The reconstructed residual error: A novel segmentation evaluation measure for reconstructed images in tomography
Introduction
In many applications of tomography [1], such as the delineation of anatomical structures (in medical imaging) and object detection (in computer vision), the reconstructed image must be segmented before the results can be analyzed. Segmentation is defined as the classification of image pixels into distinct classes, based on similarity with respect to some characteristic. Numerous methods have been proposed, such as global and local thresholding, region growing, clustering, and atlas-guided approaches [2], [3], [4].
Given a segmentation, objectively evaluating the accuracy of that segmentation is not a trivial task [5], [6]. Supervised methods evaluate a segmentation algorithm by comparing its output with gold standard segmentations. However, since such segmentations are typically not available in practice, they must often be generated manually, which is not easy and may make the evaluation subjective. Unsupervised methods (also sometimes called stand-alone methods [7]) do not need gold standards, as they evaluate the segmentation results directly, using one or more of the de facto standard criteria of Haralick and Shapiro [2]. These methods are objective and applicable to a wide variety of images, but their analysis is restricted to the segmentation result itself. A possibility that is often overlooked, in the specific case of tomography, is exploiting the available projection images, which can provide external information about the segmented image.
The current paper introduces the reconstructed residual error, which does exploit the original projection images. Our method is applicable to reconstruction problems for which the segmentation is based on the density of the scanned object, where we use the term density to refer to the particular physical property of the object of which linear projections are acquired during the scanning process (e.g., mass density, X-ray attenuation, electron beam scattering, etc.). The reconstructed residual error is an unsupervised evaluation in the terminology of [6], since it is an objective evaluation at the level of the segmentation itself that does not need a reference image [6, Fig. 1]. In contrast to the unsupervised methods surveyed in [6], the proposed method does not have to rely on the criteria of Haralick and Shapiro, since it uses the projection images as external information.
The reconstructed residual error evaluates a given segmentation by providing a spatial map of the errors. It is computed by reconstructing the difference between the recorded data and the forward projection of that segmentation. The computation of the error map is independent of the methods that were used for reconstructing the image and determining the segmentation.
The remainder of this paper is organized as follows. In Section 2, the reconstructed residual error is defined and its properties are described in detail. Section 3 reports on the results of experiments, using both simulations and experimental micro-CT data. These results are discussed in Section 4, and conclusions are drawn in Section 5.
Section snippets
Reconstructed residual error
Here, we describe the reconstructed residual error. We first present an intuitive overview of its computation, before giving a complete description of its properties.
Experiments and results
In this section, we describe the experiments, for both simulated and experimental data, that were carried out to investigate the properties of the reconstructed residual error. To improve the clarity of the presented results, the simulation experiments are based on two-dimensional slices and parallel-beam geometry. However, as was already mentioned in Section 2.3, the method is readily applicable to three-dimensional objects and other geometries such as cone beam. The application in Section 3.3
Discussion
In tomography, most techniques that are used for segmentation and segmentation evaluation do no exploit the projection data. There are a few algorithms that do exploit this information during segmentation [17], [18], however, they compute a quality measure that is a single number (the projection distance in [17] and the segmentation inconsistency in [18]). In contrast, the reconstructed residual error is a spatial map of the segmentation quality. This allows studying local variations of the
Conclusions
We have introduced the reconstructed residual error, as a way to evaluate the segmentation quality of a reconstructed image in cases where the segmentation is based on the density of the scanned object. We have used the Moore–Penrose pseudoinverse as a mathematical model for investigating the properties of the technique, and then generalized this approach to practical reconstruction algorithms.
The reconstructed residual error provides an accurate map of the errors in a segmented tomogram.
Acknowledgments
The mouse femur dataset is courtesy of Phil Salmon, Bruker microCT. This work was financially supported by the IWT TomFood project (IWT is the agency for Innovation by Science and Technology–Flanders, Belgium) and by the NWO (the Netherlands Organisation for Scientific Research – The Netherlands, Research Programme 639.072.005). This work was conducted in the framework of the Extrema COST Action MP1207.
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