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Adaptive Gaussian Mixture Models for Continuous Representation of Digital Medical Images

Final Report Summary - GMM4MEDICAL (Adaptive Gaussian Mixture Models for Continuous Representation of Digital Medical Images)

In tomographic medical imaging, voxel images are not acquired directly but sample data of statistical nature is measured from the tissue placed in the field of view. From the acquired raw data, a volumetric image is then reconstructed by computational methods. Since the data acquisition pattern depends on the device form factor and does not take into account the underlying image representation, reconstruction artifacts are likely to occur, especially when images are represented by uniform grids of voxels that do not take varying data density into account. As a consequence, images contain visible noise artifacts while the resolution is often insufficient in regions that would be supported by richer statistical information. Those regions are the focus of attention for image assessment and thus improving image resolution locally could provide a significant benefit for better detectability.

Alternatives to the classical representation of images by Cartesian grids of pixels and voxels exist but image modelling is not yet a very active field research. Fortunately, the combination of modern developments in statistical estimation methods, approximation properties of polynomial B-spline basis functions and efficient hierarchical space partitioning data structures provide both theoretical justifications and efficient computational methods for the generation of high-quality adaptive image models from limited input data.

The aim of this research project is to unveil a new way to represent continuous digital images in general. The approach of continuous image representation is totally new for medical imaging and contrasts with established image models that are usually based on histograms. With such a sparse continuous image model, the image resolution can be space-variant, the image space is not limited by sharp boundaries, and more importantly, the number of image elements, hence the resolution, can be adapted locally as a function of the amount of input information available for image reconstruction.

The techniques developed in this project can be expected have a broad impact since they can be transferred to many other stochastic reconstruction scenarios. As a prototypic sample scenario we will look at Monte-Carlo techniques for global illumination simulation in a 3D scene.

In the course of this project, we have developed algorithms for the incremental generation of sparse and continuous image representations taking raw PET-measurement data (TOF-PET list-mode data) as input. The representation consists of an unstructured superposition of Gausian kernels and is inspired from the field of parametric density estimation for Gaussian mixture models. The position, size, aspect ratio and orientation of each image element (= Gauss kernel) is optimised along with its weight through a very fast online estimation method. Furthermore, the number of mixture components, hence the image resolution, is locally adapted according to the density of the available data. The system model is represented in the same basis as the image elements and captures time of flight and positron range effects with high accuracy. Computations use apodised B-spline approximations of Gaussians and simple closed-form analytical expressions without any sampling or interpolation. In consequence, the reconstructed image never suffers from spurious aliasing artifacts. To evaluate the method, noiseless images of the XCAT brain phantom were reconstructed from simulated data.

In the context of our studies of various quasi-random sampling schemes for applications in general imaging tasks (medical and non-medical), we discovered a new technique leading to low-descrepancy sets of multidimensional samples. Most classical constructions of low-discrepancy point sets are based on generalisations of the one-dimensional binary van der Corput sequence, whose implementation requires nontrivial bit-operations. As an alternative, we introduced the quasi-regular golden ratio sequences, which are based on the fractional part of successive integer multiples of the golden ratio. By leveraging results from number theory, we showed that point sets, which evenly cover the unit square or disc, can be computed by a simple incremental permutation of a generator golden ratio sequence. We compared ambient occlusion images generated with a Monte Carlo ray tracer based on random, Hammersley, blue noise, and golden ratio point sets. We plan to make the source code of the ray tracer used for our experiments available online.