A Mumford–Shah level-set approach for the inversion and segmentation of X-ray tomography data

Abstract

A level-set based approach for the determination of a piecewise constant density function from data of its Radon transform is presented. Simultaneously, a segmentation of the reconstructed density is obtained. The segmenting contour and the corresponding density are found as minimizers of a Mumford–Shah like functional over the set of admissible contours and – for a fixed contour – over the space of piecewise constant densities which may be discontinuous across the contour. Shape sensitivity analysis is used to find a descent direction for the cost functional which leads to an update formula for the contour in the level-set framework. The descent direction can be chosen with respect to different metrics. The use of an L²-type and an H¹-type metric is proposed and the corresponding steepest descent flow equations are derived. A heuristic approach for the insertion of additional components of the density is presented. The method is tested for several data sets including synthetic as well as real-world data. It is shown that the method works especially well for large data noise (∼10% noise). The choice of the H¹-metric for the determination of the descent direction is found to have positive effect on the number of level-set steps necessary for finding the optimal contours and densities.

Introduction

In medical imaging, computerized tomography (CT) is a widely used technique for the determination of the mass density f of a sample from measurements of the attenuation of X-ray beams sent through the material along different angles and offsets. The measured data g_d are connected to the density f via the Radon Transform,

$$g_{\text{d}}(s\text{,}\omega )=\mathit{Rf}≔{\int }_{\mathbb{R}}f(s\omega +t{\omega }^{\perp })\mathrm{d}t\text{,}$$

$(s\text{,}\omega )\in \mathbb{R}\times S^1$. To compute the density distribution f the equation g_d = Rf has to be inverted. It is a well-known fact that the Radon transform is not continuously invertible e.g. as a mapping from L² into L². For this reason, regularization methods have to be used in the presence of data noise. Probably the most widely used algorithm for the inversion of tomography data is the filtered back projection method [43], [38]. In principle, this algorithm combines Fourier data smoothing techniques with the application of the inverse operator R⁻¹ restricted to a finite dimensional subspace of the data space. Other methods include the Algebraic Reconstruction Technique (ART) [20] and classical regularization methods as truncated singular value decomposition or Tikhonov regularization (see [19]). For a good survey on the analytical properties of the Radon transform and its inverse operator and on established reconstruction techniques we refer to [33].

In many practical applications one is not only interested in the reconstruction of the density distribution but also in the extraction of some specific features within the image which represents the density distribution of the sample. For example, the planning of surgery might require the determination of the boundaries of inner organs like liver or lung or the separation of cancerous and healthy tissue. To this end, image segmentation methods are applied – a posteriori – to the output of the inversion method. Besides region growing algorithms and other ideas based on local criteria for the classification of pixels according to their membership in a certain region, deformable interfaces (snakes, active contours, level-set techniques) have received considerable attention in image segmentation in the recent years. In the latter approaches a collection of curves (surfaces for 3-D data) is introduced and iteratively updated in such a way that finally the curves separate approximately homogenous regions. This is achieved by minimizing an energy functional which penalizes the occurrence of inhomogeneous features within the distinct regions where the separating contours are the optimization variables. Different energy functionals like e.g. elastic energy in connection with edge detectors [28], [9], [15], [23] region based functionals [37], [27], [26] or Mumford–Shah like functionals [11], [13], [22] have been considered and different geometric models for the curves as for instance parametrized snakes [45] or level-set techniques [35], [42] have been used. We refer also to the monographs [3], [41] for more detailed expositions of the subject.

Segmentation methods do often fail when the image is heavily contaminated with noise. For computerized tomography, the quality of reconstruction of the density function will be limited due to the data noise, and image postprocessing [47] might be necessary before segmentation. Consequently, the procedure for extracting the contour of an object in a density image usually includes the following steps:

$$\text{data}\stackrel{R^{-1}}{\to }\text{density function}\to \text{image postprocessing}\to \text{segmentation.}$$

The main drawback of this approach is that the measured data are only used for the reconstruction of the density distribution f. Image postprocessing and segmentation rely only on the density and errors in the reconstruction – due to numerical problems or a wrong choice of the regularization parameters – will inevitably tamper the segmentation.

The main goal of this paper is the development of an algorithm that gives simultaneously a reconstruction and a segmentation directly from the measured data. To this end, we consider the Mumford–Shah like functional

$$J(f\text{,}\Gamma )=\Vert \mathit{Rf}-g_{\text{d}}{\Vert }_{L^2(\mathbb{R}\times S^1)}^2+\alpha |\Gamma |\text{.}$$

Here Γ is a geometric variable which represents the set of points on or across which the density distribution f is allowed to have certain singularities. The main idea using Mumford–Shah approaches is to introduce the singularity set as an additional unknown which can be chosen within an appropriate class of sets $\mathcal{G}$. For fixed Γ the density f is chosen from a function space X(Γ) which allows singularities of a prescribed kind on or across Γ. Usually a regularization term $\Vert f{\Vert }_{X(\Gamma )}^p$ is added to the functional to ensure well-posedness of the minimization with respect to f. The norm ∥ · ∥_X(Γ) is constructed in such a way that the occurrence of singularities on or across Γ is not penalized by ∥ · ∥_X(Γ). In our approach we choose X(Γ) as the space of functions which are piecewise constant on ${\mathbb{R}}^2⧹\Gamma$. The rather low dimension of this space automatically ensures well-posedness with respect to f. Thus, a regularization term penalizing f is not necessary. The classical Mumford–Shah functional (where the operator R in (1.1) is replaced by the identity and a penalty term acting on f is added) was originally designed to identify the set of jump singularities of a given function and – simultaneously – to find a smooth approximation of the function away from the singularities (see [32], [10], [11], [22]). In our case the unknown image and the data are related via the Radon transform and not via the identity. The functional (1.1) depends on the functional variable f (the density) as well as a geometrical variable Γ. A Mumford–Shah approach for the inversion of ill-poses operator equations has been used before in [4], [29], where the problem of deblurring of a given image is considered. In these papers, the Radon transform in (1.1) is replaced by a convolution operator with a smoothing (possibly unknown) kernel. In [4] an Ambrosio-Tortorelli type approximation of the Mumford–Shah functional is used (see [1] for the origin of this approximation), whereas the authors [29] use a level-set based approach.

For medical applications, it is reasonable to restrict the reconstruction to densities f that are constant with respect to a partition of the body, as the tissues of inner organs, bones, or muscles have approximately constant density. For the following we assume that – for fixed Γ – the density f is constant on each connected component of D⧹Γ, where D denotes the domain of definition of f. The geometrical variable Γ then denotes a collection of closed curves that describe the boundaries of the regions where the function is constant, and |Γ| is the sum of the lengths of all these curves. In our work, we extend the classical piecewise constant Mumford–Shah functional as already considered by Mumford and Shah [32] and by Chan and Vese [11] in the level-set context to the situation where the identity operator is replaced by the Radon transform R. In fact, our approach has much im common with the Chan–Vese model and the analysis and implementation of our method can be performed along the same lines as for the Chan–Vese model. Replacing the identity I by R, however, yields a more complicated optimality system for the piecewise constant function values of the density and requires more involved techniques for the shape sensitivity analysis.

By minimizing the Mumford–Shah functional (1.1), the first term on the right hand side of (1.1) ensures that the reconstruction for the functional parameter f is close enough to a solution of the equation Rf = g, whereas the term α|Γ| controls the length of the boundary of the partition of the image. Consequently, if the regularization parameter is chosen properly, we obtain both a reconstruction for the density f and a segmentation of the image, represented by the curves Γ, directly from the data g_d.

The main difficulty in using a Mumford–Shah approach lies in the different structure of the geometric variable (the singularity set) and the functional variable (the reconstruction) which cannot be easily treated in a straight forward way within the framework of nonlinear optimization. Usually either the geometrical variable is eliminated – leading to phase field like formulation with non-convex cost functionals for the functional variable [6], [10] – or the functional variable is eliminated and the problem is reduced to a shape optimization problem [12], [46], [22]. Here, we follow the second approach, i.e. in

$$\underset{f\text{,}\Gamma }{\min }J(f\text{,}\Gamma )\text{,}$$

we first minimize with respect to f and we denote the corresponding solution by f(Γ). In a second step the reduced functional

$$\widehat{J}(\Gamma )=J(f(\Gamma )\text{,}\Gamma )$$

is minimized with respect to Γ. The proposed algorithm uses a preconditioned shape gradient of $\widehat{J}(\Gamma )$ for the construction of a descent direction for the functional. From the point of view of algorithmic innovation the introduction of a preconditioner which is related to a Newton type descent direction for the geometric regularization term α|Γ| is one of the major issues of our paper. The preconditioned gradient direction produces smoother intermediate geometries and allows to use larger step sizes. It is also shown experimentally that, in the preconditioned formulation, the regularizing effect of the length term α|Γ| is not nullified by the possible instability of the time-stepping procedure due to the choice of too large time-steps, as it can be the case for the L²-gradient descent.

The update of the geometry is done using the level set methodology. The idea to combine level set and shape sensitivity techniques for the solution of inverse problems was, to our knowledge, first used in [40]. Other level-set based methods for inverse problems involving shapes were considered in [30], [25], [36], [5], [8], [24], [14], [18], [17] and others.

We also address the problem of topology changes during the propagation of the interface Γ towards a minimum of (1.3). Although it is possible in the level set context for one region Ω with boundary Γ to split into two regions, and, vice versa, for two regions to merge into one, it is well known that it is usually not possible to create new regions inside already existing ones. Based on the functional gradient of (1.1) we propose a method that generates a new component of the partition of the image at locations where it is beneficial to do so. With this, we can avoid local minima at which the algorithm otherwise would get stuck.

In the last section we provide some reconstructions/segmentations for synthetic as well as for real data. It turns out that our method works particularly well if the data are very noisy. In this case, we obtain even better reconstructions for the functional parameter than standard regularization methods. The method is compared with a Tikhonov regularization approach with a total variation regularization term. Numerical studies comparing different parameter choices are presented.