A pointwise limit theorem for filtered backprojection in computed tomography

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1 A pointwise it theorem for filtered backprojection in computed tomography Yangbo Ye a) Department of Mathematics University of Iowa Iowa City Iowa Jiehua Zhu b) Department of Mathematics University of Iowa Iowa City Iowa Ge ang c) Departments of Mathematics Biomedical Engineering and Radiology University of Iowa Iowa City Iowa Received 6 September 22; accepted for publication 4 January 23; published 22 April 23 Computed tomography CT is one of the most important areas in the modern science and technology. The most popular approach for image reconstruction is filtered backprojection. It is essential to understand the it behavior of the filtered backprojection algorithms. The classic results on the it of image reconstruction are typically done in the norm sense. In this paper we use the method of ited bandwidth to handle filtered backprojection-based image reconstruction when the spectrum of an underlying image is not absolutely integrable. Our main contribution is assuming the method of ited bandwidth to prove a pointwise it theorem for a class of functions practically relevant and quite general. Further work is underway to extend the theory and explore its practical applications. 23 American Association of Physicists in Medicine. DOI:.8/ Key words: computed tomography CT image reconstruction filtered backprojection error analysis ited bandwidth pointwise it I. INTRODUCTION Computed tomography CT is one of the most important areas in modern science and technology. The most popular approach for image reconstruction is filtered backprojection. 2 Let f (xy) be an absolutely integrable function on R 2 representing an object with a finite support such as defined on a unit disk. In this context the standard procedure for computed tomogrphy CT refers to the following steps: Step. Projection: P t f xyx y sin tdx dy Step 2. Filtration: Q t S wwe 2itw dw where S w Step 3. Backprojection: P te 2iwt dt f xy Q x y sin )d. The correctness of the above procedure is usually shown by citing the Fourier slice theorem that 2 3 S w fˆw w sin f xye 2ixw yw sin dx dy. Rigorously speaking the filtered backprojection algorithm defined by 2 and 3 is valid at almost every point (xy) for an absolutely integrable function f (x y) when its Fourier transform fˆ(u) is also absolutely integrable. hen fˆ(u) is not absolutely integrable it is essential to understand the it behavior of the filtered backprojection algorithms. There are various strategies to handle this divergence problem. Examples include the method of ited bandwidth the Abel method of summability and the Shepp Logan filter technique. 8 Generally speaking different strategies have their own advantages and disadvantages. As a matter of fact CT scanner manufacturers typically implement a number of reconstruction filters to balance spatial and contrast resolution for different applications. Typically the method of ited bandwidth produces higher spatial resolution while the Abel method of summability and the Shepp Logan filtering lead to better contrast resolution. The convolution kernel associated with the method of ited bandwidth is usually referred to as the standard kernel while the smooth kernels associated with the Shepp Logan method and other similar methods may be regarded as contrast kernels. The Abel method of summability is to introduce a convergence factor e w and then take : 4 86 Med. Phys. 3 5 May Õ23Õ3 5 Õ86Õ7Õ$2. 23 Am. Assoc. Phys. Med. 86

2 87 Ye Zhu and ang: A it theorem in CT 87 Q t S wwe w e 2itw dw f xy Q x y sin d for and f xy f xy. 7 For any absolutely integrable function f (xy) it is a classical result that f (xy) tends to f (xy) in the L norm as : f xy f xydx dy. Using the concept of the Lebesque set of f (xy) we can further prove that f (xy) tends to f (xy) as almost everywhere. 3 It is our goal in the present paper to study the it behavior of the method of ited bandwidth. This method is to assume that the bandwidth is ited and replace 2 and 3 by 9 and : Q t S wwe 2itw dw f xy Q x y sin d. Then we take the it to reconstruct the image f (xy): f xy f xy. As far as the method of ited bandwidth defined in 9 is concerned a corresponding L statement does not exist. In fact 9 can be written as Q t S ww/we 2itw dw 2 where is the characteristic function of the interval which can be interpreted as the characteristic function in polar coordinates on the unit disk x 2 y 2. Consequently f (xy) is the Fourier inversion of fˆ(uv)(u/v/). By the Fourier convolution formula f xy f xy * u/v/ where ((u/v/)) is expressed as follows: u/v/ xy Here in polar coordinates u/v/ e 2ixuyv du dv 2 ˆ xy. ˆ w w sin J 2w/w 3 4 is not absolutely integrable on the plane where J is the Bessel function of the first kind of order one see 2 below. Therefore f (xy) in3 is not of L in general. e need an L 2 theory to describe the it behavior of the ited bandwidth based reconstruction in. Assume f L (R 2 )L 2 (R 2 ) then fˆl 2 (R 2 ). The Fourier transform of an L 2 function g is defined as the L 2 it of ĝ xy guvu/v/ e 2iuxvy du dv. 5 By the Plancherel theorem 3 we immediately have f xy f xy 2 dx dy. 6 Similar to 6 the classic results on the it of image reconstruction are typically done in the norm sense. The most famous results are summarized in the book by Natterer. 2 Specifically they worked with C and assumed the sup norm defined on H ( n ) as the worst-case error. The only prior information they made use of is f H ( n ) with being about /2 and moderate see pp in Ref. 2 for details. In this paper we study the method of ited bandwidth to handle filtered backprojection based image reconstruction when the spectrum of an underlying image is not absolutely integrable. Particularly assuming the method of a ited bandwidth we prove a pointwise it theorem for a class of functions practically relevant and quite general. In the second section we present our main result the pointwise it theorem. In the third section we give two analytic examples while in the last section we discuss several issues and conclude the paper. II. POINTISE LIMIT THEOREM Because the pointwise it is stronger and much more specific than the L 2 it in this section we prove a pointwise it theorem to refine 6 for a class of quite general functions f (xy). Indeed we will consider functions f (xy) that are smooth but have finitely many jumps of finite magnitude along piecewise continuous curves. Theorem pointwise it theorem : Let f(xy) be an absolutely integrable function on R 2. Assume that other than finitely many jumps of finite magnitude of the function and/or its first and second partial derivatives along piecewise continuous curves this function has absolutely integrable second partial derivatives. Then the filtered backprojection using the method of ited bandwidth converges pointwise at any point (x y ) not on a jump i.e. f x y f x y. e remark that in the scenario of medical imaging there are distinctive components in the human body including solid organs soft tissues bones and so on. ithin each compo- Medical Physics Vol. 3 No. 5 May 23

3 88 Ye Zhu and ang: A it theorem in CT 88 nent the x-ray linear attenuation coefficient varies smoothly and can be well modeled by smooth functions with continuous second derivatives. On the other hand across the boundaries among the components jumps in the x-ray linear attenuation coefficient its first and second partial derivatives can be represented by a finite number of gate-like functions. The discontinuities across the boundaries in the first and second derivatives can be relevant in contrast dynamics such as bolus-driven perfusion for cancer studies. Before we prove Theorem let us formulate this class of functions. As we will use polar coordinates centered at (x y ) in our proof we express the curves of discontinuities in the following way. Note that this expression of our function f (xy) directly depends on a given point (x y ) not on a jump. For a different point (x y ) the polar coordinates will be different and the formulas for the function f (x r y r sin ) will also be different. ithout a loss of generality let rr j () j...m be piecewise continuous curves in. e may further assume that r ()r 2 () r m ()C for any. To capture various jumps of the function and its derivatives on these curves we use a polynomial of r in the region between two adjacent curves. More precisely define jump functions: j h j ra a j r a 5 j r 5 if r j rr j otherwise 7 for j...m where as coefficients a (k) j () are bounded a (k) j ()C 2 piecewise continuous functions of. Here we use a polynomial of degree 5 because we need to use six coefficients a (k) j () to match the function values its first and second derivatives in the direction of r on the outer and inner rims of the region between r j () and r j (). e remark that our jump function in 7 generalizes the gate function s j r a j if r j rr j otherwise. This generalization allows us to control jumps not only in value magnitude but also in derivatives. For our function f (xy) in Theorem we request that in polar coordinates centered at the point (x y ) it can be written as f x r y r sin gx r y r sin h r h m r 8 such that i g(xy) is absolutely integrable on R 2 ii g(xy) has all second partial derivatives everywhere in R 2 and iii its second partial derivatives are also absolutely integrable on R 2. Note that on the right side of 8 the second differentiability of g(x y) on jump curves is achieved by suitably choosing the coefficients a j (k) () of the polynomials h j (r). Actually we may replace conditions ii and iii by a weaker condition that 2gx r y r sin tdot 9 for some fixed ast. Here we used the Fourier ine transform ĝ t grrtdr. The assumption 9 essentially means that the Fourier ine transform of g(x y) along any half-line starting at (x y ) goes to in O(t ) as t. This is a version of the Riemann Lebesque theorem. 4 If as in Theorem g(x y) has absolutely integrable second partial derivatives the bound in 9 will be O(t 2 ). The magnitude of the jumps cannot be allowed to go to infinity as controlled by a j (k) ()C 2 for j...m and k...5. The jumps cannot occur at the point (x y ) either so that r j () cannot approach to. If we denote by cc(x y ) the shortest distance from (x y ) to the nearest discontinuous curve we then have r j ()c(x y ). e emphasize that the class of the above-defined functions f (xy) in8 should be regarded as a universal model for a variety of practical applications. Technically g(x r y r sin ) is smooth and describes continuous changes in the image domain while terms h j (r) capture the structural discontinuities. Combining the two kinds of bases we can satisfactorily approximate images that are regionwise smooth. In the proof below and in later sections we will use the Bessel function J n of order n of the first kind and the Gauss hypergeometric series 2 F defined as and J n z 2 2 e iz sin tnt dt 2F ab;c;z n aa anbb bn z n. cc cnn! See Ref. 6 for details. Proof of Theorem : By 3 we have f x y f xy * 2 ˆ xy x y 2 In polar coordinates it is f x xy y ˆ xydx dy. 2 2 Medical Physics Vol. 3 No. 5 May 23

4 89 Ye Zhu and ang: A it theorem in CT 89 f x y 2 2 d f x r y r sin J 2r rdr r 2 d f x r y r sin J 2rdr 2 2 d t f x 2 y t sin 2 J tdt. 22 Recall the Parseval formula for absolute integrable functions f and g on : f tĝ tdt fˆ tgtdt. 23 It is known that 5 t; J t t t 2 t. 24 Denote the function on the right side as g(t). Then by the inversion formula of the Fourier ine transform J t 2 g t. 25 Consequently f x y 2 2 r d f x 2 y r sin 2 r d 2 f x 2 y r sin 2 t 2 gtdt 2 tdt 2 r d 2 f x 2 y r sin tdt 2 t t 2 f x y 2 2 d f x r y r sin tdt 2t t If for a fixed 2 (g(x r y r sin ) (t)d O(/t ) then we have 2 2 d gx r y r sin 2t tdt t 2 dt t t 2 27 which converges and tends to as. Consequently the contribution to 26 of the term g in the expression of f (x r y r sin ) in8 will tend to. As for the terms a j (k) ()r k in h j (r) on the right side of 8 we have the following computation: a j u j r xa j sin r jx sin r j x x a j ru j r x a j d dx r j x r j x x a 2 j r 2 u j r x a 2 j d2 sin r j x sin r jx dx 2 x a 3 j r 3 u j r x a 3 j d3 r jx r j x dx 3 x a 4 j r 4 u j r x a 4 j d4 sin r jx sin r j x dx 4 x a 5 j r 5 u j r x a 5 j d5 r j x r jx dx 5 x where u j (r) for r j ()rr j () and vanishes elsewhere. Consequently Medical Physics Vol. 3 No. 5 May 23

5 82 Ye Zhu and ang: A it theorem in CT 82 a k j r k u j r x a k j r x j k sin r j x r j k sin r j x 28 for k...5 where represents terms with x 2 or higher powers of x in the denominators. These omitted terms will contribute expressions tending to when asin 27. The leading term in 28 in turn contributes 2 a k 2 j r j k sin 2r d j t dt t 2 2 a k 2 j r j k d sin 2r j t dt t a k j r j k J 2r j d 2 2 a k j r j k J 2r j d. 29 Recall that we assumed that a j () and r j () are piecewise continuous and bounded: a j ()C 2 c(x y )r j () C for 2. Consequently using the asymptotic expansion J z 2 z z/4oz3/2 3 we obtain a bound O(C k C 2 k /c(x y )) for 29 which tends to as. The theorem then follows immediately. III. EXAMPLES A. Disk function Theorem covers the characteristic function on the unit disk f xy x2 y 2 3 otherwise. Its Fourier transform fˆw w sin S w J 2w 32 w is not absolutely integrable over the plane. Now 9 becomes According to we have f xy Q x y sin d 2 d J 2w 2wx y sin dw. 34 hen (xy)() we may express x y sin as x 2 y 2 sin( ) for a certain phase shifting. Changing variables we have f xy2 d J 2w2wx 2 y 2 sin dw d 2 J wwx 2 y 2 sin dw. Switch the order of integration f xy 2 J wdw wx 2 y 2 sin d 2 J wj wx 2 y 2 dw hen (xy)() 36 is also true. If x 2 y 2 we have the discontinuous integral of eber Schafheitlin as the it f xy J wj wx 2 y 2 dw x2 y 2 x 2 y 2. Therefore 37 f xy f xy 38 for points (xy) not on the unit circle. If x 2 y 2 we have 2 J wj wdw 2 J Hence for x 2 y 2 f xy 2 J Q t S wwe 2itw dw w w J 2we 2iwt dw 2 J 2w2twdw. 33 B. Delta function If we apply the band-ited reconstruction algorithm 9 and to the Dirac delta function f (xy)(xy) the resultant (xy) exhibits interesting singular behaviors. In fact ˆ w w sin S w. 4 Medical Physics Vol. 3 No. 5 May 23

6 82 Ye Zhu and ang: A it theorem in CT 82 If t we have Q t we 2itw dw sin2t sin2 t. 42 t 2 t 2 hen (xy)() xy Q x y sin d sin2x2 y 2 sin x 2 y 2 sin sin2 x 2 y 2 sin d 43 2 x 2 y 2 sin 2 by changing variables. Denote Ax 2 y 2 and tsin. The integrand then equals sin2at At sin2 At A 2 t 2 Integrating it termwise we obtain xy A n 2 2 n n 2At 2n. 2n!2n2 44 n A 2n n!n! A J 2A x 2 y J 2x 2 y 2 2 if (xy)(). On the other hand it is easy to find that For a given point (xy)() we may use the asymptotic expansion J z 2 z z 3 4 O z 3/2 47 and derive an asymptotic expansion of 45 as : xy x 2 y 2 3/4 2x 2 y O 5/2. 48 /2 x 2 y 2 Consequently in addition to () according to 46 (xy) will have a larger and larger magnitude of vibration at any fixed location (xy)() as. Nevertheless (xy) indeed approaches (xy) as in the following sense that for a function f (xy) asin Theorem f x xy y xydx dy f x y 49 where the point (x y ) satisfies the requirement in Theorem. This is because the integral on the left-hand side of 49 is actually equal to f (x y ) and Theorem applies. In particular 2 xydx dy d r J 2rr dr 2 J 2rdr J zdz. 5 It is interesting to compare (xy) with (xy) obtained by the Abel method of summability: 2 xy x 2 y /2 It is well known that (xy) tends to as at any point (xy)(). IV. DISCUSSIONS AND CONCLUSION There are relative merits and drawbacks associated with different it methods for handling the case that fˆ(u) is not absolutely integrable. Given comparable parameters the method of ited bandwidth tends to produce finer spatial resolution higher image noise and stronger edge ringing while the Abel method of summability may result in smoother profiles at the t of poorer spatial resolution. A heuristic way to define the comparability of the parameters with the two methods may be to request that the area after hard/soft spectral truncation be identical. Mathematically it is requested that wdw which is equivalent to we w dw or. 53 Because f (xy) is a partial sum of a Fourier series it exhibits the Gibbs phenomenon around discontinuous points. An example is a reconstructed unit disk function for as shown in Fig.. On the other hand the Gibbs phenomenon does not occur with the Abel method of summability as demonstrated by f (xy) with.2.44 in Fig. 2. However the sharpness of the slope on the unit circle is better in Fig. than that in Fig. 2 which indicates better spatial resolution in the former than that in the latter. Quantitatively the magnitude of the gradient of f (xy) on the unit circle is computed as follows: Medical Physics Vol. 3 No. 5 May 23

7 822 Ye Zhu and ang: A it theorem in CT 822 FIG.. Radial profile of a unit disk reconstructed using the method of ited bandwidth. FIG. 2. Radial profile of a unit disk reconstructed using the Abel method of summability. grad f r 2 J wj ww dw 2 J 2 ww dw J 2 2J 2J and its counterpart is grad f r e w/2 J wj ww dw e w/2 J 2 ww dw F ;3; F ;3; It can be shown that for 3 we have grad f rgrad f r The current choice by CT manufacturers is based on the method of ited bandwidth and allows various degrees of modifications on the associated ramp filter while the Abel method of summability remains an interesting theoretical possibility. The popularity of the method of ited bandwidth is due to the discrete detection technology and the Shannon sampling theorem. However it is well known that the Fourier spectrum spreads over the whole space for any spatial function finitely supported. Suppose that we can record projection data continuously or extrapolate the Fourier spectrum well beyond the Nyquist bandwidth imposed by the sampling rate the Abel method of summability may have some practical merits. In addition to the methods we have discussed other methods are also possible based on various other ways to do the intermediate integrals which should be equivalent to using approximate delta function of different types. 7 In conclusion for the method of ited bandwidth we have proved a pointwise it theorem for a general class of functions. Our results have improved the understanding on filtered backprojection in CT. Although our work has been done in the parallel beam geometry the findings are instructive for the divergent beam geometry. Further work is underway to extend the theory and explore its practical applications. ACKNOLEDGMENTS The work is partially supported by NIH Grant No. RO DC359 and a grant from the Carver Scientific Research Initative Grant Program. The authors thank Seung ook Lee John Meinel and Xiang Li with the CT/Micro-CT Lab University of Iowa for discussions and are grateful to anonymous reviewers for constructive critiques. a Electronic mail: yangbo-ye@uiowa.edu b Electronic mail: jiehua-zhu@uiowa.edu c Electronic mail: ge-wang@uiowa.edu A. C. Kak and M. Slaney Principles of Computerized Tomographic Imaging Society for Industrial and Applied Mathematics Philadelphia 2. 2 F. Natterer The Mathematics of Computerized Tomography Society for Industrial and Applied Mathematics Philadelphia 2. 3 E. M. Stein and G. eiss Introduction to Fourier Analysis on Euclidean Spaces Princeton University Press Princeton M. J. Lighthill An Introduction to Fourier Analysis and Generalized Functions Cambridge University Press Great Britain H. Bateman Tables of Integral Transforms McGraw-Hill New York 954 Vol. I. 6 H. Bateman Higher Transcendental Functions McGraw-Hill New York 953 Vol. II. 7 G.. ei Discrete singular convolution for the solution of Fokker- Plank equation J. Chem. Phys F. Natterer Numerical methods in tomography Medical Physics Vol. 3 No. 5 May 23