ 1;

1; otherwise;

�

where d ¼ gcdðx;y;NÞ: ð14Þ

The following example demonstrates how to find v6(2, 4)from v3(2, 4) and v2(2, 4).

or (a) prime FRT, v7(x, y) (b) dyadic FRT (or DPRT), v8(x, y) prime-adic


A graphical representation of these discrete lines is pre-sented in Fig. 5. The lines in this figure have been drawn inthe direction of the shortest vector in each lattice. Forexample, in the 3·3 case, the lattice generated by (2, 1),i.e. Lfð2;1Þ; ðp;0Þg, has been drawn with basis (�1, 1)which generates the same lattice, but with maximum pointdensity along the selected line direction.

4.2. Fourier inversion

The reconstruction of an image from its set of discreteperiodic projections can be performed in the frequencydomain by taking the 1D DFT of each projection (inO(N!(N) log N) operations) and mapping it onto the 2DDFT of the image, using the discrete Fourier slicetheorem (8). This will over-represent some spatial frequen-cies according to vN(u, v). Dividing the value at each spatialfrequency by vN(u, v) and applying the inverse 2D DFT tothis data in O(N2 log N) operations recovers the originalimage exactly.

It was shown in [24] for square prime sized arrays thatFourier inversion has a large constant of proportionalityand that discrete back-projection of O(N3) is actually fasterfor array sizes less than O(600). The greater efficiency ofFourier inversion with increasing array size implies itsimplementation speed overtakes that of back-projectionwith increasing N and is consistently faster for array sizesgreater than O(1800). The speed of Fourier inversiondepends on how well N can be divided to perform a fastFourier transform (FFT). Prime array sizes give the worstcase scenario and show that, for smaller array sizes, it may

x

y

1

1

1

(–1,1) (mod 3)

x

y

3 1

1 1

(0,1) (mod 2)(1,1) (mod 2)(1,0) (mod 2)

x

y

3 1

1 1

1 3

1 1

3 1

1 1

( 1,2) (mod 6)(–1,1) (mod 6)( 2,1) (mod 6)

Fig. 5. Depiction of the discrete lines presented in the example for v6(2, 4).The numbers indicate the number of discrete lines that sample each pixel.Note in the 6·6 case, the reduced basis vector (2, 1) generates the samelattice as (4, 5) and the reduced basis vector (�1, 1) generates the samelattice as (1, 5).

often be beneficial to perform image reconstruction via dis-crete back-projection. The remainder of this section detailstwo distinct methods to achieve this: multi-resolutionalcorrected back-projection and filtered back-projection.

4.3. Multi-resolutional correction for vN(x, y) in back-

projection

For the FRT, vp(x, y) is !(p) = p + 1 at the origin and 1at all other (x, y). The sum of all discrete line sums contain-ing a specific pixel (j, k), [i.e. R1,0(j) and Rm,1Æi�mjæp for0 � m < p] gives the sum of the entire image, Isum, withthe value of I at (j, k) included an additional p times, (seeexample in Fig. 4a). The over-representation can be cor-rected for by subtracting the sum of the image, Isum anddividing the result by p. The image is recovered from itsFRT as

Iðx;yÞ ¼ 1

p

Xp�1

m¼0

Rm;1hx� ymip þ R1;0ðyÞ � I sum

!; ð15Þ

where Isum can be obtained as the sum of all discrete linesums from any one projection, i.e. I sum ¼

Pp�1t¼0 Rb;aðtÞ for

any (b, a)2Hp, as all projections sample the entire imageonce.

For the prime-adic (and hence dyadic) FRT, vp(x, y) is!(pn) at the origin and gcd(x, y, pn) at all other (x, y). Thisgives an over-representation at each resolutionp� p;p2 � p2; . . . ;pn�1 � pn�1 as depicted in Fig. 4b and c.This can be corrected for using a multi-resolutionalscheme. To correct at each resolution pk·pk within a pn·pn

array, an important property must be introduced.Firstly notice that Hpj � Hpn for j < n. Since the image is

assumed to be periodic, the infinite lattice produced by tak-ing a line generated by ðb;aÞ 2 Hpj , for all discrete lineswith intercept t”0 (mod pj), (i.e. t ¼ 0;pj;2pj; . . .), gives thesame infinite lattice as that produced by ðb;aÞ 2 Hpj inthe pj·pj array, i.e.

Lfðm;1Þ;ðpj;0Þ;ðpn;0Þg � Lfðm;1Þ;ðpj;0Þg and

Lfð1;psÞ;ð0;pjÞ;ð0;pnÞg � Lfð1;psÞ;ð0;pjÞg;

for any ðm;1Þ;ð1;psÞ 2 Hpj :Therefore taking all the line sums from Hpj that include

the origin and all other parallel line sums with t”0 (mod pj)

in Z2pn (i.e.

Pn�j�1

k¼0

Rb;að0þ kpjÞ for all (b, a) in Hpj )yields the

sampling pattern for Z2pj , i.e. vpjðx;yÞ replicated (mod pj)

in the x and y directions. Examples of this for pj = 21

and pj = 22 within Z223 are depicted in Fig. 6b and d.

These replicating patterns can be used to correct forvpnðx;yÞ at resolution pj·pj. The initial back-projected value,I 0(x, y), can be corrected through a multi-resolutional pro-cess reducing the over-representation from vpn to zero overall Z2

pn n fðx;yÞg; each step, npj , corrects for the over-repre-sentation described above at resolution pj·pj. The inversionprocess is given as

Fig. 6. (a) v2(x, y) (b) All discrete lines in H2 on the 8·8 array, with intercept 0 (mod 2) yields v2(x, y) replicated (mod 2) (c) v4(x, y) (d) All discrete lines inH4 on the 8·8 array, with intercept 0 (mod 4) yields v4(x, y) replicated (mod 4).


Iðx;yÞ ¼ 1

Nnpnðx; yÞ �

Xn�1

j¼1

ðp� 1Þpj

Nnpjðx;yÞ � p

NI sum

!;

ð16Þwhere

npj ¼Xpj�1

m¼0

XNpj�1

k¼0

Rm;1ðhx� myipj þ kpjÞ

24

þXpj�1�1

s¼0

XNpj�1

j¼0

R1;sðhy� psxipj þ kpjÞ

375:

The correction at each resolution pj·pj corresponds to allfactors, F, of N, denoted F|N. Eq. (16) can be rewritten as

Iðx;yÞ ¼ 1

NnN¼pnðx;yÞ þ

XF¼pjjN

AF njðx;yÞ þ A1¼p0 I sum

!;

ð17Þwhere the fraction of contribution required at each resolu-tion, F = pj, is given as

AF¼pj ¼ pj

N

YpjNpj

�p; if gcdðp;pjÞ ¼ 1;ði:e: pj ¼ 1Þ;1� p; otherwise:

�ð18Þ

The inversion process for composite N is a naturalextension of that for N = pn. It is also undertaken by cor-recting for the over-representation at each resolution after

Fig. 7. Top left corner of (a) v45(x, y) (b) All discrete lines in H9 on the 45·45Fig. 4c) replicated (mod 9). To fully correct (a), n45

9 ðx; yÞ, n455 ðx; yÞ, n45

3 ðx; yÞ, an

back projection of pixel (x, y), which is defined as1=N

Pða;bÞHN

Rb;ahay� bxiN . As for the over-sampling function

vN(x, y) defined in the previous section, the resolutions thatrequire correction correspond to all the factors, F, of N

(F|N). For the example shown in Fig. 7a, whereN = 325 = 45, the over-sampling must be corrected for atscales 15·15, 9·9, 5·5 and 3·3. Subtracting a fraction ofIsum corrects at resolution 1·1, leaving only a multiple(N) of I(x, y).

The sampling achieved by taking the projection set foran F·F array, for some F|N, for all discrete line sums withtranslates t (mod F) yields the sampling pattern vF(x, y)replicated (mod F) in the x and y directions. An exampleof this is depicted in Fig. 7b for N = 45 and F = 9. ThisF·F back-projection sampling pattern, achieved as

nNF ðx;yÞ ¼

Xðb;aÞ2HF

XN=F�1

k¼0

Rb;ahay� bxþ kF iN ; ð19Þ

can be used to correct for the over-representation fromback-projection at the resolution of F·F (9·9 in the exam-ple) by adding a fraction, AN

F , of this back-projection,nN

F ðx;yÞ. This technique is applied to correct the back-pro-jection at each resolution, F|N.

Since the over-sampling of pixel (x, y) after back-projec-tion of the origin, vN(x, y), is a multiplicative function, thescaling factors, AN

F , for correction after back-projection at

array, with intercept 0 (mod 9), i.e. n459 ðx; yÞ, yields v9(x, y) (as shown in

d Isum are required.


each resolution F·F of the multi-resolutional back-projec-tions, nN

F ðx;yÞ are also multiplicative. Given gcd(m, n) = 1,the scaling factor required for resolution F·F in a pq·pq

array for F|pq is found as

ApqF � Am

f AnF =f for some f 2 Zm that divides F : ð20Þ

For example, in the 45·45 array, the back-projection,n45

9 ð0;0Þ, which gives the sampling shown in Fig. 7b, cor-rects for the over-representation after back-projection atresolution 9·9, by adding it A45

9 ¼ A99A5

1 ¼ 1 5� 1 ¼ �1times. The value for AN

F can therefore be found as

AF ¼FN

YpjN=F

�p; if gcdðp;F Þ ¼ 1;

1� p; otherwise:

�ð21Þ

Note that in (15), I sum ¼Pp�1

t¼0

Rb;aðtÞ for any (b, a)2Hp. If

the projection set for a 1·1 array is defined as H1 = {(0,1)}, then Isum can be found as nN

1 ðx;yÞ for any ðx;yÞ 2 Z2N .

Therefore the entire inversion process can be written as

Iðx;yÞ ¼ 1

N

XF jN

ANF nN

F ðx;yÞ: ð22Þ

The result of the correction process at each resolutionfor the inversion of the !(175)·175 FRT of a 175·175Lena image is depicted in Fig. 8.

4.4. Filtered back-projection

The original image can be reconstructed exactly fromback-projection without a multi-resolutional correction ifthe projections are filtered prior to back-projection. This is

Fig. 8. Depiction of the correction made by each resolution of the reconstructioback-projection. (b) Corrected at resolution 35·35. (c) Corrected at resolution(f) subtracting A1Isum completes all required corrections and exactly reproduc

similar to filtered back-projection from acquired projectionsfor the continuous RT, where the projections are convolvedwith a form of the Ram-Lak filter (a low pass filtered rampfunction in frequency domain). The 2D IDFT of I^ðu;vÞ,

Iðx;yÞ ¼ 1

N 2

XN�1

u¼0

XN�1

v¼0

I^ðu;vÞexp½i2pðuxþ vyÞ=N �; ð23Þ

can be rewritten as a sum over all spatial frequenciesI^ðaw;� bwÞ for w2[0, N) for each discrete line or subgroupgenerated by (b, a)2HN. Dividing by the over-samplingfunction to account for over-representation of some fre-quencies, (23) becomes

Iðx;yÞ ¼ 1

N 2

Xðb;aÞ2HN

XN�1

w¼0

1

vN ðaw;� bwÞ I^ðaw;

� bwÞexp½i2pðax� byÞw=N�: ð24Þ

By definition, gcd(a, b) = 1, therefore gcd(aw,�bw,N) = gcd(w, N) and the 2D over-sampling function,vN(aw,�bw) can be reduced to a 1D function according to

vNðwÞ ¼ dYpjd

1þ 1=p; if gcdðp;N=dÞ ¼ 1;

1; otherwise;

�

where d ¼ gcdðw;NÞ: ð25Þ

This shows the remarkable fact that all discrete lineswith basis (b, a), such that gcd(a, b) = 1, through the originof I^ðu; vÞ, have an identical over-sampling function. Anexample of this function for N = 60 is presented in Fig. 9.

From the discrete Fourier Slice theorem (8),R^

b;aðwÞ ¼ I^ðaw;�bwÞ, so (24) becomes

n of the !(175) = 240·175 FRT of a 175·175 Lena image. (a) Uncorrected25·25. (d) Corrected at resolution 7·7. (e) Corrected at resolution 5·5 andes the original image.

0

20

40

60

80

100

120

140

160

–30 –20 –10 0 10 0 30

υ N(ω

)

ω

Fig. 9. A plot of the 1D over-sampling function vN(w) forN = 22·3·5 = 60. The reciprocal of this function gives the filter requiredto correct for the over-representation in the frequency domain.


Iðx;yÞ ¼ 1

N 2

Xðb;aÞ2HN

XN�1

w¼0

1

vN ðwÞR^b;aðwÞexp½i2pðax� byÞw=N �

¼ 1

N 2

Xðb;aÞ2HN

R~b;aðax� byÞ: ð26Þ

This is simply a back-projection of filtered projections,R~b;aðtÞ. These filtered projections are found as

R~b;aðgÞ ¼XN�1

w¼0

1

vN ðwÞR^b;aðwÞexp½i2pgw=N �

¼XN�1

w¼0

1

vN ðwÞXN�1

t¼0

Rb;aðtÞexp½�i2ptw=N �exp½i2pgw=N �

¼XN�1

t¼0

Rb;aðtÞXN�1

w¼0

1

vN ðwÞexp½i2pðg� tÞw=N �

¼XN�1

s¼0

Rb;aðg� sÞXN�1

w¼0

1

vN ðwÞexp½i2psw=N �

¼XN�1

s¼0

Rb;aðg� sÞHðsÞ: ð27Þ

–15

–10

–5

0

5

10

15

20

25(a)

–30 –20 –10 0τ

10 20 30

h(τ)

Fig. 10. (a) Plot of the 1D filter obtained from the IDFT of 1/vN(w) for N = 22·by direct discrete back-projection. This filter is compared with (b) the standarfiltered back-projection.

So the multiplication of R^b;aðwÞ by 1/vN(w) in the fre-

quency domain corresponds to the 1D circular convolutionof projections, Rb,a, with kernel, H, defined as

HðsÞ ¼XN�1

w¼0

1

vNðwÞexp½i2psw=N �: ð28Þ

There are no singularities in this function (by design) asvN(w) � 1 for all w 2 ZN . An example of this filter forN = 60, based on the sampling function from Fig. 9, is pre-sented in Fig. 10a. This can be compared to the Ram–Lakfilter for N = 60 shown in Fig. 10b. While they appear verydifferent, their purpose is the same. Both filters have a max-imum at w = 0. The Ram-Lak filter has global minima at|w| = ±1 and minima of decreasing magnitude at |w| = 3,5,. . .. This accounts for the over-representation uponback-projection immediately surrounding the point beingback-projected due to the polar nature of the operation.The 1/vN(w) filter has a global minimum at |w| = 30 = N/2 and minima of decreasing magnitude at |w|”0 (mod N/3), (mod N/4), (mod N/5),. . . (at the factors of N). Thisaccounts for the multi-resolutional over-representationupon discrete back-projection about the point of back-projection.

Implementation of filtered back-projection using theresult shown in (24) achieves exact image reconstructionfrom projections. This form of back-projection is com-pleted in O(N2!(N)) operations, regardless of performingthe convolution in the frequency or spatial domain. Thedominating computational cost is that of back-projection,requiring O(!(N)) operations for each of the N2 pixels.

5. Conclusion

An FRT based on modulo arithmetic that applies toN·N arrays for N 2 N has been presented. It projects the2D image into a set of Npp|N(1 + 1/p) projections of lengthN. A method to determine the required projection set wasalso presented.

Important properties of the continuous Radon trans-form are preserved in this discrete formalism. A discrete

(b)

τ

h(τ)

–40

–20

0

20

40

60

80

100

–30 –20 –10 0 10 20 30

3·5 = 60. This enables exact image reconstruction from filtered projectionsd Ram-Lak filter used to reconstruct images from acquired projections by


form of the Fourier slice theorem and the convolutionproperty hold for this generalised FRT. These propertiesallow the FRT to be obtained in O(N2log Npp|N(1 + 1/p))operations and make it a useful image processing tool,reducing 2D problems to a set of 1D problems.

Due to the compositeness of N, it is a redundanttransform, as for the DPRT. The redundancy occurs ina multi-resolutional pattern based on the factors of N.A function, vN(x, y), which characterises this multi-reso-lutional redundancy was presented which gives the over-sampling of pixels upon back-projection of discrete linesums through the origin. Given the (x, y) symmetry ofthis function, it also gives the over-representation inthe projections of each spatial frequency, at (y, �x), inthe 2D DFT of the image.

Determination of this over-representation functionenabled the inverse FRT to be formulated. Three methodswere presented; Fourier inversion, multi-resolutional cor-rected back-projection and filtered back-projection. The2D over-representation function has the remarkable prop-erty of being identical along all 1D discrete lines of theform ax”by (mod N) for gcd(a, b) = 1, implying the filterfor each projection is the same regardless of direction.

The multi-resolutional nature of the transform mayprove useful in image analysis, particularly for matchingprojected textures and patterns. Research into this aspectand the development of an orthogonal FRT over N·N

and M·N are the subject of ongoing work. An investiga-tion into the distribution of the discrete angles required,for N·N (as compared to the p·p set) and comparison withthe uniformly distributed angle set for the continuous RT isalso the subject of ongoing research.

Acknowledgements

The preparation of this work for publication was sup-ported by a post-graduate publication award granted toAK through the Monash Research Graduate School.

References

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generalised finite radon transform for n×n images

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