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Multivalued discrete tomography involves reconstructing images composed of three or more gray levels from projections. We propose a method based on the continuous-time optimization approach with a nonlinear dynamical system that effectively utilizes competition dynamics to solve the problem of multivalued discrete tomography. We perform theoretical analysis to understand how the system obtains the desired multivalued reconstructed image. Numerical experiments illustrate that the proposed method also works well when the number of pixels is comparatively high even if the exact labels are unknown.

Multivalued (or nonbinary) discrete tomography involves the reconstruction of images composed of three or more gray levels from projections. Compared with computed tomography, it is possible to reduce the number of projections by using prior knowledge about a set of gray levels. This is important for medical use as it is the basis for identifying characteristic regions in tomographic images [

We propose two differential equations to represent an autonomous system and a nonautonomous system that have similar structures. For the autonomous system, it has been proven theoretically that the stable equilibria corresponding to the ideal solution and the undesired solution coexist and that a saddle-type equilibrium exists that plays an important role in the behavior of the solutions. We investigate the mechanism behind this behavior through a numerical example. The results of numerical experiments show that the proposed method works well even if the number of pixels is comparatively high. Further numerical experiments show that the nonautonomous system can be applied in cases in which the exact label set is not given.

Let

With the above preliminaries, the discrete tomography described in this paper involves solving the following equation for unknown vector

If the problem is consistent, a true solution of (

To solve (

We can rewrite the dynamical system in (

In (

In this paper, we treat multivalued discrete tomography as an extension of the binary tomography problems addressed in [

We rewrite (

If we choose initial value

As the system can be written as

The Jacobian or the derivative of

Each of the all-zeros and all-ones equilibria of (

From (

Let us define the set

If there exists an equilibrium in

When the equilibrium

Numerous saddle-type equilibria exist in the system, and these play an important role in separating trajectories that converge to true and false stable equilibria. We consider the two equilibrium sets

Some elements of

If

The local stability of the equilibrium

Our proposed system in (

In the system given by (

We begin with the simplest possible example, that of a

Now, we suppose a true solution as

The trajectory

We solved (

Time response of

Let us discuss the initial value dependency of (

Equilibria and attractors projected on

From Proposition

We prepared a

(a) Phantom image, (b) pixel value of each segment, (c) sinogram, and (d) reconstructed image obtained by FBP.

From the above setup, we obtained the segmented images shown in Figures

Reconstructed image. (a) Image of

Figure

Density profiles. Gray values of (a) phantom image and (b) images with proposed method and FBP. Green, red, and blue lines indicate gray values of images shown in Figures

In the previous experiments, we assumed that the exact label set was given. The proposed nonautonomous system in (

Let us provide a phantom that contains four different pixel values: 0.5, 0.6, 0.9, and 1; see Figure

(a) Phantom image, (b) pixel value of each segment, (c) sinogram, (d) reconstructed image obtained by FBP, and (e) objective composited image.

The results are shown in Figures

Reconstructed image. (a) Image of

We proposed a novel method for solving the problem of multivalued discrete tomography. Our method is based on the initial value problem of a nonlinear differential equation, which is inspired by the Lotka-Volterra competitive activity that enforces exclusivity among the state variables from which pixel values are constructed.

We proved the stability of all equilibria when the tomographic inverse problem was well posed. The equilibrium corresponding to the desired reconstructed image was stable; however, other false stable equilibria corresponding to undesired images coexisted. Therefore, a solution orbit that converged to a true or false equilibrium was determined by the initial value.

From the numerical experiments, we observed that a solution starting from the same uniform initial value converged to the true equilibrium, regardless of the pattern or the size of an image. Moreover, we proposed a modified system that is aimed at realizing self-adjusting labeling by adding a nonautonomous term. We confirmed that the nonautonomous system automatically classifies pixels that are not listed in the label set distinguished to the nearest label.

The authors declare that they have no conflicts of interest.