Part-to-whole Registration of Histology and MRI using Shape Elements

Image registration between histology and magnetic resonance imaging (MRI) is a challenging task due to differences in structural content and contrast. Too thick and wide specimens cannot be processed all at once and must be cut into smaller pieces. This dramatically increases the complexity of the problem, since each piece should be individually and manually pre-aligned. To the best of our knowledge, no automatic method can reliably locate such piece of tissue within its respective whole in the MRI slice, and align it without any prior information. We propose here a novel automatic approach to the joint problem of multimodal registration between histology and MRI, when only a fraction of tissue is available from histology. The approach relies on the representation of images using their level lines so as to reach contrast invariance. Shape elements obtained via the extraction of bitangents are encoded in a projective-invariant manner, which permits the identification of common pieces of curves between two images. We evaluated the approach on human brain histology and compared resulting alignments against manually annotated ground truths. Considering the complexity of the brain folding patterns, preliminary results are promising and suggest the use of characteristic and meaningful shape elements for improved robustness and efficiency.Comment: Paper accepted at ICCV Workshop (Bio-Image Computing

Feature correspondences From Multiple Views of Coplanar Ellipses

International audienceWe address the problem of feature correspondences in images of coplanar ellipses with objective to benefit of robust ellipse fitting algorithm. The main difficulty is the lack of projective invariant points immediately available. Therefore, our key idea is to construct virtual line and point features using the property of tangent invariance under perspective projection. The proposed method requires first a robust detection of ellipse edge points to fit a parametric model on each ellipse. The feature lines are then obtained by computing the 4 bitangents to each couple of ellipses. The points are derived by considering the tangent points and the intersection points between bitangents. Results of experimental studies are presented to demonstrate the reliability and robustness of the feature extraction process. Subpixel accuracy is easily achieved. A real application to camera self-calibration is also described

Scheimpflug Self-Calibration Based on Tangency Points

International audienceSPIV self-calibration strongly depends on the accuracy of the detection of the projection of the control points. A new family of control points and an algorithm of image detection are proposed to overcome the bias associated to the use of dot centers as control points in SPIV self-calibration

On maximally inflected hyperbolic curves

In this note we study the distribution of real inflection points among the ovals of a real non-singular hyperbolic curve of even degree. Using Hilbert's method we show that for any integers $d$ and $r$ such that $4\leq r \leq 2d^2-2d$, there is a non-singular hyperbolic curve of degree $2d$ in $\mathbb R^2$ with exactly $r$ line segments in the boundary of its convex hull. We also give a complete classification of possible distributions of inflection points among the ovals of a maximally inflected non-singular hyperbolic curve of degree $6$.Comment: 13 pages, 8 figure

Algebraic Methods for Dynamical Systems and Optimisation

This thesis develops various aspects of Algebraic Geometry and its applications in different fields of science. In Chapter 2 we characterise the feasible set of an optimisation problem relevant in chemical process engineering. We consider the polynomial dynamical system associated with mass-action kinetics of a chemical reaction network. Given an initial point, the attainable region of that point is the smallest convex and forward closed set that contains the trajectory. We show that this region is a spectrahedral shadow for a class of linear dynamical systems. As a step towards representing attainable regions we develop algorithms to compute the convex hulls of trajectories. We present an implementation of this algorithm which works in dimensions 2,3 and 4. These algorithms are based on a theory that approximates the boundary of the convex hull of curves by a family of polytopes. If the convex hull is represented as the output of our algorithms we can also check whether it is forward closed or not. Chapter 3 has two parts. In this first part, we do a case study of planar curves of degree 6. It is known that there are 64 rigid isotopy types of these curves. We construct explicit polynomial representatives with integer coefficients for each of these types using different techniques in the literature. We present an algorithm, and its implementation in software Mathematica, for determining the isotopy type of a given sextic. Using the representatives various sextics for each type were sampled. On those samples we explored the number of real bitangents, inflection points and eigenvectors. We also computed the tensor rank of the representatives by numerical methods. We show that the locus of all real lines that do not meet a given sextic is a union of up to 46 convex regions that is bounded by its dual curve. In the second part of Chapter 3 we consider a problem arising in molecular biology. In a system where molecules bind to a target molecule with multiple binding sites, cooperativity measures how the already bound molecules affect the chances of other molecules binding. We address an optimisation problem that arises while quantifying cooperativity. We compute cooperativity for the real data of molecules binding to hemoglobin and its variants. In Chapter 4, given a variety X in n-dimensional projective space we look at its image under the map that takes each point in X to its coordinate-wise r-th power. We compute the degree of the image. We also study their defining equations, particularly for hypersurfaces and linear spaces. We exhibit the set-theoretic equations of the coordinate-wise square of a linear space L of dimension k embedded in a high dimensional ambient space. We also establish a link between coordinate-wise squares of linear spaces and the study of real symmetric matrices with degenerate eigenspectrum

Computing pseudotriangulations via branched coverings

We describe an efficient algorithm to compute a pseudotriangulation of a finite planar family of pairwise disjoint convex bodies presented by its chirotope. The design of the algorithm relies on a deepening of the theory of visibility complexes and on the extension of that theory to the setting of branched coverings. The problem of computing a pseudotriangulation that contains a given set of bitangent line segments is also examined.Comment: 66 pages, 39 figure

Registration of histology and magnetic resonance imaging of the brain

Combining histology and non-invasive imaging has been attracting the attention of the medical imaging community for a long time, due to its potential to correlate macroscopic information with the underlying microscopic properties of tissues. Histology is an invasive procedure that disrupts the spatial arrangement of the tissue components but enables visualisation and characterisation at a cellular level. In contrast, macroscopic imaging allows non-invasive acquisition of volumetric information but does not provide any microscopic details. Through the establishment of spatial correspondences obtained via image registration, it is possible to compare micro- and macroscopic information and to recover the original histological arrangement in three dimensions. In this thesis, I present: (i) a survey of the literature relative to methods for histology reconstruction with and without the help of 3D medical imaging; (ii) a graph-theoretic method for histology volume reconstruction from sets of 2D sections, without external information; (iii) a method for multimodal 2D linear registration between histology and MRI based on partial matching of shape-informative boundaries

Visual perception of solid shape from occluding contours

The relative motion of object and observer induces a motion field in the observer's visual image that is smooth everywhere except along the object's occluding contours. Thus, occluding contours and smooth motion fields can be viewed as complementary and as separate sources of information about an object's shape. I studied how the human visual system perceives solid shape from the occluding contours of rotating objects and from the smooth motion field induced by moving planar surface patches.I propose a three-stage model for the perception of solid shape from the occluding contours of a rotating object. First, the object's motion is determined. I argue that this is only possible using points of correspondence and only when the object's axis of rotation is frontoparallel. In the second stage, the motion field along the contour is used to compute relative depth and surface curvature along the rim, the contour's pre-image. Third, local shape descriptors are propagated inside the figure to yield a global percept of solid shape. To determine which shape descriptors are computed by human subjects, I used a novel task in which subjects have to discriminate between flat ellipses and solid ellipsoids with varying thickness. I found that discriminability is proportional to the inverse of radial curvature but is not proportional to Gaussian or mean curvature. Certain slants of the axis of rotation decrease discriminability. Subjects who could discriminate ellipsoids and ellipses perceived the ellipsoids' angular velocity more veridically than did subjects who could not discriminate the two.Any smooth motion field can locally be described by divergence, curl, and deformation. If the motion field is induced by a rotating plane, the amount of deformation is proportional to the plane's slant and its angular velocity. Similarly, for translating planes, deformation is proportional to slant and image motion. Slant judgments of human observers were to a first-order approximation proportional to deformation per se, that is, observers do not take object motion into account. Recent psychophysical evidence suggests that human subjects need motion discontinuities for this. Thus, contours might be necessary to correctly perceive slant from smooth motion fields

Model free visual servoing in macro and micro domain robotic applications

This thesis explores model free visual servoing algorithms by experimentally evaluating their performances for various tasks performed both in macro and micro domains. Model free or so called uncalibrated visual servoing does not need the system (vision system + robotic system) calibration and the model of the observed scene, since it provides an online estimation of the composite (image + robot) Jacobian. It is robust to parameter changes and disturbances. A model free visual servoing scheme is tested on a 7 DOF Mitsubishi PA10 robotic arm and on a microassembly workstation which is developed in our lab. In macro domain, a new approach for planar shape alignment is presented. The alignment task is performed based on bitangent points which are acquired using convex-hull of a curve. Both calibrated and uncalibrated visual servoing schemes are employed and compared. Furthermore, model free visual servoing is used for various trajectory following tasks such as square, circle, sine etc. and these reference trajectories are generated by a linear interpolator which produces midway targets along them. Model free visual servoing can provide more exibility in microsystems, since the calibration of the optical system is a tedious and error prone process, and recalibration is required at each focusing level of the optical system. Therefore, micropositioning and three di erent trajectory following tasks are also performed in micro world. Experimental results validate the utility of model free visual servoing algorithms in both domains

Inflection Points of Real and Tropical Plane Curves

We prove that Viro's patchworking produces real algebraic curves with the maximal number of real inflection points. In particular this implies that maximally inflected real algebraic $M$-curves realize many isotopy types. The strategy we adopt in this paper is tropical: we study tropical limits of inflection points of classical plane algebraic curves. The main tropical tool we use to understand these tropical inflection points are tropical modifications.Comment: 29 pages, 20 figures. To be published in "Journal of Singularities