Non-regularised inverse finite element analysis for 3D traction force microscopy

The tractions that cells exert on a gel substrate from the observed displacements is an increasingly attractive and valuable information in biomedical experiments. The computation of these tractions requires in general the solution of an inverse problem. Here, we resort to the discretisation with finite elements of the associated direct variational formulation, and solve the inverse analysis using a least square approach. This strategy requires the minimisation of an error functional, which is usually regularised in order to obtain a stable system of equations with a unique solution. In this paper we show that for many common threedimensional geometries, meshes and loading conditions, this regularisation is unnecessary. In these cases, the computational cost of the inverse problem becomes equivalent to a direct finite element problem. For the non-regularised functional, we deduce the necessary and sufficient conditions that the dimensions of the interpolated displacement and traction fields must preserve in order to exactly satisfy or yield a unique solution of the discrete equilibrium equations. We apply the theoretical results to some illustrative examples and to real experimental data. Due to the relevance of the results for biologists and modellers, the article concludes with some practical rules that the finite element discretisation must satisfy.Peer ReviewedPostprint (author's final draft

Characterization of coorbit spaces with phase-space covers

We show that coorbit spaces can be characterized in terms of arbitrary phase-space covers, which are families of phase-space multipliers associated with partitions of unity. This generalizes previously known results for time-frequency analysis to include time-scale decompositions. As a by-product, we extend the existing results for time-frequency analysis to an irregular setting.Comment: 31 pages. Revised version (title slightly changed). Typos fixe

Surgery of spline-type and molecular frames

We prove a result about producing new frames for general spline-type spaces by piecing together portions of known frames. Using spline-type spaces as models for the range of certain integral transforms, we obtain results for time-frequency decompositions and sampling.Comment: 34 pages. Corrected typo

Computational medical imaging for total knee arthroplasty using visualitzation toolkit

This project is presented as a Master Thesis in the field of Civil Engineering, Biomedical specialization. As the project of an Erasmus exchange student, this thesis has been under supervision both the Universite Livre de Bruxelles and the Universitat Politecnica de Catalunya. The purpose of this thesis to put in practice all the knowledges acquired during this Master in Industrial Engineering in UPC and to be a support for medical staff in total knee arthoplasty procedures. Prof. Emmanuel Thienpont has been working for years as orthopaedic surgeon at the Hospital Sant Luc, Brussels. His years of work and research have been mainly focused on Total Knee Arthroplasty or TKA. During one of the most important steps of this procedure, the orthopaedic surgeon has to cut the head of the femur following two perpendicular cutting planes. Nevertheless, the orientation of these planes are directly dependant of the femur constitution. This Master Thesis has been conceived in order to offer the surgeon a tool to determine the proper direction planes in a previous step before the surgical procedure. This project pretends to give the surgeon an openfree computational platform to access to patient geometrical and physiological information before involving the subject in any invasive procedure

Sliding joints in 3D beams: conserving algorithms using the master-slave approach

This paper proposes two time-integration algorithms for motion of geometrically exact 3D beams under sliding contact conditions. The algorithms are derived using the socalled master–slave approach, in which constraint equations and the related time-integration of a system of differential and algebraic equations are eliminated by design. Specifically, we study conservation of energy and momenta when the sliding conditions on beams are imposed and discuss their algorithmic viability. Situations where the contact jumps to adjacent finite elements are analysed in detail and the results are tested on two representative numerical examples. It is concluded that an algorithmic preservation of kinematic constraint conditions is of utmost importance.Peer ReviewedPostprint (author's final draft

Multitaper estimation on arbitrary domains

Multitaper estimators have enjoyed significant success in estimating spectral densities from finite samples using as tapers Slepian functions defined on the acquisition domain. Unfortunately, the numerical calculation of these Slepian tapers is only tractable for certain symmetric domains, such as rectangles or disks. In addition, no performance bounds are currently available for the mean squared error of the spectral density estimate. This situation is inadequate for applications such as cryo-electron microscopy, where noise models must be estimated from irregular domains with small sample sizes. We show that the multitaper estimator only depends on the linear space spanned by the tapers. As a result, Slepian tapers may be replaced by proxy tapers spanning the same subspace (validating the common practice of using partially converged solutions to the Slepian eigenproblem as tapers). These proxies may consequently be calculated using standard numerical algorithms for block diagonalization. We also prove a set of performance bounds for multitaper estimators on arbitrary domains. The method is demonstrated on synthetic and experimental datasets from cryo-electron microscopy, where it reduces mean squared error by a factor of two or more compared to traditional methods.Comment: 28 pages, 11 figure