The Weighted Backprojection Techniques of Image Reconstruction

Ivan Kazantsev

Computing Center, 630090 Novosibirsk, Russia

Abstract . An approach is described for determining a vector of sum- mation weights which gives the best approximation in backprojection operation for image reconstruction from projections. Algorithm for the best choice of summation weights is presented and their influence on reconstruction quality is illustrated. Extension of the approach to the projection decomposed parts weighting is considered. The method sug- gested result in artefacts suppressing and reconstruction accuracy im- provement.

1 Introduction

Inside structure exploration of spatial objects with the help of data acqui- sition and processing system, using interaction of some type of penetrating beam with the object under investigation as the physical base is an important problem within many branches of medical imaging and engineering. Data got from acquisition system are integral features of object local properties; a useful ihformation is derived by data processing in correspondence with applied math- ematical model. As a result of this process an investigator gets an approximate object structure representatign in the form of digital images. The degree of re- construction approximation to the true structure is defined by a lot of factors. First of all, it is defined by a mathematical model adequacy, data completness and numerical algorithms features [6].

Because of variety of such factors and complication of their interaction, different problems of the best combination or the choice of imaging system characteristics and data processing techniques arise in computerized tomogra- phy [3]. As a rule, the aim is to enhance the image reconstruction accuracy at a reasonable time and means. This paper deals with considerations regarding the modification of well-known convolution-backprojection technique with prefer- ence to features of backprojection operator . Many attempts have been made since this method was invented to improve the quality of the reconstruction by investigating a number of convolution kernels, regularization procedures, etc. In this paper the factor of imbedding the summation coefficients in the backpro- jection formula for parallel geometry is investigated. Some aspects of optimal weighting of projections are already considered theoretically [1]. In this paper we try to answer the question: whether the reconstruction accuracy is affected by optimal choice of summation weights at all and what useful effects due to the weighting and projection decomposition for the given image could be obtained.

Htav~i~, ~ira (Eds.): CAIP '95 Proceedings, LNCS 970 �9 Springer-Vedag Berlin Heidelberg 1995

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2 Weighted Backprojection

Let D be a unit circle in the plane (x,y),L2(D) is the Lebesgue space of functions with the support D , f E L2(D). Denote J = [0,~r); a tuple w = (Wl,...,wn), w E j n = j • ... • j being a set of projection directions. We introduce P ~ - the X-ray transform that maps a function f into the set of its line integrals~ for some wj E J:

Pcoj f (s ) = p(wj, s) = f ( t . cos wj - s . sin w j, t . sin ~oj + s . cos ~j)dt.

In computerized tomography the data, as a rule, are presented in the form of collection

P, , f = (P~I f , ' " , P~n f) ,

for some n- tuple w E j n . We make an assumption that we are dealing with complete projections, i.e. that each p(wj,s) is known for all s. It is convinient for our present purpuses to use only complete projections as well as complete filtered projections and their ridge functions [4]. In general, we consider the reconstruction operator 7~ as a technique which provides us with a minimum norm solution. More precisely, the following theorem takes place [2, 4]:

T h e o r e m l . Let w = (w l , . . . ,wn) be a tuple of distinct angles. Let f E L2(D) and let h be the unique function in L2(D) of the smallest norm which satisfies Pwj f = Pwj h, j = 1 , . . . , n. Then there exist functions h i , . . . , hn E

L2([ -1 , 1], (1 - t2) - 1 / 2 ) such thai

n

h(x, y) = h j (x . r + y . sin j= l

The minimum norm solution is [2] the projection of f on F -l- , an orthogonal complement of the set :P = Njn___l Nj, Nj is null space of Pwj, and L2(D) =

. T ' ~ $ '• . Hence, the inner product (h, f - h) = 0 and it holds that

IIf - h(x, y)[[2 = i[fll2 _ [ih(x, y)H2.

The inner product is given by (f, g) = fD fgdxdy. Let us consider T~ to be the approximation of f by functions from the subspace spanned by rigid functions pj(x, y), which are backprojected filtered projections :

n

= v), (1) j= l

for some c j-s, where

pj(x, y) = f_~ OO

p(wj, t )k(x . cos ~oj + y - s in wj - t)dt.

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The convolution kernel k(t) is a regularized version of the inverse Fourier trans- form of function I~1. Discretized version of the kernel is not of special interest here, for it is seemed that overall effect of baekprojection weighting is common for different types of p--filters. In our computer experiments Shepp-Logan kernel [7] was used. The best vector of coefficients cj will minimize the distance from function f to a finite-dimensional subspace spanned by the elements Pl, . . . , Pn: [a]:

~2 = [ I f - n ( f ,w , c)[I 2 = Ilfll 2 - (2 (b , c ) - (re, c)), (2)

where b is the vector of elements bj = {f, pj}, F is the Gram matrix F = (Tij), 7ij = (P~, pj} and the scalar product is given by (b, c) = ~ bjcj. It is easy to see that

bj = (f, pj) = ; f(x, y)pj (x, y) = (qj, pj),

where qj(z,y) = p(wj, xcoswj + sinwj) is ridge function which obtained by backprojecting the projection pj. Note, that bj-s can be evaluated in one- dimensional projection space. Therefore, the inner products {f, pj} and the quantity Q(f, w, c) =_ 2(b, c) - (Yc, c) can be evaluated by means of projection data and the vector c. The greater Q, the smaller distance & It is known [5J, that for fixed f and w the functional Q has maximal value when the vector c satisfies equation Fc = b:

/ (p2,.p~)(p2,.p2) (p2,.p~) • c2 = (q~,p2> . .

\ {Pn,Pl) (Pn,P2) {Pn,Pn) c \ (qn,'Pn)

Then in the case of optimal vector e = F-lb we have, for fixed f and r

Qc(f, w) =_ (b, c) -- (b, F-lb).

The obtained functional Qc can be considered as a potentiality for the norm IIhll 2 (mentioned in Theorem i) evaluation. Let us denote u = (1 , . . . , 1)- vec- tor consisting of n units. When directions r are equidistant, approximation (1) becomes the traditional well-known backprojection technique 7~(f,~v, u) =

n ~ j = l pj(x, y). In this case we will measure the reconstruction accuracy with the help of the distance denoted by ~u which can be computed as follows:

~ = I l l - ~ ( f , ~ , u)[[ 2 : IIf[I 2 - Q ~ ( f , w) ,

where the functional Qu has a form Qu(f, w) = 2 ~jn=. 1 bj - ~ = 1 ~ j n l 7~j. In the same manner, we will denote a distance of the (optimal) weighted backpro- jection reconstruction from the image f by 6c:

~ = [If - ~ ( f , ~ , e)ll 2 = Ilfll 2 - Q ~ ( f , ~ ) ,

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Let us consider further extension of this approach to account a contribution into reconstruction not only from whole projections but also from their parts and samples. We wilt deal again with p-filtered and backprojected projections as before but splitted into sums of their parts. Let us introduce a breaking Wk, k = 1 , . . . , K of interval W = [-1, 1] with Uk Wk = W, w i n wj = 0 for i # j , and

a splitting of the function pj(x, y) into sum pj(x, y) = Zhk'=i wkj(x , y), such that

wkj(x,y) = { ~J(x,Y),f~ x 'c~ + y 's in~j E Wk for x . coswj + y . sinwj ~ Wk.

Approximation of f by the functions wkj has a form

K n c) = ckykj (x , y).

k=l j=l

Reordering the matrix (Ckj) E MK, n into the vector (ci) E R Kn, we obtain the representation similar to that in the formula (1):

Kn T4(f, ~, c) = Z ciwi(x, y). (3)

i=1

3 N u m e r i c a l S t u d i e s a n d C o n c l u s i o n s

The main aim of computer experiments is to compare reconstructions with vec- tors of weights u and c. The test image is a composition of three circles and a triangle. Projection data are mathematically generated as line integrals (128 equidistant parallel rays per view) through the image reconstructed. The views are equidistantly spaced over the full 180 ~ angular range. For interactively cho- sen n - - current number of projections and K - - a number of data groups in each projection decomposed, equidistant geometry of parallel rays was investi- gated. The Gram matrices of Kn x Kn size are calculated, numerically inverted and Qu, Qc-measures as indicators for the reconstruction accuracy are consid- ered (see Tables 1,2,3). Numerical experiments have shown that the reconstruc- tion accuracy increases (with Qc-measures) when summation elements in the baekprojection procedure are optimally weighted and decomposition of projec- tions is used. In our opinion, informative domains of the image are distorted by

Table 1. Weights summary for the test reconstruction without decomposition

6 projections (n=6), decomposition number K=I, Gram matrix of order 6 Projections: 1-st !2-nd 3-rd 4-th 5-th 6-th Projection weights for standard backprojection (ud:s) 1.00 1.00 1.00 1.00 1.00 1.00 Projection weights for optimal backprojection (cj-s : 0.69 1.02 0.67 1.05 0.69 0.98

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Table 2. Weights summary for the parts of projections decomposed

6 projections (n=6) , decomposition number K=4, Gram matrix of order 24 I Projections: 11-st 12-udl3-rdl4-thl5-th I 6:th

Vector c: Weights of the first part of the projection: 0.3010.30 0]28 1.04 0.95 1.0i Weights of the second part of the projection: 0.74 0.67 0.77 0.95 1.00 0.96 Weights of the third part of the projection: 0.75 0.79 0.76 0.86 0.87 0.87 Weights of the fourth part of the projection: 0.73 0.72 0.73 1.09 1.02 0.92

Table 3. Q-values summary for reconstructions with decomposition

16 projections (n = 6), decomposition number K, Gram matrix of order Kn] 2 UII = 2181375,Qu = 1730364

Decomposition number K: 1 4 I 8 I 12 Reconstruction accuracy evaluation Qc: 1876165 1938068 1979701 2011126

uniform weighting in formulas of the customary method. OptimM weighting and decomposition result in artefacts suppressing and have considerable influence on the reconstruction accuracy, especially in the case of data incompletness. It is possible also to indicate informative groups of the data processed after appro- priate decomposition of projections. Thus, it has been shown that based upon a set of optimally weighted and decomposed data, the convolution-backprojection operator improves its performance and becomes nonlinear procedure with fea- tures similar to algebraic reconstruction techniques.

Acknowledgments

The research described in this publication was made possible in part by Grant No. NPA000 from the International Science Foundation.

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