1. Introduction

In computed tomography (CT) projection images of the sample are recorded from different directions around a rotation axis. A 3D image of the sample can then be reconstructed using the discovery by Radon on the relationship of the projections and the Fourier transform of the sample [1]. As an implementation of this mathematical relationship, filtered backprojection (FBP) technique gives accurate solution given that sufficient number of projections, suitably distributed around the sample, is available [2]. In general, however, reconstruction has to be considered also for cases where a full set of projections is not available. The underlying reason for the missing projections may be either dose or time constraints that limit the number of angles, or the sample geometry that does not allow imaging from all necessary directions. In both cases the incomplete number of projections leads to artefacts in the reconstructed sample volume.

A relatively common case is that of flat samples, where one dimension is much smaller than the other two. For example in x-ray (micro)imaging this type of samples could be microelectronic devices, paintings or fossils; samples that have to be imaged nondestructively for example to do in-situ measurements or to preserve the potentially valuable and unique sample intact. Figure 1 illustrates two possible scanning geometries for this type of sample for the parallel-beam case. In the CT geometry the sample is rotated around an axis that is perpendicular to the beam propagation direction. As the sample is rotated, the transmitted intensity has large variations among the angles, and for some angles there may be no transmission at all.

 

Fig. 1 (a) Illustration of the two scanning geometries for the parallel-beam case. Here ω_CT is the tomographic rotation axis and ω_CL is the laminographic rotation axis. The specimen coordinate frame (x′,y′,z′) is defined with (x′,y′) spanning the in-plane direction and z′ being parallel to the specimen surface normal. (b,c) Sketch of the filling of Fourier space in CL and CT, respectively, for Ψ = 30°. (d) Sketch of the difference in the filling of the Fourier space for the two methods, with parts cut away for better visualization.

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The second geometry, called computed laminography (CL), has been developed specifically for the imaging of flat samples [3–5]. In CL the rotation axis is tilted with respect to the incoming beam, the sample surface normal being approximately parallel to the rotation axis. In CL each projection has similar overall transmission, and all the projections can be recorded. Applications for CL have recently been found in industrial nondestructive testing [6–9], x-ray micro- and nanoimaging [5,10–12], and neutron imaging [13].

Both CT and CL have missing information in the scans, but this missing information is differently distributed. For CL the rotation axis is tilted by the angle Ψ towards the optical axis. In the Fourier space this corresponds to a cone of missing information with an opening angle of 2Ψ (Fig. 1(b)) [3]. In limited-angle CT there is a wedge of missing projections at angles where no projection images could be recorded. This corresponds to a wedge in the Fourier space (Fig. 1(c)), again with an opening angle of 2Ψ. We call the angle Ψ the missing-information angle. For a given Ψ the missing information cone of CL is completely contained within the missing information wedge of CT. Due to the different orientation of the scans the information at high frequencies is also differently available in the two methods as illustrated in (Fig. 1(d)).

As already discussed into greater detail [14,15] the filtering of CL is designed in such a way that the resulting modulation transfer function is 1 in the complete region of the Fourier domain and 0 in the non-filled regions. The same is valid for the CT geometry and in particular for limited-angle CT inside the scanning range. This means that the shape of the artefacts directly depends on the coverage of the Fourier domain for both scanning schemes.

Despite the popularity of CL, not much has been published about the nature of the artefacts as compared to those from CT on a similar sample. The advantages of CL have been illustrated on several occassions [16–18], but there have not been quantitative studies on the conditions under which CL offers benefit over limited-angle CT. To address this question we made a systematic comparison of the artefacts arising in the two methods. The artefacts arising from missing angles are scale invariant, and their general shape is given only by the missing wedge or cone in the Fourier space. The relevant factors that determine the image quality are thus the relative sizes and shapes of the features in the sample, and the relative contrasts of these features. We used both experimental and simulated images of various types of samples, allowing us to identify the essential features of the artefacts, and thus to compare CT and CL in terms of image quality.

For the purpose of comparing the intrinsic properties of the two methods, we used the same Ψ for both methods. The volume reconstructions were done using the filtered backprojection (FBP) algorithm. Hence the two methods were on an equal ground on the basis of the reconstruction algorithm. Methods other than FBP could have been used for reconstruction, for example iterative [19, 20] or statistical [21] reconstruction methods that work better in case of missing angular data. However, our purpose was to compare the inherent properties of the two imaging geometries, and we left the consideration of the optimal reconstruction technique out of the question by limiting ourselves to the FBP. It has to be noted that FBP can actually introduce artefacts beyond those arising just from the missing angles and therefore is not particularly well suited for missing-angle data. However, we chose this method over the others because it is the one that would be most used in practice, especially for large data sets such as those arising from high-resolution absorption or phase-contrast CT or CL.

We investigate two different cases: i) complete unavailability of some projection angles, and ii) tradeoff between noise level and the missing projection angles. The first case corresponds to having a Ψ > 0. In this case the result of reconstruction is not unambiguously defined, and the missing information usually means that the true reconstruction cannot be achieved. Reconstruction from such data is therefore not a well-posed problem because the solution is not unique. The two scanning geometries produce different results in this respect, and we evaluate the methods to see how the results deviate from the true reconstruction. Our results indicate that in this case CL has a clear advantage over CT when Ψ is a significant fraction of 90 degrees. The second case explores the connection between the uniqueness of the solution and the stability of reconstruction with respect to noise. In this case all projection angles are accessible, but some of the angles only at the expense of significantly increased noise level. We found out that CT offers the better results in this respect, as it allows to choose the tradeoff between the noise and the missing data after imaging has taken place.

2. Experimental techniques and computer simulations

2.1. Experimental techniques

In order to compare CL and CT experimentally we imaged a Siemens star resolution test pattern with the two techniques. The test pattern is lithographically manufactured 200 nm high gold structure on a silicon substrate (essentially a two dimesional test structure), with a range of features at different sizes from 50 nm to 800 nm. The experiments were done at the ESRF nanoimaging endstation ID22-NI, which allows resolutions of below 100 nm to be reached with hard x-rays [22–24]. On this beamline, 17 keV x-rays are focused to a point of less than 100 nm in diameter by using multilayer coated Kirkpatrick-Baez mirrors, giving the flux of more than 10¹² photons/s in the focus.

Images with 60 nm pixel size were recorded to obtain CT and CL data sets. For the CL data the angle Ψ was 30 degrees, and 1199 projections were recorded around a 360 degree sample rotation. For the CT the sample was rotated around an axis in the sample plane (see Fig. 1(a)), and 1499 projections were obtained for the 360 degrees of rotation. Projection angles where the sample surface normal was at an angle of more than 60° with respect to the optical axis were discarded to get the same Ψ for the two methods. Therefore total of 999 projections were used for the CT reconstruction. The exposure times were chosen so as to have similar counting statistics in the incoming beam (on the average 1900 counts/pixel for CL, and 1980 counts/pixel for CT). The imaging setup used magnified projection geometry to obtain small pixel size on the images. However, the beam opening angle was only 2 mrad, and the data was treated subsequently with the assumption of parallel beam geometry. The projection microscopy geometry produced images with strong propagation based phase contrast, and a phase retrieval algorithm based on contrast transfer functions was used to recover the phase maps for each projection before 3D reconstruction [25].

2.2. Computer simulations

We simulated x-ray attenuation for test patterns made from 3D arrangements of spheres and rectangular boxes. Projections were calculated simulating the actual geometries of CT and CL, and assuming fully parallel beam. The images were calculated on a pixel grid with the smallest objects being about 10 pixels in size. Therefore the effects from the discrete nature of the images were considered negligible. Furthermore, completely idealized conditions were assumed concerning accuracy of alignment and mechanical stability of the rotation axis. Images without noise and with added Poisson statistics noise were simulated to study the general properties of the methods and the noise characteristics, respectively.

3. Results

3.1. Experimental results

As a starting point for our investigations we imaged a Siemens star resolution test pattern using CT and CL. The results for an in-plane slice (x′,y′) are shown in Fig. 2. In the horizontal direction of the images the CT result has clearly weakened contrast. This direction is perpendicular to the CT rotation axis (y′-direction). The ring separating different parts of the star pattern (marked with arrow 1 in Fig. 2(a)) seems to disappear for the CT result when close to vertical direction. This is due to non-uniform contrast that depends on the direction of the structure. For the CL result the contrast from this ring is uniform in all directions. On the other hand, the CL result has an artefact that extends beyond the star pattern as a bright halo (marked with arrow 2 in Fig. 2(b)). For CT this artefact is reduced in the vertical direction (x′-direction). CT has this advantage because along the CT rotation axis the different planes are independently reconstructed, and the relation between two planes does not depend on the missing angles. In the through-plane slices shown in Fig. 2(c) and Fig. 2(d) both of the methods produce similar results, and the artefacts arising from missing angular data are clearly visible in these slices. Figures 2(e) and 2(f) show through-plane slices in the vertical direction, and here the CT result has clearly more artefacts than the CL result. Figure 2(g) shows profile plots along the half circle illustrated on the star pattern. This clearly shows that the contrast from CL is uniform, while the contrast from CT has a large directional dependence.

 

Fig. 2 Experimental images of a Siemens star test pattern recorded with CT and CL. The central in-plane slices shown for CT (a) and CL (b). (c,d) Through-plane slices along the horizontal line shown in (a) and (b). (e,f) Through-plane slices along the vertical line shown in (a) and (b). (g) A profile plot along the semi circle depicted in (a) and (b) for CT (dashed line) and CL (solid line). All of the images were equally normalized for good visualization. The white numbered arrows in (a) and (b) indicate features that are discussed in the text. The missing information angle Ψ was 30°.

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3.2. General shape of the artefacts

Single isolated objects offer the possibility to study the intrinsic properties of the artefacts. To this end we simulated images from a single sphere located on the rotation axis. The missing-information angle Ψ was varied from 0 to 45 degrees to examine how the artefacts develop as more and more of the Fourier space becomes undefined. Generally the shape of the artefacts for the two methods is similar in planes that contain axis z′, as seen in Fig. 3. In the (x′,y′)-plane, however, CT shows anisotropy whereas CL is isotropic. Along the z′-axis the artifacts change the apparent height of the sphere by a factor proportional to 1/cosΨ, a relation which is closely followed by the results of CL. However, for CT the reconstructed height is slightly larger, and CT gives on the average 5% larger error than CL.

 

Fig. 3 (a) Illustration showing the sample geometry and the two planes that we choose for inspection. The substrate extends infinitely in the x′ and y′-directions, but only a portion of the substrate is shown. (b) Through-plane and in-plane slices of reconstructed single sphere for Ψ = 30°. (c) A plot of the profiles through the middle of the sphere along the CL rotation axis for Ψ = 25°. (d) A plot of measured height, as a full-width half-maximum (FWHM) of the central vertical profile, as a fraction of the real radius of the sphere for CT (crosses) and CL (open circles).

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3.3. Effect of artefacts on spatial resolution

When two or more objects are close together, the artifacts corresponding to an individual object superimpose with the surrounding objects, therefore influencing the distinguishability of different objects. The shape of the objects plays a role in the artifacts, but we limit ourselves to the case of stacked rectangular boxes of different aspect ratios. We consider boxes with height h and a square cross-section with depth and width given by W. The ratio of h/W and the spacing between boxes were varied to illustrate different degrees of effect from the artefacts. We construct a linear resolution test grating from boxes stacked on top of each other, with the positions x_i and heights h_i given by

Single slices and profile plots of the reconstructed test patterns are shown in Fig. 4. In the case where the test pattern is parallel to the CT rotation axis (Fig. 4(a)) CT produces perfect results in terms of resolution. Thus the aspect ratio of the boxes does not affect the resolution in this direction. Rotating the test pattern by 90 degrees while still keeping it in the sample plane (Fig. 4) produces results where CT has strong artefacts, resulting in large errors in contrast. In both of these two orientations CL produces results that are somewhere in between the two results of CT, and the CL results are independent of the orientation of the test pattern. When the test pattern is place parallel to the CL rotation axis (and thus perpendicular to the sample surface) we see the biggest deterioration of the resolution due to the lateral size of the boxes (Fig. 4). In this direction CL can distinguish objects for smaller h/W than what CT can. This result is consistent with the one obtained for the single sphere, and agrees with the theory presented in Fig. 1. It has to be noted that for both methods the resolution in the direction along the CL rotation axis is highly degraded as compared to the resolution for the two directions in the sample plane. This is consistent with the fact that information in the Fourier space is mostly missing in the z′-direction.

 

Fig. 4 Resolution test pattern aligned along the CT rotation axis (a), perpendicular to the CT rotation axis in the sample plane (b), and along the sample surface normal (c). On the left are shown the reconstructed central slices of the test pattern, in the middle are shown plots through the middle of the test pattern, and on the right are illustrations of the geometry. Note that the profile plot in (c) has a different scale on the horizontal axis than the other two profile plots.

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3.4. Effects of local tomography

Although in practical applications the effects of local tomography seem to have only a minor role, there are some cases where these effects can be important. One such case is a detail that extends beyond the field of view, such as a long wire on a microchip. In this case the orientation of the wire can have a large effect when reconstructing from a missing-angle CT scan [16]. If the wire is oriented along the CT rotation axis in the sample plane, then the wire can be reconstructed without problem. On the other hand, if the wire is oriented near the perpendicular direction of the CT rotation axis (still in the sample plane), then the CT reconstruction will have major artefacts. This is explained by CT being unable to quantify the depth coordinate of the object because the wire is not seen “head on” in any of the recorded projections. When considering a single slice containing just the wire, each angle in the sinogram has a constant value. Therefore the best reconstruction for that slice is a constant whose value is compatible with the average value of the sinogram. We demonstrate this dramatic effect by using thin long boxes that run perpendicular to the CT rotation axis. The results are shown in Fig. 5, and it is seen that the long wire practically disappears from the images reconstructed from limited-angle CT data. Especially the through-plane slices show the inferior quality of the CT reconstruction in this case.

 

Fig. 5 An arrangement of boxes where the two larger boxes go beyond the field of view. Result for CL (a), for CT when the long structures are perpendicular to the rotation axis (b), for CT after rotating the structure by 5° (c). The images at the top show an in-plane slice, and the images at the bottom show a through-plane slice taken along the dashed lines depicted in the top images.

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3.5. Noise-limited case

The previous calculations have been done assuming sufficient counting statistics, so that the noise in the images is not a limiting factor. If counting statistics become a limiting factor, then the choice of the missing information angle becomes a case of optimization between the artifacts and the noise in the image. Figure 6(a) shows how the transmitted intensity evolves as a function of Ψ for samples with different transmittance at normal incidence. We see that for a flat sample the absorption increases dramatically as the angle Ψ approaches 0. This means that for a given available counting statistics in the incoming beam (set by the experimental boundary conditions) we have to choose Ψ so that the counting statistics after the sample still allow the signal from the objects of interest to be above the noise level.

 

Fig. 6 The transmitted intensity as a function of the angle Ψ for differently transmitting flat objects (a). Simulated images of the resolution test pattern with noise for CT (b) and CL (c) at Ψ = 30°. Simulated images of the resolution test pattern with noise for CT (d) and CL (e) at Ψ = 1.8°. The values for signal-to-noise ratios have been calculated for the feature indicated by the arrow in panel (b).

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To quantify the effects of noise we used similar resolution test pattern as in section 3.3. Photon counting statistics were chosen so that the SNR of the test structure at normal incidence projection image (Ψ = 90°) was about 15. The substrate transmitted 95.5% of the photons at normal incidence. Reconstructed results for Ψ = 30° are shown in Figs. 6(b), 6(c). Boxes with different W are shown in the five columns, and it is clear for large W the results get severely distorted due to the strong artefacts. The SNR value reported was calculated for the feature shown with the arrown in Fig. 6(b) by comparing the gray value in the center of the feature to the gray value in the background. Figures 6(d), 6(e) show results for Ψ = 1.8°, being close to conditions of full CT. However, the transmitted intensity at this angle is small, so that the noise level in the projections is increased.

We can observe that the larger Ψ gives a better SNR than the smaller Ψ for both of the techniques. This is caused by the average SNR in projections being higher for larger Ψ. For the large Ψ the CT result has a smaller SNR, due to the stronger artefacts that reduce the contrast. For the smaller Ψ the artefacts are greatly reduced in both methods. Even though the artefacts are diminished, the SNR also decreases because of the decreased counting statistics in the 2D projections. In CL every projection has a similar SNR determined by the transmission at the chosen angle Ψ. In CT the SNR of the projections varies, being equal to the one for CL when the angle Ψ is reached, but generally being much higher for the projection angles that are closer to the normal incidence. The average SNR of the CT projections is thus higher than in CL, and this gives the reconstructed CT results better SNR as well.

4. Discussion

The choice of the optimum missing-information angle is determined either by mechanical factors (small Ψ not accessible due to sample geometry or mechanics of the imaging setup) or by consideration of the SNR. In the first case it is clear that the smallest possible Ψ should be used in order to minimize the artifacts. In the second case there is always a trade off between signal to noise ratio and the artefacts. Generally then the Ψ should be chosen as small as possible, so that the objects of interest have sufficient SNR to be still visible. However, it is not possible to always know in advance what is the suitable Ψ and this may make choosing the best Ψ difficult. For the CT geometry the problem is easier, as the choice of the limit angle Ψ can be deferred to the reconstruction stage, giving possibility to find a balance between the SNR and the artefacts.

CL brings benefits over CT mainly in situations where there is a fundamental limitation to go beyond angles smaller than certain threshold Ψ. These limitations could arise from the sample itself, for example some highly absorbing structure moving into the field of view after certain tilt angle, or some structures starting to overlap so that below a certain angle the absorption suddenly jumps up. Otherwise the limitations could arise from the measurement setup, for example a bulky sample environment [26], or the necessity to bring the sample close to the source for high resolution imaging. In this type of situations CL has clear benefits over CT in terms of lower amount of artefacts present.

Here we have considered only the case of absorption contrast of x-rays. For other contrast mechanisms CL has some additional advantages that have not been dealt with in this paper. For x-ray phase contrast imaging in local CT mode, the phase retrieval is not quantitative. Therefore projections may not get correctly weighed, whereas in the CL geometry each projection sees similar thickness of the sample, allowing projections to be normalized to the same average level. Furthermore, x-ray phase contrast imaging does not work well with large phase jumps (for example due to sample edges) in the image, and this can give strong artefacts from the angles where Ψ is too small.

One of the most promising imaging modes for CL is fluorescence laminography. In fluorescence CT some angles would result in situation where the outgoing photons have to travel through much of the sample to go to the detector. For low energy photons this would be detrimental in quite large range of sample orientations, limiting the achievable filling of the Fourier space. Additionally, at some angles a lot of the signal would come from regions outside the region of interest, because the incoming beam would traverse a long distance within the sample. For CL however, the geometry can be designed so that the fluorescence photons go to the detector along the sample surface normal, minimizing the path these photons have to travel inside the sample. Furthermore, the incoming beam traverses the sample equally in all projections, giving similar count rates in all projections. A study is in progress on the advantages of fluorescence laminography in 3D imaging.

5. Conclusions

We have shown that CL performs better than CT when there is a limiting angle Ψ beyond which the sample cannot be imaged. Specifically, the artefacts in the direction of the sample surface normal limit the achievable resolution more strongly in CT than in CL. Furthermore in the plane of the sample surface, the artefacts from CT are non-isotropic and create stronger features in the images than the isotropic artefacts from CL. For microelectronic and micromechanical devices, and other specimens containing long straight structures, the orientation of the sample was shown to be critical when applying CT. In this type of samples, structures in some of the directions could not be reconstructed well with CT, whereas CL reconstructed structures in every direction equally well. When limited only by counting statistics, the CT method has an advantage over CL of being able to choose between SNR and artefacts after the data has been recorded. The benefits of CL make it the method of choice for a number of cases where the missing-angle range for limited-angle CT is large. Utilizing the laminographic principle in other tomographic modalities besides x-rays, such as visible light or electron based methods [17,27], brings benefits in cases where the projection angles close to the sample surface are not accessible.