An investigation of the Mach Phenomenon and its relation to visual image evaluation

The Mach Phenomenon*(as perceived on photographic film) is herein investigated as a function of edge acutance, contrast and illumination level for the purpose of determining its probable effect in the area of visual image evaluation. The results indicate that dark Mach band widths increase as acutance decreases and dark band width is independent of contrast and illumination level while light band widths remain constant, over the ranges investigated. Using these findings as a basis, a dark band width predictor equation was found, taking the form; dark band width (in microns) =K/acutance, when average edge gradient is the measure of acutance. The probable effects (which increase in importance as acutance decreases) of the Mach bands on visual image evaluation is then discussed. * a visual sensation of density bands near edge

Lattice width directions and Minkowski's 3^d-theorem

We show that the number of lattice directions in which a d-dimensional convex body in R^d has minimum width is at most 3^d-1, with equality only for the regular cross-polytope. This is deduced from a sharpened version of the 3^d-theorem due to Hermann Minkowski (22 June 1864--12 January 1909), for which we provide two independent proofs.Comment: 1 figure, 10 page

Consistent and invertible deformation vector fields for a breathing anthropomorphic phantom: a post-processing framework for the XCAT phantom.

Breathing motion is challenging for radiotherapy planning and delivery. This requires advanced four-dimensional (4D) imaging and motion mitigation strategies and associated validation tools with known deformations. Numerical phantoms such as the XCAT provide reproducible and realistic data for simulation-based validation. However, the XCAT generates partially inconsistent and non-invertible deformations where tumours remain rigid and structures can move through each other. We address these limitations by post-processing the XCAT deformation vector fields (DVF) to generate a breathing phantom with realistic motion and quantifiable deformation. An open-source post-processing framework was developed that corrects and inverts the XCAT-DVFs while preserving sliding motion between organs. Those post-processed DVFs are used to warp the first XCAT-generated image to consecutive time points providing a 4D phantom with a tumour that moves consistently with the anatomy, the ability to scale lung density as well as consistent and invertible DVFs. For a regularly breathing case, the inverse consistency of the DVFs was verified and the tumour motion was compared to the original XCAT. The generated phantom and DVFs were used to validate a motion-including dose reconstruction (MIDR) method using isocenter shifts to emulate rigid motion. Differences between the reconstructed doses with and without lung density scaling were evaluated. The post-processing framework produced DVFs with a maximum [Formula: see text]-percentile inverse-consistency error of 0.02 mm. The generated phantom preserved the dominant sliding motion between the chest wall and inner organs. The tumour of the original XCAT phantom preserved its trajectory while deforming consistently with the underlying tissue. The MIDR was compared to the ground truth dose reconstruction illustrating its limitations. MIDR with and without lung density scaling resulted in small dose differences up to 1 Gy (prescription 54 Gy). The proposed open-source post-processing framework overcomes important limitations of the original XCAT phantom and makes it applicable to a wider range of validation applications within radiotherapy

Respiratory motion modelling for MR-guided lung cancer radiotherapy: model development and geometric accuracy evaluation.

Objective.Respiratory motion of lung tumours and adjacent structures is challenging for radiotherapy. Online MR-imaging cannot currently provide real-time volumetric information of the moving patient anatomy, therefore limiting precise dose delivery, delivered dose reconstruction, and downstream adaptation methods.Approach.We tailor a respiratory motion modelling framework towards an MR-Linac workflow to estimate the time-resolved 4D motion from real-time data. We develop a multi-slice acquisition scheme which acquires thick, overlapping 2D motion-slices in different locations and orientations, interleaved with 2D surrogate-slices from a fixed location. The framework fits a motion model directly to the input data without the need for sorting or binning to account for inter- and intra-cycle variation of the breathing motion. The framework alternates between model fitting and motion-compensated super-resolution image reconstruction to recover a high-quality motion-free image and a motion model. The fitted model can then estimate the 4D motion from 2D surrogate-slices. The framework is applied to four simulated anthropomorphic datasets and evaluated against known ground truth anatomy and motion. Clinical applicability is demonstrated by applying our framework to eight datasets acquired on an MR-Linac from four lung cancer patients.Main results.The framework accurately reconstructs high-quality motion-compensated 3D images with 2 mm3isotropic voxels. For the simulated case with the largest target motion, the motion model achieved a mean deformation field error of 1.13 mm. For the patient cases residual error registrations estimate the model error to be 1.07 mm (1.64 mm), 0.91 mm (1.32 mm), and 0.88 mm (1.33 mm) in superior-inferior, anterior-posterior, and left-right directions respectively for the building (application) data.Significance.The motion modelling framework estimates the patient motion with high accuracy and accurately reconstructs the anatomy. The image acquisition scheme can be flexibly integrated into an MR-Linac workflow whilst maintaining the capability of online motion-management strategies based on cine imaging such as target tracking and/or gating

On orbits of the automorphism group on a complete toric variety

Let X be a complete toric variety and Aut(X) be the automorphism group. We give an explit description of Aut(X)-orbits on X. In particular, we show that Aut(X) acts on X transitively if and only if X is a product of projective spaces.Comment: 10 pages, 4 figure

Parametric Polyhedra with at least kk Lattice Points: Their Semigroup Structure and the k-Frobenius Problem

Given an integral $d \times n$ matrix $A$, the well-studied affine semigroup \mbox{ Sg} (A)=\{ b : Ax=b, \ x \in {\mathbb Z}^n, x \geq 0\} can be stratified by the number of lattice points inside the parametric polyhedra $P_A(b)=\{x: Ax=b, x\geq0\}$. Such families of parametric polyhedra appear in many areas of combinatorics, convex geometry, algebra and number theory. The key themes of this paper are: (1) A structure theory that characterizes precisely the subset \mbox{ Sg}_{\geq k}(A) of all vectors b \in \mbox{ Sg}(A) such that $P_A(b) \cap {\mathbb Z}^n$ has at least $k$ solutions. We demonstrate that this set is finitely generated, it is a union of translated copies of a semigroup which can be computed explicitly via Hilbert bases computations. Related results can be derived for those right-hand-side vectors $b$ for which $P_A(b) \cap {\mathbb Z}^n$ has exactly $k$ solutions or fewer than $k$ solutions. (2) A computational complexity theory. We show that, when $n$, $k$ are fixed natural numbers, one can compute in polynomial time an encoding of \mbox{ Sg}_{\geq k}(A) as a multivariate generating function, using a short sum of rational functions. As a consequence, one can identify all right-hand-side vectors of bounded norm that have at least $k$ solutions. (3) Applications and computation for the $k$-Frobenius numbers. Using Generating functions we prove that for fixed $n,k$ the $k$-Frobenius number can be computed in polynomial time. This generalizes a well-known result for $k=1$ by R. Kannan. Using some adaptation of dynamic programming we show some practical computations of $k$-Frobenius numbers and their relatives

From conformal embeddings to quantum symmetries: an exceptional SU(4) example

We briefly discuss several algebraic tools that are used to describe the quantum symmetries of Boundary Conformal Field Theories on a torus. The starting point is a fusion category, together with an action on another category described by a quantum graph. For known examples, the corresponding modular invariant partition function, which is sometimes associated with a conformal embedding, provides enough information to recover the whole structure. We illustrate these notions with the example of the conformal embedding of SU(4) at level 4 into Spin(15) at level 1, leading to the exceptional quantum graph E4(SU(4)).Comment: 22 pages, 3 color figures. Version 2: We changed the color of figures (ps files) in such a way that they are still understood when converted to gray levels. Version 3: Several references have been adde

Few smooth d-polytopes with n lattice points

We prove that, for fixed n there exist only finitely many embeddings of Q-factorial toric varieties X into P^n that are induced by a complete linear system. The proof is based on a combinatorial result that for fixed nonnegative integers d and n, there are only finitely many smooth d-polytopes with n lattice points. We also enumerate all smooth 3-polytopes with at most 12 lattice points. In fact, it is sufficient to bound the singularities and the number of lattice points on edges to prove finiteness.Comment: 20+2 pages; major revision: new author, new structure, new result

AQFT from n-functorial QFT

There are essentially two different approaches to the axiomatization of quantum field theory (QFT): algebraic QFT, going back to Haag and Kastler, and functorial QFT, going back to Atiyah and Segal. More recently, based on ideas by Baez and Dolan, the latter is being refined to "extended" functorial QFT by Freed, Hopkins, Lurie and others. The first approach uses local nets of operator algebras which assign to each patch an algebra "of observables", the latter uses n-functors which assign to each patch a "propagator of states". In this note we present an observation about how these two axiom systems are naturally related: we demonstrate under mild assumptions that every 2-dimensional extended Minkowskian QFT 2-functor ("parallel surface transport") naturally yields a local net. This is obtained by postcomposing the propagation 2-functor with an operation that mimics the passage from the Schroedinger picture to the Heisenberg picture in quantum mechanics. The argument has a straightforward generalization to general pseudo-Riemannian structure and higher dimensions.Comment: 39 pages; further examples added: Hopf spin chains and asymptotic inclusion of subfactors; references adde

Understanding the Benefit of Magnetic Resonance-guided Adaptive Radiotherapy in Rectal Cancer Patients: a Single-centre Study.

AIMS: Neoadjuvant chemoradiotherapy followed by surgery is the mainstay of treatment for patients with rectal cancer. Standard clinical target volume (CTV) to planning target volume (PTV) margins of 10 mm are used to accommodate inter- and intrafraction motion of target. Treating on magnetic resonance-integrated linear accelerators (MR-linacs) allows for online manual recontouring and adaptation (MRgART) enabling the reduction of PTV margins. The aim of this study was to investigate motion of the primary CTV (CTVA; gross tumour volume and macroscopic nodes with 10 mm expansion to cover microscopic disease) in order to develop a simultaneous integrated boost protocol for use on MR-linacs. MATERIALS AND METHODS: Patients suitable for neoadjuvant chemoradiotherapy were recruited for treatment on MR-linac using a two-phase technique; only the five phase 1 fractions on MR-linac were used for analysis. Intrafraction motion of CTVA was measured between pre-treatment and post-treatment MRI scans. In MRgART, isotropically expanded pre-treatment PTV margins from 1 to 10 mm were rigidly propagated to post-treatment MRI to determine overlap with 95% of CTVA. The PTV margin was considered acceptable if overlap was >95% in 90% of fractions. To understand the benefit of MRgART, the same methodology was repeated using a reference computed tomography planning scan for pre-treatment imaging. RESULTS: In total, nine patients were recruited between January 2018 and December 2020 with T3a-T4, N0-N2, M0 disease. Forty-five fractions were analysed in total. The median motion across all planes was 0 mm, demonstrating minimal intrafraction motion. A PTV margin of 3 and 5mm was found to be acceptable in 96 and 98% of fractions, respectively. When comparing to the computed tomography reference scan, the analysis found that PTV margins to 5 and 10 mm only acceptably covered 51 and 76% of fractions, respectively. CONCLUSION: PTV margins can be reduced to 3-5 mm in MRgART for rectal cancer treatment on MR-linac within an simultaneous integrated boost protocol