This paper proposes a new level-set-based shape recovery approach that can be applied to a wide range of binary tomography reconstructions. In this technique, we derive generic evolution equations for shape reconstruction in terms of the underlying level-set parameters. We show that using the appropriate basis function to parameterize the level-set function results in an optimization problem with a small number of parameters, which overcomes many of the problems associated with the traditional level-set approach. More concretely, in this paper, we use Gaussian functions as a basis function placed at sparse grid points to represent the parametric level-set function and provide more flexibility in the binary representation of the reconstructed image. In addition, we suggest a convex optimization method that can overcome the problem of the local minimum of the cost function by successfully recovering the coefficients of the basis function. Finally, we illustrate the performance of the proposed method using synthetic images and real X-ray CT projection data. We show that the proposed reconstruction method compares favorably to various state-of-the-art reconstruction techniques for limited-data tomography, and it is also relatively stable in the presence of modest amounts of noise. Furthermore, the shape representation using a compact Gaussian radial basis function works well.

Computed tomography (CT) is a non-invasive imaging technique that reconstructs images of non-accessible or non-visible objects from a set of projection data [

Unfortunately, reducing the number of projection data results in artifacts in the reconstructed image. There exist two classes of reconstruction algorithms in CT. The first class of reconstruction algorithms, known as analytical reconstruction methods, such as Filtered Back Projection (FBP) [

Recently, reconstruction algorithms using sparsity of magnitude of intensity gradient as prior knowledge have been used with the name of Total-Variation (TV) to handle sparse-view and limited-angle CT reconstruction [

To reduce computing time or to increase reconstruction quality for a given computation time, we consider a parametric level-set function that may be defined using a collection of basic functions. This parametric level-set function is likely to behave well since there is no need for re-initialization, unlike the classical level-set method, and we can easily derive the corresponding evolution equation based on the underlying parameters. According to Kilmer et al. [

In this paper, we propose a novel representation of the parametric level-set (PALS) for shape-based inverse problems. There are some similarities to the previous methods in our approach, such as the use of a basis function to represent the parametric level-set function [

Besides the above advantages, the proposed technique is based totally on convex optimization, so it has a consistent performance with a small number of projection data by solving the issue of the cost function’s local minimum. Moreover, the number of parameters in the level-set function is often far smaller than the number of image pixels. Finally, the results of numerical experiments with synthetic images and real X-ray CT projection data show that the proposed method compares favorably to other state-of-the-art reconstruction techniques, and that it is also relatively stable in the presence of modest amounts of noise.

This paper is organized as follows. First, we describe how to formulate the shape-based reconstruction problems in tomography dealt with in this paper. Second, we present the mathematical basis of the parametric level-set technique, and various practical issues are discussed. Then, we give the experimental results of our method for some examples and real data, comparing it to other possible reconstruction methods. Finally, we conclude the paper.

In this study, we consider the discrete tomography reconstruction problem, which may be expressed mathematically as follows. To begin, any image is defined on

The image reconstruction problem is an inverse problem in which the image

In many cases of CT imaging,

For example, to solve this optimization problem, the following Newton-like algorithm can be used to find the image

In the areas of geometric inverse problems, such as shape optimization, image segmentation, and interface tracking, level-set methods have received a lot of attention due to their ability to adapt to topological changes in the region structures. In [

In a large class of shape-based inverse problems [

By using

In many existing shape-based approaches, the level-set function

A PALS function is an example of a basis expansion using known basis functions and unknown weights. As a result of this statement, the level-set function

In this section, we describe the method for reconstructing the image using the parametric level-set and how it evolves when parameters

Next, we discuss how to select the Gaussian width parameter

To calculate the gradient of

In

In our implementation, we solve the minimization problem of

The final reconstruction algorithm is summarized in Algorithm below.

From

However,

A final remark on solving the under-determined inverse problems is as follows. Our original cost function to solve the under-determined CT reconstruction is the standard least-squares error of

In this section, we describe experimental results for image reconstruction in sparse-view CT and limited-angle CT. The experiments were performed by using synthetic images and real X-ray CT projection data of a carved cheese slice [

For the experiments with synthetic images, we used four images shown in

There are several reconstruction methods for tomography. We are focusing on the following four methods, which have been developed for binary tomography aiming at reconstructing nice images with limited projection data.

In

In addition to the synthetic images, we applied the proposed method to real X-ray CT projection data of a carved cheese slice [

In the case of limited-angle CT, the problem becomes much more difficult, but the proposed method still works well, and the reconstructed images were very close to the true images. For example,

In

For the real data, in Figs.

We evaluated the performance of the proposed method in the presence of three different levels of noise in the projection data. We added Gaussian noise with ratio 0.1%, 5%, and 10% to the projection data, from which image reconstruction was performed. In

In this paper, we proposed a novel approach using a parametric level-set method for image reconstruction in binary tomography. The basic formulation of the problem is summarized as follows. Shapes of objects are represented via a level-set function, which is represented by using a Gaussian radial basis function (GRBF). We are using an appropriate parametrization of the level-set function, which can reduce the problem dimension and essentially regularizes the problem in a successful way. Based on this modeling, the level-set function is more likely to behave well, so it doesn’t need re-initialization like the traditional methods. Furthermore, unlike the use of traditional level-set methods, the problem to be solved for image reconstruction becomes a tractable convex optimization. We solved the optimization problem using a Newton-type method because the number of underlying parameters in a parametric level-set approach is usually much smaller than the number of pixels that we finally would like to get the image by discretizing the level-set function. We tested the proposed method on synthetic images and real data. According to our experiments, the proposed approach compares favorably with some state-of-the-art reconstruction methods, and it is also relatively stable in the presence of modest amount of noise.

Real projection data used in this study is available from the members of the X-ray lab at the Faculty of Science, University of Helsinki, Finland upon request