ADAPTIVE ALGEBRAIC RECONSTRUCTION OF CT IMAGES IN COMPARISON TO FREQUENCY DOMAIN METHODS. Bernhard Bundschuh Universität-GH-Siegen, Germany

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1 ADAPIVE ALGEBRAIC RECONSRUCION OF C IMAGES IN COMPARISON O FREQUENCY DOMAIN MEHODS Bernhard Bundschuh Universität-GH-Siegen German Abstract his paper presents a new algebraic algorithm for the reconstruction of computer tomographic (C) images. Since the reconstruction is an ill posed problem a least squares approach with adaptivel weighted linear shift invariant regularization functions is used. If the weighting functions are properl matched to the regularization functions the adaptive algorithm simultaneousl leads to proper noise suppression and enhanced resolution of discontinuities. Simulations demonstrate the reconstruction capabilities of the new algorithm in comparison to filtered backprojection. 1. Statement of the Problem Fig. 1 shows the data collection process in C using parallel beam projections. It can be described as the application of a twodimensional linear shift variant filter [1] to the original image. Basicall fan beam projections lead to the same considerations [1]. he generation of the measured data g(tϑ) can be described using a line integral. In x-ra-c the noise n(tϑ) is Poisson distributed []. ( ϑ) = ( ) δ ( cosϑ + sin ϑ ) + ( ϑ) g t s x x t dxd n t (1) Using a different point of view eq. defines the linear shift variant point spread function of the twodimensional linear sstem which generates the measured data. ( ϑ) = ( ) ( ϑ) + ( ϑ) g t s x h x t dxd n t ( ϑ) = δ( cosϑ + sinϑ ) h x t x t ()

2 he discrete equivalent of eq. can be written using components: ( d ϑ ) = ( x ) ( x d ϑ ) + ( d ϑ ) g i i s i i h i i i i n i i ix i (3) or vector matrix notation: g = H s + n (4) emitters x-ras h(i x i i d i ϑ ) = A/ x original image s(x) x x A g(t ϑ ) t ϑ detectors projection Figure 1: Parallel beam C-scanner. Regularized Algebraic Reconstruction Since H represents a low pass sstem and the measured data g are nois the reconstruction of s is an ill posed problem [3]. he quadratic error term E Q includes a group of linear shift invariant regularization functions which are represented b the block oeplitz matrices F1 F K F M. Since we intend to develop an adaptive algorithm we introduce the inverse weighting matrices V1 V K V A M and. ( ) ( ) ( M M M ) = g Hs$ A g Hs$ + s$ F V F + F V F + K + F V F s$ (5) E Q he solution vector $s is found b minimization of the error E Q. Setting the derivative of E Q with respect to the solution vector to zero leads to a linear sstem of equations.

3 E Q $s = 0 (6) ( M M M ) $s = H A H + F V F + F V F + K + F V F H A g (7) V1 V K V M and A are optimized b minimization of the expectation D Q of the squared distance between the original signal s and the estimated signal $s. D Q ( $ ) ( $ ) = s s s s (8) DQ V m = 0 m DQ A = 0 (9) Straightforward application of matrix differentiation rules and matrix algebra leads to M+1 identical matrix equations so that there is an infinite number of solutions. 1 1 ( 1 1opt 1 opt M Mopt M ) F V F + F V F + K + F V F ss H = H A nn (10) Since each solution minimizes D Q we are free to select the simplest one: opt Aopt = nn (11) Fm VmoptFm ss H mh = µ Aopt nn m (1) Vmopt = µ m 1 Fm ss F m m (13) where: µ 1 + µ + K+ µ M = 1 0 µ m 1 m he matrices V1 opt V opt K V Mopt and Aoptare determined b the autocorrelation matrices in other words b the statistics of the original signal and the noise. If the are inserted into eq. 7 the resulting estimate is similar to the estimate provided b a shift variant Wiener Filter [34]. If the signal and the noise are Gaussian distributed it is also similar to a Maximum a Posteriori estimate [3]. here are two reasons wh this algorithm is not applicable to most practical problems. First the autocorrelation matrices must be known ver accuratel. Usuall this is no problem for the noise. But a reliable estimate of the autocorrelation matrix of the original signal is hardl available. Second the size of the matrices is prohibitivel large especiall for multidimensional signals. he autocorrelation matrix of an image which consists of not more than pixels contains coefficients!

4 3 Suboptimal Diagonal Weighting Matrices In order to derive a suboptimal but applicable algorithm we do not longer search for optimal matrices in the general form but for optimal diagonal matrices. E Q ( ) D ( ) ( 1 D1 1 D M DM M ) = g Hs$ A g Hs$ + s$ F V F + F V F + K + F V F s$ (14) ( D 1 D1 1 D M DM M ) $s = H A H + F V F + F V F + K + F V F H A g (15) V V K V and A are again optimized b minimization of D Q (eq. 8). D1 D DM D he simplest solution is: DQ vdm = 0 m i DQ ad = 0 k (16) ii kk D ad kkopt = nn vdm = µ m 1 Fm ss Fm (17) kk iiopt ii We onl take into account the diagonal elements of the previousl determined general weighting matrices. Obviousl VDmopt and ADopt are the diagonal matrices which are closest to Vmopt and Aopt. If the noise is white the optimal diagonal matrix A D equals the general matrix A. he local variances of the Poisson distributed noise which can be estimated from the measured data provide the diagonal elements of A Dopt. ( ) adopt = σ n id i (18) ϑ Now we can calculate the diagonal elements of the weighting matrix V Dm. vdm = µ 1 m fm( jx j ) s( ix jx i j ) ii (19) jx j he reciprocals determine the optimal weighting function wm( ix i ). w i i f j j s i j i j jx j ( ) = µ ( ) ( ) m x m m x x x (0) Eq. 0 indicates that each optimal weighting function is matched to the corresponding regularization function in a remarkabl simple wa [4]. he selection of a group of suitable regu-

5 larization functions [4] which is not critical completel determines the optimal weighting functions. EQ = n id i g i i s i i h i i i i i i i i d ϑ ( ) ( d ) $ ( x ) ( x d ) σ ϑ ϑ ϑ + wm ix i f j j s i j i j m i i x jx j x ( ) m( x ) $ ( x x ) + (1) ( ) de ds$ q q = 0 q q () Q x x Basicall the solution of the resulting linear sstem of equations can be found b matrix inversion. But usuall an iteration is much faster. he weighting functions lead to strong smoothing in continuous regions and to modest smoothing at discontinuities [4]. ix i ( ) s$ ix i σ n ( id iϑ ) h( ix i id iϑ ) h( qx q id iϑ ) + i i d ϑ + w j j f j i j i f j q j q m j j x ( ) ( ) ( ) m x m x x m x x = ( id i ) g( id i ) h( qx q id i ) = σ n ϑ ϑ ϑ id iϑ (3) Unless a pure simulation is performed w m (i x i ) is not known a priori. herefore the first guess of the unknown signal e.g. the result of filtered backprojection is also used to obtain a first estimate of w m (i x i ). If we assume that during the iteration the estimated signal converges to the original signal improved estimates of the weighting functions can be determined successivel from the intermediate results of the iteration. Simulation experiments showed that even in the case of a starting guess of 0 this trick approach converges without problems. So we obtain an algorithm which is self updating at least up to a certain degree of accurac [4]. Since the general matrices V m were replaced b the diagonal matrices V D the application of the regularization functions to the original signal should create signals which are as "white" as possible. On the other hand the number of coefficients of f m (i x i ) is limited. Otherwise the number of computations required at each step of the iteration would be too large. Examples for appropriate regularization functions are the differential operators mentioned in part 4 of this paper. A more detailed but preliminar investigation can be found in [4].

6 4. Results of Simulation Figure shows the widel used Shepp and Logan head phantom [1] which provides a standard test signal for the qualit assessment of C reconstruction algorithms. It consists of a collection of ellipses of different size brightness and orientation. Figure : est signal Figure 3 shows the result of a reconstruction using filtered backprojection [1]. he noise which is superimposed to the projections is white and Poisson distributed. he signal to noise ratio is 30 db. A cos -window in the frequenc domain suppresses high frequenc noise which is emphasized too strongl b the filter operation before the backprojection is performed. Figure 3: Result of filtered backprojection

7 Figure 4: Adaptive algebraic reconstruction Figure 4 shows a reconstruction using the adaptive algebraic approach. Since the basic element of the head phantom is an ellipse four regularization and weighting functions are used two in direction of the coordinate axes and two in diagonal direction. ( x ) = δ( x ) δ( x ) ( x ) = δ( x ) δ( x ) ( x ) = δ( x ) δ( x ) ( x ) = δ( x ) δ( x + ) f j j j j j j 1 1 f j j j j j j 1 f 3 j j j j j 1 j 1 f 4 j j j j j 1 j 1 he result of filtered backprojection is used as a starting guess for the iteration. he result of algebraic reconstruction is less grain than that of filtered backprojection and the contours appear much clearer. On the other hand using a general purpose computer the filtered backprojection is calculated within a few seconds while the algebraic reconstruction lasts several minutes for a pixel image. Using a special purpose hardware processor it should be possible to reduce the time for an algebraic reconstruction b several orders of magnitude. Since the algebraic reconstruction method is less sensitive to noise its main advantage seems to be the potential reduction of the required radiation dose. 5. Conclusions he adaptive algebraic algorithm provides clearer reconstructions than filtered backprojection and it is less sensitive to noise. herefore it offers the potential for a considerable reduction of the required radiation dose. he reduction of the required computer power is the main problem which remains to be solved. A special purpose reconstruction hardware should allow a reduction b several orders of magnitude. 6. Acknowledgement I would like to thank Ulrich Wieczorek Detlev Herrmann and Uwe Wiemann for their support and for the valuable discussions. Also I would like to thank the Zentrum für Sensorssteme and the ministr of science of Nordrhein Westfalen who funded this work. References [1] A.C. Kak M. Slane Principles of Computerized omographic Imaging IEEE Press New York 1987 [] G.. Herman Image Reconstruction from Projections the Fundamentals of Computerized omograph Academic Press New York London oronto Sdne San Francisco 1980 [3] H.C. Andrews B.R. Hunt Digital Image Restoration Prentice-Hall Englewood Cliffs N.J. 1977

8 [4] B. Bundschuh Adaptive Image Restoration and Reconstruction hird International Seminar on Digital Image Processing in Medicine Remote Sensing and Visualization of Information Riga April 199