Efficient implementation of a local tomography reconstruction algorithm

Local tomography and artifacts

The most common local tomography reconstruction method is extrapolating the sinogram before computing the filtered backprojection (FBP), hereafter denoted padded FBP. The extrapolation is usually done by replicating the sinogram boundary values. This prevents truncation artifact (Gibbs phenomenon) from occurring, and often provides acceptable results [10].

However, this technique can fail when the ROI is surrounded by anisotropic and/or strongly absorbing material or when the reconstruction has intrinsically low contrast (for example different parts with the same linear absorption coefficient).

The notable local tomography artifact is the cupping effect. On a reconstructed image, local tomography artifacts appear as a varying contrast. The gray values are typically higher far from the center than close to the center, forming a “cup.” The cupping is also visible when plotting an image line passing through the center, as a function of the pixel location. Such lines are hereby called profiles, for example, the vertical line profile is the vertical line of the image passing through the center.

Figure 2 shows the Shepp–Logan phantom with a region of interest. Figure 3 shows the reconstruction with padded FBP, and Fig. 4 shows line profile of the reconstruction. The cupping effect is clearly visible in both the reconstruction image and profile. This cupping effect can be detrimental for the post-reconstruction analysis, for example, segmentation.

In this work, we examine a family of exact reconstruction methods based on a known subregion. We implement a method handling a reduced number of unknowns by expressing the image in a coarse basis in order to correct the cupping effect.

Iterative reconstruction

In this context, the reconstructed slice x is an image of support \(N \times N = N^2\), where N is the number of pixels of the detector horizontally. The sinogram d support is \(N \times N_p\), where \(N_p\) is the number of projections. Thus, the projector is theoretically an operator of dimensions \((N \times N_p, \, N^2)\), assuming that slices are stacked as one-dimensional \(N^2\) vectors, and sinograms are stacked as one-dimensional \(N \times N_p\) vectors.

Efficient implementations of the projection and backprojection operators enable to solve problem (3). The ASTRA toolbox [11], for example, has versatile geometry capabilities and built-in algorithms for solving (3) for \(\phi (x) = 0\).

In this work, we consider the case where a subregion is known. This prior knowledge on the volume can be used to constrain the sets of solutions. A uniqueness theorem was stated in [6] along with a reconstruction algorithm based on differentiated backprojection and projection onto convex sets to invert the finite Hilbert transform. This algorithm, however, is difficult to implement, and no implementation is readily available for experiments.

We focus on a simpler approach based on formalism (3). In this formulation, the prior knowledge can be encoded in several ways. The first is to enforce the values of \(\tilde{x}\) in the known region, for example, using an indicator function. The second is to add a term penalizing the distance between the values of \(\tilde{x}\) in the known region and the actual values. We adopt the latter approach, which was proposed, for example, in [12].

Let \(\Omega\) denote the domain where the values of the volume are known. It is a subset (possibly a union of subsets) of the image support \(N^2\), and we denote \(N_\Omega\) its cardinality, that is, the total number of known pixels. Let \(x_{|\Omega }\) denote the values of x inside the known region. The prior knowledge is encoded by \(\phi (x) = \lambda \left\| x_{|\Omega } - u_0 \right\| _2^2\), where \(u_0\) denotes the known values inside \(\Omega\) and \(\lambda \ge 0\) is a parameter weighting the fidelity to the known zone. Both \(x_{|\Omega }\) and \(u_0\) have \(N_\Omega\) components.

Figures 5 and 6 show the reconstruction result on the Shepp–Logan phantom with such choice of \(\phi (x)\). The cupping effect is almost removed, but the reconstructed slice is also noisy, which is a known effect of least squares minimization on an ill-posed problem when running too many iterations [13]. On the other hand, many iterations are required to reduce the cupping effect.

A workaround on this problem is adding a regularization term to stabilize the solution. A popular regularization is Total Variation (TV), promoting piecewise-constant solutions. The function \(\phi (x)\) in (3) can then be written \(\phi (x) = \lambda \left\| x_{|\Omega } - u_0 \right\| _2^2 + \beta \left\| \nabla x \right\| _1\), where \(\beta \ge 0\) weights the regularization. Figures 7 and 8 show the result of reconstruction with this method. The reconstruction is much more accurate and bears almost no difference with respect to the ground truth, which is an illustration of the uniqueness theorem stated in [6].

This approach, however, has two drawbacks. The first is using a prior which might not be accurate: in this example, Total Variation promotes piecewise-constant images and is thus not adapted for complex samples. The second drawback is on the computational side. Adding a non-differentiable prior involves to change the optimization algorithm for another probably less efficient in the sense that more iterations are required to reach convergence. In the examples, the preconditioned Chambolle–Pock algorithm described in [14] was used for the TV minimization. Approximatively, 3000 iterations are required to approximately get rid of the cupping effect (when approximatively 500 are required in the case of a complete scan), and more than 10,000 iterations are required to get the line profiles shown in Fig.  8. This approach is impracticable for modern datasets with increasing amount of data: on the one hand, projection and backprojection become costly operations, while on the other hand, even more iterations are required due to the higher number of variables.

The main contribution of this work is an efficient implementation of the method described in [9]. The method is based on the following observation: the padded FBP reconstruction yields acceptable reconstruction of features of the ROI [15], but can suffer from a low-frequency bias (cupping effect). On the other hand, iterative algorithms converge slowly due to the high indeterminacy of the problem, even with a known subregion. For these reasons, a refinement of the initial reconstruction is computed rather than the complete solution.

Correction of the low-frequency bias

Projection a of Gaussian tiling

The choice of a Gaussian basis for a coarse representation of the correction term is based on a characteristic of the Gaussian kernel: it is both rotationally invariant and separable [16]. These two properties provide a computational advantage: the order of projection and convolution can somehow be exchanged.

More precisely, given an image y consisting of the Gaussian coefficients evenly placed with a spacing s, the standard way to compute \(\tilde{P} G y\) is first performing the convolution Gy defined by (8) and then projecting with \(\tilde{P}\). An equivalent computation, however, can be done by first projecting the image of isolated points y, and then convolving each line of the resulting sinogram by a one-dimensional Gaussian function. This is illustrated in Fig. 9.

This latter approach has two advantages. First, the two-dimensional convolution (or two series of one-dimensional convolution in this separable case) is replaced by a series of one-dimensional convolutions. Secondly, the projection here only consists in projecting isolated points. This operation can be optimized by designing a point-projector based on look-up tables.

LUT-based point-projector

As previously discussed, the operators involved in forward model (10) are a cropping operator, a one-dimensional convolution, and a projector. The convolution can be efficiently implemented, either in the Fourier space or in the direct space when one of the functions has a small support. Therefore, a fast projector is essential for solving (10) in an iterative fashion. In our case, the object to project has a very special structure, as it consists in points spaced by several pixels. Thus, standard projectors of tomography softwares can be replaced by a more efficient implementation, hereby called point- projector, based on look-up tables.

In the remainder, the following notations are used: The support of the original image is \(N^2\). The number of projections is \(N_p\), so the acquired sinogram has size \(N \times N_p\). The size of the extended image is \(N_2 ^2\) where \(N_2 \ge N\). The number of Gaussian functions used to tile the support is \(N_g\). The spacing between Gaussian blobs on the image is s ; thus we have \(N_g \simeq \left( \frac{N_2}{s} \right) ^2\) in a first approximation. We also use the following indexes convention : Gaussian coefficients are numbered with \(i \in [0, N_g]\), and sinogram indexes are numbered with \(k \in [0, N_s]\) where \(N_s = N_2 \times N_p\) is the size of the (extended) sinogram.

Each Gaussian coefficient number \(i \in [0, N_g[\) is projected on (at most) \(N_p\) positions in the sinogram. Therefore, a look-up table J is built so that for each i, J[i] is the “list” of locations in sinogram hit by this point after projection. The LUT J is an array of size \(N_g \times N_p\). Each entry \(J_{i,j}\) corresponds to a position, in the sinogram, that is hit by a projected point \(i \in [0, N_g]\). For example, entry \(J_{0,2}\) is an index in the sinogram that is hit by point 0 ; and entry \(J_{5,j}\) are an indexes hit by point 5 for all j. This is illustrated in Figs. 10, 11.

When computing the sinogram, however, the look-up table J is best accessed “backward”: for a given position \(k \in [0, N_s[\) in the sinogram, we have to determine which points are hitting it through projection. To this end, two look-up tables J and \(\mathrm {Pos}\) are built. For \(k \in [0, N_s]\), \(\mathrm {Pos}[k]\) indicates a position in LUT J, and \(J[p_k]\) is a coefficient number \(i \in [0, N_g]\) being projected at position k. Therefore, the LUT J does not contain sinogram indexes anymore, but rather coefficient indexes. This is illustrated in Fig. 12. The LUT J is re-ordered such that the interval \([p_k, p_{k+1}-1]\) gives access to an indexes range in J ; this index range is the set of all coefficients indexes being projected on sinogram index k.

The point-projector is described by Algorithm 1. The matrix W, indexed in the same way as J, contains the weights of the projections: depending on the position of a point in the image and the projection angle, its projection does not exactly fall into a sinogram pixel. The matrix W thus encodes the geometric contribution of the projection of the points.

This projection scheme basically consists in storing the explicit projection matrix \(\tilde{P}\) with a Compressed Sparse Row (CSR) format [17], where LUT J corresponds to “col_ind,” LUT \(\mathrm {Pos}\) corresponds to “row_ptr,” and matrix W contains the values. Storing the entire “linear-algebra” projection matrix without compression would entail to store \((N_2^2)\times (N_2 \cdot N_p)\) elements, which is impracticable (for example, more than one terabyte is required for a \(1024^2\) slice). However, as each slice point is projected on at most \(N_p\) sinogram positions, this matrix actually has at most \(N_2^2 \times N_p\) non-zero elements. Additionally, as the slice is reduced on a coarse basis, there are \(\left( \frac{N_2}{s} \right) ^2 \times N_p\) non-zero values to store in this case. The format described above is used to store these elements. Algorithm 1 is thus no more than a matrix-vector multiplication with a sparse matrix in CSR format.

This approach for computing the point-projector is friendly in a memory-write point of view: after accumulating the contributions of all coefficients projected on position k, the sinogram at index k, \(\mathrm {sino}[k]\), is updated accordingly. This is especially important for GPU implementation, as consecutive threads access contiguous memory locations, which is a coalesced access pattern. On GPUs, each memory transaction actually entails accessing L bytes, so coalesced access to 32 bits scalars results in a read or write of L / 4 addresses in a single transaction (for example, \(L = 128\) for modern NVidia GPUs).

Implementation of the adjoint operators

As a gradient-based optimization algorithm is used for solving (10), the adjoint of operator \(C \tilde{P} G\) has to be computed. This operator \(G^T \tilde{P}^T C^T\) consists in extending the sinogram with zeros, point-backprojecting and retrieving the Gaussian components from the backprojected image. As mentioned above, the operator G can be described as \(G = H_\sigma U\) where U is an upsampling operator (here with a factor s), and \(H_\sigma\) is the convolution with 2D Gaussian kernel (7). Thus, \(G^T = H_\sigma ^T U^T\) which is a downsampling followed by a convolution with kernel (7). The actual computation is then \(G^T \tilde{P}^T C^T = H_\sigma ^T U^T \tilde{P}^T C^T = U^T \tilde{P}^T H_\sigma ^1 C^T\) where \(H_\sigma ^1\) is a one-dimensional convolution on the sinogram rows.

As previously, these operations can be merged. As \(G^T \tilde{P}^T C^T\) returns a Gaussian coefficients vector from a sinogram, only the coefficients are of interest here. Therefore, the point-backprojector \(\tilde{P}^T\) is merged with the downsampling \(U^T\) as previously. For a given coefficient, we have to find which sinogram entries backproject on the coefficient position. This approach avoids to compute useless \(N_g \times (s-1)^2\) backprojections points on the image, as it is downsampled afterward.

The point-backprojector is implemented, as previously, with a LUT \(J_2\) of size \(N_g \times N_p\) and a LUT \(\mathrm {Pos2}\) of size \(N_g +1\). The matrix \(\mathrm {J_2}\) is re-ordered so that for all \(i \in [0, N_g[\), the interval \([\mathrm {Pos2}[i], \, \mathrm {Pos2}[i+1]-1]\) corresponds to an index range in LUT \(J_2\). This is illustrated in Fig. 13.

The point-backprojector is given by Algorithm 2. Again, the backprojection from a sinogram to a Gaussian coefficients vector corresponds to a matrix-vector multiplication with a matrix in CSR format. The matrix \(W_2\), containing the geometric weights of the backprojector, can be viewed as the Column Sparse Storage (CSC) version of the matrix W.

Parallel implementation

In modern experiments carried on X-ray light sources, the data volumes, produced by new generations of detectors, always overwhelm the computing power. Simply waiting for more powerful machines is of little hope, as advances in detectors overrun the Moore’s law. Instead, an algorithmic work has to be accomplished to exploit parallelism of modern architectures. In the last decade, the advent of general-purpose GPU (GPGPU) computing was advantageously used, especially in tomography.

The proposed method has been implemented in the PyHST2 software [18] used at ESRF for tomographic reconstruction, with the CUDA language targeting Nvidia GPUs. The point-projector and point-backprojector, which are the most time-consuming operators, are implemented as efficient CUDA kernels. As for Algorithms 1 and 2, the CUDA point-projector and point-backprojector are implemented as matrix-vector multiplication with a matrix in CSR format.

We describe here the implementation of the point-projector, i.e., the computation of the sinogram values \(\mathrm {sino}[k]\) for \(k \in [0, N_s]\). The point-backprojector follows the same principle. To compute the sinogram value \(\mathrm {sino}[k]\), the LUT J has to be accessed from \(p_k\) to \(p_{k+1}-1\) as illustrated in Fig. 13. This memory range is accessed in parallel by threads of the many-cores GPU with the following principle. Each thread reads \(m \ge 1\) values in the LUT. With these values J[j], where \(j = p_k, p_k +1, \ldots\), the coefficients vector is accessed at \(\mathrm {coeffs}[J[j]]\). The threads are grouped in blocks, and each thread updates a temporary array in shared memory with the contributions read in \(\mathrm {coeffs}[J[j]]\). Then, in each block, the shared array is accumulated by one thread. The result is added to \(\mathrm {sino}[k]\). This is illustrated in Fig. 14.

The parallelization is done on the read of matrix J, as it is the biggest data structure of the method. As it has been re-arranged so that the interval \([\mathrm {pos}[k], \mathrm {pos}[k+1]-1]\) is a contiguous memory range in J, the described implementation has an efficient memory access pattern.

Multi-resolution Gaussian basis

The correction term \(x_e\) in model (4) is a tiling of Gaussian functions : \(x_e = G c\), where c is the vector of coefficients in the Gaussian basis, and G is the operator previously described. In a first approach, all the Gaussian functions (7) have the same variance \(\sigma ^2\), so that operator G is linear and problem (10) is convex. The coefficients are placed on a support of size \(N_2^2\) before being (theoretically) convolved with a 2D Gaussian kernel. The spacing between points is s, so that the number of required coefficients is approximately \(N_g \simeq \left( \frac{N_2}{s} \right) ^2\).

Implementation of (11) is straightforward. The coefficients in vector c are classified according to the distance to the center, forming subsets of coefficients \(c^1, c^2, \ldots\). Each subset is point-projected and line-convolved with the corresponding \(\sigma _1, \sigma _2, \ldots\). The resulting sinograms are summed to obtain the projection of Gc.

This representation of correction features as Gaussian blobs is actually not a basis in the mathematical sense: some images cannot be represented by a linear combination of Gaussians. However, this representation is very close to a basis for \(\sigma \simeq s\) [20]. In our case, we choose \(s = 0.65 \times \sigma\), meaning that there is a significant overlap between the Gaussians. The discrete Gaussian kernel is truncated at \(3\sigma\), so its length is \(\lceil 6\sigma +1 \rceil\) samples.

Optimization algorithm

Efficient optimization algorithms can be used to solve the quadratic problem (10). We use the conjugate gradient (CG) algorithm, requiring the computation of the adjoint of the involved operators previously described. CG also entails matrix-vector multiplications, which are efficiently implemented with the CSR representation of point-projector and backprojector.

In the GPU implementation, all the involved arrays are single precision (float 32 bits) as most GPUs are relatively not efficient with 64 bits operations. However, the conjugate gradient algorithm involves scalar products. These operations are implemented by dedicated kernels returning double precision values, as error accumulation is noticeable when accumulating on large arrays in single precision.