Fast iterative algorithm forreconstruction from divergent-ray

projectionsC.E. Goutis, B.Sc, M.Sc, Ph.D., Mem.I.E.E.E., and S.N. Drossos, B.Sc

M.Sc, A.M.I.E.E.

Indexing terms: Algorithms, Divergent-ray projections

Abstract: A convergent algorithm is introduced which solves the exact system relating the Lagrange multipliersand the projections using the successive relaxation method without any approximation. The system gives theminimum-energy solution which is different from that of the convolution algorithm, but it is the same as theART reconstruction except that it discretises the Lagrange multipliers instead of the image. Its reconstructionsare substantially better than those of the convolution algorithm. Although the new algorithm uses the exactmatrix of this system, its speed is very high because it utilises the special matrix structure; namely the areas ofequal elements. The statistics of the error between the 'true' projections and the projections corresponding tothe reconstruction can be made the same as the noise statistics in the projection data, but the algorithm doesnot aim to produce the minimum norm solution to the problem of fitting noisy data. The number of projectionsN should be related to the number of measurements per projection P by P = vN/2, where v is an integer.Computed results verifying these conclusions are included. The algorithm can also use the object boundaries toimprove its reconstruction.

1 Introduction

The finite number of projections available in applicationsdo not fully specify the object (image) for reconstruction.In fact, there are an infinite number of objects which couldhave produced the same set of projection measurements.This necessitates the use of assumptions in selecting thereconstruction. The optimisation techniques introduced inReferences 1-3 consider the projections as constraints andfind the reconstruction by optimising a cost criterionsubject to these constraints. It was shown there that theART methods also optimise the variance of the image [1],but they have to use serious approximations to achieve anacceptable computational time, which in turn reduces thequality of the reconstruction. This is overcome in the algo-rithm introduced here as it first estimates a number of one-dimensional functions, the Lagrange multiplier functions,instead of the two-dimensional image itself which is com-puted from the Lagrange multipliers afterwards, thusreducing the dimensionality of the reconstruction problem.

The optimisation of a general cost function is given inReference 3. Applying this to specific cost functions, anumber of models were produced which relate the two-dimensional image with the one-dimensional Lagrangemultiplier functions for parallel- and divergent-ray projec-tions. The extension of models in many dimensions wasalso given in References 1-3. The model derived from theminimum-energy criterion produces a linear system rela-ting the measured projections and the digitised Lagrangemultiplier functions. Based on this, a direct algorithm anda very fast iterative algorithm were given for parallel-rayprojections [2, 3]. Their performance is much better thanthat of the convolution algorithm; the iterative solutionscoping very well with noise. The projection-slice theoremwas extended to divergent-ray geometry in Reference 1,where a direct algorithm involving circulant matrices wasalso presented.

These constrained optimisation techniques wereextended to the general case of curved-line projections in

Paper 3089E (C2) first received 28th March 1983 and in revised form 25th October1983The authors are with ihe Department of Electrical & Electronic Engineering, MerzLaboratories, University of Newcastle upon Tyne, Newcastle upon Tyne NE1 7RU,England

Reference 4, including the projection slice theorem and theART methods. There the reconstruction problem wastreated as an optimisation problem in Hibert spaces, toshow that the reconstruction obtained by using theminimum-energy model for the general projections (whichincludes the parallel- and divergent-ray projections) is thesame as the reconstruction of the additive ART method,except that the latter discretises the two-dimensional imagewhile the former discretises the one-dimensional Lagrange-multiplier functions.

The iterative algorithm presented here is based on thefan-beam back-projection model derived using the objectenergy as cost function. A brief outline of the derivation ofthe above is given below; the details can be found in Refer-ence 1. The one-dimensional fan-beam projection, at anglei//fc may be expressed as

9k(0k)= I f(x,y)drkJll(Ok)

/ c = l , 2 , ..., (1)

where rk, 6k are the polar coordinates with origin thesource position Ok (Fig. 1), f(x, y) is the two-dimensionalimage to be reconstructed defined in the region D, ^ and l2

are the limits of the image as specified by D (if D is a circle

detectors

Fig. 1 Divergent ray geometry with strip numbering scheme AThe group of strips shown forms the k\h projectionThe strips are numbered clockwise

IEE PROCEEDINGS, Vol. 131, Pi. E, No. 3, MAY 1984 89

radius R, then /x = 0 and l2 = 2R) and N denotes thenumber of projections.

The Euler-Lagrange method for the constraints of eqn.1 and the energy-cost criterion/2/2 express the reconstruc-tion problem as the minimisation of

f*dxdy+fc= 1 JO

In

(2)

where {Xk(0k)} are the one-dimensional Lagrange-multiplier functions. Taking the integral over rk in eqn. 2

and noting that dx dy = rk drk d9k, eqn. 2outside,becomes

/ = \ f2

dxdy (3)

where <5(-) is the delta function. Calculating the first varia-tion of / with respect to / and setting it equal to zero, itgives the model (see Appendix of Reference 1 for details)

/(*, y) = I (4)

It is worth noting that if the derivative with respect to / o fthe quantity within the doubie integral is set to zero, eqn. 4is produced.

In the following the projections will be taken equi-spaced; i.e. i//fc + 1 — \\jk = 2n/N for all k, where ij/k denotesthe projection angle.

Adopting this model the ith measurement of the kthprojection may be expressed as

gu-Qi l\ k=l

i = 1, 2, . . . , P K=l,2,...,N (5)

where £>, = (j - 1) A0 and A0 = n/P. A9 is the stripangular width which is assumed to be constant and Pdenotes the number of measurements per projection.

Discretising each {Xk(6k)} into P values, it gives

where Xki corresponds to the /c,-strip with origin at 0k andthe index i is specified as in Fig. 1; i.e. angle 9k is measuredclockwise. If the projection measurements are indexed inthe same way, eqn. 5 may be rewritten in the form of alinear system; namely,

(7)8 =

where

k'T:

g'T-

B'k'

= [AV

= [£'i7

r yr

iT. * 2 ,

and B' is an NP x NP matrix. Note that the prime in allthese symbols denotes the indexing used (Fig. 1); i.e. thesubvector k'k is produced by discretisation of the kth con-tinuous Lagrange multiplier /-k{6k) only; similarly for g'k. Adifferent convention will be used in the following for fullutilisation of the special properties of matrix B'.

In Section 2 the properties of a rearranged version ofthe block matrix B' are established, especially the areas of

equal elements. The algorithm, its convergence and itscomputational and storage requirements are given inSection 3. The performance of the new algorithm and itscomparison with the convolution algorithm are examinedin Section 4.

2 Matrix structure

The elements of matrix B' (eqn. 7) will be rearranged toobtain B. This will be achieved by introducing a new num-bering scheme for the divergent-ray strips which, in effect,rearranges the elements of k' and g'.

Considering the value of Xk(9k) to be constant withineach strip, the contribution of Xnj into gki is specified by

, ki — rb drb ddL (8)

where 5 denotes the intersection area of the ki and nj stripslying within D.

It is shown in Appendix 8 that if the areas lie within Dand the vertex of each strip does not lie within the otherstrip, hnj ki depends on the angle between the dichotomesof the nj and ki divergent-ray strips and the strip angularwidth Ad only. Hence any other pair of strips withdichotomes parallel to the ones of the nj and ki strips,respectively, may produce an equal matrix element. This isan important feature which results in areas of equalelement in B, and it is due to the factor \/rk in theminimum energy model (eqn. 4) produced by the con-strained optimisation.

Two divergent-ray strips are called codirectional if theircorresponding boundary lines are parallel and have thesame direction; e.g. strips 0 4 and O5 are codirectional, butOl and 06 are not, neither are Ox and O7. The matrixelements corresponding to intersections of codirectionalstrips with another strip are equal if the intersection areaslie within D.

The number of codirectional strips is required to belarge as this results in large areas of equal elements whichare used to reduce the computational requirements. Toachieve this, the number of equispaced projections shouldbe related to the number of strips of equal width per pro-jection as

(9)

where v is an integer. This relationship generates vNgroups of codirectional strips, each group forming a re-arranged projection. Fig. 2 shows one such group forN = 8, P = 4 and v = 1; namely strips 1-4 in this Figure.

A new numbering scheme is introduced to facilitate theelement rearrangement (strip regrouping) in A and g. Fig. 1shows scheme A (used in eqns. 6 and 7) which is the con-ventional manner of indexing divergent ray strips forproducing a fan-beam projection, the clockwise notationfor index i being used for convenience. Fig. 2 showsscheme B; i.e. how a rearranged projection/Lagrangemultiplier subvector may be formed by grouping togetherall the codirectional strips. Each group consists of N/2codirectional strips: the first strip being the same in bothschemes, e.g. (4, 1) in Fig. 2. The second strip in scheme Bis taken from the next projection, e.g. (5, 2) in Fig. 2. Ingeneral, the (k, i) and (k + 1, i + I) are parallel if1 < k < N and 1 < i < P. This is due to eqn. 9 whichmakes ij/k + l — ij/k = AiJ/, for all K.

The (k, i) strip of scheme A is renumbered in scheme Bas mj where j = i and m = </c + i + 1>; the symbol < > is

90 IEE PROCEEDINGS, Vol. 75/, Pt. E, No. 3, MAY 1984

defined as <S> = S ± LN, where L is an integer such that1 < S ± LN < N.

Fig. 3 is produced from Fig. 2 by partitioning each stripinto three strips of equal angular width and shows a group

how scheme A can be converted into scheme B and viceversa. As the dichotomes of all codirectional strips belong-ing to a group are parallel, the matrix elements (eqn. 8)corresponding to intersections of any pair of strips from apair of rearranged groups are equal, provided that theassociated areas lie within D.

Table 1: Conversion between numbering schemes A and B

Scheme AFig. 1

k,i

k = <m + y - 1/ = v(y - 1) +

Scheme Bv = 1 /J = 1 Fig. 2

m = (k - i + 1 >l = i

> m, ju

Scheme BFig. 3

/n = <*-y + 1>/ - 1

/; = | / - 1mtg

uod.\v = 2P/7Vm, j, //, v

+ 1

+ 1

Fig. 2 Strip numbering scheme B

The group of strips I 4 form what is called here a rearranged projection of all stripswhich are codirectional

N = 8, P = 4, v = I,/i = 1

strips 1 2 3 4scheme A (4, 1) (5, 2) (6, 3) (7, 4)scheme B (4, 1) (4, 2) (4, 3) (4,4)

of codirectional strips for N = 8, P = 12 and v = 3 pro-duced by grouping the second strip (n = 2) of each parti-tion. Scheme B uses three indices (m, j , n), where eachgroup, i.e. the rearranged projection/Lagrange multiplier,is specified by (m, n) and the elements in a projection/Lagrange multiplier are numbered by j . Table 1 specifies

<S> =S ±LN where L is an integer such that 1 < S ± LN ^ N.Also | • \intg denotes the integer part.

First the linear system for v = 1 will be considered.Rearranging the elements of g as specified in Table 1,namely

gl = [ 0 1 1 0 2 2 ' • • 0(JV/2)lO

Sm —

and defining {Afc} using {k'k} in identical manner, the linearsystem (eqn. 7) may be rewritten as

. 2 9 <(N/2) + m -

= Bk (10)

or as

gs/2

-gN

B\B0

pT-°JV/2 - 1

N/2-:

N/2-:

BN/2 - 1

- 1 BNj2

BlBjBl

Bo BN/2

pTDN/2-l

N/2

1 Bo \\_ kN _

(11)

Fig. 3 Effect of partitioning each strip of Fig. 2 into v = 3 strips

The numbered codirectional strips (4, 1, 2) to (4, 4, 2) form a rearranged projection(Table 1, m = 4, /i = 2)N = 8, P = 12. v = 3, 1 «£/<=$ 3

where

gT = \jg\gl •

and D is assumed to be a circle for convenience.The contribution of subvector kn to gk, is specified by

submatrix Bkn, which depends on {n — k) only, i.e. Bkn =fi(fc_n). Also if (n — /c) is negative Bkn = BN + n_k whichmeans that the last block of each row of matrix B is equalto the first block of the next row. Hence the matrix isblock circulant, also B{ = 6^_,. The contributions of kmJ

to gk, which corresponds to the Arth group of codirectionalstrips, are given by the7th column of the B|m_k| submatrix.

As can be seen from Fig. 4 these contributions corre-sponding to the dotted areas within the (mj) strip areequal; the contribution of the shaded area being differentas part of the area lies outside D. The value of these equalelements is constant in each submatrix, and the elementsassociated with the shaded areas vary between zero andthis constant value. Furthermore, the (j + l)th column ofB\m-k\ ' s v e ry similar to the jth column (Fig. 4). In general,the pattern of zero-constant-variable-zero elements

IEE PROCEEDINGS, Vol. 131, Pt. E, No. 3, MAY 1984 91

appears. This feature leads to the structure of B|m_fc[ shownin Fig. 5, where three areas exist. Their elements are zero,

kP

Fig. 4 The contribution of Xmj and /.m(J+,, into the rearranged projectiongk, vv/m7i is associated with the dotted areas, is equal and corresponds tothejth and (j + \)th column of the B]k_m\ submatrix, respectively

(1

variable

( N . N )

Fig. 5 Submatrix structure

The area of variable element partially separate the areas with zero and constantelements

constant and variable, respectively. Note that the zero andconstant elements in Fig. 5 are not always separated byvariable elements as the intersection areas near the vertexof the associated strip, e.g. the {mj) strip in Fig. 4, always liewithin D, in contrast to areas on the other end of the strip.

The approximate shape of these areas varies as follows.Submatrix Bo specifies the contribution of kk iogk for all kand is diagonal. Starting with Bo and moving to BN_l 5 thearea of constant elements starts from a line, the first diago-nal, a degenerated ellipse, becomes an ellipse whose majoraxis is the first diagonal, then it takes the form of a circlewhen the two sets of codirectional strips are perpendicular,and subsequently it takes the form of an ellipse whosemajor axis is the second diagonal at BN/2. This procedureis repeated in the reverse order to finish with a near diago-nal submatrix BN_y = B\. If a transparent region withinthe object exists, an area of zero elements and a transition-al area will appear within the region of constant elementsin Fig. 5.

3 Recursive algorithm

The block successive relaxation method is employed forthe solution of the system given by eqn. 11, and the

92

properties of matrix B are utilised to drastically reduce thecomputational requirements so that the algorithmbecomes practically useful. The existence of areas of equalelements or equivalent, the fact that each subrow differsfrom the next subrow of the same submatrix in a few ele-ments only, is the crucial feature in reducing the computa-tional requirements without any approximations. Byadopting no approximations in the solution of the systemgiven by eqn. 11, the algorithm introduced here converges,in contrast to ART methods which oscillate due to theeffect of the approximations on the critical eigenvalues.

The algorithm is based on the observation that theproduct of the ith subrow and a subvector is equal to theproduct of the previous (i — l)th subrow and the sub-vector, plus the product of the difference subrow and thesubvector; the ith difference subrow being generated bysubtracting each element of the (i — l)th subrow from thecorresponding element of the ith subrow. The number ofnonzero elements in the difference subrow is very small asthe subrows differ in a few elements only; in fact theaverage number is /? = 3 approximately. Hence theproduct of the ith subrow and a subvector requires /?(instead of P) arithmetic operations, if the product of the(i — l)th row and the subvector is available. As the differ-ence subrow instead of the subrow is used, storage of thedifference submatrices {A£fc} instead of {Bk} is required.{ABk} is produced from {Bk} by replacing each subrow bythe difference of the previous subrow from the one to bereplaced. The first subrow is not replaced. This reduces thestorage requirement of the algorithm drastically.

Using numbering scheme B (Fig. 2 and Table 1) andassuming circular D for convenience, the steps of the algo-rithm are as follows:

(i) Calculate the Bk, k = 0, 1, 2, . . . , N/2. The {mj)element is given by eqn. 8 where the integral is taken andthe intersection area of (lw) and {k + 1, j) strips, lyingwithin D

(ii) Derive the difference submatrices AJ5fc, k = 0, 1, 2,. . . , N/2; i.e. in each {Bk} replace the ith subrow by thedifference of the (i — l)th subrow from the ith subrow, fori = 2 , 3 , . . . P.

(iii) Store every nonzero elements of ABk, k = 0, 1, 2,. . . , N/2 and also store the necessary indices to make theseelements identifiable.

(iv) Compute the initial estimate of the Lagrange multi-plier using Aki = gki/B0{i, i), for all ki.

(v) Estimate g"ki, i = 1, 2, . . . , P from k" and the differ-ence matrix. Note that the product of a subvector and theith subrow is equal to the sum of its products with the(i — l)th subrow and the ith difference subrow, respec-tively.

(vi) Update kk as

; n + l _ ; n ,p t: :\tio{l, I)

\-Qki — dkU l — l , •£>••••> r

where y is the relaxation factor. Steps (v) and (vi) form asubcycle. Repeat the subcycle for all projections, preferablyin random order to complete one cycle of the algorithm.Also repeat the cycle until the statistics of the errorbetween the estimated and measured projections {eki = gki

— gki) becomes approximately equal to the statistics of thenoise in the 'true' projections. This was achieved at thefourth iteration (cycle) in the examples used here.

(vii) Rearrange k to obtain k' (Table 1) and backprojectthe latter using eqn. 4 to produce the reconstruction.

Steps (i)—(iii) are performed once only for the precomputa-tion of {ABk} which is stored for use in all future recon-

IEE PROCEEDINGS, Vol. 131, Pt. E, No. 3, MAY 1984

struction as N and P remain constant in many 4 Computed resultsapplications.

Since submatrix Bo is diagonal, i.e. each projectionpoint {yki} is independent of Xkj,j ± i, the blockwise iter-ation gives an identical result to the element-wise iteration.For y = 1, the successive relaxation procedure becomes theGauss-Seidel iteration method.

In the general case where v ^ 1 in P = v(N/2) the algo-rithm is modified as follows. The numbering scheme B(Fig. 3 and Table 1) is used which produces subvectors g{£\k{£\ m = 1, . . . , N and n= 1, . . . , v. Introducing the N(P/v) x 1 vectors

The system of eqn. 10 may be rewritten as

B(vv)kv

where each submatrix B{fts\ n, s = 1, . . . , v, is an N(P/v) x N{P/v) matrix the structure of which is the same asthat of B (eqn. 10). Given N and P, the geometry for v ^ 1is obtained from that of v = 1 by partitioning each stripinto v strips (Fig. 3 for v = 3). The set of strips obtained bykeeping the first of v strips, produced by partitioning, cor-responds to g(l) and k(1\ and the contribution of kis) intog{fl) is specified by B(MS). Considering this for all sets it isdeduced that

£ B{fts) = Bk,s=\

where B is the matrix for v = 1 and the same P and N.The submatrices of each matrix {B{fiS)} have size

P/v x P/v and possess the areas of equal elements shownin Fig. 5. Hence the algorithm given for v = 1 has astraightforward extension where the difference matrices{ABitiS)}n, s = 1,. . . , v instead of {AB} are used.

The number of arithmetic operations required for step(v) is equal to the number of nonzero elements in AB. Thetotal number of operations required for the algorithm isc(ivN2P + 6NP2 approximately, where v is as in eqn. 9, J3is the average number of nonzero elements per subrow andc denotes the number of iterations for reconstruction;6NP2 operations are allowed for backprojection.

Storage of the block circular difference matrix ABrequires (iNP/2 words approximately. For a general objectshape, the submatrices {Aflj£s)} will depend on k and n,and they should all be stored. However the number ofarithmetic operations will not change substantially provid-ed that the transparent regions, if any, are not used. In thelatter case the computational requirements will depend onthe number and size of transparent regions.

The matrix B is semipositive definite since kTBk ^ 0 forX j= 0, where 0 is the null vector. This is verified by

\k=l JO

JV

'fc Jdrh d0u ^ 0

where Xk(6k) and gk(dk) are assumed to be constant withineach strip for convenience here. Hence the algorithm con-verges for 0 < y < 1.

The algorithm above was implemented and compared withthe convolution algorithm in Reference 5. Real imageswere digitised using a microdensitometer to produce250 x 250 matrices whose circular part with diameter 250points was retained here. The pixel intensities varybetween 0 and 255. Each simulated projection data {gk}was computed by partitioning the circular image into theassociated P = 49 divergent-ray strips. The contribution ofeach pixel to a projection measurement was taken equal tothe product of the pixel intensity and the common area ofthis pixel with the corresponding divergent strip, dividedby the distance of the pixel from the source. The division isnecessary as the ray flux is proportional to the inverse ofthe distance from the source.

The estimated projection {g"} was computed from {X"}using system of eqn. 11 as specified in step (v) of the algo-rithm. The reconstruction, which is a 49 x 49 matrix here,is obtained from the final estimate of the Lagrange multi-pliers using eqn. 4; linear interpolation being used toproduce the continuous Lagrange multipliers. By averag-ing a 250 x 250 matrix, the 49 x 49 discretised originalimage was obtained. Plots of some corresponding rows (orcolumns) of the circular original and reconstruction imagesare shown for comparison of the algorithms. Whenevernoisy projections are used the added noise is Gaussianwith signal to noise ratio SNR = 20 dB.

Figs. 6A and B show the same row of the matricesassociated with the original image and the reconstructionof the convolution algorithm and that of the iterative algo-rithm obtained at the fourth iterating with y = 0.8, P = 49and N = 15 projections with and without noise. Thereconstruction of the iterative algorithm produced issmoother, follows the original better and avoids the spu-rious peaks which produce artifacts. Note the very highspurious peak of the convolution algorithm on the left ofthe graph. As in practice the projections always have noise,the features in the reconstructions with noise are moreimportant. These features are also observed in Fig. 7,where a column of the same original and reconstructionsare plotted.

The profile of the reconstruction of the convolutionalgorithm in Fig. 7B shows serious spurious peaks which

240 -f\

Fig. 6A Plot of a row from the original and the reconstructions of theconvolution and the new iterative algorithm

original• convolution

iterative

IEE PROCEEDINGS, Vol. 131, Pt. E, No. 3, MAY 1984 93

degrade the reconstruction as they lead to artifacts.Overall the quality of the reconstruction of the iterative

240

200

160

c~ 120

80

0 5 10 15 20 25 30 35 40 45x

Fig. 6B Plot of the same row from the original and the reconstructionsusing noisy projections with SNR = 20 (IB

original• convolution

iterativeThe iterative algorithm approximately matches the statistics of the Gaussian noiseadded to the projections with those of the errors between the 'true' projections andthe estimated ones from the Lagrange multipliers. Many spurious peaks appear inthe row of the convolution algorithm which seriously reduce the quality of thereconstruction.

algorithm introduced here is substantially better than thatof the convolution algorithm. Fig. 8 shows the distributionof the eigenvalues of matrix B for three sizes; which areproduced by taking P = 15 and N = 30, 60 and 120,respectively. The eigenvalues are scaled so that themaximum value is always 0.1. (N — 1) eigenvalues areclearly shown to be zero (Fig. 8). Furthermore there is asubstantial drop in the values of eigenvalues at aroundX = 150.

The convergence of the algorithm was verified computa-tionally for 0 < y < 2. However, when the projections havenoise, the final reconstruction must not satisfy them

240

200

150

80

40h

00 5 10 15 20 25 30 35 40 45

x

Fig. 7 A Plot of a column, same condition as Fig. 6 A

original• convolution

iterative

exactly as this results in seriously degraded reconstruc-tions. Figs. 9a and 9b shows profiles of the same row and

240

200

160

c~ 120a-

| 80

40

0 5 10 15 20 25 30 35 40 45x

Fig. 7B Plot of a column, same conditions as Fig. 6B

— original• • - - • • c o n v o l u t i o n

- • • i t e r a t i v e

Note the spurious peaks of the convolution algorithm

column of the original and the reconstructions of the iter-ative algorithm; one of which satisfies the noisy projec-tions exactly and the other is the same as in Figs. 6 and 7and satisfies the statistics of the noise used. Specifically theerrors between the projection data and the estimated pro-jections, corresponding to the Lagrange multipliers used toobtain the reconstruction (step (vi) of the algorithm), havethe same mean, variance and distribution with the Gauss-ian noise added to the projections approximately. Themean value of the projection errors above was approx-imately zero and its variance is shown in Fig. 10 where thevariance of the noise added to the projection data is alsogiven. The theoretical Gaussian distribution along with thedistribution of actual noise samples and the projectionerror samples for N = 15 and P = 49 are shown in Fig. 11.

0 200 400 600 800 1000 1200 1400 1600x

Fig. 8 Distribution of the eigenvalues of the matrix B (10)

N =120-. . . -_. N = 60,P= 15

N = 30The (N - 1) zero eigenvalues are very clear in all three matrix sizes

94 1EE PROCEEDINGS, Vol. 131, Pt. E, No. 3, MAY 1984

240 -

0 5 10 15 20 25 30 35 40 45

25 30 35 40 45

Fig. 9 The same row and column of the original, and two reconstruc-tions of the iterative algorithm; one satisfying the noisy projections exactlyand the other approximately matching (at the fourth iteration here) thestatistics of the Gaussian noise added to the projections (SNR = 20 dB)

original• iterative satisfying ys

iterative (4th iteration)The importance of matching the statistics is clearly demonstrated

10.5

10.0

9.5

9.0

8.5

8.0

7.5

701 10 20 30 40 50iteration number K

Fig. 10 Plot of the energy of the error between the 'true' projections andthe projections computed from the Lagrange multipliers of the kth iteration.

200r

180

160

140

120

100

8 0

60

40

20

Ob=

— \

\

hX\

-18635.8 18635.8

Fig. 11 Gaussian distribution and two histograms showing the distribu-tions of the noise samples and the errors between the 'true' projections andthe ones computed from the Lagrange multipliers corresponding to thereconstruction shown in Fig. 6 and Fig. 7

noiseGaussianerrors

G = noise varianceNoise variance is also shown

The resemblance of the distributions is clearly shown; infact, the #2-test of significance for goodness of fit gaveacceptance of the hypothesis with 95% confidence.

5 Conclusions

A fast algorithm was introduced which gives theminimum-energy reconstruction complying with the avail-able divergent-ray projections. Using the block successiverelaxation method it solves the linear system relating therearranged (defined later) projections and Lagrange multi-pliers without any approximation and exploits its specialmatrix structure, namely the areas of equal elements.Divergent-ray strips are called codirectional here if theircorresponding boundary lines have the same direction; theprojection measurements corresponding to all codirection-al strips form a rearranged projection. The rearrangedLagrange multipliers are defined similarly. The boundaryelements of the regions of equal elements in the sub-matrices are required only by the algorithm for the exactimplementation of the successive relaxation method; thusdrastically reducing the computational requirements. Thisiterative procedure differs from the additive ART in that itdiscretises the image instead of the Lagrange multipliers.

The convergence of the algorithm was proven.However, for Gaussian noise, the iterative procedure is ter-minated when the variance of the error between the projec-tion data and the projection corresponding to thereconstruction is equal to the variance of the projectionnoise, and the number of iterations is sufficiently large tomake the distribution of the above error Gaussian approx-imately. It should be noted here that this algorithm doesnot give the minimum-norm solution to the problem offitting noisy data to within an error criterion. There aremany sets of Lagrange multipliers and consequently manyimages which could give the same error power. The selec-tion of the reconstruction by minimum norm, which can bedefined using the Lagrange multipliers or image, results ina different approach than that of this algorithm. Computa-tional results, including the eigenvalue distribution andcomparison of the new algorithm with the convolutionalgorithm in Reference 5, have also been given.

IEE PROCEEDINGS, Vol. 131, Pt. E, No. 3, MAY 1984 95

6 Acknowledgment

Mr. Drossos would like to thank the 'A.S. Onassis publicbenefit foundation' for the scholarship awarded to him.

7 References

1 GOUTIS, C.E., and DURRANI, T.S.: 'Constrained optimisation algo-rithms for divergent ray tomography', IEEE Trans., 1981, NS-28, pp.3620-3627

2 GOUTIS, C.E.: 'Constraint optimisation algorithms for digital imagereconstructions from projections'. Ph.D. Thesis, Southampton Uni-versity, England, 1978.

3 DURRANI, T.S., and GOUTIS, C.E.: 'Optimisation techniques fordigital image reconstructions from their projections', IEE Proc. E,Comput. & Digital Tech., 1980,127, (5), pp. 161-169

4 GOUTIS, C.E., and DURRANI, T.S.: Tomographic algorithms forgeneral line integrals', IEEE Trans., 1982, NS-29, pp. 1399-1404

5 HERMAN, G.T., LAKSMINARAYAN, A.V., and NAPARSTEK, A.:'Convolution reconstruction techniques for divergent beams', Comput.Biol. & Met!., 1976, 6, pp. 259-271

8 Appendix

It is shown here that the hki nj matrix element (eqn. 8)depends on the angle if/' between the dichotomes of the njand ki strips only, as the angular width A9 is taken to beconstant.

Eqn. 8 may be written as

nnj, ki —

V+A0/2 fh 1

-drkd0krn

A0/2

-Ae/2 sin 9)dd> (12)

Fig. 12 Divergent-ray strip intersectionrk = OkA, rn = OnA

where rk, 6k, rn, /l5 l2 and </> are as in Fig. 12. Integratingwith respect to 6 gives

hkl.nl = l n ) a A f l \ # (13)

which depend on \j/' only, as A6 is taken to be constant.

Costas E. Goutis graduated from the Uni-versity of Athens, Greece, in 1967, andreceived the M.Sc. degree in Electronicsfrom Heriot-Watt University, Edinburgh,in 1974 and the Ph.D. degree fromSouthampton University, England, in 1979.He has worked for the University of Athensand the Greek Telecommunications Organ-isation as a Technical Manager. From 1976to 1979 he worked as a Research Assistantand Research Fellow in the University of

Strathclyde. In 1979 he joined the Department of Electrical andElectronic Engineering of the University of Newcastle uponTyne, where he is a lecturer. His research is in the areas of signaland image processing, and implementation of algorithms usingVLSI. Dr. Goutis is a member of IEEE.

Stavros N. Drossos was born in Athens,Greece, in 1958. He received the B.Sc.degree with Honours in Electrical and Elec-tronic Engineering from Sunderland Poly-technic in 1979 and the M.Sc. degree in thesame field from the University of Newcastleupon Tyne in 1980. He is currently a Doc-toral candidate at the University of New-castle upon Tyne and has been a recipientof an 'Onassis Foundation' Scholarship.His research interests are in algorithms in

image reconstruction from projections. Mr. Drossos is an associ-ate member of IEE and a student member of IEEE.

96 IEE PROCEEDINGS, Vol. 131, Pi. E, No. 3, MAY 1984