New reconstruction algorithm, distance dependent exact, with reduced statistical α error, dedicated to emission tomography using a collimator
with large and long holes
Christian Jeanguillaume, Richard Simonnet, Herv´e Rakotonirina and Jean Jacques Loeb
Abstract— SPECT (single photon emission computerized to- mography) is physically one of the worst medical imaging modalities. Despite a considerable and still increasing medical impact, its spatial resolution (beyond 1 cm) and its sensitivity (less than 10−4) are both awful. This situation is mainly due to the use of a thin parallel hole collimator. In addition the application of the unfitted radon-transform worsens the figure.
We already suggested a different approach of collimation called CACAO in French standing for computer aided colli- mation gamma camera. This approach uses a large and long hole collimator, a different sequence of acquisition with a linear scanning motion and a dedicated reconstruction program taking full account of the depth dependent response function.
Up to now, however, the CACAO project has failed to convince the scientific community of its superiority over the conventional thin hole collimator. This is due of a lack of a good reconstruction algorithm. In this paper we depict a new tomographic reconstruction algorithm for the CACAO problem.
In addition to the former cited advantages, this new algorithm is exact, it takes full account of the finite geometry of the collimator holes, it reduces the type 1 statistical error (false positive) and reduces the hindering effect of points with large errors (outliers). Example of reconstruction with exact data and with a limited number of detected photons are provided.
Comparison with MLEM algorithm is provided.
I. INTRODUCTION
A complete description of the CACAO project can be found in [1]. It is to be noted that attempts to enlarge the hole of the collimator, in radionuclide imaging have been very limited. The work of Lodge [2] must be noted.
The bottleneck of the thin hole collimator (THC) must not be underestimated: it is a no-go situation : increasing the diameter of the holes leads to a loss of spatial resolution, diminishing the diameter of the holes leads to a further loss of sensitivity. The recent progress on semiconductor detectors does not solve the problem: a 100µm resolution would lead to an unbearable 10−6 ratio of sensitivity. Fur- thermore the depth dependence of the spatial resolution with the collimator to source distance is not negligible.
However most tomographic reconstruction algorithms treat that effect as an approximation. For all these reasons we decided to get rid of the thin hole collimator principle. We
C. Jeanguillaume, R Simmonet are with LISA : Laboratoire d’ing´enierie des syst`emes automatis´es. Universit´e d’Angers 62 Avenue Notre Dame du Lac 49000 Angers France
C. Jeanguillaume, H Rakotonirina are with Service de M´edecine nucl´eaire CHU d’Angers rue Larrey 49933 Angers France
J.J. Loeb is with LAREMA Laboratoire Angevin de REcherche en MAth´ematiques 2 Bd Lavoisier Facult´e des Sciences d’Angers. 49045 Angers France
first demonstrated that the accuracy of the reconstructed image obtained by CACAO, will be potentially better than those obtained using the thin hole collimator technic. This superiority was demonstrated in the theoretical side, in a manner independent of the reconstruction algorithm and a good reconstruction algorithm was still to be found.
II. PRINCIPLE, DIRECT PROBLEM
1) Analytical description: . The CACAO project will be presented here in a 2D reduction. To obtain a sufficient amount of data we choose to add a linear scanning motion in the acquisition sequence. The standard orbital tomographic movement around the patient is conserved. Fig 1 depicts the variables used. A transverse section of the patient is represented by the density of radio-active sources %(x, y).
The collimator has been simplified to a unique hole, for the sake of clarity. This hole can rotate around the patient and can also perform a linear motion along the u axis. The coordinates system x,y is considered at rest while the rotating axes u,w follows the rotation of the detector, measured by φ.
Fig. 1. Scheme of the notations used to describe the direct problem
Photons impinging the detector are localized by the po- sition in the hole ν, the position of the hole in the u axis, measured by χ and the angle of rotation φ. Therefore the description of the direct problem aims to define the acqui- sition functiong(χ, ν, φ). Elementary geometry considering only photons traveling in the air (perfect attenuation model in the lead collimator, no attenuation in the patient), leads to an integral equation of the direct problem. This equation gives g(χ, ν, φ) in terms of %(x, y) or rather %(u, w, φ).
It is obtained by considering the integration in a polygon
bounded by the 2 extreme rays touching the 2 corners of the collimator entrance hole, considering the finite extent of the patient (≤τ) and the entrance face of the collimator as the nearest distance of observation (≥p).
g(χ, ν, φ) = Z τ
p
Z χ+ν−(wP)(ν−D2)
χ+ν−(wP)(ν+D2) cosθ
d2
ρ(ucosφ+wsinφ,−usinφ+wcosφ)dudw (1) Where
-g(χ, ν, φ) represents the intensity of the detected photons with χ measuring the position of the camera head, ν the co-ordinate on the detector surface.
-%(u, w, φ) represents the distribution of the radioactive source in the rotating(u, w)co-ordinates.
-φmeasures the inclination of the camera head in the(x, y) co-ordinates.
-θrepresents the angle between the gamma ray direction and the perpendicular of the surface of the detector (directionw) -dis the source to detector distance.
-cosθ/d2 represents the law of illumination.
-pis the depth of the collimator,
-p, τ are the limits of integration in the directionw -τ represents the farthest limit of integration, it is given by the diameter of the camera orbit.
-D is the width of the collimator hole.
Fig. 2. Example of the acquisition datab, obtained from a point source D=5, p=8, 4 different values ofφ. Image representation in the planeχ,(ν, φ).
2) Discrete description: Due to a finite spatial resolution of the detector, we can discretize this problem. The patient or the object to visualize is split into pixels, and all the pixels are numbered to produce a 1D vector z. Usually the pixels are numbered from left to right (x) and top to bottom (y). In the same way the variables χ, ν, φ are discretized to form the vector b Fig 2. Note that ν is limited by the diameter of the collimator hole (D) andχmust be larger than the extent of the patient, both values depend on the spatial resolution of the detector. The different values taken by φ can be very limited, in this article we will only consider 4 different values (00,900,1800,2700). This discretization of the problem leads to a matrix A linking the vector of the acquisition data b to the vector of the original object z in accordance to the linearity of the integral (1). This matrix linking the acquisition vectorbto the object vectorz is our direct problem.
b=A.z (2)
The inverse problem is : findzknowingAand a measure bin a reasonably robust way. It is well known that constraints must be added to regularized the problem. Most often this kind of problem is treated by a least square minimization to fit the data and another constraint like a condition of smoothing on the solution. Very often, conventional methods of regularization lead to oscillation of the solution. This oscillation may let us to find radio-active source in a place where it is not.
III. ALGORITHMS
All these algorithms have been written in the M athematicaR environment.
A. Algorithm MISO ; Diminution of type I error: minimal solution
The first and the following algorithms are dedicated to reducing the type I (orα )error (false positive), by looking at a minimal solution.
M inimize δ=b−A.zmin
Subject to δ≥0 And zmin≥0 (3) It is obvious that if the data are equal to zero the product A.zmin will be also equal to zero, leading to a reduction of the type I error. Such problems can be efficiently solved by the class of algorithm called linear programming. [3], [4]
B. Algorithm DIMIBES ; Distance Minimization Between Extreme Solutions
As the minimal solution always underestimates the real values, we tried to extend the preceding algorithm with a maximal solutionzmax. In order to work only in the set of the solutions of the problem, we constructed this solution by adding a constant to the previously foundzmin solution
zmax=zmin+κ.ψ (4) whereψis a vector constant andκis a variable placed here to withstand the minimization of the distance between the 2 extreme solutions. To stay in the set of possible solutions for the problem, the vector ψ is calculated by a linear combination of the column of the transfer matrix (A). To calculate the 2 extreme solutions that encased the data we solve the following optimization problem :
M inimize δ=b−A.zmin
and η=A.(zmin+κ.ψ)−b
Subject to δ≥0, η≥0, zmin≥0, κ≥0 (5) C. Algorithm IECP ; Iterative Elimination of Contact Points A complete description of this algorithm can be found in [5]. The equation 5 gives 2 extreme solutions which encase the data and touch the data by at least 2 points. We will call these points “contact points” in the following. It is obvious that we can find a better solution by going past these contact points. It will be not demonstrated in this short article, but these contact points are often the points with the
largest deviation from the expected solution (outliers). We define 2 sets of points called respectively I and S for the point which has either touched the minimal or the maximal extreme solution defined in (5). A third set M is composed of the set of point which has not yet touched an extreme solution. We expect the points belonging to I and S to be respectively inferior or superior to the expected solution for the problem. The iterative algorithm is defined by applying the following minimization iteratively.
M inimize δ=bSM−ASM.zmin
and η=AM I.(zmin+κ.ψ)−bM I
Subject to δ≥0, η≥0, zmin≥0, κ≥0 (6) Where
bSM ={b\(b∈S∪M)}
bM I={b\(b∈M ∪I)} (7) And respectively ASM and AM I are the reduction of the matrix A to the rows corresponding to the 2 set (S∪M) and (M ∪I). At each step, at least 2 contact points are removed from the set M to grow the sets I and S. The algorithm is stopped when there are no more points in M.
This corresponds to a low value ofκand an equality between zmin andzmax.
D. Algorithm MLEM-E ; Maximum Likelihood Expectation Maximization for Emission :
As a comparison we used 200 iterations of the MLEM-E update described in [6], [7]. As our matrix focuses on the only the 4 angles, the difference with the OSEM algorithm [8] must be very thin, if at all.
IV. RESULTS A. A. Exact simulation
The simple image test of Fig.3 has been calculated. The matrix is 8x8. It represents a sparse object. Sparse objects are not uncommon on radionuclide imaging (ex: bone scan) and on the control of workers in the nuclear field.
A CACAO acquisition without noise has been calculated.
All values are given in pixel width. The parameters of the collimator hole are D=5, p=8, 4 angles of acquisition with a radius of acquisition of 15 (distance from the center of the figure to the detector plane). Thirty different values ofχare necessary, so the dimensions of the matrix A are 8x8=64 columns and 30x5x4=600 rows.
The algorithm MLEM-E has been used to calculate a reconstruction image. This image is given in figure 4. A lot of noise is observed due to the limited number of iteration (200) and the bad conditioning of the matrix. The SNR calculation gives2.2. A better value of 3.7 can be obtained after 500 iterations.
The algorithm MISO (minimal solution) has been used to calculate a reconstruction image. This image is given in figure 5. No difference from the original image can be visually found. The SNR calculation gives5.9 106
Fig. 3. Image test
Fig. 4. CACAO reconstruction perform by MLEM-E algorithm object : Image test of Fig.3
B. B. Noisy Data
Poisson noise was simulated on the acquisition to test the robustness of the algorithms. A total of 105 detected pho- tons were simulated. As previously described, the MLEM-E algorithm has been applied to the noisy data (Fig 6). The SNR calculation of the reconstructed image is2.1
The algorithm IECP (iterative elimination of contact points) has been applied to the noisy data. Sixty-five steps were sufficient to dispatch the 600 values of b into the set I and S. The residual value ofκwas 9.4 10−9. The resulting image is depicted in Fig. 7. The SNR calculation of the reconstructed image is4.7.
V. DISCUSSION
It is well known that stopping the iterations of the MLEM- E type algorithm [6]–[8] leads to a smoother image. This is an effective means of regulation of inverse problems in tomographic reconstruction. In contrasted sparse images, this can be a disadvantage, worsened by ill-conditioned matrices.
Our algorithm with a fixed stopping criteria, seems to present
Fig. 5. CACAO reconstruction of the image test of Fig.3 by the minimal solution (Algorithm MISO)
Fig. 6. CACAO reconstruction performed by MLEM-E algorithm; object : Image test of Fig.3 Poisson Noise : 100 000 photons SNR=2.1
an advantage in these conditions. The convex support of the object is in addition, well recovered by our algorithm. This is not the case for the MLEM-E algorithm. There remain some pixels, zero in the Image test exhibiting faint values in the reconstructed image of Fig 7. They are few (7) and we hope to remove them by either increasing the number of acquisition angles or using a multi-hole collimator [9].
It is to be noted that due to the rather large collimator hole used (D=5 pixels), these reconstructed images should be compared, with simulation built with less than 20,000 photons in the thin hole collimator model.
VI. CONCLUSIONS AND FUTURE WORKS These limited results have been obtained, for a very small (8x8) matrix image, and for a sparse object. Further works need to be done to confirm these results in more realistic
Fig. 7. CACAO reconstruction performed by algorithm IECP object : Image test of Fig.3 Poisson Noise : 100 000 photons SNR=4.7
conditions. A comparison with conventional thin hole con- figuration, and the optimization of the parameters of the collimator (D,p) need also to be performed. These incomplete results however, lead us to think that the association of CACAO with the semi-conductor detector, will soon greatly improve the images of nuclear medicine. [10].
VII. ACKNOWLEDGMENTS
The authors gratefully thank the Universit´e d’Angers.
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