Region-of-Interest, Phase Contrast & Limited Data
18 to 20 May 2015 at ESRF, Grenoble - France

MOTIVATION:
Over the past decade X-ray tomography has made significant progress concerning the spatial, temporal and density resolution that can be obtained with state of the art instrumentation. Nevertheless, the community is facing a number of recurring challenges, related to

1. “Large” objects exceeding the detector field of view (region-of-interest tomography)
2. Combined phase retrieval and tomographic reconstruction
3. Tomography from highly limited data: low-dose acquisition, continuous rotation, flat objects

At the same time, the community dealing with the mathematical and computational aspects of image reconstruction has evolved in its own right, developing both theory and algorithms applicable to the above-mentioned challenges.

During the workshop people from this community will be exposed to the acquisition of experimental data and are invited to suggest solutions to the above challenges on real data.

Key Challenge 1

1. Region-of-Interest tomography: how to make iterative/prior-based methods work in this case.

Problem Description: Region of interest (ROI) tomography is a very common scenario in synchrotron tomography. We distinguish two fundamentally different cases:

(i) Full projection data of the object are available, but one only wants to reconstruct a particular ROI, in order to save computation time.

(ii) Projections of the object under study are truncated. In the worst case, the boundary of the object is not within the Field-of-View (FOV) for any of the projections. The goal is to accurately reconstruct the ROI, which is defined as the region that is covered by all the projections.

In this challenge, we focus on the second type: ROI tomography from truncated projections. Currently, Filtered Backprojection is used almost exclusively for such reconstructions. The projection data are padded with the value reached at the boundary, or sometimes with a smoothly declining function, after which FBP is applied. In some special cases, where the boundaries of the object are visible in some of the projections, special inversion formulas exist based on the Hilbert transform that can yield exact reconstruction. For the case where all projections are truncated, it is well known in inverse problems theory that exact reconstruction is not possible, yet FBP is applied extensively, and quite successfully, in practice, even though certain artefacts can be observed in this case.

Reconstruction methods that incorporate prior knowledge, such as certain sparsity assumptions, can potentially alleviate the non-uniqueness problem for ROI reconstruction, also in type (ii). However, most available reconstruction algorithms for such priors are iterative, and therefore require the ability to compute a forward projection (FP) of the object volume. For ROI tomography this poses a major obstacle, as the full volume data is not readily available during reconstruction.

Several basic strategies can be explored to deal with this problem:

1. ignore the exterior region; simply pretend that the full object is contained within the FOV and has no external contributions

2. represent the full exterior region at high resolution; this can potentially make the reconstruction problem more consistent, at the expense of a vast increase in memory and computation time.

3. represent the full exterior region at coarse resolution;

4. use a model-based approach to determine the contributions of each of the projections to the interior region vs. exterior region; perhaps some coarse shape model of the object can work for this;

As part of this challenge, datasets will be made available for which the full ground truth reconstruction (from many projections, without truncation) is available and where the ROI is made increasingly smaller. The exterior region should contain a mixture of densities (perhaps some very high densities?), which may lead to artefact in the interior region. The goal is to develop an approach where iterative methods (standard ART, SIRT, and also sparsity promoting methods such as FISTA) can be applied to the interior region, yielding (i) a quality as similar as possible to application of the same algorithm for the full volume to the ground-truth dataset, observing the final contents of the ROI; (ii) a computation time that is as similar as possible as applying the basic iterative method just on the size of the ROI.

Corresponding datasets:

Link to Problem descriptions

and to Data Repository

back to top of the page

Key Challenge 2

2. Phase contrast tomography: incorporating phase retrieval in the tomography reconstruction.

    Problem Description: When performing in-line (propagation-based) phase-contrast X-ray tomographic (PCT) measurements, images are obtained with both absorption and refraction contribution. This allows to enhance the contrast by resolving 3D variations of the (electron) density in the sample. Quantitative phase tomography aims at the 3D reconstruction of the complex refractive index. The recorded phase-contrast radiographs are determined by the projection of the real and imaginary part of the refractive index, but the relationship is in general non-linear and not straightforwardly invertible. The inversion problem is furthermore ill-posed for the low spatial frequencies and for well determined frequencies. For a given tomography angle, radiographs can be recorded at a single or multiple distances. If a single distance is used, phase retrieval is most often based on the homogeneity assumption proposed by Paganin et al. In the case of multiple distances (holotomography), the problem is regularly linearised by using contrast transfer functions.

The common reconstruction approach foresees a two-steps procedure:

1) A phase retrieval procedure is firstly applied to all the acquired radiographs, each angle being processed individually. The parameters of the chosen phase retrieval algorithm are optimized on the projection data.

2) The 3D tomographic reconstruction is then performed using as input the phase-retrieved projections.

As an alternative, the use of iterative methods could merge these two steps allowing to simultaneously perform the phase retrieval operation and the Radon inversion. One possible approach is based on regularized Newton methods allowing to incorporate the phase retrieval step in the tomographic reconstruction.

The choice of the combined approach has both advantages and drawbacks with respect to the more common procedure. The combined method allows to apply prior knowledge such as discreteness or positivity in the object domain rather than the projection domain. This may allow to alleviate the low spatial frequency problem. Furthermore, the effect of parameters of the process can be evaluated better on the 3D reconstruction. On the other hand, the combined approach is heavier from a computational point of view and less flexible than the 2-steps procedure. For example, different tomographic reconstruction algorithms can be tested on the same set of phase-retrieved projections, without having to re-calculate the phase retrieval in between. Moreover, not all phase-retrieval methods are suitable for the combined approach.

More details can be found at the following links:

http://www.fis.unical.it/files/fl161/2267PhaseretrievalinXrayphasecontrast.pdf

http://arxiv.org/pdf/1502.05073v1.pdf                                                

Link to : Problem description ESRF01  and Data Repository
and to : ¨Problem description SLS01 and Data Repository

back to top of the page

Key Challenge 3

3. Tomography from highly limited data: low-dose (noisy projection data, living organisms), continuous rotation, flat objects (limited angle)

Problem Description: When performing a tomographic measurement the instrument parameters such as exposure time, angular scan range need to be tuned according to the experiment and desired image quality. For dynamic CT, the common interest is to minimize sample motion, reduce the dose to the sample or environment induced sample changes during data collection. For example for in vivo low-dose imaging especially of small animals fast scans are required to overcome motion artifacts. Hence, the amount of projections available is still significant (i.e. several thousand) but each of them with a rather poor signal-to-noise ratio due to very short exposure times. Decision on how to set these parameters is often based on the beamline scientist instrument and sample experience and include the following:

Exposure time: when collecting tomographic data of fast evolving samples the data collection is set the highest speed possible that is compatible with sample movement and environment condition stability and/or the dose to the sample. In these cases, to prevent sample motion (or to reduce them below the pixel resolution during the full data collection time) and/or degradation, the exposure time is often set to be very short generating very noisy data (the noise problem is furthermore enhance by the fact that fast detectors itself operate with a higher noise level than for example a slow-scan CCD).

Angular scan range: In some situations, complete projections are not available (projections are missing at certain orientations) due to physical limitations in the data acquisition process (environment cell surrounding the sample, flat objects, presence of metal artifacts requiring the exclusion of some portions of the projection data). The projections could also not be distributed uniformly in angle. The environment cell adds an overall background (larger than the field of view), in many cases white fields includes portion of the environment cells of different thickness and shape, in other situation (load cell) many views are blocked (40 deg or more).

Angular integration: Projections acquired with fast, continuous gantry rotation may suffer from radial blurring, depending on the rotation speed and the exposure time of each projection.

Fly scan vs stop & go: once the exposure time and pixel resolution are set a decision is made about collecting data in fly scan (setting an appropriate rotary stage speed and let the camera take images at specific angular position) or to move the rotary stage in angular increments and collect images while the sample is still.

Number of projections:

When collecting a tomographic data set, usually an extensive set of sample projections is acquired over a total angular range of 180° (parallel X-ray beam geometry) or 360° (cone beam geometry or extended field of view). The total number of projection is typically selected as ~ /2 x detector width (Nyquist) in case of FBP is used for tomographic reconstruction. A widely used method to reduce the dose delivered to the sample or the acquisition time for fast in situ experiments consists in reducing the number of projections by widening the projection angle interval (sparse projection data) or by collecting a large amount of noisy projections (see above: Exposure time).

Link to Problem Description APS02 and to Data Repository

and to Problem description APS05 and to Data Repository

and to Problem Description ESRF02 and to Data Repository

back to top of the page