Analysis of and workarounds for element reversal for a finite element-based algorithm for warping triangular and tetrahedral meshes

We consider an algorithm called FEMWARP for warping triangular and tetrahedral finite element meshes that computes the warping using the finite element method itself. The algorithm takes as input a two- or three-dimensional domain defined by a boundary mesh (segments in one dimension or triangles in two dimensions) that has a volume mesh (triangles in two dimensions or tetrahedra in three dimensions) in its interior. It also takes as input a prescribed movement of the boundary mesh. It computes as output updated positions of the vertices of the volume mesh. The first step of the algorithm is to determine from the initial mesh a set of local weights for each interior vertex that describes each interior vertex in terms of the positions of its neighbors. These weights are computed using a finite element stiffness matrix. After a boundary transformation is applied, a linear system of equations based upon the weights is solved to determine the final positions of the interior vertices. The FEMWARP algorithm has been considered in the previous literature (e.g., in a 2001 paper by Baker). FEMWARP has been succesful in computing deformed meshes for certain applications. However, sometimes FEMWARP reverses elements; this is our main concern in this paper. We analyze the causes for this undesirable behavior and propose several techniques to make the method more robust against reversals. The most successful of the proposed methods includes combining FEMWARP with an optimization-based untangler.Comment: Revision of earlier version of paper. Submitted for publication in BIT Numerical Mathematics on 27 April 2010. Accepted for publication on 7 September 2010. Published online on 9 October 2010. The final publication is available at http://www.springerlink.co

Tracking Cell Signals in Fluorescent Images

In this paper we present the techniques for tracking cell signal in GFP (Green Fluorescent Protein) images of growing cell colonies. We use such tracking for both data extraction and dynamic modeling of intracellular processes. The techniques are based on optimization of energy functions, which simultaneously determines cell correspondences, while estimating the mapping functions. In addition to spatial mappings such as affine and Thin-Plate Spline mapping, the cell growth and cell division histories must be estimated as well. Different levels of joint optimization are discussed. The most unusual tracking feature addressed in this paper is the possibility of one-to-two correspondences caused by cell division. A novel extended softassign algorithm for solutions of one-to-many correspondences is detailed in this paper. The techniques are demonstrated on three sets of data: growing bacillus Subtillus and e-coli colonies and a developing plant shoot apical meristem. The techniques are currently used by biologists for data extraction and hypothesis formation

DeepMatching: Hierarchical Deformable Dense Matching

We introduce a novel matching algorithm, called DeepMatching, to compute dense correspondences between images. DeepMatching relies on a hierarchical, multi-layer, correlational architecture designed for matching images and was inspired by deep convolutional approaches. The proposed matching algorithm can handle non-rigid deformations and repetitive textures and efficiently determines dense correspondences in the presence of significant changes between images. We evaluate the performance of DeepMatching, in comparison with state-of-the-art matching algorithms, on the Mikolajczyk (Mikolajczyk et al 2005), the MPI-Sintel (Butler et al 2012) and the Kitti (Geiger et al 2013) datasets. DeepMatching outperforms the state-of-the-art algorithms and shows excellent results in particular for repetitive textures.We also propose a method for estimating optical flow, called DeepFlow, by integrating DeepMatching in the large displacement optical flow (LDOF) approach of Brox and Malik (2011). Compared to existing matching algorithms, additional robustness to large displacements and complex motion is obtained thanks to our matching approach. DeepFlow obtains competitive performance on public benchmarks for optical flow estimation