Generalized Filtered Backprojection for Digital Breast Tomosynthesis Reconstruction

Generalized Filtered Backprojection for Digital Breast Tomosynthesis Reconstruction Klaus Erhard a, Michael Grass a, Sebastian Hitziger b, Armin Iske b and Tim Nielsen a a Philips Research Europe Hamburg, Germany; b University of Hamburg, Germany ABSTRACT Filtered backprojection (FBP) has been commonly used as an efficient and robust reconstruction technique in tomographic X-ray imaging during the last decades. For standard geometries like circle or helix it is known how to efficiently filter the data. However, for geometries with only few projection views or with a limited angular range, the application of FBP algorithms generally provides poor results. In digital breast tomosynthesis (DBT) these limitations give rise to image artifacts due to the limited angular range and the coarse angular sampling. In this work, a generalized FBP algorithm is presented, which uses the filtered projection data of all acquired views for backprojection along one direction. The proposed method yields a computationally efficient generalized FBP algorithm for DBT, which provides similar image quality as iterative reconstruction techniques while preserving the ability for region of interest reconstructions. To demonstrate the excellent performance of this method, examples are given with a simulated breast phantom and the hardware BR3D phantom. Keywords: Digital Breast Tomosynthesis, FBP, ART, generalized FBP 1. INTRODUCTION Filtered backprojection (FBP) has been commonly used as an efficient and robust reconstruction technique in tomographic X-ray imaging during the last decades. One reason for its efficiency is that it allows for region of interest (ROI) reconstructions and only requires one backprojection per view, which can be parallelized easily. For limited angle tomography acquisitions such as digital breast tomosynthesis (DBT), however, standard FBP reconstruction algorithms provide poor results and give rise to image artifacts due to the limited angular range and the coarse angular sampling. Several modification of the ramp filter have been proposed 1 3 to improve the image quality of FBP applied to limited angular range tomography, but these filter modifications are controlled with free parameters, which have to be optimized and adapted to the particular acquisition geometry. Iterative reconstruction algorithms are often used in DBT since they potentially yield a reconstructed image that is in better accordance with the measured data. Compared to FBP, iterative algorithms are more stable against noise and can handle arbitrary geometries. However, the improvement in image quality and flexibility has to be paid with computational complexity and typically longer computation times. In this work, a generalized FBP algorithm is presented, which uses the filtered projection data of all acquired views for backprojection along one direction in order to compute a high quality image similar to iterative techniques. The proposed method requires the computation of geometry-dependent filter kernels but can still be formulated as a generalized filtered backprojection scheme. Within this generalized FBP formulation, the filtering step incorporates computations across multiple views while only one backprojection per view-angle is needed. Therefore, parallelization of the backprojection step is similar to the standard FBP algorithm. Hence, the proposed method yields an efficient reconstruction algorithm with an accuracy comparable to iterative techniques, which will be demonstrated on both simulated breast tomosynthesis data and measurements of the BR3D phantom on an experimental X-ray system. Further author information: (Send correspondence to K.E.) K.E.: E-mail: klaus.erhard@philips.com, Telephone: ++49 40 5078 4539

2. METHODS While FBP methods provide fast and accurate reconstruction algorithms for tomographic X-Ray imaging whenever the source trajectory is complete and sufficiently many projections are acquired, 4 they perform worse on limited angle tomographic data where both the trajectory is incomplete and the sampling is coarse. In digital breast tomosynthesis, for example, typical examination protocols acquire only 10-30 X-ray projections over a limited angular range of 15-60. 5 Here, a common FBP image reconstruction suffers from severe artefacts such as the loss of the low frequency information value and edge sharpening along the source trajectory. 6, 7 These artefacts can be weakened with the use of iterative reconstruction techniques since these kind of algorithms successively update the reconstructed image in order to reduce the mismatch between measured projection data and the re-projections generated from the current image. 2.1 Motivation of the filter design A similar reconstruction quality as for example with the algebraic reconstruction technique (ART) cannot be achieved with a view-by-view filtering of the projection data in standard FBP reconstruction since the update step of ART implicitly involves information from all projection angles prior to backprojecting from one single view. To mimic this feature of iterative reconstruction algorithms, cross-view projection filtering has to be enabled within the FBP framework. Therefore, a more general filtered backprojection is introduced by x = BF y, (1) with image x, measured projection data filter matrix and the backprojection operator y = (y 1,..., y N ) T, (2) F 11... F 1N F =..... (3) F N1... F NN B = 1 N (B 1,..., B N ). (4) Here, y i denotes the measured data for the source position i, i.e. P x = y with the projection operator P = (P 1,..., P N ) T, (5) defined via the projection operators P i for each view position i. Note, that in contrast to standard FBP algorithms, the off-diagonal filter matrices F ij, i j, acting on the measured data y j and contributing to the filtered view i prior to backprojecting from source position i, are non-zero for the proposed generalized FBP method. Computation of an image x that is consistent with the measured data can be achieved with the knowledge of the generalized inverse P + of the projection operator P via x = P + y. (6) However, direct computation of the generalized inverse is not feasible due to the complexity of P and therefore iterative methods are commonly applied for solving Eq. (6). Comparing Eq. (1) with the following observation P + = B(P B) + shows that filtering with F = (P B) + (7) in the generalized FBP formula (1) yields the same result x as applying the generalized inverse P + to the measured data y.

2.2 Computation of filter kernels The computation of the filter kernels for the generalized filtered backprojection algorithm given by Eq. (1) and Eq. (7) will be demonstrated in the following for a DBT geometry. The acquisition geometry is defined by a stationary detector and an X-ray source moving on a straight line at fixed height above the detector parallel to its columns, see Fig. 1 (a). Therefore, each detector column consisting of M detector elements can be processed separately. However, the computation of the filter F for one particular column still requires the evaluation of the generalized inverse (P B) + of the matrix P B = P 1 B 1... P 1 B N..... P N B 1... P N B N which consists of N N blocks of matrices of the size M M., (8) To analyze the combined operation P i B j, explicit definitions of the projection and backprojection operators (assuming an infinite continuous detector for analytical purposes) are given by D (P i x)(ξ) = 1 ( x ξ + η ) D 0 H (ξ i ξ), η dη, (9) ( ) Hξ ηξi (B i y)(ξ, η) = y. (10) H η Here, ξ is the spatial coordinate along the trajectory direction and η is perpendicular to it (0 η D), ξ i is the ξ-component of the source position for view i. Furthermore, H denotes the constant source to image distance (SID) and D is the compression thickness, see Fig. 1 (a). The value of P i x at the detector position ξ is the average absorption coefficient on a line connecting ξ with the X-ray source position ξ i. The backprojection is obtained by distributing a projection value along the line which connects the corresponding detector coordinate with the X-ray source. The combined operation P i B j can be derived from the definitions (9) and (10). For i = j it is the identity operation: P i B i y = y. For i j is given by: P i B j y(ξ) = H D D ji H D 0 ji y(ξ z) dz, (11) ( ji + z) 2 where ji = ξ j ξ i is the distance between source positions for views i and j. From Eq. (11), it is clear that P i B j y is a convolution of y with a shift invariant kernel which we will abbreviate with ψ ij. It also follows from Eq. (11) that the kernels for P i B j and P j B i are symmetrical, i.e. ψ ij (ξ) = ψ ji ( ξ) for i j. For typical values of M = 2048 and N = 17 in digital breast tomosynthesis the numerical complexity of a direct calculation of (P B) + is O(N 3 M 3 ) and hence too large for practical purposes. To overcome this problem, the operator (P B) acting on the projection data is analyzed in the Fourier domain. Each block matrix P i B j describes a convolution on the projection data with a shift-invariant kernel of limited width depending on the angle between the source positions i and j. Hence, the block matrices P i B j can be described as a diagonal matrix in the frequency domain with diagonal elements given by the frequency components of the Fourier transformed convolution operator P i B j, i.e. ( ) (1) (M) P i B j = diag P i B j,..., Pi B j, (12) and P B can be reordered w.r.t. the frequency components k = 1,..., M, yielding a block-diagonal shape of M blocks of size N N each: ( P B = diag P B (1),..., P B (M)), (13)

(a) (b) (c) Figure 1. (a) Acquisition geometry of the simulated DBT system. (b) Circular arc geometry of the experimental X-ray setup. The mapping from the radial source positions onto the virtual line introduces an approximation error of the cone-beam geometry indicated with the dashed and solid fans. (c) Axial slice through simulated breast phantom with anatomical background noise and ellipsoidal lesion. with P B = P 1 B 1. P N B 1... P1 B N....... PN B N. (14) Now, the calculation of the filter ˆF ) + = ( P B can be performed separately for each frequency component k = 1,..., M, which reduces the complexity from O(N 3 M 3 ) to O(MN 3 ) and enables the practical implementation of the proposed method. The matrices ) ( P B can be calculated analytically for the given acquisition geometry from the Fourier transform ˆψ ij of ψ ij. 2.3 Phantoms For the simulation study a series of breast phantoms have been generated from dynamic contrast-enhanced MRI images, acquired on a Philips MR Intera Achieva 1.5 T scanner at the University of Chicago Hospitals, by segmentation into three tissue compartments representing adipose, glandular and skin tissue and application of a compression model as previously described in. 8 To further refine these software breast phantoms, anatomic noise background structure has been added with a random signal following a power law behavior. To this end, a random white noise signal has been Fourier transformed and multiplied with a power-law function H(f) = B 0.5 /f 1.5 in frequency domain such that the resulting structure noise 9 admits the power spectrum P (f) = B/f 3. An exemplary breast phantom with an additional ellipsoidal lesion and a resolution of 200 µm in each direction is depicted in Fig. 1 (c). Furthermore, the method has been demonstrated using the BR3D phantom (CIRS Inc., Norfolk, VA) as depicted in Fig. 2 (a). The phantom of size 100 mm 180 mm 10 mm comprises five background slabs of 1 cm height, which consist of a mixture of adipose and glandular tissue equivalent material. An additional target slab containing an assortment of micro-calcifications, fibrils and masses has been positioned between the second and the third background slab. The semicircular shaped slabs provide a heterogeneous background pattern and obscure the embedded detail structures on the projection data, see Fig. 2 (b).

(a) (b) (c) (d) (e) Figure 2. (a) BR3D hardware phantom comprising five background slabs and one target slab positioned between background slab two and three. (b) Central projection through all slabs of the BR3D phantom. (c) - (e) Reconstruction of one slice through a background slab within the BR3D phantom using FBP algorithm with a ramp filter (c), ART with 10 iterations (d), and the proposed generalized FBP algorithm (e), respectively. (a) (b) (c) (d) Figure 3. Axial slice reconstructions with (a) standard FBP, (b) proposed generalized FBP and (c) ART with 10 iterations. (d) shows profiles of the reconstructed attenuation coefficient along the line indicated in (a) - (c). 3. RESULTS Simulated projection data have been generated from breast phantoms for a DBT system with a stationary detector with 100 µm pixel pitch. The tube is moving parallel to the detector columns at constant height of SID = 665 mm above the detector sampling N = 17 projections at equidistant angles between ±16 measured against the detector normal, see Fig. 1 (a). Fig. 3 (a) - (c) show a comparison of an axial slice through the reconstructed volumes for FBP with a ramp filter, the proposed generalized FBP, and ART with 10 iterations. Fig. 3 (d) shows a profile of the attenuation values along the line indicated in the images. Obviously, the FBP algorithm with a standard ramp filter yields a loss of the low spatial frequency information and even negative values occur in the reconstructed attenuation coefficients. On the other hand, the generalized FBP and the iteratively computed ART reconstruct an attenuation function that better fits to the measured data. Both methods exhibit a very similar appearance and the line profiles shown in Fig. 3 (d) almost coincide. For an experimental validation of these simulated results, a BR3D phantom with five background slabs and one target slab has been examined on a laboratory setup (Philips Healthcare, Best, The Netherlands). This system enables a rotational movement of the X-ray tube around a pivot point, which is located 80 mm above a stationary X-ray detector with 50 µm pixel size. The source to iso-center distance was 590 mm and 25 projections have been acquired along a circular arc of 48 at equidistant angular positions with a total mean glandular dose of 2.1 mgy. The deviation of the laboratory geometry from the linear geometry depicted in Fig. 1 (a) results in convolution kernels, which are not shift-invariant. Nevertheless, the generalized FBP algorithm can be applied approximately in this situation by mapping the radial source positions onto a virtual line with constant distance from the stationary detector. This approximation results in a geometric mismatch of the true and virtual geometry as illustrated in Fig. 1 (b).

(a) (b) (c) (d) Figure 4. Region of interest through the target slab of the BR3D phantom reconstructed with (a) standard FBP, (b) proposed generalized FBP and (c) ART with 10 iterations. (d) shows profiles of the reconstructed attenuation coefficient along the line indicated in (a) - (c).

Fig. 4 compares the reconstruction of the target slab using the ramp-filtered backprojection with the proposed method and an ART reconstruction after 10 iterations. The FBP algorithm reconstructs the attenuation values up to a constant bias, see Fig. 4 (d), hence the level and window settings have been adapted for a harmonized presentation of each reconstruction. Both the ART and the generalized FBP are more stable against noise, which is demonstrated in Fig. 4 (a) - (c). The proposed generalized FBP method provides a high image quality comparable to the ART reconstruction. A shift in the reconstructed attenuation values could be observed with the generalized FBP method, which is contributed to an approximation in the filter kernel computation due to the circular arc geometry. 4. DISCUSSION The proposed method yields a computationally efficient generalized FBP algorithm for digital breast tomosynthesis, which provides similar image quality as iterative reconstruction techniques while preserving the ability for region of interest reconstructions. Both a small number of views and a limited angular range can be handled by the generalized FBP with a decent image quality in contrast to the standard FBP reconstruction, which yields a severe loss of the low frequency information of the object. Moreover, due to the filter computation as the pseudo-inverse operator, the reconstructed image provides an optimal solution in the least-squares sense, which minimizes the error between the measured data and the re-projections of the reconstructed image. The computational performance of the generalized FBP is comparable to standard FBP algorithms. The filtering step requires additional cross-view computations and therefore the backprojection step can only be started after the data acquisition has been completed. The presented comparison of the proposed method with FBP and ART both in a simulation study and with measurements on a laboratory X-ray system demonstrate a comparable image quality and robustness against noise in the projection data as standard iterative reconstruction techniques such as ART. ACKNOWLEDGMENTS The authors would like to thank Lex Alving and Robert Hofsink, Philips Healthcare, Best, The Netherlands, for performing the tomosynthesis acquisition of the BR3D phantom on an experimental X-ray system. REFERENCES [1] Lauritsch, G. and Haerer, W. H., Theoretical framework for filtered back projection in tomosynthesis, Medical Imaging 1998: Image Processing 3338(1), 1127 1137, SPIE (1998). [2] Zhao, B. and Zhao, W., Three-dimensional linear system analysis for breast tomosynthesis., Med Phys 35, 5219 5232 (Dec 2008). [3] Ludwig, J., Mertelmeier, T., Kunze, H., and Härer, W., A novel approach for filtered backprojection in tomosynthesis based on filter kernels determined by iterative reconstruction techniques, in [IWDM 08: Proceedings of the 9th international workshop on Digital Mammography], 612 620, Springer-Verlag, Berlin,Heidelberg (2008). [4] Kak, A. C. and Slaney, M., [Principles of Computerized Tomographic Imaging], IEEE Press, New York (1987). [5] Dobbins, J. T. and Godfrey, D. J., Digital x-ray tomosynthesis: current state of the art and clinical potential., Phys Med Biol 48, R65 106 (Oct 2003). [6] Hamaker, C., Smith, K. T., and Wagner, D. C. S. S. L., The divergent beam x-ray transform, Rocky Mountain Journal of Mathematics 6, 253 283 (1980). [7] Louis, A. K. and Rieder, A., Incomplete data problems in x-ray computerized tomography, 56, 371 383 (1989). [8] Erhard, K., Grass, M., and Nielsen, T., A second pass correction method for calcification artifacts in digital breast tomosynthesis, Medical Imaging 2011: Physics of Medical Imaging 7961(1), 796119, SPIE (2011). [9] Burgess, A. E., Jacobson, F. L., and Judy, P. F., Human observer detection experiments with mammograms and power-law noise., Med Phys 28, 419 437 (Apr 2001).