IOP PUBLISHING
PHYSICS IN MEDICINE AND BIOLOGY
Phys. Med. Biol. 53 (2008) 3841–3861
doi:10.1088/0031-9155/53/14/009
An optimization-based method for geometrical calibration in cone-beam CT without dedicated phantoms D Panetta1,2, N Belcari1,2, A Del Guerra1,2 and S Moehrs1,2 1 2
Department of Physics ‘E. Fermi’, University of Pisa, L.go B. Pontecorvo, 3-I-56127 Pisa, Italy INFN, Pisa, L.go B. Pontecorvo, 3-I-56127 Pisa, Italy
E-mail:
[email protected]
Received 10 December 2007, in final form 5 May 2008 Published 26 June 2008 Online at stacks.iop.org/PMB/53/3841 Abstract In this paper we present a new method for the determination of geometrical misalignments in cone-beam CT scanners, from the analysis of the projection data of a generic object. No a priori knowledge of the object shape and positioning is required. We show that a cost function, which depends on the misalignment parameters, can be defined using the projection data and that such a cost function has a local minimum in correspondence to the actual parameters of the system. Hence, the calibration of the scanner can be carried out by minimizing the cost function using standard optimization techniques. The method is developed for a particular class of 3D object functions, for which the redundancy of the fan beam sinogram in the transaxial midplane can be extended to cone-beam projection data, even at wide cone angles. The method has an approximated validity for objects which do not belong to that class; in that case, a suitable subset of the projection data can be selected in order to compute the cost function. We show by numerical simulations that our method is capable to determine with high accuracy the most critical misalignment parameters of the scanner, i.e., the transversal shift and the skew of the detector. Additionally, the detector slant can be determined. Other parameters such as the detector tilt, the longitudinal shift and the error in the source–detector distance cannot be determined with our method, as the proposed cost function has a very weak dependence on them. However, due to the negligible influence of these latter parameters in the reconstructed image quality, they can be kept fixed at estimated values in both calibration and reconstruction processes without compromising the final result. A trade-off between computational cost and calibration accuracy must be considered when choosing the data subset used for the computation of the cost function. Results on real data of a mouse femur as obtained with a small animal micro-CT are
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shown as well, proving the capability of the proposed calibration method. In principle, the method can be adapted to other cone-beam imaging modalities (e.g., single photon emission computed tomography).
1. Introduction The demand for high-precision methods for the calibration of high-resolution micro-CT scanners is motivated by the very high dependence of the scanner performances to the mechanical positioning of the various components. An uncorrected displacement of the axis of rotation (AoR) in the order of magnitude of the detector pitch may result in a significant image blurring, and consequently in a loss of image spatial resolution. Displacements are continuously added in the system by many factors, including mechanical instability, vibrations, thermal drift of bearings and components, etc. Many methods have been proposed for the measurement of the misalignment parameters from the analysis of the acquisition data, both for the 2D fan beam geometry (Gullberg et al 1984, Azevedo et al 1990) and 3D cone-beam geometry (Noo et al 2000, Bequ´e et al 2003, von Smekal et al 2004, Yang et al 2006). In the fan beam case, the most important parameter affecting image resolution is the transversal shift of the detector array, i.e., the distance from the center of the detector array to the projection of the center of rotation. In Gullberg et al (1984), the transversal shift of the detector array is estimated from the centroids of the projection data of a point-like source, by using the Levenberg–Marquardt method. Azevedo et al (1990) argued that, for a generic object, the center of mass of its projection at each angle coincides with the projection of the center of mass of the object; hence, they determine the transversal shift of the detector by analyzing the center of mass of a generic object, instead of using a real point object. In the conebeam geometry, the calibration problem is more complicated because of the greater number of degrees of freedom of the misalignment. At the best of our knowledge, the possibility of extending the object-independent method of Azevedo to the cone-beam geometry was not yet investigated. Noo et al (2000), Bequ´e et al (2003), von Smekal et al (2004) and Yang et al (2006) proposed methods of calibration based on the analysis of the projection data of dedicated phantoms, composed by a certain number of point-like objects. The above methods are capable of detecting many parameters with good accuracy, but in most cases the precise knowledge of the phantom shape and positioning is required. In this paper, we propose a new method of geometrical calibration for cone-beam scanners that is different from all the methods cited above. We measure the misalignment parameters of a cone-beam scanner by minimizing a geometry-dependent cost function. The cost function is computed from the projection data of a generic object, hence, no a priori knowledge of the object shape or positioning is required. The method is capable to determine with good accuracy the transversal shift and the skew of the detector with respect to the AoR and the focal point. The detector slant can be determined as well, but with less accuracy. As stated in von Smekal et al (2004) and Yang et al (2006), transversal shift and skew are the two most critical parameters which influence the resolution of the reconstructed images. Since our calibration method is nearly independent from the object, it can be integrated in the analysis/reconstruction process in order to continuously monitor the geometrical misalignment of the scanner. This cannot be carried out by using the methods cited above. The requirement of the proposed calibration method is that all or part of the projection data must be redundant. In section 3 we will clarify under which hypotheses redundancy occurs in the cone-beam geometry.
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2. Definition of the geometry The geometry of the cone-beam CT scanner is shown in figure 1. The coordinate system xyz is fixed with respect to the phantom, while the primed coordinate system x y z is fixed with respect to the source-detector system (i.e., the gantry). The two coordinate systems differ only by a rotation θ about the z ≡ z axis, which by convention is coincident with the Axis of Rotation (AoR) of the scanner. The gantry angle θ is positive when the gantry coordinate system x y z is rotated anticlockwise with respect to the phantom coordinate system xyz. We will assume that the focus of the x-ray source describes a perfect circular orbit around the AoR (which is supposed to be stable in time) and that, for each projection, the corresponding gantry angle θ is exactly known. The y axis is defined as the axis passing through the center of the x-ray focal spot and perpendicular to the AoR. Once the y axis is defined, all other axes of the two coordinate systems are univocally determined by the gantry angle θ . We can now define the coordinate system on the detector plane. We assume a perfectly planar detector surface without distorsions. However, all of the results can be adapted to other detector shapes (e.g., cylindric detectors). Throughout the text, we will refer to the detector panel as the finite rectangle representing the active surface of the flat-panel detector, and to the detector plane as the infinite plane on which the detector panel lies. We denoted the source-to-axis distance (SAD) and the source-to-detector distance (SDD) with the symbols d and D, respectively. We will distinguish between the two coordinate systems of the actual (misaligned) detector and of the perfectly aligned detector. In our notation, u and v represent the coordinates in the plane of the ideal aligned detector; w is the coordinate perpendicular to the uv plane. The axes u and v coincide with the projections of the axes x and z on the aligned detector plane, respectively. The origin of the coordinate system uvw is placed at the geometric center of the detector panel, with the w axis pointing toward the AoR (see figure 1). Similar coordinates are also defined for the misaligned detector and are denoted by the symbols ut , vt , wt , where the subscript t stands for transformed coordinates. In the real detector, the ut and vt are no longer coincident with the projection of the x and z axis. The exact correspondence between the two coordinate systems uvw and ut vt wt will be clarified in section 3.1. Let us now parametrize the overall misalignment of the scanner. The total number of degrees of freedom is 6 for the detector array and 3 for the (point-like) x-ray focal spot3 . Because of the freedom of choice for the origin of the z axis and for the origin of the gantry angle θ , two of these 9◦ of freedom are removed. Hence, a complete description of the scanner misalignment involves seven parameters (Bequ´e et al 2003). We have chosen to assign six of these seven parameters to the misalignment of the detector panel with respect to the x-ray source and the AoR. The remaining parameter will describe the deviation of the distance d from the nominal SAD. Using the above parametrization, no misalignments are attributed to the positioning of the x-ray source and the AoR. Figure 1 illustrates the geometrical significance of the seven misalignment parameters. Starting from the position of the ideal detector, three rotations and one translation are applied in the following order to obtain the misaligned detector: a rotation by an angle λ about the u axis (tilt); a rotation by an angle η about the v axis (slant); a rotation by an angle φ about the w axis (skew); a rigid translation by a vector (δu, δv, −δD)T in the uvw reference frame. The minus sign in front of δD reflects the fact that when D is increased, the detector panel moves in the negative direction of the w axis. The seventh parameter, δd, is the deviation of d from the nominal SAD. If uncorrected, it does not cause image quality deterioration. This misalignment can be thought as a parallel translation of the AoR along the y axis, causing a variation of 3 The x-ray focal spot can be considered to be point-like if we assume that the detector plane is uniformly irradiated. All beam inhomogeneities are supposed to be removed via flat-field correction.
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Figure 1. Geometry of the cone-beam CT scanner. Top left: perspective view. Top right: front view. Bottom left: top view. Bottom right: side view. The AoR coincides with the z axis by convention. The y axis is defined as the axis passing through the source and the AoR, perpendicular to the AoR. All other axes are defined univocally with respect to y , given the AoR and the gantry angle θ . The SAD and the SDD are denoted by d and D, respectively. The coordinates of an ideal aligned detector (dotted lines) are denoted by u, v and w, while the corresponding coordinates of the misaligned detector are indicated with ut , vt and wt . The misalignment with respect to the ideally aligned system is parametrized with the seven parameters δu (transversal shift), δυ (longitudinal shift), δD (error in SDD), φ (skew), η (slant), λ (tilt), δd (error in SAD). Our method of calibration is independent on the parameter δd.
the magnification factor m = D/d and hence a variation of the voxel size of the reconstructed images. Our method is insensitive to the parameter δd, so we will ignore it in the following discussion. This means that, within our method, a very accurate calibration of the voxel size will require a scan of an object with well-known physical dimension. However, we believe that this step is somewhat unnecessary: in fact, the uncertainty on voxel size is dominated by the error on m, which can be easily kept less than 1–2% by simple direct measurements of the SAD and the SDD. In the following discussion, we will focus our attention to the six degrees of freedom characterizing the position of the detector with respect to the point-like x-ray focal spot and the AoR, whose positions are assumed to be known. 3. Theory The main idea behind the proposed calibration method is to analyze the redundancy of the projection data, in order to define a cost function which depends on the misalignment parameters. In this section, we will develop the theory of our method in the circular cone-beam
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Figure 2. Example showing the proposed calibration method in a 2D parallel-beam geometry for a simple (point-like) object. (a) If the linear detector array of width W is perfectly aligned, the subtraction of the function pθ+π (−u) (redundant data) from the function pθ (u) yields the null function in the interval [0, π ), as stated by equation (1). (b) If the detector is misaligned, the subtraction of pθ+π (−ut ) from pθ (ut ) no longer yields the null function; equation (2) must be used instead of equation (1), and the correct value of δu can be found iteratively until the subtraction approaches the null function.
geometry, for a particular class of objects for which redundancy of the projection data is exact over the whole Field of View (FoV), even for wide cone angles; successively, we will apply the same methodology to more generic objects for which the data redundancy is approximated and holds only for small cone angles. Even in the case of approximated redundancy and depending on the shape of the specific object, a sufficient amount of data can be analyzed in order to determine the most critical misalignment parameters with good precision and accuracy. Before developing the theory in its general form, we will discuss a very simple example in the 2D parallel beam geometry that is useful in understanding the idea of the proposed calibration method. Let us consider the parallel beam sinogram of a point-like object, as shown in figure 2. We assume for simplicity that only the transversal shift δu of the linear detector array is present. A common method to determine δu in this simple case is to perform a nonlinear fit of the sinusoidal sinogram, which allows us to find the optimum value of the sinusoid baseline offset in the least-squares sense. An alternative way to find a good estimate of δu is based on the analysis of the redundancies of the sinogram. More specifically, if we denote by pθ (u) the parallel beam sinogram of a generic object function f (x, y), the
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well-known periodicity relation of the function p (in the case of perfect alignment) is given by Kak and Slaney (1988): pθ (u) = pθ+π (−u),
(1)
for all u, θ and for a generic object function f . As stated before, it is assumed that the angle θ is known exactly. It can be easily shown that, if δu = 0, the above equation can be generalized as follows: pθ (ut ) = pθ+π (−ut − 2δu).
(2)
Equation (2) is simply verified observing that, for a given ut = u∗ , we have from equation (1): pθ (ut = u∗ ) = pθ (u = u∗ + δu) = pθ+π (u = −u∗ − δu) = pθ+π (ut = −u∗ − 2δu).
(3)
According to equation (2), the actual transversal shift of the detector array can be found by varying δu until the function eθ (ut ) = pθ (ut ) − pθ+π (−ut − 2δu) becomes null for all ut , θ . More specifically, the transversal shift can be computed by solving the following equation with respect to δu: π W/2 dθ dut [pθ (ut ) − pθ+π (−ut − 2δu)]2 = 0, (4) 0
−W/2
where W is the width of the detector array. The left-hand side of equation (4) can be seen as a cost function which might be solved with an iterative optimization method, starting at an initial guess δu0 and iterating i times until a (local) minimum is found at some point δubest . Because real data are discrete and affected by noise, in all practical cases the value of the cost function at the minimum is greater than zero. As we will see below, under certain conditions, that minimum will be still located near the true value of the misalignment parameter. The problem of the geometrical calibration of the scanner was formulated here as an optimization problem with one parameter (the transversal shift δu), for which the left-hand side of equation (4) is the cost function. Note that no assumption was made about the object function f (x, y). Furthermore, a big amount of data was used to compute the cost function, so we expect robustness of the method with respect to the noise. In the following subsections, we will generalize the above defined cost function to be used for the calibration of cone-beam scanners. 3.1. Symmetric objects In general, the periodicity relation of 2D sinograms (parallel and fan beam) cannot be extended to cone-beam projection data of a generic object. In fact, in cone-beam geometry only the restriction of the projection data to the transaxial midplane z = 0 is rigorously redundant, because it is coincident with the 2D fan beam sinogram of the object function. In this section we will distinguish two classes of 3D object functions for which the periodicity relation of the 2D fan beam sinogram can be exactly extended to the entire cone-beam dataset, even for wide cone angles. The calibration cost function for the cone-beam geometry will be defined for objects that belong to those classes. Let us now introduce some notations that will be useful in the development of the theory. In the following we will use homogeneous coordinates in the detector space, so we will denote by u = (u, v, w, 1)T a generic point in the coordinate system of the aligned detector, and by ut = (ut , vt , wt , 1)T a generic point in the coordinate system of the misaligned detector. First,
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let us assume a perfect alignment. We will denote by gθ (u) the 3D divergent x-ray transform of f (x, y, z), i.e., the cone-beam projection of f for a given gantry angle θ . More precisely, the value of gθ (u) coincides with the line integral of f along a line passing through the point u and the x-ray focal spot. If the point u lies on the plane w = 0, the two coordinates (u, v) describe the location of the detector element at which the function g will be evaluated. In general, the point u does not lie on the detector panel and must be projected onto the plane w = 0 before evaluating the data. The restriction of g to the intersection of the two planes v = 0 and w = 0 is the fan beam sinogram of f , which has the following property (Kak and Slaney 1988): gθ (u)|v,w=0 = gθ+π−2γ (−u)|v,w=0 , where
(5)
u γ = γ (u) = arctan . D−w
(6)
Now, we will give a condition for which equation (5) can be generalized to all v = 0. For this purpose, let us introduce the operator S of reflection about the u = 0 plane: ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ u −1 0 0 0 u −u ⎜ v ⎟ ⎜ 0 1 0 0⎟ ⎜ v ⎟ ⎜ v ⎟ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ S⎜ (7) ⎝w⎠ = ⎝ 0 0 1 0⎠ ⎝w ⎠ = ⎝ w ⎠ . 1
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Theorem 1. Let f = f (x) be a nontrivial object function in the 3D space, where x = (r, ϑ, z), and let the function f satisfies at least one of the two properties P1. ∂f/∂z = 0, P2. ∂f/∂ϑ = 0. Then the 3D divergent x-ray transform gθ (u) of f satisfies the periodicity relation gθ (u) = gθ+π−2γ (Su)
(8)
for all u, θ in the acquisition range of g. We will give an heuristic proof of the above theorem. For objects that satisfy the property P1, the function gθ can be factorized as follows (Feldkamp et al 1984) us 12
ν2 gθ (u) |v=0 . (9) gθ (u) = 1 + (D − w)2 + u2 The above equation states that each restriction of g to a certain ν = 0 is equal to the fan beam sinogram of f , multiplied by a function which does not depend on θ and is invariant with respect to the transformation S. The proof of theorem 1 follows by substituting equation (9) in equation (8) and observing that the result obtained is the extension of equation (5) to a generic w. If f satisfies the property P2, it is trivial to verify that ∂g/∂θ = 0 is satisfied as well; hence, each projection at a given gantry angle is equal to the projection at any other angle. Now it must be verified that gθ (u) = gθ (Su)
(10)
for all u, θ . Equation (10) is satisfied if the function f is symmetric with respect to reflections about the plane x = 0, for all θ . This latter property, in turn, is verified if and only if f is rotationally invariant, which is true by hypothesis. Hence theorem 1 is proven.
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Throughout this paper, we will refer to objects satisfying the hypotheses of theorem 1 as symmetric objects. If f is symmetric, for each cone-beam projection at a given angle θ , a corresponding mirror projection can be computed from redundant data using equation (8). Ideally, in the case of noiseless data in the continuous domain, the difference of direct and mirror projections will always yield the null function. Let us now consider the more realistic case in which the detector array is misaligned. Using the above notation, the data collected on the misaligned detector are denoted by gθ (ut ). We want to generalize equation (8) to this case. If all misalignment parameters of the detector are known, the operator T which transforms the coordinates of the aligned detector to those of the misaligned detector is defined as Tu = ut .
(11)
We also denote with P the operator of projection along the x-ray direction of a generic point ut onto the detector plane. The explicit dependence of the above operators upon the misalignment parameters of the system will be clarified below. Using this formalism, we can write the generalization of equation (8) for the misaligned detector as follows: gθ (ut ) = gθ+π−2γ (Hut ),
(12)
H = PTST−1
(13)
where and
u(ut ) . γ = γ (ut ) = arctan D − w(ut )
(14)
In equation (14), u(ut ) and w(ut ) are components of the vector u = T−1 ut . Equation (12) simply states that, in order to compute the mirror projection of gθ (ut ) for the misaligned detector, the reflection operator S in equation (8) must be replaced by a more general transformation operator H which is a function of the misalignment parameters. Note that equation (12) is satisfied only if the parameters used to build the operator H are coincident with the actual parameters of the system. This is the key concept of the proposed calibration method. In fact, the misalignment parameters of the system are always unknown in practice: the aim of our method is to vary iteratively those parameters (and hence H) until equation (12) is satisfied. Let us denote with δ = (δu, δv, δD, φ, η, λ) the generic 6-tuple of misalignment parameters, and let Bδ ⊂ R6 be a suitable convex subset of the six-dimensional space of the misalignment parameters. The 6-tuple of the actual parameters is be denoted by δtrue = (δutrue , δvtrue , δDtrue , φtrue , ηtrue , λtrue )T , and the subset Bδ is chosen such that δtrue ∈ Bδ . We define the following cost function θmax c(δ ) = dθ dut [gθ (ut ) − gθ+π−2γ (Hut )]2 , (15) 0
c : Bδ → R,
Det
(16)
where the integral in dut is extended to the whole detector panel. The angle θmax in equation (15) should be taken in a way that, for each direct projection gθ (ut ), the corresponding mirror projection gθ+π−2γ (Hut ) can be correctly computed from the available scan data. For a full-scan acquisition, in order to maximize the amount of data included in the computation, we can use θmax = π − 2γmax , where γmax = arctan(W/2D) is the angular half-width of the detector. A lower value of θmax will reduce the computation time, but will reduce also the robustness of the method. For a short-scan acquisition, redundant data are available only for
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the first gantry angle θ0 (Kak and Slaney 1988) and hence the integration over θ is not possible. In that case, the cost function must be computed for θ = θ0 only. It follows directly from equation (15) that the above cost function is non-negative, i.e., c(δ ) 0 for all δ ∈ Bδ . By bringing together equations (12) and (15), it is trivial to verify that c(δtrue ) = 0.
(17)
Hence the solution of the calibration problem is the 6-tuple of parameters for which the cost function (15) is at its minimum. Rigorously speaking, the above equality holds only in the case of noiseless data in the continuous domain. The minimum of c can be found iteratively, starting at some initial value δ 0 . At the ith iteration, the cost function is computed as follows: θmax i dθ dut [gθ (ut ) − gθ+π−2γ i (Hi ut )]2 , (18) c(δ ) = 0
Det
where δ i ∈ Bδ is the ith solution of the iterative minimization process, depending on the optimization method in use. In the above equation, the angle γ i and the operator Hi were built using the 6-tuple of misalignment parameters δ i . As discussed in section 4.3, we use the simplex method (Nelder and Mead 1965) to minimize the cost function (15). An issue on using equation (15) to find the misalignment parameters is that, in general, the solution of the problem c(δ ) = 0 might not be unique. Although theorem 1 ensures the existence of the minimum of the above-defined cost function at δ = δtrue for symmetric objects, it is insufficient to guarantee the uniqueness of that minimum. In fact, depending on the specific form of the object function, equation (17) might be verified in a subset f ⊂ Bδ , with δtrue ∈ f . The subscript f indicates the dependence of the solution space on the object function. We will see in section 4.2 that, in most practical cases, this issue can be overcome by restricting the minimization process to a suitable subset of parameters. In fact, using simulated data, we observed that in the neighborhood of δtrue , a unique local minimum of the cost function can be found in the subspace of the three parameters δu, φ and η. Hence, the most critical misalignment parameters of the system can be determined by the minimization of the cost function (15). In this section we have formulated the calibration problem for the cone-beam scanner as a multidimensional optimization task. The cost function c(δ ) can be easily computed from the projection data, as described in section 4.1. 3.2. Generic objects If the object function does not respect the hypotheses of theorem 1, only the restriction g|v=0 to the midplane has exactly redundant data. In fact, all lines of projection not lying on the v = 0 plane have some tilt angle with respect to the transaxial midplane; therefore, such lines of projection never lie on the same plane of their ‘redundant’ counterpart. As a consequence we will not find, in general, a minimum of c(δ ) > 0 at δ = δtrue if the cost function is integrated over the whole detector panel as indicated in equation (15). This problem can be handled in two ways. First of all, we can restrict the integration in equation (15) to a narrow region of interest (RoI), denoted by R0 and centered at the v = 0 axis. In this way, we will include in the cost function only the portion of the data which has approximated redundancy. The extent of R0 in the v direction should be small enough to neglect the tilt angle of the corresponding lines of projection, but also wide enough to include a sufficient amount of data in the computation of the cost function. In the next section, we will discuss in which cases the axial angular extent of R0 could be considered negligible for our purposes. Another aspect to take into account is that, in practice, most objects functions can be
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decomposed as a ‘superposition’ of symmetric and asymmetric functions (e.g., a mouse lying on a z-invariant couch, or a specimen encapsulated in a container cylinder). For a generic object function f , we can write f = f (s) + f (a) ,
(19)
where f (s) and f (a) are the symmetric and asymmetric components of f , respectively. For the linearity of the x-ray transform, we can also write g = g (s) + g (a) . (s)
(20) (a)
(s)
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In the above equation, g and g are the x-ray transforms of f and f , respectively. In the following discussion, we will make the assumption that g (s) is not null. Let us choose a set of RoIs Rj in the detector panel, with j = 1, . . . , M, defined in such a way that g (a) |Rj = 0 for all θ ∈ [0, θmax ]. In other words, the union of all those RoIs must be a subset of the detector plane on which, for all angles included in the analysis, only the symmetric component of f is projected. Then, we can compute the calibration cost function as follows: c(δ ) =
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(21)
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where the summation was extended also to the central RoI R0 , as defined above. As we will see below, the modified cost function just defined can be used to calibrate the scanner using objects which do not respect the hypotheses of theorem 1. In the worst case in which f is such that g (s) is null (i.e., if f is completely asymmetric), the calibration is possible by computing the cost function over R0 only. Of course, we expect that the reduction of the amount of data included in the analysis could have an impact on the accuracy of the calibration. 4. Implementation of the algorithm 4.1. Computation of the cost function Let us discuss our own implementation of the proposed calibration method. First of all, we will give the explicit expression of the compound operator H defined in equation (13), required to compute mirror projections. The suboperator T defined in equation (11) represents a rototranslation of the detector coordinate system and is in turn a compound operator: T(δ ) = O(δu, δv, δD)Rw (φ)Rv (η)Ru (λ),
(22)
where O represents a translation by the vector (δu, δv, −δD, 1)T and Ru , Rv , Rw represent the rotations about the u, v and w axes, respectively. More explicitly, ⎛ ⎞ ⎛ ⎞ 1 0 0 δu cos φ sin φ 0 0 ⎜0 1 0 ⎜−sin φ cos φ 0 0 ⎟ δv ⎟ ⎟ ⎟ Rw = ⎜ O=⎜ ⎝0 0 1 −δD ⎠ ⎝ 0 0 1 0⎠ 0 0 0 1 ⎛ cos η 0 −sin η ⎜ 0 1 0 Rv = ⎜ ⎝ sin η 0 cos η 0 0 0
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(23)
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For completeness, we also report the expression of the inverse operator T−1 : −1 −1 −1 T−1 = R−1 u Rv Rw O ,
(24)
where R−1 u (λ) = Ru (−λ)
R−1 v (η) = Rv (−η)
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O−1 (δu, δv, δD) = O(−δu, −δv, −δD).
(25)
The projection operator P is already discussed in Karolczak et al (2001), and is given by ⎛ ⎞ foc 1 0 −ufoc 0 t /wt ⎜ ⎟ ⎜0 1 −vtfoc /wtfoc 0⎟ (26) P=⎜ ⎟ ⎝0 0 0 0⎠ 0 0 0 −1/wtfoc where
⎛
⎞
⎛ ⎞ 0 ⎜ v foc ⎟ ⎜ 0⎟ ⎜ t ⎟ ⎟. = Tufoc = T ⎜ ⎜ ⎟ = ufoc t ⎝ D⎠ ⎝w foc ⎠ t 1 1 ufoc t
(27)
is the position of the x-ray focal spot in the coordinates of the misaligned detector. Now, we are able to compute the cost function at a given point δ by using a discrete version of (21). First of all, we have to define a set of RoIs Rj , j = 0, . . . , M as discussed in section 3.2. Given a gantry angle θ , the mirror projection gθ+π−2γ (Hut ) inside those RoIs can be computed by using algorithm 1. Algorithm 1 Compute the mirror projection at a given gantry angle θ Require: the set of cone-beam projections gθ [ut ]; Require: the set of RoIs Rj , j = 0, . . . , M; Require: the gantry angle θ ; Require: the set of misalignment parameters δ = (δu, δv, δD, φ, η, λ); 1: compute the matrix T using equations (22) and (23) T 2: compute the vector ufoc t = T(0, 0, D, 1) −1 3: compute the matrices P and T using equations (26) and (27) 4: compute the matrix H = PTST−1 5: for all selected Rj , j = 0, . . . , M do 6: for all points ut in Rj do 7: compute the vector u = T−1 ut 8: compute the fan angle γ (u) = arctan[u/(D − w)] 9: if θ + π − 2γ ∈ {angular range of acquisition } then 10: compute the vector Hut 11: if Hut ∈ {spatial limits of the detector} then 12: evaluate gθ+π−2γ (Hut ) using trilinear interpolation 13: else 14: ignore ut 15: end if
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16: else 17: ignore ut 18: end if 19: end for 20: end for Let us now define the error projection, ej , as the difference of the direct and mirror projection inside Rj : ej [ut ] = gθ [ut ]|Rj − gθ+π−2γ [Hut ].
(28)
The square brackets denote the discrete nature of the data. In the previous equation, the value of gθ+π−2γ [Hut ] is obtained via trilinear interpolation if it is not directly available from the data. Finally, the cost function can be computed as 1 c(δ ) = e2 [ut ], (29) Nu Nθ u ,v ,θ,j j t
t
where Nu is the total number of points analyzed at each angle and Nθ is the number of angles analyzed. Until now, we have not considered yet the contribution of sampling and noise to the cost function. These two factors will contribute to the cost function in a way that is not correlated to the geometry of the system; hence, a shift of the minimum could appear, with a consequent error in the determination of the misalignment parameters of the system. The error caused by the sampling is supposed to be reduced, in practice, by the interpolation involved in the computation of the mirror projection. On the other hand, the contribution of the quantum noise can be reduced by including as much data as possible in the computation. For this reason, the set of RoIs and the angular range to be used in the computation of the cost function should be chosen so as to have a trade-off between calibration accuracy and computational complexity. 4.2. Dependence of the cost function on misalignment parameters The ability of the proposed method to identify the correct values of the misalignment parameters is highly related to the curvature of the cost function (29) in the neighborhood of the point δ = δtrue . Ideally, we would have a sharp minimum of the cost function in all six dimensions of the parameter space, but, as it can be expected, not all parameters have the same impact on c(δ ). In order to study the behavior of the proposed cost function we have simulated cone-beam data from a numerical phantom, composed by five ellipsoids of various densities embedded in a long water cylinder of 3 cm in diameter. The object was chosen in such a way that both the symmetric and the asymmetric components are not null. The simulation was performed by using known misalignment parameters, and adding poisson noise so as to get more realistic data. The level of the simulated noise corresponds to an average of 5000 detected photons per pixel, without attenuation. The mass attenuation coefficient of all ellipsoids is µ/ρ = 0.268 cm2 g−1 , corresponding to the mass attenuation coefficient of water at 40 keV; thus, the number of detected photons is reduced by a factor of about 2 along a diameter of the cylinder, and by a factor of 7 along a line of maximum attenuation. A Gaussian smoothing of radius 0.5 pixel was also applied to the projection images (successively to the noise) to simulate the finite spatial resolution of the detector. The phantom is shown in figure 3; in table 1 all the geometrical and physical parameters of the phantom are summarized, and in table 2 the geometrical parameters of the simulated acquisition are reported. Figure 4 gives a better insight of the calibration method. As can be seen from the figure, the variance of the error projection is strongly reduced if the correct misalignment parameters
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Figure 3. The numerical phantom used to validate the calibration method. It is composed by five ellipsoids enclosed in a cylinder. The geometrical parameters of the ellipsoids are reported in table 1. The object as a whole is not symmetric, but it does not have a not null symmetric component (the container cylinder and the lower central disc are both symmetric objects).
Table 1. Geometrical parameters of the ellipsoids composing the simulated object in figure 3. cx,y,z are the coordinates of the center of each ellipsoid, rx,y,z are the half-length of the three principal axes, α is the rotation angle about the principal axis parallel to z, expressed in degrees, and ρ is the density in g cm−3 . All lengths are expressed in cm. The mass attenuation coefficient of all ellipsoids is µ/ρ = 0.268 cm2 g−1 , corresponding to the mass attenuation coefficient of water at 40 keV. cx 0.45 −0.75 −0.75 −0.9 0 0.09 a
cy
cz
0.06 −0.09 0.75 0.9 0 −0.12
−0.15 0.15 1.05 1.05 −1.5 0
rx
ry
rz
0.75 0.3 0.06 0.03 0.9 1.5
0.3 0.6 0.06 0.03 0.9 1.5
0.9 0.12 0.06 0.03 0.03 ∞a
α
ρ
−30 15 0 0 0 0
3 5 10 10 3 1
The container cylinder was simulated as an ellipsoid with rz = 1012 pixels.
are used. In fact, that variance is strictly related to the value of the cost function (29), which is at a minimum for δ = δtrue . As expected, only the asymmetric components of the phantom remain visible in the error projection; as stated above, the RoIs should be chosen in such a way that the asymmetric component is excluded from the computation of the cost function. The highlighted central RoI subtends a cone angle of about 3◦ . In that RoI we note that the direct and mirror projections of the ellipsoids match with good approximation, but this is due in part to the axial invariance of the ellipsoids in that position. Hence, in this particular case, an angle of less than 3◦ could be considered negligible for our purposes, because in a RoI subtending that angle we have a well-approximated redundancy of the projection data. This could not be the case for other, more strongly asymmetric, objects. In general, we think that a visual inspection of the whole error projection is advisable before choosing the axial extent of the central RoI R0 . Figure 4 also shows that, besides the asymmetric component, the main
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u
utrue
Figure 4. Direct (left) and error (center and right) projections from simulated cone-beam data from the numerical phantom, at angle θ = 0. The error projection was computed using algorithm 1, for two different 6-tuples of misalignment parameters. It can be seen that the variance of the error image is strongly reduced if the correct misalignment parameters are used. Note that in the error projection computed with the correct parameters, the symmetric component of the phantom (i.e., the container cylinder and the lower disc) has disappeared. The highlighted central RoI subtends an angle of 3◦ (axially). Table 2. Geometrical parameters used in the simulated acquisition. The parameters δutrue , δvtrue and δDtrue are expressed in unit of detector pixel, while the three angles φtrue , ηtrue and λtrue are in degrees. Misalignment parameters Detector size 5122
Pixel size 0.2 mm
Angles
Step
SAD
SDD
δutrue
δvtrue
δDtrue
φtrue
ηtrue
λtrue
480
0.75◦
15 cm
30 cm
11.754
−12.43
13.543
1.49
2.5443
1.325
contribution to the variance of the error image is given by the edges of the object. This fact could be exploited to restrict the computation of the cost function to RoIs which enclose those edges, and hence reducing the computation time without significantly compromising the result of the calibration. Nevertheless, one should ensure that the selected RoIs enclose the desired edges for all the angles included in the analysis. Now, we will study the sensitivity of the cost function with respect to the various parameters. In figure 5 we reported six 1D profiles of the cost function, passing through the point δ = δtrue and along the directions parallel to the six coordinate axes of the misalignment parameter space. The profiles were computed for various sets of RoIs and the number of projections. As we can see in the graphs of figure 5, the relative curvature of the cost function along the δu and φ axes is considerably greater than the curvature along the other axis. This means that the proposed calibration method will be very sensitive in detecting such critical parameters, even with a single RoI as shown in figures 5(a)–(c). There is also a curvature along the directions of the less important parameters, but its relative magnitude is 2–3 orders of magnitude lower; only the curvature along the η direction seems to be sufficient to identify a sharp minimum, but this depends strongly on the choice of the RoIs. By adding other RoIs, as in figure 5(d), the curvature along φ is increased of 1–2 orders of magnitude, while remaining practically unchanged along the other directions. Comparing figures 5(d)–(f) with figures 5(g)–(i), it seems that the shape of c(δ ) is not significantly modified by computing the cost function on a single angle instead of integrating it over 200 angles. However, this could be fortuitous; as stated above, an excessive reduction of the amount of data included in the computation could lead to important errors in the estimation of the parameters, especially
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10000
3700 2
7000 6000 5000
c(δ′) (arb. un.)
δD′, ∂2c = 4.9e+001
φ′, ∂ c = 3.5e+003
3600
2
δυ′, ∂ c = 2.8e+001
8000
c(δ′) (arb. un.)
2
δu′, ∂ c = 3.6e+004
9000
4000
2
η′, ∂ c = 9.2e+002
3500
λ′, ∂2c = 1.6e+001
3400 3300 3200 3100
3000
3000
2000
0
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0.1 0.2 0.3 0.4 0.5
0
δ′ (pixel)
(a) 1 RoI, 200 angles, 3.6 . 106 points
(b)
(c)
8000
14000
δu′, ∂2c = 2.9e+004
φ′, ∂2c = 5.9e+004 12000
2
δυ′, ∂ c = 2.0e+001 2
δD′, ∂ c = 5.5e+001
6000 5000 4000
c(δ′) (arb. un.)
c(δ′) (arb. un.)
7000
2
η′, ∂ c = 6.4e+002 2
λ′, ∂ c = 1.1e+002
10000 8000 6000
3000
4000
2000
0
2000
0.1 0.2 0.3 0.4 0.5
0
δ′ (pixel)
(d) 5 RoI’s, 200 angles, 5.7 . 106 points
(f)
4
4
x 10
9
x 10
2
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δu′, ∂ c = 2.7e+005 6
δD′, ∂2c = 5.6e+002
5 4 3
φ′, ∂ c = 4.2e+005
8
2
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c(δ′) (arb. un.)
c(δ′) (arb. un.)
0.1 0.2 0.3 0.4 0.5
δ′ (deg)
(e) 7
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2
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6 5 4 3
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0
(g) 5 RoI’s, 1 angle, 2.8 . 104 points
1
0.1 0.2 0.3 0.4 0.5
0
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(h)
(i)
4
5
4
x 10
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x 10
2
φ′, ∂ c = 1.2e+005 4
2
δυ′, ∂ c = 1.1e+002 δD′, ∂2c = 2.9e+002
4 3.5 3 2.5
c(δ′) (arb. un.)
c(δ′) (arb. un.)
2
δu′, ∂ c = 1.5e+005
4.5
(j) 2 RoI’s, 200 angles, 5.6 . 106 points
0.1 0.2 0.3 0.4 0.5
δ′ (deg)
3 2.5
2
2 1.5
0.1 0.2 0.3 0.4 0.5
λ′, ∂2c = 1.6e+002
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0
2
η′, ∂ c = 4.7e+003
0
0.1 0.2 0.3 0.4 0.5
δ′ (pixel)
δ′ (deg)
(k)
(l)
Figure 5. 1D profiles of the cost function through the point δ = δtrue , along the six axes of the misalignment parameter space, computed for various sets of RoIs and number of angles. The highlighted rectangles in the error projections show the position of the RoIs in each study. All plots are drawn as functions of the variable δ = δ − δtrue . For each profile, the mean value of the second derivative in the interval |δ | ∈ [−0.25, 0.25] was reported. It can be seen that the two most critical parameters, δu (transversal shift) and φ (skew) have a higher influence on the cost function. Also the slant could be determined. The cost function presents minima also along the direction of the other parameters, but with much smaller curvatures. For such small curvatures, the contribution of the noise and sampling seems to be dominant with respect to the desired contribution of the system geometry.
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at high noise levels. A trade-off between computational complexity and robustness should always be considered. For noise levels comparable to that of the present simulation, we think that the integration over 10–30 angles would be sufficient, depending on the position and size of the selected RoIs. In figures 5(j)–(l), we added a RoI which encloses the lower central disc. Due to the high axial non-uniformity of that detail, an increase of sensitivity in δv and λ could be expected. However, we can see from the graph that it is not the case. Anyway, we cannot exclude the existence of objects which increase the sensitivity of our method also for the longitudinal shift, the tilt and the error in SDD. By now, we will consider restricting the optimization process to the three parameters δu, φ and η; the other three parameters shall be kept fixed at approximated values, estimated upon mechanical measurements or other calibration modalities. 4.3. Minimization of the cost function We used the Nelder–Mead simplex algorithm (Nelder and Mead 1965) for the minimization of the cost function (29). The advantage of this algorithm is that it is a direct search optimization algorithm, i.e., it does not require the knowledge of the analytic expression of the cost function and of its derivatives. We have seen that Nelder–Mead gives reasonable results in many empirical trials; in principle, one can use other optimization methods to calibrate the scanner within the proposed method. In our implementation, the search of the minimum is terminated when the simplex size falls below 10−3 units, computed in the metrics of the δ -space. In that space, the linear coordinates are expressed in units of the detector pixel and the angular coordinates are expressed in degrees. At each iteration, the point δ i at which the cost function is evaluated can move in the subspace of the three parameters δu, φ and η. Given an initial guess δ 0 , for all iterations i = 1, . . . , N the following constraints are posed on δ i : |δui − δu0 | 50 pixel; δv i = δv 0 ; δD i = δD 0 ; |φ i − φ 0 | 5 degrees |ηi − η0 | 5 degrees λi = λ0 . In order to reduce the computation time, our software searches the minimum angle by angle; at each angle, the starting point is updated with the result obtained at the previous angle. This reduces considerably the number of iterations required to find the minimum, just after the first 2–3 angles. At the end of the optimization process, the best estimates of the misalignment parameters are computed as the average of the best estimates obtained over all the projections analyzed. 5. Results 5.1. Validation of the algorithm with simulated data Based on the results of the previous subsection, we carried out the calibration of our simulated scanner by searching for a local minimum of the cost function from the numerical phantom of figure 3. The calibration was done at three different noise levels: noiseless (infinite number of photons per pixel), medium (5000 photons per pixel without attenuation, as in
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Table 3. Calibration results of the simulated scanner, expressed as mean value ± standard deviation over the 20 projections analyzed. The RoIs used in the analysis are shown in figure 5(d). For all noise level, the total number of image points analyzed was 5.7 × 105 ; the starting point was δ 0 = (0, −10, 10, 0, 0, 0)T . The algorithm was executed on a PC with Intel Pentium M750, 1.86 GHz. All lengths are expressed in units of the detector pixel size and all angles are expressed in degrees. No. of iterations Noise level
δubest
φbest
ηbest
First view
Average ± SD
Time (total)
Noiseless Medium High
11.974 ± 0.011 11.976 ± 0.017 11.994 ± 0.036
1.410 ± 0.097 1.410 ± 0.094 1.406 ± 0.094
2.619 ± 0.101 2.650 ± 0.103 2.601 ± 0.349
340 281 213
54 ± 70 60 ± 56 63 ± 40
42 s 50 s 52 s
Table 4. Calibration result of a second run in which, for each noise level, the starting point was taken at the result point of the first run. Both precision and accuracy are slightly increased with respect to those obtained in the first run, demonstrating a weak dependence of the method upon the initial conditions. No. of iterations Noise level
δubest
φbest
ηbest
First view
Average ± SD
Time (total)
Noiseless Medium High
11.971 ± 0.007 11.973 ± 0.010 11.986 ± 0.026
1.432 ± 0.002 1.432 ± 0.003 1.426 ± 0.013
2.606 ± 0.098 2.649 ± 0.110 2.672 ± 0.282
54 51 51
37 ± 7 42 ± 10 56 ± 17
34 s 45 s 56 s
the previous section) and high (500 photons per pixel without attenuation); for each noise level, the cost function was extracted by the analysis of five RoIs as shown in figure 5(d), over 20 projections at steps of 7.5◦ . The starting point was δ 0 = (0, −10, 10, 0, 0, 0)T . The calibration was also repeated on a second run, in which the starting point was placed at the result point obtained in the first run. The results of the first and the second run are shown in Tables 3 and 4, respectively. The estimated parameters are close to the true parameters with good approximation: for all noise levels, we obtained |δutrue − δubest | < 0.25 pixel, |φtrue − φbest | < 0.1◦ and |ηtrue − ηbest | < 0.1◦ . This demonstrates the accuracy of our method and its robustness with respect to noise. In figure 6, we show a set of reconstructions of the details of the simulated object, obtained with a modified Feldkamp algorithm with perspective correction (Karolczack 2001). We note that the images reconstructed with the estimated parameters have no appreciable artifacts, and are practically indistinguishable from the corresponding images reconstructed with the true parameters. Only a small discrepancy in the location of the reconstructed objects can be observed. As can be seen in table 3, the analysis of the first angles, in which the starting point is far from the minimum, takes a somewhat big number of iterations (300–400). This number decreases by a factor of 5–10 just in the first three angles analyzed, then remains almost constant on 40–50 iterations for all other angles. On the second run, as one can expect, the overall number of iterations is reduced because the starting point is already near the minimum. Comparing the results of the first and second run, we note only a small dependence of precision, accuracy and computation time on the starting point. 5.2. Results with real data The proposed algorithm was used to calibrate our prototype micro-CT scanner, from the scan data of a mouse femur. The main components of the prototype scanner are a microfocus
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Figure 6. Reconstructions of the details of the simulated object, on a transaxial plane passing through the center of the two small spheres at z = 1.05 cm. All images are 64 × 64, with a voxel size of 0.1 mm. The visualized grayscales (min; max) are (−1500 HU; 9500 HU) for the reconstructions and (−1000 HU; 1000 HU) for the difference images. No artifacts are visible on the images reconstructed using the evaluated parameters δbest , even after the first run. The difference images show only a small discrepancy on the positions of the reconstructed objects with respect to the original.
(a)
(b)
Figure 7. (a) Projection image of the mouse femur acquired with the micro-CT scanner prototype, and (b) the corresponding error projection computed at δ = δ 0 . The RoIs used for the computation of the cost function were highlighted in (b).
x-ray source (Hamamatsu 60KVMFX) with a fixed tungsten anode, a maximum accelerating voltage up to 60 kV, a maximum power of 10 W and a focal spot size of 7 µm; a flat-panel CMOS detector (Rad-icon RadEye 4) composed by 2048 × 1024 pixels of 482 µm2 ; a rotating stage fixed to the specimen holder, which is also capable of translating along the z axis in order to select the longitudinal position of the FoV inside the specimen. In the acquisition presented here, we used a beam quality of 40 kV with 1 mm Al filtration; 480 projections were acquired over 360◦ , and an exposure time of 1.6 s per angle was used. Successively to the scanning, an angular rebinning was applied in order to reduce the number of angles to 240. The geometrical parameters were SAD = 110 mm and SDD = 535 mm, corresponding to a magnification factor of 4.9. The cost function was computed on five RoIs as shown in figure 7(b), over 20 projections at steps of 7.5◦ . The starting point was δ 0 = (−70, 0, 0, The result of 0, 0, 0)T ; the total number of points analyzed was 2.1 × 106 .
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(a) Sagittal
(b) Coronal
Figure 8. Images of the mouse femur, reconstructed with the estimated misalignment parameters computed with the proposed calibration method. The size of the isotropic voxel is 10 µm.
Table 5. Calibration results for the micro-CT scanner prototype, obtained from the scan data of a mouse femur. The total number of image points analyzed was 2.1 × 106 ; the starting point was δ = (−70, 0, 0, 0, 0, 0)T . As one could expect, the time spent for the calibration in this case is increased with respect to the simulation study; this is due to the greater number of image points analyzed at each iteration. No. of iterations δubest
φbest
ηbest
First view
Average ± SD
Time (total)
−67.645 ± 0.385
−0.360 ± 0.010
−1.116 ± 0.187
167
82 ± 28
246 s
the calibration is reported in table 5, while in figure 8 we show the tomographic images reconstructed using the estimated misalignment parameters. To demonstrate the accuracy of our calibration method with real data, we compared the reconstruction obtained using the estimated parameters δbest with those obtained using a slightly different set of parameters. The results are shown in figure 9, where we can see that the reconstruction at δ = δbest is always better than the others. The differences caused by the variation of the slant are not appreciable and were not reported in figure 9; in fact, the reconstructed detail is placed too close to the center of the FoV where the effect of this parameter is less important. 6. Discussion and conclusions We have presented a novel method for the geometric calibration of cone-beam scanners. The method does not require any a priori knowledge of the shape and positioning of the scanned object, and allows the measurement of the three critical parameters δu, φ and η with good precision and accuracy. The results reported in tables 3 and 4 show that the method is robust with respect to noise and has a small dependence on the initial condition. In principle, any object can be used to calibrate the scanner. Thus, the data acquired during a routine work session can be used to perform the calibration. A trade-off between calibration accuracy and
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Figure 9. Details of the coronal image of figure 8, reconstructed with small variations of the misalignment parameters from those obtained with our calibration method. The images reconstructed at δ = δbest presents sharper edges, thus proving the validity of our method.
computational cost must be taken into account when choosing the subset of data to include in the computation of the cost function. A drawback of the method is the very poor sensitivity with respect to the longitudinal shift (δv), the error in SDD (δD) and the tilt (λ). As a consequence, these three parameters must be kept fixed at some estimated parameters in the optimization process. However, it is known that the three parameters above are also the lesser critical parameters with respect to the image quality. The approximated validity of the method in the case of asymmetric objects does not seem to compromise the accuracy of the calibration, as reported in sections 5.1 and 5.2. The profiles in figure 5 show that the choice of RoIs can influence the sensitivity of the method (i.e., the curvature of the cost function) with respect to the various parameters. A rectangular RoI intersecting with the midplane should always be included in the computation of the cost function; it is also important to exploit the symmetry of some details in the object (e.g., the cylindrical container for specimens, or the edges of a couch) to include more data in the computation and increase the sensitivity of the method. In general, a check of the RoI prior to start the calibration process is always advisable. It is worth noting that our method is also independent on beam-hardening effects. In fact, a pair of redundant data points is obtained by traversing the object in the two senses along the same line of projection (or along two equivalent lines); the hardening of the x-ray spectrum along that line is the same in the two senses, hence the redundancy is preserved also for polychromatic data.
Acknowledgments We would like to thank Professor Michel Defrise for his precious observations and corrections. Patent Pending, Application n. IT PI2007A000129.
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