A local structural descriptor for image matching via normalized graph laplacian embedding

This paper investigates graph spectral approaches to the problem of point pattern matching. Specifically, we concentrate on the issue of how to effectively use graph spectral properties to characterize point patterns in the presence of positional jitter and outliers. A novel local spectral descriptor is proposed to represent the attribute domain of feature points. For a point in a given point-set, weight graphs are constructed on its neighboring points and then their normalized Laplacian matrices are computed. According to the known spectral radius of the normalized Laplacian matrix, the distribution of the eigenvalues of these normalized Laplacian matrices is summarized as a histogram to form a descriptor. The proposed spectral descriptor is finally combined with the approximate distance order for recovering correspondences between point-sets. Extensive experiments demonstrate the effectiveness of the proposed approach and its superiority to the existing methods

Twisted higher index theory on good orbifolds and fractional quantum numbers

The twisted Connes-Moscovici higher index theorem is generalized to the case of good orbifolds. The higher index is shown to be a rational number, and in fact non-integer in specific examples of 2-orbifolds. This results in a non-commutative geometry model that predicts the occurrence of fractional quantum numbers in the Hall effect on the hyperbolic plane.Comment: 47 pages, Late

Twisted index theory on good orbifolds, I: noncommutative Bloch theory

This paper, together with Part II, expands the results of math.DG/9803051. In Part I we study the twisted index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant under a projective action of the orbifold fundamental group. We apply these results to obtain qualitative results on real and complex hyperbolic spaces in 2 and 4 dimensions, related to generalizations of the Bethe-Sommerfeld conjecture and the Ten Martini Problem, on the spectrum of self adjoint elliptic operators which are invariant under a projective action of a discrete cocompact group.Comment: 34 pages, LaTe

A Perfect Match Condition for Point-Set Matching Problems Using the Optimal Mass Transport Approach

We study the performance of optimal mass transport--based methods applied to point-set matching problems. The present study, which is based on the L2 mass transport cost, states that perfect matches always occur when the product of the point-set cardinality and the norm of the curl of the nonrigid deformation field does not exceed some constant. This analytic result is justified by a numerical study of matching two sets of pulmonary vascular tree branch points whose displacement is caused by the lung volume changes in the same human subject. The nearly perfect match performance verifies the effectiveness of this mass transport--based approach.Read More: http://epubs.siam.org/doi/abs/10.1137/12086443

Molecular Predictors of 3D Morphogenesis by Breast Cancer Cell Lines in 3D Culture

Correlative analysis of molecular markers with phenotypic signatures is the simplest model for hypothesis generation. In this paper, a panel of 24 breast cell lines was grown in 3D culture, their morphology was imaged through phase contrast microscopy, and computational methods were developed to segment and represent each colony at multiple dimensions. Subsequently, subpopulations from these morphological responses were identified through consensus clustering to reveal three clusters of round, grape-like, and stellate phenotypes. In some cases, cell lines with particular pathobiological phenotypes clustered together (e.g., ERBB2 amplified cell lines sharing the same morphometric properties as the grape-like phenotype). Next, associations with molecular features were realized through (i) differential analysis within each morphological cluster, and (ii) regression analysis across the entire panel of cell lines. In both cases, the dominant genes that are predictive of the morphological signatures were identified. Specifically, PPARγ has been associated with the invasive stellate morphological phenotype, which corresponds to triple-negative pathobiology. PPARγ has been validated through two supporting biological assays

Instanton Expansion of Noncommutative Gauge Theory in Two Dimensions

We show that noncommutative gauge theory in two dimensions is an exactly solvable model. A cohomological formulation of gauge theory defined on the noncommutative torus is used to show that its quantum partition function can be written as a sum over contributions from classical solutions. We derive an explicit formula for the partition function of Yang-Mills theory defined on a projective module for arbitrary noncommutativity parameter \theta which is manifestly invariant under gauge Morita equivalence. The energy observables are shown to be smooth functions of \theta. The construction of noncommutative instanton contributions to the path integral is described in some detail. In general, there are infinitely many gauge inequivalent contributions of fixed topological charge, along with a finite number of quantum fluctuations about each instanton. The associated moduli spaces are combinations of symmetric products of an ordinary two-torus whose orbifold singularities are not resolved by noncommutativity. In particular, the weak coupling limit of the gauge theory is independent of \theta and computes the symplectic volume of the moduli space of constant curvature connections on the noncommutative torus.Comment: 52 pages LaTeX, 1 eps figure, uses espf. V2: References added and repaired; V3: Typos corrected, some clarifying explanations added; version to be published in Communications in Mathematical Physic

Biaxial Nematic Order in Liver Tissue

Understanding how biological cells organize to form complex functional tissues is a question of key interest at the interface between biology and physics. The liver is a model system for a complex three-dimensional epithelial tissue, which performs many vital functions. Recent advances in imaging methods provide access to experimental data at the subcellular level. Structural details of individual cells in bulk tissues can be resolved, which prompts for new analysis methods. In this thesis, we use concepts from soft matter physics to elucidate and quantify structural properties of mouse liver tissue. Epithelial cells are structurally anisotropic and possess a distinct apico-basal cell polarity that can be characterized, in most cases, by a vector. For the parenchymal cells of the liver (hepatocytes), however, this is not possible. We therefore develop a general method to characterize the distribution of membrane-bound proteins in cells using a multipole decomposition. We first verify that simple epithelial cells of the kidney are of vectorial cell polarity type and then show that hepatocytes are of second order (nematic) cell polarity type. We propose a method to quantify orientational order in curved geometries and reveal lobule-level patterns of aligned cell polarity axes in the liver. These lobule-level patterns follow, on average, streamlines defined by the locations of larger vessels running through the tissue. We show that this characterizes the liver as a nematic liquid crystal with biaxial order. We use the quantification of orientational order to investigate the effect of specific knock-down of the adhesion protein Integrin ß-1. Building upon these observations, we study a model of nematic interactions. We find that interactions among neighboring cells alone cannot account for the observed ordering patterns. Instead, coupling to an external field yields cell polarity fields that closely resemble the experimental data. Furthermore, we analyze the structural properties of the two transport networks present in the liver (sinusoids and bile canaliculi) and identify a nematic alignment between the anisotropy of the sinusoid network and the nematic cell polarity of hepatocytes. We propose a minimal lattice-based model that captures essential characteristics of network organization in the liver by local rules. In conclusion, using data analysis and minimal theoretical models, we found that the liver constitutes an example of a living biaxial liquid crystal.:1. Introduction 1 1.1. From molecules to cells, tissues and organisms: multi-scale hierarchical organization in animals 1 1.2. The liver as a model system of complex three-dimensional tissue 2 1.3. Biology of tissues 5 1.4. Physics of tissues 9 1.4.1. Continuum descriptions 11 1.4.2. Discrete models 11 1.4.3. Two-dimensional case study: planar cell polarity in the fly wing 15 1.4.4. Challenges of three-dimensional models for liver tissue 16 1.5. Liquids, crystals and liquid crystals 16 1.5.1. The uniaxial nematic order parameter 19 1.5.2. The biaxial nematic ordering tensor 21 1.5.3. Continuum theory of nematic order 23 1.5.4. Smectic order 25 1.6. Three-dimensional imaging of liver tissue 26 1.7. Overview of the thesis 28 2. Characterizing cellular anisotropy 31 2.1. Classifying protein distributions on cell surfaces 31 2.1.1. Mode expansion to characterize distributions on the unit sphere 31 2.1.2. Vectorial and nematic classes of surface distributions 33 2.1.3. Cell polarity on non-spherical surfaces 34 2.2. Cell polarity in kidney and liver tissues 36 2.2.1. Kidney cells exhibit vectorial polarity 36 2.2.2. Hepatocytes exhibit nematic polarity 37 2.3. Local network anisotropy 40 2.4. Summary 41 3. Order parameters for tissue organization 43 3.1. Orientational order: quantifying biaxial phases 43 3.1.1. Biaxial nematic order parameters 45 3.1.2. Co-orientational order parameters 51 3.1.3. Invariants of moment tensors 52 3.1.4. Relation between these three schemes 53 3.1.5. Example: nematic coupling to an external field 55 3.2. A tissue-level reference field 59 3.3. Orientational order in inhomogeneous systems 62 3.4. Positional order: identifying signatures of smectic and columnar order 64 3.5. Summary 67 4. The liver lobule exhibits biaxial liquid-crystal order 69 4.1. Coarse-graining reveals nematic cell polarity patterns on the lobulelevel 69 4.2. Coarse-grained patterns match tissue-level reference field 73 4.3. Apical and basal nematic cell polarity are anti-correlated 74 4.4. Co-orientational order: nematic cell polarity is aligned with network anisotropy 76 4.5. RNAi knock-down perturbs orientational order in liver tissue 78 4.6. Signatures of smectic order in liver tissue 81 4.7. Summary 86 5. Effective models for cell and network polarity coordination 89 5.1. Discretization of a uniaxial nematic free energy 89 5.2. Discretization of a biaxial nematic free energy 91 5.3. Application to cell polarity organization in liver tissue 92 5.3.1. Spatial profile of orientational order in liver tissue 93 5.3.2. Orientational order from neighbor-interactions and boundary conditions 94 5.3.3. Orientational order from coupling to an external field 99 5.4. Biaxial interaction model 101 5.5. Summary 105 6. Network self-organization in a liver-inspired lattice model 107 6.1. Cubic lattice geometry motivated by liver tissue 107 6.2. Effective energy for local network segment interactions 110 6.3. Characterizing network structures in the cubic lattice geometry 113 6.4. Local interaction rules generate macroscopic network structures 115 6.5. Effect of mutual repulsion between unlike segment types on network structure 118 6.6. Summary 121 7. Discussion and Outlook 123 A. Appendix 127 A.1. Mean field theory fo the isotropic-uniaxial nematic transition 127 A.2. Distortions of the Mollweide projection 129 A.3. Shape parameters for basal membrane around hepatocytes 130 A.4. Randomized control for network segment anisotropies 130 A.5. The dihedral symmetry group D2h 131 A.6. Relation between orientational order parameters and elements of the super-tensor 134 A.7. Formal separation of molecular asymmetry and orientation 134 A.8. Order parameters under action of axes permutation 137 A.9. Minimal integrity basis for symmetric traceless tensors 139 A.10. Discretization of distortion free energy on cubic lattice 141 A.11. Metropolis Algorithm for uniaxial cell polarity coordination 142 A.12. States in the zero-noise limit of the nearest-neighbor interaction model 143 A.13. Metropolis Algorithm for network self-organization 144 A.14. Structural quantifications for varying values of mutual network segment repulsion 146 A.15. Structural quantifications for varying values of self-attraction of network segments 148 A.16. Structural quantifications for varying values of cell demand 150 Bibliography 152 Acknowledgements 17