A **Riemannian manifol**d is a mathematical concept that generalizes curved surfaces to higher dimensions, giving a precise meaning to concepts like angle, length, area, volume and curvature. A glimpse of the consequences of a non-ﬂat geometry is given on the sphere, where the shortest path between two points – a geodesic – is along a great circle. Different from Euclidean space, the angle sum of geodesic triangles on the sphere is always larger than 180 degrees. Signals and data found in applied research are sometimes naturally described by such curved spaces. This dissertation presents basic research and tools for the analysis, processing and visualization of such manifold-valued data, with a particular emphasis on future applications in medical imaging and visualization.Two-dimensional manifolds, i.e. surfaces, enter naturally into the geometric modelling of anatomical entities, such as the human brain cortex and the colon. In advanced algorithms for processing of images obtained from computed tomography (CT) and ultrasound imaging (US), images themselves and derived local structure tensor ﬁelds may be interpreted as two- or three-dimensional manifolds. In diffusion tensor magnetic resonance imaging (DT-MRI), the natural description of diffusion in the human body is a second-order tensor ﬁeld, which can be related to the metric of a manifold. A ﬁnal example is the analysis of shape variations of anatomical entities, e.g. the lateral ventricles in the brain, within a population by describing the set of all possible shapes as a manifold.Work presented in this dissertation include: Probabilistic interpretation of intrinsic and extrinsic means in manifolds. A Bayesian approach to ﬁltering of vector data, removing noise from sampled manifolds and signals. Principles for the storage of tensor ﬁeld data and learning a natural metric for empirical data.The main contribution is a novel class of algorithms called LogMaps, for the numerical estimation of logp (x) from empirical data sampled from a low-dimensional manifold or geometric model embedded in Euclidean space…

*Contents*

1 Introduction

1.1 Motivations

1.2 Potential impact

1.3 Dissertation overview

1.4 Contributions

1.5 Publications

1.6 Abbreviations

1.7 Mathematical Notation

2 Mathematics

2.1 Linear algebra

2.1.1 Vector spaces

2.1.2 Linear maps

2.1.3 The dual vector space

2.1.4 The Einstein summation convention

2.1.5 Coordinate changes

2.1.6 Inner products and metrics

2.2 Tensors

2.2.1 Outer products

2.2.2 Cartesian tensors

2.2.3 Index gymnastics

2.3 Manifolds

2.3.1 Charts and atlases

2.3.2 The tangent space

2.3.3 Geodesic length and distance

2.3.4 Further reading

3 Dimension reduction and manifold learning

3.1 Machine learning

3.1.1 Dimensionality reduction

3.1.2 Manifold learning

3.1.3 Laplacian eigenmaps

3.1.4 Isomap – isometric feature mapping

3.1.5 A brief historical timeline

4 Diffusion tensor MRI

4.1 Diffusion imaging

4.1.1 Diffusion

4.1.2 Estimating diffusion tensors

4.1.3 Diffusion in the human brain

4.1.4 Applications of DT-MRI

4.2 Processing diffusion tensor data

4.2.1 Scalar invariants

4.2.2 Fiber tracking

4.2.3 Fiber tract connectivity

4.2.4 Segmentation of white matter

4.3 Visualization of streamline data

4.3.1 Local and global features in DT-MRI

4.3.2 Visualization of ﬁber tract connectivity

5 Empirical LogMaps

5.1 Introduction

5.2 Related work

5.2.1 Programming on manifolds

5.2.2 Previous work on Riemannian normal coordinates

5.3 The LogMap algorithm

5.4 Mathematical properties of RNC and LogMaps

5.4.1 The LogMap formula

5.4.2 On the optimality of LogMaps

5.5 Experiments

5.5.1 The Swiss roll

5.5.2 The torus

5.5.3 Local phase

5.5.4 Blob-shapes

5.5.5 Conclusion

6 LogMap texture mapping

6.1 Introduction

6.2 Previous work

6.3 The LogMap method

6.4 Computing geodesic distance

6.5 Experiments

6.5.1 The Stanford bunny

6.5.2 Plane with a bump

6.5.3 A model problem

6.6 Conclusions and future work

7 Estimating skeletons from LogMap

7.1 Algorithm

7.2 Experiments

7.3 Conclusion

8 Geodesic glyph warping

8.1 Introduction

8.2 Related work

8.3 Index notation

8.4 The metric and metric spheres

8.5 The geodesic equation and geodesic spheres

8.6 The exponential map and Riemannian normal coordinates

8.7 Solving the geodesic equation

8.8 Geodesic spheres and warped coordinate systems

8.9 The logarithmic map

8.10 Experiments

8.11 Conclusion

9 Natural metrics for parameterized image manifolds

9.1 Introduction

9.2 Related work

9.3 A model for image manifolds

9.4 An experiment: Intrinsic geometry in DWI

9.5 Conclusions

10 Intrinsic and extrinsic means

10.1 Introduction

10.1.1 The intrinsic mean

10.1.2 The extrinsic mean

10.2 Modeling noise by Brownian motion

10.2.1 Means as ML estimates in Rn

10.2.2 Intrinsic means as ML estimates in S1

10.2.3 Extrinsic means as ML estimates in S1

10.3 Experiments

10.4 Discussion

11 Bayesian feature space ﬁltering

11.1 Introduction

11.2 Previous work

11.3 The Bayesian method

11.3.1 Noise models

11.3.2 Signal models for images

11.3.3 Signal models for N-D data sets

11.3.4 Estimation

11.4 Importance sampling

11.4.1 Proper samples

11.4.2 Importance sampling

11.5 Implementation

11.5.1 Vector-valued images

11.5.2 Unordered N-D data

11.6 Experiments

11.6.1 Scalar signals

11.6.2 Vector-valued signals

11.6.3 Unordered N-D data

11.7 Conclusion

12 Storing regularly sampled tensor charts

12.1 Introduction

12.2 Related work

12.3 Geometric arrays

12.4 Scalar array data storage

12.5 Tensor array data storage

12.6 The tensor array core

12.6.1 Storing array data

12.7 Examples

12.8 Discussion

13 Summary and outlook

13.1 Future Research

Author: Brun, Anders

Source: Linkoping University

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