Automatic extraction of the mid-facial plane for cranio-maxillofacial surgery planning

Recently developed computer applications provide tools for planning cranio-maxillofacial interventions based on 3-dimensional (3D) virtual models of the patient's skull obtained from computed-tomography (CT) scans. Precise knowledge of the location of the mid-facial plane is important for the assessment of deformities and for planning reconstructive procedures. In this work, a new method is presented to automatically compute the mid-facial plane on the basis of a surface model of the facial skeleton obtained from CT. The method matches homologous surface areas selected by the user on the left and right facial side using an iterative closest point optimization. The symmetry plane which best approximates this matching transformation is then computed. This new automatic method was evaluated in an experimental study. The study included experienced and inexperienced clinicians defining the symmetry plane by a selection of landmarks. This manual definition was systematically compared with the definition resulting from the new automatic method: Quality of the symmetry planes was evaluated by their ability to match homologous areas of the face. Results show that the new automatic method is reliable and leads to significantly higher accuracy than the manual method when performed by inexperienced clinicians. In addition, the method performs equally well in difficult trauma situations, where key landmarks are unreliable or absent.

Holomorphic families of nonequivalent embeddings and of holomorphic group actions on affine space

We construct holomorphic families of proper holomorphic embeddings of \mathbb {C}^{k} into \mathbb {C}^{n} (0\textless k\textless n-1), so that for any two different parameters in the family, no holomorphic automorphism of \mathbb {C}^{n} can map the image of the corresponding two embeddings onto each other. As an application to the study of the group of holomorphic automorphisms of \mathbb {C}^{n}, we derive the existence of families of holomorphic \mathbb {C}^{*}-actions on \mathbb {C}^{n} (n\ge5) so that different actions in the family are not conjugate. This result is surprising in view of the long-standing holomorphic linearization problem, which, in particular, asked whether there would be more than one conjugacy class of \mathbb {C}^{*}-actions on \mathbb {C}^{n} (with prescribed linear part at a fixed point).

Infinite Transitivity on Affine Varieties

In this note we survey recent results on automorphisms of affine algebraic varieties, infinitely transitive group actions and flexibility. We present related constructions and examples, and discuss geometric applications and open problems.

Frequency of Sobolev and quasiconformal dimension distortion

We study Hausdorff and Minkowski dimension distortion for images of generic affine subspaces of Euclidean space under Sobolev and quasiconformal maps. For a supercritical Sobolev map f defined on a domain in RnRn, we estimate from above the Hausdorff dimension of the set of affine subspaces parallel to a fixed m-dimensional linear subspace, whose image under f has positive HαHα measure for some fixed α>mα>m. As a consequence, we obtain new dimension distortion and absolute continuity statements valid for almost every affine subspace. Our results hold for mappings taking values in arbitrary metric spaces, yet are new even for quasiconformal maps of the plane. We illustrate our results with numerous examples.

Antenna Simulations and Measurements of Focal Plane Array Facet Reflectors

We present a conceptual prototype model of a focal plane array unit for the STEAMR instrument, highlighting the challenges presented by the required high relative beam proximity of the instrument and focus on how edge-diffraction effects contribute to the array's performance. The analysis was carried out as a comparative process using both PO & PTD and MoM techniques. We first highlight general differences between these computational techniques, with the discussion focusing on diffractive edge effects for near-field imaging reflectors with high truncation. We then present the results of in-depth modeling analyses of the STEAMR focal plane array followed by near-field antenna measurements of a breadboard model of the array. The results of these near-field measurements agree well with both simulation techniques although MoM shows slightly higher complex beam coupling to the measurements than PO & PTD.

Uniqueness of the Embedding Continuous Convolution Semigroup of a Gaussian Probability Measure on the Affine Group and an Application in Mathematical Finance

Let {μ(i)t}t≥0 ( i=1,2 ) be continuous convolution semigroups (c.c.s.) of probability measures on Aff(1) (the affine group on the real line). Suppose that μ(1)1=μ(2)1 . Assume furthermore that {μ(1)t}t≥0 is a Gaussian c.c.s. (in the sense that its generating distribution is a sum of a primitive distribution and a second-order differential operator). Then μ(1)t=μ(2)t for all t≥0 . We end up with a possible application in mathematical finance.

On the signature of positive braids and adjacency for torus knots

The objects of study in this thesis are knots. More precisely, positive braid knots, which include algebraic knots and torus knots. In the first part of this thesis, we compare two classical knot invariants - the genus g and the signature σ - for positive braid knots. Our main result on positive braid knots establishes a linear lower bound for the signature in terms of the genus. In the second part of the thesis, a positive braid approach is applied to the study of the local behavior of polynomial functions from the complex affine plane to the complex numbers. After endowing polynomial function germs with a suitable topology, the adjacency problem arises: for a fixed germ f, what classes of germs g can be found arbitrarily close to f? We introduce two purely topological notions of adjacency for knots and discuss connections to algebraic notions of adjacency and the adjacency problem.

Virtual tissue alignment and cutting plane definition--a new method to obtain optimal longitudinal histological sections.

Histomorphometric evaluation of the buccal aspects of periodontal tissues in rodents requires reproducible alignment of maxillae and highly precise sections containing central sections of buccal roots; this is a cumbersome and technically sensitive process due to the small specimen size. The aim of the present report is to describe and analyze a method to transfer virtual sections of micro-computer tomographic (CT)-generated image stacks to the microtome for undecalcified histological processing and to describe the anatomy of the periodontium in rat molars. A total of 84 undecalcified sections of all buccal roots of seven untreated rats was analyzed. The accuracy of section coordinate transfer from virtual micro-CT slice to the histological slice, right-left side differences and the measurement error for linear and angular measurements on micro-CT and on histological micrographs were calculated using the Bland-Altman method, interclass correlation coefficient and the method of moments estimator. Also, manual alignment of the micro-CT-scanned rat maxilla was compared with multiplanar computer-reconstructed alignment. The supra alveolar rat anatomy is rather similar to human anatomy, whereas the alveolar bone is of compact type and the keratinized gingival epithelium bends apical to join the junctional epithelium. The high methodological standardization presented herein ensures retrieval of histological slices with excellent display of anatomical microstructures, in a reproducible manner, minimizes random errors, and thereby may contribute to the reduction of number of animals needed.

Which side of the focal plane are you on?

Defocus blur is an indicator for the depth structure of a scene. However, given a single input image from a conventional camera one cannot distinguish between blurred objects lying in front or behind the focal plane, as they may be subject to exactly the same amount of blur. In this paper we address this limitation by exploiting coded apertures. Previous work in this area focuses on setups where the scene is placed either entirely in front or entirely behind the focal plane. We demonstrate that asymmetric apertures result in unique blurs for all distances from the camera. To exploit asymmetric apertures we propose an algorithm that can unambiguously estimate scene depth and texture from a single input image. One of the main advantages of our method is that, within the same depth range, we can work with less blurred data than in other methods. The technique is tested on both synthetic and real images.