Implicit networks are a class of neural networks whose outputs are defined by the fixed point of a parameterized operator. They have enjoyed success in many applications including natural language processing, image processing, and numerous other applications. While they have found abundant empirical success, theoretical work on its generalization is still under-explored. In this work, we consider a large family of implicit networks defined parameterized contractive fixed point operators. We show a generalization bound for this class based on a covering number argument for the Rademacher complexity of these architectures.

Cell tracking is an essential tool in live-cell imaging to determine single-cell features, such as division patterns or elongation rates. Unlike in common multiple object tracking, in microbial live-cell experiments cells are growing, moving, and dividing over time, to form cell colonies that are densely packed in mono-layer structures. With increasing cell numbers, following the precise cell-cell associations correctly over many generations becomes more and more challenging, due to the massively increasing number of possible associations. To tackle this challenge, we propose a fast parameter-free cell tracking approach, which consists of activity-prioritized nearest neighbor assignment of growing (expanding) cells and a combinatorial solver that assigns splitting mother cells to their daughters. As input for the tracking, Omnipose is utilized for instance segmentation. Unlike conventional nearest-neighbor-based tracking approaches, the assignment steps of our proposed method are based on a Gaussian activity-based metric, predicting the cell-specific migration probability, thereby limiting the number of erroneous assignments. In addition to being a building block for cell tracking, the proposed activity map is a standalone tracking-free metric for indicating cell activity. Finally, we perform a quantitative analysis of the tracking accuracy for different frame rates, to inform life scientists about a suitable (in terms of tracking performance) choice of the frame rate for their cultivation experiments, when cell tracks are the desired key outcome.

Hyperspectral images provide a rich representation of the underlying spectrum for each pixel, allowing for a pixel-wise classification/segmentation into different classes. As the acquisition of labeled training data is very time-consuming, unsupervised methods become crucial in hyperspectral image analysis. The spectral variability and noise in hyperspectral data make this task very challenging and define special requirements for such methods. Here, we present a novel unsupervised hyperspectral segmentation framework. It starts with a denoising and dimensionality reduction step by the well-established Minimum Noise Fraction (MNF) transform. Then, the Mumford-Shah (MS) segmentation functional is applied to segment the data. We equipped the MS functional with a novel robust distribution-dependent indicator function designed to handle the characteristic challenges of hyperspectral data. To optimize our objective function with respect to the parameters for which no closed form solution is available, we propose an efficient fixed point iteration scheme. Numerical experiments on four public benchmark datasets show that our method produces competitive results, which outperform three state-of-the-art methods substantially on three of these datasets.

Parallelism patterns (e.g., map or reduce) have proven to be effective tools for parallelizing high-performance applications. In this article, we study the recursive registration of a series of electron microscopy images - a time consuming and imbalanced computation necessary for nano-scale microscopy analysis. We show that by translating the image registration into a specific instance of the prefix scan, we can convert this seemingly sequential problem into a parallel computation that scales to over thousand of cores. We analyze a variety of scan algorithms that behave similarly for common low-compute operators and propose a novel work-stealing procedure for a hierarchical prefix scan. Our evaluation shows that by identifying a suitable and well-optimized prefix scan algorithm, we reduce time-to-solution on a series of 4,096 images spanning ten seconds of microscopy acquisition from over 10 hours to less than 3 minutes (using 1024 Intel Haswell cores), enabling derivation of material properties at nanoscale for long microscopy image series.

We propose a novel method for multi-phase segmentation of images based on high-dimensional local feature vectors. While the method was developed for the segmentation of extremely noisy crystal images based on localized Fourier transforms, the resulting framework is not tied to specific feature descriptors. For instance, using local spectral histograms as features, it allows for robust texture segmentation. The segmentation itself is based on the multi-phase Mumford-Shah model. Initializing the high-dimensional mean features directly is computationally too demanding and ill-posed in practice. This is resolved by projecting the features onto a low-dimensional space using principle component analysis. The resulting objective functional is minimized using a convexification and the Chambolle-Pock algorithm. Numerical results are presented, illustrating that the algorithm is very competitive in texture segmentation with state-of-the-art performance on the Prague benchmark and provides new possibilities in crystal segmentation, being robust to extreme noise and requiring no prior knowledge of the crystal structure.

In this paper the space of images is considered as a Riemannian manifold using the metamorphosis approach (see [M. I. Miller and L. Younes, Int. J. Comput. Vis., 41 (2001), pp. 61--84; A. Trouvé and L. Younes, SIAM J. Math. Anal., 37 (2005), pp. 17--59; and A. Trouvé and L. Younes, Found. Comput. Math., 5 (2005), pp. 173--198]), where the underlying Riemannian metric simultaneously measures the cost of image transport and intensity variation. A robust and effective variational time discretization of geodesics paths is proposed. This requires minimizing a discrete path energy consisting of a sum of consecutive image matching functionals over a set of image intensity maps and pairwise matching deformations. For square-integrable input images the existence of discrete, connecting geodesic paths defined as minimizers of this variational problem is shown. Furthermore, $\Gamma$-convergence of the underlying discrete path energy to the continuous path energy is proved. This includes a diffeomorphism property for the induced transport and the existence of a square-integrable weak material derivative in space and time. A spatial discretization via finite elements combined with an alternating descent scheme in the set of image intensity maps and the set of matching deformations is presented to approximate discrete geodesic paths numerically. Computational results underline the efficiency of the proposed approach and demonstrate important qualitative properties.

The problem of motion estimation and restoration of objects in a blurred video sequence is addressed in this paper. Fast movement of the objects, together with the aperture time of the camera, result in a motion-blurred image. The direct velocity estimation from this blurred video is inaccurate. On the other hand, an accurate estimation of the velocity of the moving objects is critical for restoration of motion-blurred video. Therefore, restoration needs accurate motion estimation and vice versa, and a joint process is called for. To address this problem we derive a novel model of the blurring process and propose a Mumford-Shah type of variational framework, acting on consecutive frames, for joint object deblurring and velocity estimation. The proposed procedure distinguishes between the moving object and the background and is accurate also close to the boundary of the moving object. Experimental results both on simulated and real data show the importance of this joint estimation and its superior performance when compared to the independent estimation of motion and restoration.

This paper presents a new algorithm based on the Mumford-Shah model for simultaneously detecting the edge features of two images and jointly estimating a consistent set of transformations to match them. Compared to the current asymmetric methods in the literature, this fully symmetric method allows one to determine one-to-one correspondences between the edge features of two images. The entire variational model is realized in a multiscale framework of the finite element approximation. The optimization process is guided by an estimation minimization-type algorithm and an adaptive generalized gradient flow to guarantee a fast and smooth relaxation. The algorithm is tested on T1 and T2 magnetic resonance image data to study the parameter setting. We also present promising results of four applications of the proposed algorithm: interobject monomodal registration, retinal image registration, matching digital photographs of neurosurgery with its volume data, and motion estimation for frame interpolation.