Fan-beam and cone-beam image reconstruction via filtering the backprojection image of differentiated projection data

Transcription

1 INSTITUTE OF PHYSICS PUBLISHING Phys. Med. Biol. 49 (2004) PHYSICS IN MEDICINE AND BIOLOGY PII: S (04) Fan-beam and cone-beam image reconstruction via filtering the backprojection image of differentiated projection data Tingliang Zhuang 1, Shuai Leng 1, Brian E Nett 1 and Guang-Hong Chen 1,2 1 Department of Medical Physics, University of Wisconsin-Madison, WI 53704, USA 2 Department of Radiology, University of Wisconsin-Madison, WI 53792, USA [email protected] Received 16 August 2004, in final form 14 October 2004 Published 6 December 2004 Online at stacks.iop.org/pmb/49/5489 doi: / /49/24/007 Abstract In this paper, a new image reconstruction scheme is presented based on Tuy s cone-beam inversion scheme and its fan-beam counterpart. It is demonstrated that Tuy s inversion scheme may be used to derive a new framework for fanbeam and cone-beam image reconstruction. In this new framework, images are reconstructed via filtering the backprojection image of differentiated projection data. The new framework is mathematically exact and is applicable to a general source trajectory provided the Tuy data sufficiency condition is satisfied. By choosing a piece-wise constant function for one of the components in the factorized weighting function, the filtering kernel is one dimensional, viz. the filtering process is along a straight line. Thus, the derived image reconstruction algorithm is mathematically exact and efficient. In the cone-beam case, the derived reconstruction algorithm is applicable to a large class of source trajectories where the pi-lines or the generalized pi-lines exist. In addition, the new reconstruction scheme survives the super-short scan mode in both the fan-beam and cone-beam cases provided the data are not transversely truncated. Numerical simulations were conducted to validate the new reconstruction scheme for the fan-beam case. (Some figures in this article are in colour only in the electronic version) 1. Introduction Recently, development of novel image reconstruction methods has attracted a lot of attention and many important discoveries have been made in both the fan-beam case and the cone-beam case. The major breakthroughs in the fan-beam case include the discovery of super-short /04/ $ IOP Publishing Ltd Printed in the UK 5489

2 5490 T Zhuang et al scan mode (Noo et al 2002, Kudo et al 2002, Chen 2003a) and the novel solutions for certain data truncation problems (Clackdoyle et al 2004, Clackdoyle and Noo 2004, Noo et al 2004). These new discoveries were partially inspired by the breakthrough made in solving the longstanding cone-beam image reconstruction problem for helical scanning trajectories. The most recent quantum leap in cone-beam reconstruction methods was made by Katsevich (2002a, 2002b) who proposed a shift-invariant filtered backprojection (FBP) cone-beam reconstruction algorithm for the helical source trajectory. A shift-invariant FBP reconstruction formula has also been derived for the general source trajectory (Katsevich 2003, Chen 2003b). Another important breakthrough was made by Zou and Pan (2004) who proposed a new reconstruction scheme via filtering the backprojection image of differentiated cone-beam projection data (FBPD) along the pi-lines of the helical source trajectory. The essential idea of the cone-beam FBPD was also exploited in the fan-beam case by Noo et al (2004) to solve a class of fan-beam data truncation problems. In this paper, we generalize the Zou Pan FBPD method for a conventional helical source trajectory to a general source trajectory that satisfies the Tuy data sufficiency condition. This is based on our previous work (Chen 2003a, 2003b) in developing a shift-invariant FBP algorithm for fan-beam or cone-beam projections acquired from a general source trajectory. The starting point of our derivation is Tuy s cone-beam inversion formula (Tuy 1983), and its fan-beam counterpart (Chen 2003a). One of the key observations used in this paper is that an extra condition on the weighting function must be imposed to derive an FBPD-type algorithm. Specifically, the general weighting function w( x, ˆk; t) should be constructed to satisfy the following factorization property: w( x, ˆk; t) = w 1 ( x, ˆk)w 2 ( x,t)sgn[ˆk y (t)]. This condition is required in both the fan-beam case and the cone-beam case. In contrast, there is no additional condition on the weighting function in the shift-invariant fan-beam FBP algorithm (Chen 2003a). In the cone-beam case, there is indeed an extra constraint on the weighting function (Chen 2003b). However, the condition is significantly different from the aforementioned factorization property. The main results of this paper are as follows. First, we demonstrate that the filtering kernel is one dimensional if a piece-wise constant function is chosen for the component w 2 ( x,t) of the weighting function w( x, ˆk; t). Second, in the cone-beam case, a mathematically exact FBPD image reconstruction algorithm is derived for a large class of source trajectories including: conventional helical trajectories with N-pi data acquisition (Proksa et al 2000), dynamical pitch helical trajectory (Ye et al 2004a, Katsevich et al 2004), variable radius helical trajectory (Ye et al 2004b), and the saddle trajectory (Pack et al 2004). Third, in the fan-beam case, we derived a new FBPD algorithm for the super-short scan mode. Our algorithm invokes an explicit derivative with respect to the source parameter t. This feature dictates that the new fan-beam FBPD super-short scan algorithm is different from the algorithm recently discovered by Noo et al (2004). Finally, a new term was derived for calculating the backprojection image of differentiated projection data. This result is slightly different from the original Zou Pan formulation (Zou and Pan 2004), and is also different from the results of Noo et al (2004). Preliminary numerical simulations were conducted to validate the new FBPD fan-beam algorithm. 2. Derivation of the general fan-beam and cone-beam FBPD reconstruction method for a general source trajectory We assume f( x) is the image function to be reconstructed. The function f( x) is compactly supported. A general source trajectory is parametrized by a single parameter t,t = [t i,t f ]. Here t i and t f are assumed to be the view angles corresponding to the starting and ending points of an open source trajectory. A point on the source trajectory is measured in a

3 Fan-beam and cone-beam image reconstruction via filtering the backprojection image 5491 laboratory system and denoted as y(t). The divergent fan-beam and cone-beam projections from a source point y(t) are defined as g[ r, y(t)] = + 0 dsf[ y(t) + s r]. (2.1) An intermediate function G D [ k, y(t)] is defined as the Fourier transform of the above divergent beam projections g[ r, y(t)] with respect to the variable r, viz., G D [ k, y(t)] = d rg[ r, y(t)]e i2π k r. (2.2) R D The letter D labels the dimensions of a Euclidean space. Thus, D = 2 corresponds to the fan-beam case, and D = 3 corresponds to the cone-beam case. Throughout the remainder of the paper, the convention of decomposing a vector into its magnitude and a unit direction vector will be used, i.e. r = r ˆr and k = kˆk. Using the above two definitions, the following properties of the projections and intermediate functions can be readily demonstrated (Chen 2003a, 2003b): g[ r, y(t)] = 1 ḡ[ˆr, y(t)], r (2.3) G D [ k, y(t)] = 1 k ḠD[ˆk, y(t)], D 1 (2.4) Im G D [ k, y(t)] = Im G D [ k, y(t)], (2.5) where Im G D [ k, y(t)] is the imaginary part of the intermediate function G D [ k, y(t)], it is an odd function with respect to k (2.5) and related to the function G odd D ( k, y(t)) in [9] by G odd D ( k, y(t)) =iimg D [ k, y(t)]. In addition, ḡ[ˆr, y(t)] and Ḡ D [ˆk, y(t)] arethe angular components of the projection g[ r, y(t)] and the intermediate function G D [ k, y(t)], respectively. With the above definitions and mathematical convention in hands, the image function f( x) may be reconstructed using the modified Tuy-like formula (Chen 2003a, 2003b, Tuy 1983): f( x) = dˆk w( x, ˆk; t j ) S D 1 2π ˆk y k j (t j ) q Im Ḡ D [ˆk, y(q)] q=tj. (2.6) Here w( x, ˆk; t j ) is a weighting function used in order to take into account the multiple solutions of the following equation: ˆk [ x y(t j )] = 0. (2.7) In the cone-beam case, the unit vector ˆk is the normal vector of a plane that contains the line connecting a given image point ( x), and a source point ( y(t j )). While, in the fan-beam case the unit vector ˆk is the normal vector to the line connecting a given image point ( x), and a source point ( y(t j )). This equation dictates the well-known Tuy data sufficiency condition: any plane (cone-beam case) or straight line (fan-beam case) that passes through the region of interest (ROI) should intersect the source trajectory at least once. In equations (2.6) and (2.7), y(t j ) labels the jth intersection point between the plane containing the image point x and the source trajectory y(t). The set which is the union of all these solutions is labelled as T( x, ˆk), viz., t j T( x, k) ={t ˆk [ x y(t)] = 0}. (2.8)

4 5492 T Zhuang et al The weighting function w( x, ˆk; t) satisfies the following normalization condition: w( x, ˆk; t j ) = 1. (2.9) t j T( x,ˆk) In addition, using property (2.5) and the inversion formula (2.6), the weighting function possesses the following symmetry: w( x, ˆk; t) = w( x, ˆk; t). (2.10) In order to proceed, we use the following well-known formula (Bracewell 1986) δ(t t j ) t j ˆk y (t j ) = δ{ˆk [ x y(t)]} (2.11) to turn the summation over t j into an integral over the continuous variable t in equation (2.6). Consequently, we obtain (Chen 2003a, 2003b) f( x) = 1 2π dt S D 1 k dˆkw( x, ˆk; t)sgn[ˆk y (t)] q Im Ḡ D [ˆk, y(q)] q=t δ[ˆk ( x y(t))]. (2.12) Starting with the above equation, as shown in Chen (2003a, 2003b), one can derive a shiftinvariant FBP reconstruction formula for the fan-beam and cone-beam projections respectively. However, in order to derive the desired FBPD formula, it is insightful to use the following integration representation of the delta function (Bracewell 1986): δ[ˆk ( x y(t))] = + dk e i2πkˆk ( x y(t)) (2.13) to rewrite equation (2.12) as follows: f( x) = 1 + dt dk dˆkw( x, ˆk; t)sgn[ˆk y (t)] 2π 0 S D 1 q Im Ḡ D [ˆk, y(q)] q=t e i2πkˆk [ x y(t)] dt dk dˆkw( x, ˆk; t)sgn[ˆk y (t)] 2π S D 1 q Im Ḡ D [ˆk, y(q)] q=t e i2πkˆk [ x y(t)]. (2.14) By changing the variables k k and ˆk ˆk in the second term and using equations (2.10) and (2.5), we obtain f( x)= 1 + dt dk dˆkw( x, ˆk; t)sgn[ˆk y (t)] π 0 S D 1 q Im Ḡ D [ˆk, y(q)] q=t e i2πkˆk [ x y(t)]. (2.15) In this equation, the dummy variable k takes only non-negative values. Thus, the unit vector ˆk and the non-negative value k can be combined as a complete vector, i.e., k = kˆk. Multiplying and dividing the angular component Im G D [ˆk, y(t)] by a factor k D 1 and using property (2.4), we obtain f( x) = 1 dt d kw( x, ˆk; t)sgn[ˆk y (t)] π R D q Im G D[ k, y(q)] q=t e i2πkˆk [ x y(t)], (2.16) k where we have used the volume element d k = dkk D 1 dˆk in a D-dimensional Euclidean space.

5 Fan-beam and cone-beam image reconstruction via filtering the backprojection image 5493 Figure 1. The meaning of the modified projections ḡ A [ ˆβ x, y(t)]. For a given source position and a given view angle t, the modified projection ḡ A [ ˆβ x, y(t)] = g[ ˆβ x, y(t)] for the geometric configuration shown in (a); however, ḡ A [ ˆβ x ] = g[ ˆβ x, y(t)] for the geometric configuration shown in (b). In the following, we use the definition of the intermediate function G D [ k, y(t)] to calculate the derivative with respect to the source parameter: q Im G D[ k, y(q)] q=t = Im d r R D q g[ r, y(q)] q=t e i2π k r = d r R D q g[ r, y(q)] q=t Im[e i2π k r ] = d r R D q g[ r, y(q)] 1 q=t 2i (e i2π k r e i2π k r ) = 1 d r 2i R D q {g[ r, y(q)] g[ r, y(q)]} q=t e i2π k r. (2.17) Note that the variable r is defined in a local coordinate system, i.e., r = x y(t), (2.18) where x is a point in the backprojection image space. Therefore, for the fan-beam and cone-beam projections at a given source position y(t), the substitution of equation (2.18)into equation (2.17) and the application of property (2.3) yield: q Im Ḡ D [ k, y(q)] q=t = 1 d x 1 qḡa [ ˆβ 2i R D x x, y(q)] q=t e i2π k [ x y(t)], (2.19) y(t) where the unit vector ˆβ x is defined as ˆβ x = x y(t) x y(t). (2.20) The modified fan-beam and cone-beam data ḡ A [ ˆβ x, y(t)] are defined as ḡ A [ ˆβ x, y(t)] = ḡ[ ˆβ x, y(t)] ḡ[ ˆβ x, y(t)]. (2.21) The meaning of ḡ A [ ˆβ x, y(t)] will be elaborated upon later (figure 1). After substituting equation (2.19) into equation (2.16) and switching the order of integrations, we obtain f( x) = 1 d x d k e i2π k ( x x ) dt w( x, ˆk; t)sgn[ˆk y (t)] qḡa [ ˆβ 2πi R D R D k x x, y(q)] q=t. y(t) (2.22)

6 5494 T Zhuang et al In the above equation, the integrations over the variables t and k are coupled to one another through the factor w( x, ˆk; t)sgn[ˆk y (t)]. A key observation in this paper is to impose the following constraint, i.e., a factorization property, on the choice of the weighting functions: w( x, ˆk; t) = w 1 ( x, ˆk)w 2 ( x,t)sgn[ˆk y (t)]. (2.23) In other words, we require that the weighting function is factorable so that the integrations over the variables t and k can be decoupled. Under this condition, the reconstruction formula (2.22) can be further simplified as f( x) = d x K( x, x )Q( x, x ), (2.24) R D where K( x, x ) = 1 2πi R D k d kw 1 ( x, ˆk) e i2π k ( x x ), (2.25) Q( x, x w 2 ( x,t) ) = dt qḡa [ ˆβ x x, y(q)] q=t. (2.26) y(t) In equation (2.24), the image reconstruction consists of two major steps. In the first step, the fan-beam or cone-beam projections are weighted by a weighting function w 2 ( x,t), and then processed by backprojecting the differentiated projection data to obtain the function Q( x, x ). In the second step, the function Q( x, x ) is multiplied by a kernel function K( x, x ), and then integration is performed over the entire Euclidean space spanned by the variable x, to obtain the reconstructed image value at the point x. It is important to elaborate on the meaning of the modified fan-beam and cone-beam projections ḡ A [ ˆβ x, y(t)]. As shown in figure 1, the modified projections ḡ A [ ˆβ x, y(t)] are generally non-zero in the whole Euclidean space. Therefore, one potential problem in the derived image reconstruction method is that we have to numerically compute an integral over an infinite space. In addition, without an explicit choice of the weighting function, it is not yet clear whether the kernel function may possess an explicit and simple form. Therefore, up to this point, it is not certain whether equation (2.24) (2.26) provide a computationally efficient and practical image reconstruction algorithm. Fortunately, although we require the weighting function to satisfy the factorization property (2.23), we still have sufficient degrees of freedom to choose an appropriate weighting function that will turn equations (2.24) (2.26) into a practical and efficient algorithm. Namely, it is possible to obtain a one-dimensional filtering kernel. In this case, although the function Q( x, x ) is not band limited along the filtering lines, one can still use the data from a finite range to reconstruct the image (Noo et al 2004, Milkhlin 1957). In the following sections, we will show how to choose the weighting function to make this kernel function simple and easy to implement in practice. Before we discuss specific examples of cone-beam and fan-beam image reconstruction via the FBPD, further clarification upon the properties of the weighting function is beneficial. Since the weighting function satisfies the normalization condition (2.9), the components w 1 ( x, ˆk) and w 2 ( x,t) in equation (2.23) are not completely independent of one another. In fact, given a choice of the component w 2 ( x,t), the component w 1 ( x, ˆk) is determined by the following equation: w 1 ( x, ˆk)w 2 ( x,t j ) sgn[ˆk y (t j )] = 1. (2.27) t j T( x,ˆk)

7 Fan-beam and cone-beam image reconstruction via filtering the backprojection image 5495 Since the function w 1 ( x, ˆk) does not depend on t, the summation over t j for the last two terms on left-hand side of the above equation can be calculated by converting the summation into an integral: w 2 ( x,t j ) sgn[ˆk y (t j )] t j T( x,ˆk) where = = dt w 2 ( x,t)sgn[ˆk y (t)] ˆk y (t) δ[ˆk ( x y(t))] dt + + dlw 2 ( x,t)ˆk y (t) e i2πlˆk [ x y(t)] = 1 dl dt ei2πlˆk [ x y(t)] w 2 ( x,t) 2πi l t = dt w 2 ( x,t) t h( x, ˆk,t), (2.28) h( x, ˆk, t) = 1 2πi + dl ei2πlˆk [ x y(t)] l = 1 2 sgn[ˆk ( x y(t))]. (2.29) The function w 1 ( x, ˆk) is the inverse of the above integral value in equation (2.28). 3. Mathematically exact cone-beam image reconstruction via FBPD along pi-lines and generalized pi-lines For the conventional helical source trajectory with 1-pi data acquisition, there is one and only one pi-line for each point in the ROI defined by the cylindrical volume contained within the helix Defrise et al (2000). However, in the case of an N-pi data acquisition scheme (Proksa et al 2000), there are multiple different lines that pass through each image point in the ROI and intersect the helical trajectory at two different points. Recently, the pi-line concept has also been generalized to the case of a helical trajectory with dynamical helical pitch (Ye et al 2004a, Katsevich et al 2004) and the case of a helical trajectory with a variable radius (Ye et al 2004b). For these generalized helical trajectories, under certain conditions, there is one and only one pi-line for each point in the ROI inside a helix (Ye et al 2004a, 2004b, Katsevich et al 2004). The pi-line concept has been generalized to the saddle trajectory (Pack et al 2004). An important feature of the saddle trajectory is that the pi-lines are not unique for a point within the ROI. However, a common feature of the aforementioned trajectories is the existence of at least one pi-line or one generalized pi-line. The pi-line provides us a natural geometric guideline in choosing the weighting function. If we denote the corresponding arc of a pi-line associated with an object point x as I( x) = [ t πa ( x),t πb ( x) ], we have a convenient choice for the component w 2 ( x,t) in the weighting function: w 2 ( x,t) = { 1, if t I( x), 0, otherwise. With this choice of w 2 ( x,t), the function w 1 ( x, ˆk) can be calculated as follows. (3.1) Using

8 5496 T Zhuang et al equation (2.28), we have w 2 ( x,t j ) sgn[ˆk y (t j )] = t j T( x,ˆk) I( x) dt t h( x, ˆk,t) = h( x, ˆk,t πa ) h( x, ˆk,t πb ) Since t πa and t πb are the endpoints of the pi-line I( x),wehave = 1 2{ sgn [ˆk ( x y ( t πa ))] sgn [ˆk ( x y ( t πb ))]}. (3.2) sgn [ˆk ( x y ( t πa ))] sgn [ˆk ( x y ( t πb ))] = 2sgn [ˆk ( y ( t πb ) y ( tπa ))]. (3.3) Therefore, the function w 1 ( x, ˆk) can be determined using equation (2.27): w 1 ( x, ˆk) = sgn[ˆk ( y(t πb ) y(t πa ))]. (3.4) Using the explicit form of the component w 1 ( x, ˆk) in equation (3.4), the kernel function K( x, x ) can be calculated from equation (2.25)as K( x, x ) = 1 d k sgn[ˆk e π ( x)]e i2π ˆk ( x x ), (3.5) 2πi R 3 k where e π ( x) = y(t πb ) y(t πa ) has been introduced. If we align the z-axis of the vector k along the pi-line, we obtain K( x, x ) = 1 1 δ(x 2π 2 z π z π π x π )δ(y π y π ), (3.6) where x and x are now along pi-lines, viz. x = (x π,y π,z π ) and x = (x π,y π,z π ). Therefore, we obtain an explicit one-dimensional filtering kernel. This is the same kernel function first obtained by Zou and Pan (2004) for a conventional helical trajectory. Here we show that the same kernel can be used in any source trajectory provided that the concept of a pi-line is meaningful. Therefore, the filtering kernel is a shift-invariant Hilbert kernel along the pi-line. In fact, the same form of the function w 2 ( x,t) as given in equation (3.1) may be chosen to reconstruct the points on an arbitrary straight line that passes one point and intersects the source trajectory at two different points t a and t b. As long as the form of w 2 ( x,t) is chosen as given in equation (3.1), the filtering kernel given in equation (3.5) is applicable. The backprojection function Q( x, x ) is given by Q( x, x 1 ) = dt qḡa [ ˆβ x x, y(q)] q=t. (3.7) y(t) I( x) The image point, x, dependence of this function shows up through the pi-line. The function ḡ A [ ˆβ x, y(t)] is defined in equation (2.21), which is different from the function D( r 0 (q), ˆβ( r,λ))used by Zou and Pan in their original publication (equation 11) (Zou and Pan 2004). In summary, we have derived a mathematically exact image reconstruction algorithm for any source trajectory provided that the concept of a pi-line or a generalized pi-line is meaningful and the source trajectory satisfies the Tuy data sufficiency condition. In this algorithm, an image is reconstructed via filtering the backprojection image of differentiated projection data using a one-dimensional Hilbert kernel as given by equations (3.7) and (3.6).

9 Fan-beam and cone-beam image reconstruction via filtering the backprojection image 5497 Figure 2. For a given image point x, a horizontal line intersects with the arc at two points t a and t b. The backprojection image values are calculated using the projection data acquired over the angular range [t a,t b ]. The same straight line is used to reconstruct all the points on it. 4. Fan-beam FBPD image reconstruction for a circular source trajectory In this section, we consider image reconstruction from fan-beam projections on an arc source trajectory with radius r. The source trajectory is parametrized as y(t) = r(cos t,sin t), t [t i,t f ], (4.1) where t i and t f are view angles corresponding to the starting and ending points on the scanning path. For this scanning path, there are infinitely many lines that pass through an object point x and intersect the source trajectory at two points. In order to derive an explicit FBPD reconstruction algorithm, we discuss two possible choices of the component w 2 ( x,t) of the weighting function w( x, ˆk; t). Case 1. Weighting function chosen for filtering along parallel lines The intersection points of this horizontal line with the source trajectory are labelled by y(t a ( x)) and y(t b ( x)). The first choice of the function w 2 ( x,t) is { 1, if t [ta,t b ], w 2 ( x,t) = (4.2) 0, otherwise. Using equation (2.27) and (2.28), we obtain w 1 ( x, ˆk) = sgn[ˆk ( y(t b ) y(t a ))]. (4.3) Substituting the above equation into equation (2.25), we obtain K( x, x ) = 1 1 2π 2 x x δ(y y ). (4.4) Here we have aligned the vector k along the horizontal x-direction. Thus, the filtering is a Hilbert transform conducted along the horizontal direction. In the case of an equi-angular detector, the derivative filtering backprojection function Q( x, x ) is given by Q( x, x ) = = tb ( x) t a ( x) tb ( x) t a ( x) 1 dt qḡa [ ˆβ x x, y(q)] q=t, y(t) ( 1 dt x y(t) t γ ) [g m (γ, t) γ =γ x g m (γ, t) γ =γ x +π], (4.5) where g m (γ, t) is the measured fan-beam projection data at view angle t and fan angle γ.the values of γ x and ˆβ x are determined by γ x = φ x t + π, ˆβ x = (cos φ x, sin φ x ), (4.6) as demonstrated in figure 3. The final reconstruction formula is given by f( x) = 1 2π dx 1 dy x x δ(y y )Q( x, x ). (4.7)

10 5498 T Zhuang et al y x β x φx γ x x Figure 3. Illustration of the geometric meaning of angles γ x and φ x. Figure 4. The second choice of the component w 2 ( x,t) in the weighing function along the three arcs labelled by T 1 ( x),t 2 ( x) and T 3 ( x). For a given reconstruction point x = (x, y), the backprojection function Q( x, x ) should be calculated over an infinite horizontal line which passes through x. Namely, x = (x,y ) in equation (4.5) extends along an infinite one-dimensional line. If the image function is compactly supported, a modified Hilbert transform may be used to perform the filtering process which only requires data over a finite range. This procedure will be discussed in detail in the numerical simulations section given below. The implementation steps are summarized below: Calculate the backprojection image values of differentiated fan-beam data on a Cartesian grid. Filter the backprojection image with the Hilbert kernel along horizontal lines (figure 2). The advantage of this method is that the image reconstruction can be conducted directly on a Cartesian grid. However, the disadvantage is that, for different filtering lines, varying amounts of the projection data are used in the derivative backprojection step. Therefore, this choice of weighting function potentially leads to non-uniform noise distribution. Case 2. Weighting function chosen for filtering along intersecting lines In this case, the objective is to use the same amount of projection data to reconstruct all the points within an ROI. For a given image point x, two lines are used to connect the image point x with the two ending points at the view angles t i and t f. As shown in figure 4 these two straight lines intersect with the scanning path at the view angles t a ( x) and t b ( x), respectively (figure 4). Consequently, the scanning path is divided into the three arcs: T 1 ( x) ={t t i <t<t a ( x)}, T 2 ( x) ={t t a ( x)<t<t b ( x)}, (4.8) T 3 ( x) ={t t b ( x)<t<t f }.

11 Fan-beam and cone-beam image reconstruction via filtering the backprojection image 5499 The function w 2 ( x,t) is chosen to be a, if t T 1 ( x), w 2 ( x,t) = 1, if t T 2 ( x), (4.9) b, if t T 3 ( x), where constants a and b satisfy the condition: a + b = 1, a b. (4.10) Using the above function w 2 ( x,t), the integral in equation (2.28) can be directly calculated to yield: w 2 ( x,t j ) sgn[ˆk y (t j )] = dt w 2 ( x,t) t h( x, ˆk,t) t j T( x,ˆk) = a sgn[ˆk ( y(t b ) y(t i ))]+b sgn[ˆk ( y(t a ) y(t f ))], (4.11) where in the last step, the fact that the point x lies at the intersection between the line passing through ( y(t i ), y(t b )) and the line passing through ( y(t f ), y(t a )) has been used. The inverse of the above equation provides the factor w 1 ( x, ˆk) in the weighting function. The result is w 1 ( x, ˆk) = a b a sgn[ˆk b ( y(t b ) y(t i ))]+ b a sgn[ˆk ( y(t a ) y(t f ))]. (4.12) For simplicity, we introduce the following notation: ê i ( x) = y(t b) y(t i ) y(t b ) y(t i ), ê f ( x) = y(t a) y(t f ) y(t a ) y(t f ). (4.13) Substituting equation (4.12) into equation (2.25), we obtain, { } K( x, x 1 a ) = δ[( x x ) ê 2π 2 (b a) ( x x i ) ê ]+ b δ[( x x ) ê i ( x x f ) ê ], f (4.14) where { ê i, ê {ê } i} and f, ê f are two sets of orthonormal bases, viz., ê i ê i = 0, ê f ê f = 0. (4.15) Therefore, the filtering kernel is still the Hilbert kernel. However, the filtering lines are along the two directions ê i and ê f respectively. In this case, the backprojection operation is given by [ ] Q( x, x ) = a + + b dt T 1 ( x) T 2 ( x) T 3 ( x) ( 1 x y(t) t ) [g m (γ, t) γ =γ x g m (γ, t) γ =γ x +π], (4.16) γ where γ x is determined by equation (4.6) and the weighting factors a and b satisfy equation (4.10). The final reconstruction formula is given by 1 + { + f( x) = dx dy a δ[ ] ( x x ) ê i + b δ [ ]} ( x x ) ê f Q( x, x ). 2π 2 (b a) ( x x ) ê i ( x x ) ê f (4.17) For a given reconstruction point x = (x, y), the backprojection function Q( x, x ) should be calculated over two infinite lines which pass through x. Namely, x = (x,y ) in equation (4.16) extends along two infinite one-dimensional lines, as shown in figure 4.

12 5500 T Zhuang et al The implementation steps are summarized below: Calculate the backprojection image values of differentiated fan-beam data on a Cartesian grid. Filter the backprojection image with the Hilbert kernel along the lines labelled by ê i and ê f (figure 4). Sum the weighted and filtered contributions from the two filtering lines. The advantage of this method is that the same amount of projection data is used for all the reconstructed points within an ROI. However, the disadvantage is that, for a single point filtering process has to be performed along two filtering lines. 5. Numerical simulations In this section, we present the numerical simulations to illustrate the validity of the proposed fan-beam FBPD image reconstruction algorithm. For simplicity, the filtering lines were chosen to be along the horizontal direction, and were equally spaced. The filtering kernel and the backprojection operation for differentiated fan-beam data are given in equations (4.4) and (4.5), respectively. In the new image reconstruction framework presented in this paper, the backprojection image of differentiated projection data is well defined for any point x in the Euclidean space. Thus, the Hilbert transform used to reconstruct image points along the filtering lines is mathematically well defined. However, it is a demanding task to numerically conduct the Hilbert transform along the filtering line in that the function Q( x, x ) is not band limited. Fortunately, when the target function is compactly supported data from a finite range may be used as input for a modified Hilbert transform to reconstruct the target function (Noo et al 2004, Milkhlin 1957). For the purpose of these simulations the image to be reconstructed is non-zero only within a circle of radius R 0, and the filtering operation was performed along horizontal lines. Under these conditions, the following formula was used to reconstruct image points ( x): f(x,y 0 ) = 1 2π 2 1 R2 x 2 [ +R R ] dx R2 x 2 Q( x, x ) + C, (5.1) x x where x = (x, y 0 ), x = (x,y 0 ), R > R 0 and C is a constant which is filtering line dependent. In this formula, only the data in the range x [ R,+R] of the function Q( x, x ) are needed in the image reconstruction. To determine the constant C in equation (5.1), one can use the fact that image values should be zero when the image point is outside the support. Therefore, if a point x c = ( x,y 0 ) is outside the image support, then the constant C can be determined as +R C = dx R2 x 2 Q( x R x c, x ). (5.2) x In practice, several points along the filtering lines which were also outside the region of compact support were chosen to calculate the constant C. The mean value of several calculated C values was used in (5.1) to determine the image values for points along the same filtering line. A standard Shepp Logan phantom (Kak and Slaney 1987) was used to generate the projection data from an arc source trajectory. The fan angle was chosen to be γ m = π to avoid 4 the data truncation. The sampling rate of the equi-angular detector was chosen as γ = γ m π The sampling rate of the view angles was chosen to be t = In computing the backprojection image of differentiated projection data, the derivatives were calculated by a three-point formula. The images were reconstructed by filtering the 727.

13 Fan-beam and cone-beam image reconstruction via filtering the backprojection image Theoretical Values Reconstructed Values Theoretical Values Reconstructed Values (a) (b) (c) Figure 5. The FBPD image reconstruction of the Shepp Logan phantom using projection data acquired from a short-scan trajectory. (a) The reconstructed image with a display window of [0.99, 1.06]. (b) The intensity plots from the central horizontal line of both the theoretical and reconstructed image values. (c) The intensity plots from the central vertical line of both the theoretical and reconstructed image values Theoretical Values Reconstructed Values Theoretical Values Reconstructed Values (a) (b) (c) Figure 6. The FBPD image reconstruction of the Shepp Logan phantom using projection data acquired in the super-short-scan mode. (a) The reconstructed image with a display window of [0.99, 1.06]. (b) The intensity plots from the central horizontal line of both the theoretical and reconstructed image values. (c) The intensity plots from the central vertical line of both the theoretical and reconstructed image values. backprojection image along evenly spaced horizontal lines. A Hann window was used to suppress the high frequency fluctuations in the reconstructed images. Figure 5(a) demonstrates an image which was reconstructed from a short-scan trajectory, the angular range was [t i,t f ] = [ π 8, ] 9π 8. Figure 5(b) shows the intensity plots along the central horizontal line. Figure 5(c) presents the intensity plots along the central vertical line. Figure 6(a) demonstrates an image reconstructed from the projection data acquired along a super-short scan trajectory, the angular range of the scanning path was chosen to be [t i,t f ] = [0,π]. Figure 6(b) shows the intensity plots along the central horizontal line. Figure 6(c) demonstrates the intensity plots along the central vertical line. 6. Conclusions and discussion In this work, an FBPD inversion scheme has been formulated for any general source trajectory that fulfils the Tuy data sufficiency condition. The starting point of our derivation was the

14 5502 T Zhuang et al cone-beam Tuy inversion scheme and its fan-beam counterpart. After imposing a factorization property on the weighting function, it was shown that the cone-beam Tuy inversion formula and its fan-beam counterpart can be generically written as an integral of a product of two functions, viz., K( x, x ) and Q( x, x ). The function Q( x, x ) was the weighted backprojection operation on the differentiated projection data. The function K( x, x ) was the Fourier transform of one of the factors, w 1 ( x, ˆk), of the factorized weighting function w( x, ˆk; t). We have shown that a numerically efficient reconstruction scheme can be derived if the second component w 2 ( x,t)of the weighting function w( x, ˆk; t) is chosen to be piecewise constant. In this case, the filtering kernel is reduced to a Hilbert kernel or a linear combination of Hilbert kernels along different filtering lines. In the cone-beam case, we have generalized Zou Pan s reconstruction formula for a conventional helical trajectory to a large class of source trajectories for which the existence of pi-lines has been established by other investigators (Ye et al 2004a, 2004b, Katsevich et al 2004, Pack et al 2004). In the fan-beam case, we derived a new image-reconstruction algorithm via filtering the backprojection image of differentiated projection data for both the short and super-short scanning modes. Preliminary numerical results were presented to validate the new fan-beam FBPD algorithm for both the short and super-short scanning modes. Future research will include the study of implementation issues such as optimal data flow, and computational efficiency. The applications of this image FBPD reconstruction formulation to other important cone-beam and fan-beam scanning modes will also be further explored in the future. Acknowledgments This work was partially supported by National Institute of Health grants 1R21 EB , 1R21 CA , a startup grant from University of Wisconsin-Madison and a grant from GE Healthcare Technologies. References Bracewell R N 1986 The Fourier Transform and Its Applications 2nd edn (New York: McGraw-Hill) Chen G-H 2003a A new framework of image reconstruction from fan beam projections Med. Phys Chen G-H 2003b An alternative derivation of Katsevich s cone-beam reconstruction formula Med. Phys Clackdoyle R and Noo F 2004 A large class of inversion formulae for the 2D Radon transform of functions of compact support Inverse Problems Clackdoyle R, Noo F, Guo J and Roberts J 2004 A quantitative reconstruction from truncated projections in classical tomography IEEE Trans. Nucl. Sci Defrise M, Noo F and Kudo H 2000 A solution to the long-object problem in helical cone-beam tomography Phys. Med. Biol Kak A C and Slaney M 1987 Principles of Computerized Tomographic Imaging (New York: IEEE Press) Katsevich A 2002a Theoretically exact filtered backprojection-type inversion algorithms for spiral CT SIAM J. Appl. Math Katsevich A 2002b Analysis of an exact inversion algorithm for spiral cone-beam CT Phys. Med. Biol Katsevich A 2003 A general scheme for constructing inversion algorithm for cone beam CT Int. J. Math. Math. Sci Katsevich A, Basu S and Hsieh J 2004 Exact filtered backprojection reconstruction for dynamic pitch helical cone beam computed tomography Phys. Med. Biol Kudo H, Noo F, Defrise M and Clackdole R 2002 New super-short scan algorithms for fan-beam and conebeam reconstruction Conference Record of the 2002 IEEE Nuclear Science Symposium and Medical Imaging Conference (Norfolk, VA) (IEEE Service Centre) vol 2, pp Milkhlin S G 1957 Integral Equations (New York: Pergamon) pp Noo F, Clackdoyle R and Pack J D 2004 A two-step Hilbert transform method for 2D image reconstruction Phys. Med. Biol

15 Fan-beam and cone-beam image reconstruction via filtering the backprojection image 5503 Noo F, Defrise M, Clackdoyle R and Kudo H 2002 Image reconstruction from fan-beam projections on less than a short-scan Phys. Med. Biol Pack J D, Noo F and Kudo H 2004 Investigation of saddle trajectories for cardiac CT imaging in cone-beam geometry Phys. Med. Biol Proksa R, Köhler Th, Grass M and Timmer J 2000 The N-pi-Method for Helical Cone-beam CT IEEE Trans. Med. Imaging Tuy H K 1983 An inversion formula for cone-beam reconstruction SIAM J. Appl. Math Ye Y, Zhu J and Wang G 2004a Minimum detection windows, PI-line existence and uniqueness for helical cone-beam scanning of variable pitch Med. Phys Ye Y, Zhu J and Wang G 2004b Geometrical studies on variable radius spiral cone-beam CT Med. Phys Zou Y and Pan X 2004 Exact image reconstruction on PI-lines from minimum data in helical cone-beam CT Phys. Med. Biol