1 Introduction
Computed tomography (CT) is a non invasive method that is widely used in many applications to reveal inner structure of an unknown target using propagation of Xray through the target from multiple view or projections. To reconstruct the attenuation values of the target, standard methods such as filtered backprojection (FBP) or FeldmannDavidKress (FDK) have been well implemented buzug2008computed ; natterer1986mathematics ; natterer2001mathematical ; avinash1988principles . However, these methods are well understood only when the target is not moving (static) and dense Xray measured data from full angles are available.
In recent years, dynamic Xray tomography, in which the target is nonstatic or changing, has received particular attention in CT, and several ideas have been proposed lately, for example bonnet2003dynamic ; Roux2004 ; Katsevich2010 ; hahn2014reconstruction ; hahn2014efficient ; hahn2015dynamic ; bubba2017shearlet ; burger2017variational ; niemi2015dynamic . The goal here is to reconstruct the target in space and time from severely undersampled measurements.
We assume that the data measured at any single time instance is too scarce for reconstructing the target properly. Our motivation stems from applications where the radiation dose needs to be minimized, fast acquisition is required, big datasets need to be avoided, or the imaging geometry is restricted. Also, patient movement during a medical CT scan results in a dynamic dataset.
We are also interested in multisource arrangements where at each time step we have measurements only from a few angular directions. Such arrangements have received so far only little attention in the literature berninger1980multiple ; robb1983high ; stiel1993digital ; liu2001half ; perhaps so because filtered backprojection type methods are not wellsuited for the resulting sparse datasets. However, progress in the mathematics of inverse problems now allows novel reconstruction approaches.
In this paper we propose the dimension reduction Kalman filter for undersampled dynamic Xray tomography. We demonstrate the method with both simulated and real measurement data. In comparison to many other methods, ours is computationally very light, and can compute the solution online as the measurement takes place without the need of solving all the reconstructions at the same time. The dimension reduction Kalman filter is also based on direct computations, i.e., the solution is not obtained iteratively.
We also show how the dimension reduction can be applied in Rauch–Tung–Striebel (RTS) smoother that can be used for postprocessing the Kalman filter results. RTS smoothing improves the quality of the reconstructions in particular in the beginning of the simulation.
This new dimension reduction version of the Kalman filter was recently introduced by solonen2016 and has been motivated and tested for (chaotic) geophysical applications, where it performed well against the standard methods like extended Kalman filter and ensemble Kalman filter. In this paper, we show that it can be used also for dynamic Xray tomography where the application of the standard version of the Kalman filter would be too memoryconsuming.
2 Materials and methods
2.1 Dynamic Xray tomography
In Xray tomographic imaging, the aim is to collect information about an unknown attenuation function using available measurement data as a collection of line integrals through it. Mathematically, the Radon transform represents the measurement data buzug2008computed ; natterer2001mathematical ; SiltanenBook , so the problem can be modeled as
(1) 
In dynamic Xray tomography, the attenuation function is also a function of the time .
Using a discrete notation, the model at time step can be written as
(2) 
where
is the measurement vector called sinogram,
is the vectorized Xray image, is the measurement matrix and is the noise vector.The aim of the static Xray tomography, is to construct an image so that the distance between and is minimized in some statistical sense. Especially when the measurement data is scarce, additional a priori information needs to be added in the reconstruction process as there are many images giving close to minimal distance between and .
In the Bayesian framework, the inverse problem is recast as a stable statistical inference problem. Both the unknown image and the measurement are modelled as random vectors. Measurement information is included in the process via likelihood model, and all prior information we have available on the unknown is built into a prior model
. These two models, or probability distributions, are combined into a
posterior distribution using the Bayes formula. Finally one tries to find the maximum a posteriori estimate, i.e., an image that maximize the posterior distribution given the sinogram.In dynamic Xray tomography, the measurement function can be substantially different at every time step, because one often wants to work in the situation where only few angular directions are available at each time step , and the quality of the reconstruction is simply not good enough using the measurements from the same time step only. In dynamic systems one is not limited to finding solution , but rather tries to find an online (filtering) solution using all the available observations at time , or an offline (smoothing) solution where one first collect all the observations until the final time index .
Thus, in dynamic Xray tomography the idea is to reconstruct a set of changing Xray images in space and time. Let us first, however, look how we can solve this inverse problem in static case.
2.2 Priorbased dimension reduction
There are several ways to solve the static version of the linear inverse problem (2) (see, e.g., the textbook SiltanenBook for introduction). Here we discuss two ways that will later serve us in numerical computations.
In Tikhonov regularization one finds the minimum of the following functional
(3) 
where is the chosen Tikhonov matrix. The solution can obtained from the formula
(4) 
In Bayesian framework one assumes that we have a prior probability distribution for our quantities of interest. Once the measurements are taken, the prior distribution can be updated, and this new updated distribution is called the posterior. If we assume Gaussian prior
and observation error , then the posterior density can be written as(5) 
Now in this linear case also the posterior is Gaussian and the mean and its error covariance matrix can be obtained from
(6)  
(7) 
Remark. Even for low resolution Xray images of the size , the direct usage of the Bayes and Tikhonov matrix formulas is cumbersome as one would need to invert a matrix of the size . Next we show how this problem can be circumvented by using priorbased dimension reduction.
In priorbased dimension reduction (see, e.g., Marzouk20091862 ) the idea is to constrain the problem onto a subspace that contains most of the variability allowed by the prior. In recent paper solonen2016 , this concept was extended to Kalman filtering.
In dimension reduction, we start by parameterizing the state
(8) 
using the prior mean and the projection matrix that projects parameters from the low dimensional subspace onto the full state space.
The aim is now find the posterior distribution for the parameters :
(9) 
In priorbased dimension reduction, is constructed from the prior covariance matrix
using singular value decomposition (SVD). The covariance matrix
is factorized as, where unitary matrix
contains singular vectors as columns, and matrix has the singular values in the diagonal. Using the leading singular values, the projection matrix is defined as(10) 
Now it is easy to see that
(11) 
Thus this selection of the projection matrix whitens the prior, and posterior distribution (9) further simplifies to
(12) 
) the maximum a posteriori estimate for the parameter vector
and its error covariance matrix can be obtained from(13)  
(14) 
Although we now work within Bayesian framework, an interesting alternative is to use Tikhonov regularization with the same projection matrix and . Now the Tikhonov solution (4) can be expressed as
(15) 
We note that this Tikhonov solution only uses the prior covariance matrix to create the projection matrix, and the covariance matrix in itself is not used in the solution.
Remark. The key idea behind priorbased dimension reduction is that the prior covariance matrix is selected so that only small amount of singular vectors are needed to be considered. In context of Xray images of the size
, one now only needs to invert matrix of the size
. If now can be selected much smaller than , the direct evaluation of the above formulas is feasible.Remark. In order to use solutions (13–14) and (15) we do not actually need to compute the Radon matrix explicitly. Matrix representation of the measurement function
is typically created by applying some builtin Radon function to the columns of the identity matrix of the size
. In dimension reduction, we just need to apply this builtin function times (instead of times) to the columns of .2.3 Priorbased dimension reduction in Kalman filtering
So far at every time step we have only considered the observations made at the same time. In linear filtering, one considers the following pair
(16)  
(17) 
Thus in addition to the standard linear inverse problem (2), we have also the dynamic prediction model that moves the state from the time step to . Errors terms and are typically assumed zeromean and Gaussian: and .
In filtering one has two steps: prediction and update. In the prediction step the estimate of the mean and its covariance matrix of the previous time step is moved to the next time step’s prior. In Kalman filtering (e.g., SarkkaBook ), one calculates
(18)  
(19) 
Now using this prior in the update step, the posterior distribution can be written as
(20) 
and the posterior mean and covariance matrix can be obtained from the formulas similar to (6–7), often written in the so called “mform,” as in many Kalman filtering applications the number of observations is typically smaller than the number of states, see e.g. Clive .
In dynamic Xray tomography, regardless of which way we write the formulas (6–7), the direct application of the Kalman filter become unfeasible even for modest size Xray images. Next we follow solonen2016 and discuss how the dimension reduction can be used here.
Let us start by parametrizing our state as
(21) 
Now the prior mean and the covariance matrix in Kalman filter prediction step can be written as
(22)  
(23) 
and the posterior distribution for the parameters comes
(24) 
The posterior mean and covariance can be obtained from
(25)  
(26) 
In principle, the above formulas can be applied as is, however the direct computation of in (26) is not (often) feasible. In order to efficiently compute , we write
(27) 
where , and is the matrix square root . Now, applying the ShermanMorrisonWoodbury (SMW) matrix inversion lemma yields
(28) 
which can be evaluated as long as is easy to compute. In many Kalman filtering application this is the case, and often the model error covariance matrix is selected to be diagonal.
Remark. In many applications, the prediction model is a natural part of the system. In dynamic Xray tomography, however, this is not often the case. If no such model is not naturally available, one straightforward option is to model the transport from one image to another simply with identity . This option, of course, assumes that the underlying motion is small. Another option is based on optical flow methods that try to calculate the motion between two consecutive image frames. For example this option was considered in a recent study burger2017variational with variational methods. In filtering context, one could for example calculate the displacement field from images and , and then use this field to predict .
Remark. As in static case, where we do not need to calculate explicitly, we also can circumvent the evaluation of . Again we just need to apply the codelevel function of to the columns of ( times).
Remark. In this paper, we have presented the dimension reduction Kalman filter only for linear models. The method can also be applied to nonlinear filtering, see, e.g., the original paper solonen2016 for more complex cases.
2.3.1 Dimension reduction Rauch–Tung–Striebel smoother
Kalman filter solves dynamically the probability distribution given at time step all the observation . The Rauch–Tung–Striebel smoother RTS is a fixed interval smoother that solves the probability distribution given all the observation for every . The forward pass of the smoother is the same as in Kalman filter, and it can be seen as postprocessing tool for the Kalman filtering results.
In the backward pass, one starts from the final Kalman filtering estimate and operates via recursion. The smoothed state in RTS smoother can be obtained from
(29) 
and its error covariance matrix from
(30) 
Recursive solution (29) can be efficiently evaluated by using the SMW formula for like in Eq. (28), and writing .
Writing now also in Eq. (30) yields
(31)  
(32)  
(33) 
and further more
(34)  
(35)  
(36) 
This can be rearragend to
(37)  
(38)  
(39) 
where . This can again be evaluated efficiently using the SMW formula.
3 Numerical experiments
Here we discuss how the dimension reduction can be applied to 2D static and 3D dynamic Xray tomography. The purpose is to highlight different features in different methods. We start by applying the ideas to a static problem and then move to dynamic problems with real and simulated data.
3.1 Tomographic Xray data of middle slice of a walnut
In order to study how the effect of dimension reduction in static Xray tomography, we use the FIPS (Finnish Inverse Problems Society) openaccess data set of tomographic Xray data of a walnut 2015arXiv150204064H .
We start by creating a prior covariance matrix for image using standard Gaussian covariance function
(40) 
where
is the variance parameter,
is the correlation length and is the Euclidean distance between pixels and . For practical purposes we select and . We calculate the SVD for , and form using Eq. (10). Finally, we linearly interpolate each column of
to the target size .We start by using dimension reduction Tikhonov regularization. As a Tikhonvov matrix we set with . Now the formula (15) can be written as
(41) 
Figure 1 illustrates results using dimension reduction Tikhonov regularization with different number of basis vectors (, 1000, 500 or 100). We can see that lowering the number of basis vectors smoothens the reconstruction. However, the shape of the walnut is clearly recognizable even with , although the full image size is substantially larger .
Figure 2 illustrates reconstructions of a walnut using variational Tikhonov regularization 2015arXiv150204064H and dimension reduction Tikhonov regularization () with 120 and 20 projection angles. By visually comparing the results, we observe that nearly all the same features are observable, and the results are only slightly smoother with dimension reduction. Moving from 120 to 20 projection angles provides much more drastic change. We also illustrate Bayesian estimate using prior . In comparison to Tikhonov solution, we observe some additional artifacts, but the results are still very much acceptable.
3.2 Dynamic Xray tomography: stop motion 3D emoji data
For the real dynamic Xray dataset we use the 3D emoji dataset measured at the University of Helsinki, see meaney2018tomographic for details. The stop motion emoji data is available at 33 time steps with 60 angular directions. The target size is .
Again, we use standard Gaussian covariance function (40) and set and . Both observation and model error covariance matrices are set to . With this rather arbitrary setup, we get visually good results. One option for tuning these parameters, is to use marginal filter likelihood, see, e.g., npg219192014 for more discussion.
As our aim is to minimize the acquisition time, storing big datasets and preferably the radiation dose (as motivated by sampling in vivo samples), we perform filtering only with four (rotating) angular directions. This means that in the beginning, the Kalman filter reconstruction is expected to be rather poor, and gradually improve as more data becomes available. Flowing purisha2017accelerated , we also enforce nonnegativity of the Xray attenuation and set all negative pixels to zero after each time step. As a “ground truth reconstruction” we use static dimension reduction () Tikhonov solution from all 60 angular directions. Finally, we postprocess the Kalman filtering results using RTS smoother.
First, we select identity forward model . Figure 3 illustrates stop motion emoji results at four time steps , 5, 10 and 25. A GIF animation with all 33 time step is available at https://dl.dropboxusercontent.com/s/159388jcu2mpljm/EmojiAnimation.gif. As expected, in the first steps the Kalman filtering reconstruction is rather poor, but already at step five, the shape of the emoji starts to be visible. The RTS smoother solution is visually closer to the “ground truth reconstruction,” although, in the begging the right eye seems to be a little too open. One option could be rerunning the Kalman filter from the RTS smoother initial values. At time step 25, the visual quality of all the reconstructions seem be rather equal, although, Kalman filter seems to be lagging behind (e.g., left eye is little bit too close), because we only use 4 angular directions and identity forward model.
We test this setup also using large displacement optical flow 5551149 forward model, where we compute the displacement field between and , and then use this field to predict . The idea works technically well, but with just four angular directions the quality of the images are not good enough for calculating reliable flow, and also rotating angular directions are confused with physical motion. If we increase the number of angular directions to 10, the results with optical flow Kalman filter become more robust. As an example, Figure 4 shows a snapshot of this setup at time step 15. Optical flow Kalman filter is started from time step 10. We observe that the shapes of the eyes and the mouth (left eye, in particular) are now much closer to those of the ground truth reconstruction with optical flow Kalman filter than with the identity version or even with RTS smoother.
3.3 Dynamic Xray tomography: simulated chest phantom
Finally, in order to test the approach in simulated environment, we use Matlab’s 3D chest dataset (https://se.mathworks.com/help/images/segmentlungsfrom3dchestmridata.html). CT scan has been used to evaluate the blockages, injuries, tumors or other lesions, or other health problems gohagan2004baseline ; puybasset1998computed . One application is also to determine the pulmonary artery diameter (MPAD) tan1998utility , and our proposed method seems to be promising to provide better reconstruction.
In detail, we set our target resolution to , and in order to avoid inverse crime (see, e.g., SiltanenBook ), we simulate the sinograms with much higher resolution, i.e., , and add 1% noise. Parallel beam geometry is used in the computation. To create data we use 60 equally spaced angles out of 180.
For Kalman filtering covariance matrices etc., we use the very same setup as with emoji data, and start the simulations using identity forward model . An animation of the simulation with four angular directions is available at https://drive.google.com/file/d/1tloIXYErEXeH2WAM9Bywzir0GcB9ILZ6/view?usp=sharing. Figure 6 illustrates relative errors calculated against true phantom. We can see that after about 15 time steps dimension reduction () Kalman filter and RTS smoother yield about the same relative error with four angles as the dimension reduction Tikhonov solution with 60 angles. After time step 75 all relative errors become higher as the shape of the lungs become visible. The relative errors obtained with Kalman filter and RTS smoother with four angular directions are also higher than those with obtained with 60angle Tikhonov. RTS yields systematically lower relative error than Kalman filter.
Like with emoji data, also in this example, just four angular directions are not enough to provide robust results for optical flow Kalman filter. If we increase the number of angular directions to 10, the results with optical flow Kalman filter become robust. An animation with this setup, is available at https://drive.google.com/file/d/1Wt1BfsnmswoKu1jxYkIeAjneIBNKrwO2/view?usp=sharing, and Figure 5 shows a snapshot at time step 200. From the relative errors illustrated in Figure 6, we can note that the relative errors obtained with 10 angular directions are lower than those obtained with four angular directions, as expected. Dimension reduction Kalman filter with optical flow provides the lowest relative errors, and they very close to the the 60angle Tikhonov solution relative errors.
4 Concluding remarks
In recent years, several new methods have been proposed for dynamic Xray tomography. In this paper, we discussed how the priorbased dimension reduction methods, developed recently for Kalman filtering solonen2016 , can be applied for the task. The methods discussed here are computationally relatively light, and can be evaluated without variational computations. In Kalman filtering, the reconstructions can be calculated online without solving all the reconstructions at once. The same is true also for the Rauch–Tung–Striebel smoother, as the solution can be calculated recursively after the Kalman filtering has been performed.
In order to give some idea of computation times, we mention that in emoji example, one time step of Matlab implemented dimension reduction Kalman filter takes about two seconds and one RTS smoother back pass (without calculating the error covariance matrix) about 0.6 seconds (MacBook Pro, 2016, 2.7 GHz Intel Core i7, RAM 16 Gt). The optical flow, calculated using large displacement optical flow Matlab code (https://www.cs.cmu.edu/~katef/LDOF.html) adds about five seconds per time step.
The numerical results, presented in this paper are promising for future applications. The dimension reduction Kalman filter with identity forward model, seem to be extremely robust, and provide acceptable results even with just four angular directions. The setup for the filter was also rather arbitrary, and no careful tuning for the filter parameters was needed. The filtering results can be further smoothed with dimension reduction version of the RTS smoother and it improved the quality of the reconstructions especially in the beginning of the simulation. Slightly more complicated version of the dimension reduction Kalman filter can be obtained using optical flow methods. However, the quality of the reconstructions has to be good enough in order for it to work properly. For example, we could not get this version of the dimension reduction Kalman filter to work properly with just four angular directions. When the number of angular directions was increased the method became robust, and performed better (no lagging, lower relative error) than the Kalman filter and the RTS smoother with identity forward model.
Future work is to implement the strategy in in vivo samples or cardiac imaging and to implement in limited angles or random angles tomography problems. Reconstruct a human chest or lung from real measured data using the proposed method would be worth to try as well. Finally we note that the dimension reduction filter and smoother performed just as expected, and the main limitation factor was the number of data points at each time step. The dimension reduction in itself did not seem to harm the solution too much. This was even further tested with the static walnut example where dimension reduction method with just 3000 basis vectors provided visually nearly the same quality as the full inversion with the number of unknowns of . This feature is of course very much case dependent.
Acknowledgements
Authors acknowledge support from TEKES/Business Finland “Heartbeat” project 6614/31/2016, the Academy of Finland through the Finnish Centre of Excellence in Inverse Problems Research 20122017 (decision number 284715), the Finnish Centre of Excellence in Inverse Modelling and Imaging 2018–2025, decision numbers 312119, 312125 and 312122. Janne Hakkarainen thanks Marko Laine for fruitful discussions on practical implementation of dimension reduction RTS smoother.
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