Towards Grey Scale-Based Tensor Voting for Blood Vessel Analysis
Tensor Voting is a technique that uses perceptual rules to group points in a set of input data. Its main advantage lies in its ability to robustly extract geometrical shapes like curves and surfaces from point clouds even in noisy scenarios. Following the original formulation this is achieved by exploiting the relative positioning of those points with respect to each other. Having this in mind, it is not a straight forward task to apply original tensor voting to greyscale images. Due to the underlying voxel grid, digital images have all data measurements at regularly sampled positions. For that reason, the pure spatial position of data points relative to each other does not provide useful information unless one considers the measured intensity value in addition to that.
To account for that, previous approaches of employing tensor voting to scalar images have followed mainly two ideas. One is to define a subset of voxels that are likely to resemble a desired structure like curves or surfaces in the original image in a preprocessing step and to use only those points for initialisation in tensor voting. In other methods, the encoding step is modified e.g. by using estimations of local orientations for initialisation.
In contrast to these approaches, another idea is to embed all information given as input, that is position in combination with intensity value, into a 4D space and perform classic tensor voting on that. In doing so, it is neither necessary to rely on a preprocessing step for estimating local orientation features nor is it needed to employ assumptions within the encoding step as all data points are initialised with unit ball tensors. Alternatively, the intensity dimension could be partially included by considering it in the weighting function of tensor voting while still employing 3D tensors for the voting. Considering the advantage of a shorter computation time for the latter approach, it is of interest to investigate the differences between these two approaches.
Although different methods have employed an ND implementation of tensor voting before, the actual interpretation of its output, that is the estimation of a local hyper surface at each point, depends on the actual application at hand. As we are especially interested in the analysis of blood vessels in CT angiography data, we study the feasibility of detecting tubular structures and the estimation of their orientation totally within the proposed framework and also compare the two mentioned approaches with a special focus on these aspects.
In this chapter we first provide the formulation of both approaches followed by the application-specific interpretations of the shape of 4D output tensors. Based on that, we compare the information inferred by both methods from both synthetic and medical image data focusing on the application of blood vessel analysis.
We thank Hortense Kirisli, Theo van Walsum and Wiro Niessen for providing the CTA data used in the experiments. This research has been supported by the Swedish Research Council (VR), grants no. 2014-6153 and 2012-3512, and the Swedish Heart-Lung Foundation (HLF), grant no. 2011-0376.
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