Regularized theta liftings and periods of modular functions.

In this paper, we use regularized theta liftings to construct weak Maass forms of weight 1/2 as lifts of weak Maass forms of weight 0. As a special case we give a new proof of some of recent results of Duke, Toth and the third author on cycle integrals of the modular j-invariant and extend these to any congruence subgroup. Moreover, our methods allow us to settle the open question of a geometric interpretation for periods of j along infinite geodesics in the upper half plane. In particular, we give the `central value' of the (non-existent) `L-function' for j. The key to the proofs is the construction of a kind of simultaneous Green function for both the CM points and the geodesic cycles, which is of independent interest

Structured learning of assignment models for neuron reconstruction to minimize topological errors

© 20xx IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.Structured learning provides a powerful framework for empirical risk minimization on the predictions of structured models. It allows end-to-end learning of model parameters to minimize an application specific loss function. This framework is particularly well suited for discrete optimization models that are used for neuron reconstruction from anisotropic electron microscopy (EM) volumes. However, current methods are still learning unary potentials by training a classifier that is agnostic about the model it is used in. We believe the reason for that lies in the difficulties of (1) finding a representative training sample, and (2) designing an application specific loss function that captures the quality of a proposed solution. In this paper, we show how to find a representative training sample from human generated ground truth, and propose a loss function that is suitable to minimize topological errors in the reconstruction. We compare different training methods on two challenging EM-datasets. Our structured learning approach shows consistently higher reconstruction accuracy than other current learning methods.Peer ReviewedPostprint (author's final draft

Large Scale Image Segmentation with Structured Loss based Deep Learning for Connectome Reconstruction

We present a method combining affinity prediction with region agglomeration, which improves significantly upon the state of the art of neuron segmentation from electron microscopy (EM) in accuracy and scalability. Our method consists of a 3D U-NET, trained to predict affinities between voxels, followed by iterative region agglomeration. We train using a structured loss based on MALIS, encouraging topologically correct segmentations obtained from affinity thresholding. Our extension consists of two parts: First, we present a quasi-linear method to compute the loss gradient, improving over the original quadratic algorithm. Second, we compute the gradient in two separate passes to avoid spurious gradient contributions in early training stages. Our predictions are accurate enough that simple learning-free percentile-based agglomeration outperforms more involved methods used earlier on inferior predictions. We present results on three diverse EM datasets, achieving relative improvements over previous results of 27%, 15%, and 250%. Our findings suggest that a single method can be applied to both nearly isotropic block-face EM data and anisotropic serial sectioned EM data. The runtime of our method scales linearly with the size of the volume and achieves a throughput of about 2.6 seconds per megavoxel, qualifying our method for the processing of very large datasets

TED: a tolerant edit distance for segmentation evaluation

© . This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/In this paper, we present a novel error measure to compare a computer-generated segmentation of images or volumes against ground truth. This measure, which we call Tolerant Edit Distance (TED), is motivated by two observations that we usually encounter in biomedical image processing: (1) Some errors, like small boundary shifts, are tolerable in practice. Which errors are tolerable is application dependent and should be explicitly expressible in the measure. (2) Non-tolerable errors have to be corrected manually. The effort needed to do so should be reflected by the error measure. Our measure is the minimal weighted sum of split and merge operations to apply to one segmentation such that it resembles another segmentation within specified tolerance bounds. This is in contrast to other commonly used measures like Rand index or variation of information, which integrate small, but tolerable, differences. Additionally, the TED provides intuitive numbers and allows the localization and classification of errors in images or volumes. We demonstrate the applicability of the TED on 3D segmentations of neurons in electron microscopy images where topological correctness is arguable more important than exact boundary locations. Furthermore, we show that the TED is not just limited to evaluation tasks. We use it as the loss function in a max-margin learning framework to find parameters of an automatic neuron segmentation algorithm. We show that training to minimize the TED, i.e., to minimize crucial errors, leads to higher segmentation accuracy compared to other learning methods.Peer ReviewedPostprint (author's final draft