Jakob Andreas Bærentzen

David Brander, J. Andreas Bærentzen, Ann-Sofie Fisker et al.

We study the problem of identifying those cubic Bézier curves that are close in the L2 norm to planar elastic curves. The problem arises in design situations where the manufacturing process produces elastic curves; these are difficult to work with in a digital environment. We seek a sub-class of special Bézier curves as a proxy. We identify an easily computable quantity, which we call the lambda-residual, that accurately predicts a small L2 distance. We then identify geometric criteria on the control polygon that guarantee that a Bézier curve has lambda-residual below 0.4, which effectively implies that the curve is within 1 percent of its arc-length to an elastic curve in the L2 norm. Finally we give two projection algorithms that take an input Bézier curve and adjust its length and shape, whilst keeping the end-points and end-tangent angles fixed, until it is close to an elastic curve.

Kasra Arnavaz, Oswin Krause, Kilian Zepf et al.

Motivated by the challenging segmentation task of pancreatic tubular networks, this paper tackles two commonly encountered problems in biomedical imaging: Topological consistency of the segmentation, and expensive or difficult annotation. Our contributions are the following: a) We propose a topological score which measures both topological and geometric consistency between the predicted and ground truth segmentations, applied to model selection and validation. b) We provide a full deep-learning methodology for this difficult noisy task on time-series image data. In our method, we first use a semisupervised U-net architecture, applicable to generic segmentation tasks, which jointly trains an autoencoder and a segmentation network. We then use tracking of loops over time to further improve the predicted topology. This semi-supervised approach allows us to utilize unannotated data to learn feature representations that generalize to test data with high variability, in spite of our annotated training data having very limited variation. Our contributions are validated on a challenging segmentation task, locating tubular structures in the fetal pancreas from noisy live imaging confocal microscopy. We show that our semi-supervised model outperforms not only fully supervised and pre-trained models but also an approach which takes topological consistency into account during training. Further, our approach achieves a mean loop score of 0.808 for detecting loops in the fetal pancreas, compared to a U-net trained with clDice with mean loop score 0.762.

Martin O. Elingaard, Niels Aage, J. Andreas Bærentzen et al.

This paper presents a deep learning-based de-homogenization method for structural compliance minimization. By using a convolutional neural network to parameterize the mapping from a set of lamination parameters on a coarse mesh to a one-scale design on a fine mesh, we avoid solving the least square problems associated with traditional de-homogenization approaches and save time correspondingly. To train the neural network, a two-step custom loss function has been developed which ensures a periodic output field that follows the local lamination orientations. A key feature of the proposed method is that the training is carried out without any use of or reference to the underlying structural optimization problem, which renders the proposed method robust and insensitive wrt. domain size, boundary conditions, and loading. A post-processing procedure utilizing a distance transform on the output field skeleton is used to project the desired lamination widths onto the output field while ensuring a predefined minimum length-scale and volume fraction. To demonstrate that the deep learning approach has excellent generalization properties, numerical examples are shown for several different load and boundary conditions. For an appropriate choice of parameters, the de-homogenized designs perform within $7-25\%$ of the homogenization-based solution at a fraction of the computational cost. With several options for further improvements, the scheme may provide the basis for future interactive high-resolution topology optimization.

Jakeoung Koo, Anders B. Dahl, J. Andreas Bærentzen et al.

In computed tomography, the reconstruction is typically obtained on a voxel grid. In this work, however, we propose a mesh-based reconstruction method. For tomographic problems, 3D meshes have mostly been studied to simulate data acquisition, but not for reconstruction, for which a 3D mesh means the inverse process of estimating shapes from projections. In this paper, we propose a differentiable forward model for 3D meshes that bridge the gap between the forward model for 3D surfaces and optimization. We view the forward projection as a rendering process, and make it differentiable by extending recent work in differentiable rendering. We use the proposed forward model to reconstruct 3D shapes directly from projections. Experimental results for single-object problems show that the proposed method outperforms traditional voxel-based methods on noisy simulated data. We also apply the proposed method on electron tomography images of nanoparticles to demonstrate the applicability of the method on real data.