A machine learning approach to statistical shape models with applications to medical image analysis

Statistical shape models have become an indispensable tool for image analysis. The use of shape models is especially popular in computer vision and medical image analysis, where they were incorporated as a prior into a wide range of different algorithms. In spite of their big success, the study of statistical shape models has not received much attention in recent years. Shape models are often seen as an isolated technique, which merely consists of applying Principal Component Analysis to a set of example data sets. In this thesis we revisit statistical shape models and discuss their construction and applications from the perspective of machine learning and kernel methods. The shapes that belong to an object class are modeled as a Gaussian Process whose parameters are estimated from example data. This formulation puts statistical shape models in a much wider context and makes the powerful inference tools from learning theory applicable to shape modeling. Furthermore, the formulation is continuous and thus helps to avoid discretization issues, which often arise with discrete models. An important step in building statistical shape models is to establish surface correspondence. We discuss an approach which is based on kernel methods. This formulation allows us to integrate the statistical shape model as an additional prior. It thus unifies the methods of registration and shape model fitting. Using Gaussian Process regression we can integrate shape constraints in our model. These constraints can be used to enforce landmark matching in the fitting or correspondence problem. The same technique also leads directly to a new solution for shape reconstruction from partial data. In addition to experiments on synthetic 2D data sets, we show the applicability of our methods on real 3D medical data of the human head. In particular, we build a 3D model of the human skull, and present its applications for the planning of cranio-facial surgeries

Finite element surface registration incorporating curvature, volume preservation, and statistical model information

We present a novel method for nonrigid registration of 3D surfaces and images. The method can be used to register surfaces by means of their distance images, or to register medical images directly. It is formulated as a minimization problem of a sum of several terms representing the desired properties of a registration result: smoothness, volume preservation, matching of the surface, its curvature, and possible other feature images, as well as consistency with previous registration results of similar objects, represented by a statistical deformation model. While most of these concepts are already known, we present a coherent continuous formulation of these constraints, including the statistical deformation model. This continuous formulation renders the registration method independent of its discretization. The finite element discretization we present is, while independent of the registration functional, the second main contribution of this paper. The local discontinuous Galerkin method has not previously been used in image registration, and it provides an efficient and general framework to discretize each of the terms of our functional. Computational efficiency and modest memory consumption are achieved thanks to parallelization and locally adaptive mesh refinement. This allows for the first time the use of otherwise prohibitively large 3D statistical deformation models

Posterior shape models

We present a method to compute the conditional distribution of a statistical shape model given partial data. The result is a "posterior shape model", which is again a statistical shape model of the same form as the original model. This allows its direct use in the variety of algorithms that include prior knowledge about the variability of a class of shapes with a statistical shape model. Posterior shape models then provide a statistically sound yet easy method to integrate partial data into these algorithms. Usually, shape models represent a complete organ, for instance in our experiments the femur bone, modeled by a multivariate normal distribution. But because in many application certain parts of the shape are known a priori, it is of great interest to model the posterior distribution of the whole shape given the known parts. These could be isolated landmark points or larger portions of the shape, like the healthy part of a pathological or damaged organ. However, because for most shape models the dimensionality of the data is much higher than the number of examples, the normal distribution is singular, and the conditional distribution not readily available. In this paper, we present two main contributions: First, we show how the posterior model can be efficiently computed as a statistical shape model in standard form and used in any shape model algorithm. We complement this paper with a freely available implementation of our algorithms. Second, we show that most common approaches put forth in the literature to overcome this are equivalent to probabilistic principal component analysis (PPCA), and Gaussian Process regression. To illustrate the use of posterior shape models, we apply them on two problems from medical image analysis: model-based image segmentation incorporating prior knowledge from landmarks, and the prediction of anatomically correct knee shapes for trochlear dysplasia patients, which constitutes a novel medical application. Our experiments confirm that the use of conditional shape models for image segmentation improves the overall segmentation accuracy and robustness

Gaussian Process Morphable Models

Statistical shape models (SSMs) represent a class of shapes as a normal distribution of point variations, whose parameters are estimated from example shapes. Principal component analysis (PCA) is applied to obtain a low-dimensional representation of the shape variation in terms of the leading principal components. In this paper, we propose a generalization of SSMs, called Gaussian Process Morphable Models (GPMMs). We model the shape variations with a Gaussian process, which we represent using the leading components of its Karhunen-Loeve expansion. To compute the expansion, we make use of an approximation scheme based on the Nystrom method. The resulting model can be seen as a continuous analogon of an SSM. However, while for SSMs the shape variation is restricted to the span of the example data, with GPMMs we can define the shape variation using any Gaussian process. For example, we can build shape models that correspond to classical spline models, and thus do not require any example data. Furthermore, Gaussian processes make it possible to combine different models. For example, an SSM can be extended with a spline model, to obtain a model that incorporates learned shape characteristics, but is flexible enough to explain shapes that cannot be represented by the SSM. We introduce a simple algorithm for fitting a GPMM to a surface or image. This results in a non-rigid registration approach, whose regularization properties are defined by a GPMM. We show how we can obtain different registration schemes,including methods for multi-scale, spatially-varying or hybrid registration, by constructing an appropriate GPMM. As our approach strictly separates modelling from the fitting process, this is all achieved without changes to the fitting algorithm. We show the applicability and versatility of GPMMs on a clinical use case, where the goal is the model-based segmentation of 3D forearm images

Morphable Face Models - An Open Framework

In this paper, we present a novel open-source pipeline for face registration based on Gaussian processes as well as an application to face image analysis. Non-rigid registration of faces is significant for many applications in computer vision, such as the construction of 3D Morphable face models (3DMMs). Gaussian Process Morphable Models (GPMMs) unify a variety of non-rigid deformation models with B-splines and PCA models as examples. GPMM separate problem specific requirements from the registration algorithm by incorporating domain-specific adaptions as a prior model. The novelties of this paper are the following: (i) We present a strategy and modeling technique for face registration that considers symmetry, multi-scale and spatially-varying details. The registration is applied to neutral faces and facial expressions. (ii) We release an open-source software framework for registration and model-building, demonstrated on the publicly available BU3D-FE database. The released pipeline also contains an implementation of an Analysis-by-Synthesis model adaption of 2D face images, tested on the Multi-PIE and LFW database. This enables the community to reproduce, evaluate and compare the individual steps of registration to model-building and 3D/2D model fitting. (iii) Along with the framework release, we publish a new version of the Basel Face Model (BFM-2017) with an improved age distribution and an additional facial expression model

Error-Controlled Model Approximation for Gaussian Process Morphable Models

Gaussian Process Morphable Models (GPMMs) unify a variety of non-rigid deformation models for surface and image registration. Deformation models, such as B-splines, radial basis functions, and PCA models are defined as a probability distribution using a Gaussian process. The method depends heavily on the low-rank approximation of the Gaussian process, which is mandatory to obtain a parametric representation of the model. In this article, we propose the use of the pivoted Cholesky decomposition for this task, which has the following advantages: (1) Compared to the current state of the art used in GPMMs, it provides a fully controllable approximation error. The algorithm greedily computes new basis functions until the user-defined approximation accuracy is reached. (2) Unlike the currently used approach, this method can be used in a black-box-like scenario, whereas the method automatically chooses the amount of basis functions for a given model and accuracy. (3) We propose the Newton basis as an alternative basis for GPMMs. The proposed basis does not need an SVD computation and can be iteratively refined. We show that the proposed basis functions achieve competitive registration results while providing the mentioned advantages for its computation

Probabilistic Joint Face-Skull Modelling for Facial Reconstruction

We present a novel method for co-registration of two independent statistical shape models. We solve the problem of aligning a face model to a skull model with stochastic optimization based on Markov Chain Monte Carlo (MCMC). We create a probabilistic joint face-skull model and show how to obtain a distribution of plausible face shapes given a skull shape. Due to environmental and genetic factors, there exists a distribution of possible face shapes arising from the same skull. We pose facial reconstruction as a conditional distribution of plausible face shapes given a skull shape. Because it is very difficult to obtain the distribution directly from MRI or CT data, we create a dataset of artificial face-skull pairs. To do this, we propose to combine three data sources of independent origin to model the joint face-skull distribution: a face shape model, a skull shape model and tissue depth marker information. For a given skull, we compute the posterior distribution of faces matching the tissue depth distribution with Metropolis-Hastings. We estimate the joint faceskull distribution from samples of the posterior. To find faces matching to an unknown skull, we estimate the probability of the face under the joint faceskull model. To our knowledge, we are the first to provide a whole distribution of plausible faces arising from a skull instead of only a single reconstruction. We show how the face-skull model can be used to rank a face dataset and on average successfully identify the correct match in top 30%. The face ranking even works when obtaining the face shapes from 2D images. We furthermore show how the face-skull model can be useful to estimate the skull position in an MR-image

Variational Image Registration Using Inhomogeneous Regularization

We present a generalization of the convolution-based variational image registration approach, in which different regularizers can be implemented by conveniently exchanging the convolution kernel, even if it is nonseparable or nonstationary. Nonseparable kernels pose a challenge because they cannot be efficiently implemented by separate 1D convolutions. We propose to use a low-rank tensor decomposition to efficiently approximate nonseparable convolution. Nonstationary kernels pose an even greater challenge because the convolution kernel depends on, and needs to be evaluated for, every point in the image. We propose to pre-compute the local kernels and efficiently store them in memory using the Tucker tensor decomposition model. In our experiments we use the nonseparable exponential kernel and a nonstationary landmark kernel. The exponential kernel replicates desirable properties of elastic image registration, while the landmark kernel incorporates local prior knowledge about corresponding points in the images. We examine the trade-off between the computational resources needed and the approximation accuracy of the tensor decomposition methods. Furthermore, we obtain very smooth displacement fields even in the presence of large landmark displacements

Efficient computation of low-rank Gaussian process models for surface and image registration

Gaussian Process Morphable Models (GPMMs) are a unifying approach to non-rigid surface and image registration, where a deformation prior is defined using a Gaussian process. By a simple exchange of the covariance function we can formulate a wide variety of different deformation priors, such as spline-based models, free-form deformations or statistical shape and deformation models. How well the method works in practical applications depends crucially on how well a low-rank approximation of the Gaussian process can be computed. In this article we propose the use of the pivoted Cholesky decomposition for this task. This method makes it possible to efficiently compute a low-rank approximation for very large point sets, such as given by 3D meshes or 3D image grids, with a rigorously controlled approximation error. Compared to the current state of the art, which is based on the Nystro ̈m method, the approximation error is controllable and can be specified by a user-defined threshold. Further we propose a computationally more efficient and greedy alternative to currently used Karhunen-Loève expansion. This makes it possible to compute more accurate model approximations at the same computational costs. Detailed experiments from the registration of high quality human face scans and medical CT images containing the forearm with Ulna and Radius demonstrate the efficiency of the method and the computational advantages over the Nyström method

Automated, 3-D and Sub-Micron Accurate Ablation-Volume Determination by Inverse Molding and X-Ray Computed Tomography.

Ablation of materials in combination with element-specific analysis of the matter released is a widely used method to accurately determine a material's chemical composition. Among other methods, repetitive ablation using femto-second pulsed laser systems provides excellent spatial resolution through its incremental removal of nanometer thick layers. The method can be combined with high-resolution mass spectrometry, for example, laser ablation ionization mass spectrometry, to simultaneously analyze chemically the material released. With increasing depth of the volume ablated, however, secondary effects start to play an important role and the ablation geometry deviates substantially from the desired cylindrical shape. Consequently, primarily conical but sometimes even more complex, rather than cylindrical, craters are created. Their dimensions need to be analyzed to enable a direct correlation with the element-specific analytical signals. Here, a post-ablation analysis method is presented that combines generic polydimethylsiloxane-based molding of craters with the volumetric reconstruction of the crater's inverse using X-ray computed tomography. Automated analysis yields the full, sub-micron accurate anatomy of the craters, thereby a scalable and generic method to better understand the fundamentals underlying ablation processes applicable to a wide range of materials. Furthermore, it may serve toward a more accurate determination of heterogeneous material's composition for a variety of applications without requiring time- and labor-intensive analyses of individual craters