A mathematical theory of resolution limits for super-resolution of positive sources

The superresolving capacity for number and location recoveries in the super-resolution of positive sources is analyzed in this work. Specifically, we introduce the computational resolution limit for respectively the number detection and location recovery in the one-dimensional super-resolution problem and quantitatively characterize their dependency on the cutoff frequency, signal-to-noise ratio, and the sparsity of the sources. As a direct consequence, we show that targeting at the sparest positive solution in the super-resolution already provides the optimal resolution order. These results are generalized to multi-dimensional spaces. Our estimates indicate that there exist phase transitions in the corresponding reconstructions, which are confirmed by numerical experiments. Our theory fills in an important puzzle towards fully understanding the super-resolution of positive sources.

Meta Knowledge Condensation for Federated Learning

Existing federated learning paradigms usually extensively exchange distributed models at a central solver to achieve a more powerful model. However, this would incur severe communication burden between a server and multiple clients especially when data distributions are heterogeneous. As a result, current federated learning methods often require a large number of communication rounds in training. Unlike existing paradigms, we introduce an alternative perspective to significantly decrease the communication cost in federate learning. In this work, we first introduce a meta knowledge representation method that extracts meta knowledge from distributed clients. The extracted meta knowledge encodes essential information that can be used to improve the current model. As the training progresses, the contributions of training samples to a federated model also vary. Thus, we introduce a dynamic weight assignment mechanism that enables samples to contribute adaptively to the current model update. Then, informative meta knowledge from all active clients is sent to the server for model update. Training a model on the combined meta knowledge without exposing original data among different clients can significantly mitigate the heterogeneity issues. Moreover, to further ameliorate data heterogeneity, we also exchange meta knowledge among clients as conditional initialization for local meta knowledge extraction. Extensive experiments demonstrate the effectiveness and efficiency of our proposed method. Remarkably, our method outperforms the state-of-the-art by a large margin (from $74.07\%$ to $92.95\%$) on MNIST with a restricted communication budget (i.e. 10 rounds).

Nearly optimal resolution estimate for the two-dimensional super-resolution and a new algorithm for direction of arrival estimation with uniform rectangular array

In this paper, we develop a new technique to obtain nearly optimal estimates of the computational resolution limits introduced in Appl. Comput. Harmon. Anal. 56 (2022) 402-446; IEEE Trans. Inf. Theory 67(7) (2021) 4812-4827; Inverse Probl. 37(10) (2021) 104001 for two-dimensional super-resolution problems. Our main contributions are fivefold: (i) Our work improves the resolution estimate for number detection and location recovery in two-dimensional super-resolution problems to nearly optimal; (ii) As a consequence, we derive a stability result for a sparsity-promoting algorithm in two-dimensional super-resolution problems (or Direction of Arrival problems (DOA)). The stability result exhibits the optimal performance of sparsity promoting in solving such problems; (iii) Our techniques pave the way for improving the estimate for resolution limits in higher-dimensional super-resolutions to nearly optimal; (iv) Inspired by these new techniques, we propose a new coordinate-combination-based model order detection algorithm for two-dimensional DOA estimation and theoretically demonstrate its optimal performance, and (v) we also propose a new coordinate-combination-based MUSIC algorithm for super-resolving sources in two-dimensional DOA estimation. It has excellent performance and enjoys many advantages compared to the conventional DOA algorithms. The coordinate-combination idea seems to be a promising way for multi-dimensional DOA estimation.

Dynamic super-resolution in particle tracking problems

Particle tracking in biological imaging is concerned with reconstructing the trajectories, locations, or velocities of the targeting particles. The standard approach of particle tracking consists of two steps: first reconstructing statically the source locations in each time step, and second applying tracking techniques to obtain the trajectories and velocities. In contrast, the dynamic reconstruction seeks to simultaneously recover the source locations and velocities from all frames, which enjoys certain advantages. In this paper, we provide a rigorous mathematical analysis for the resolution limit of reconstructing source number, locations, and velocities by general dynamical reconstruction in particle tracking problems, by which we demonstrate the possibility of achieving super-resolution for the dynamic reconstruction. We show that when the location-velocity pairs of the particles are separated beyond certain distances (the resolution limits), the number of particles and the location-velocity pair can be stably recovered. The resolution limits are related to the cut-off frequency of the imaging system, signal-to-noise ratio, and the sparsity of the source. By these estimates, we also derive a stability result for a sparsity-promoting dynamic reconstruction. In addition, we further show that the reconstruction of velocities has a better resolution limit which improves constantly as the particles moving. This result is derived by an observation that the inherent cut-off frequency for the velocity recovery can be viewed as the total observation time multiplies the cut-off frequency of the imaging system, which may lead to a better resolution limit as compared to the one for each diffraction-limited frame. It is anticipated that this observation can inspire new reconstruction algorithms that improve the resolution of particle tracking in practice.

A measurement decoupling based fast algorithm for super-resolving point sources with multi-cluster structure

We consider the problem of resolving closely spaced point sources in one dimension from their Fourier data in a bounded domain. Classical subspace methods (e.g., MUSIC algorithm, Matrix Pencil method, etc.) show great superiority in resolving closely spaced sources, but their computational cost is usually heavy. This is especially the case for point sources with a multi-cluster structure which requires processing large-sized data matrix resulted from highly sampled measurements. To address this issue, we propose a fast algorithm termed D-MUSIC, based on a measurement decoupling strategy. We demonstrate theoretically that for point sources with a known cluster structure, their measurement can be decoupled into local measurements of each of the clusters by solving a system of linear equations that are obtained by using a multipole basis. We further develop a subsampled MUSIC algorithm to detect the cluster structure and utilize it to decouple the global measurement. In the end, the MUSIC algorithm was applied to each local measurement to resolve point sources therein. Compared to the standard MUSIC algorithm, the proposed algorithm has comparable super-resolving capability while having a much lower computational complexity.

Active Learning for Point Cloud Semantic Segmentation via Spatial-Structural Diversity Reasoning

The expensive annotation cost is notoriously known as a main constraint for the development of the point cloud semantic segmentation technique. In this paper, we propose a novel active learning-based method to tackle this problem. Dubbed SSDR-AL, our method groups the original point clouds into superpoints and selects the most informative and representative ones for label acquisition. We achieve the selection mechanism via a graph reasoning network that considers both the spatial and structural diversity of the superpoints. To deploy SSDR-AL in a more practical scenario, we design a noise aware iterative labeling scheme to confront the "noisy annotation" problem introduced by previous dominant labeling methods in superpoints. Extensive experiments on two point cloud benchmarks demonstrate the effectiveness of SSDR-AL in the semantic segmentation task. Particularly, SSDR-AL significantly outperforms the baseline method when the labeled sets are small, where SSDR-AL requires only $5.7\%$ and $1.9\%$ annotation costs to achieve the performance of $90\%$ fully supervised learning on S3DIS and Semantic3D datasets, respectively.

Mathematical foundation of sparsity-based multi-illumination super-resolution

It is well-known that the resolution of a traditional optical imaging system is limited by the so-called Rayleigh limit, which is of several hundreds of nanometers. By employing fluorescence techniques, modern microscopic methods can resolve point scatterers separated much lower than the Rayleigh limit. Localization-based fluorescence subwavelength imaging techniques such as PALM and STORM can achieve a spatial resolution of several tens of nanometers. However, these techniques have limited temporal resolution as they require tens of thousands of exposures. Employing sparsity-based models and recovery algorithms is a natural way to reduce the number of exposures and hence obtain high temporal resolution. Recently, a new multi-illumination imaging technique called Brownian Excitation Amplitude Modulation microscopy (BEAM) is introduced. BEAM achieves a threefold resolution improvement by applying a compressive sensing recovery algorithm over only few frames. Motivated by BEAM, our aim in this paper is to pioneer the mathematical foundation for sparsity-based multi-illumination super-resolution. We consider several diffraction-limited images from samples exposed to different illumination patterns and recover the source by considering the sparsest solution. We estimate the minimum separation distance between point scatterers so that they could be stably recovered. By this estimation, we reveal the dependence of the resolution on the cut-off frequency of the imaging system, the SNR, the sparsity of point scatterers, and the incoherence of illumination patterns. Our theory particularly highlights the importance of the high incoherence of illumination patterns in enhancing the resolution. It also demonstrates that super-resolution can be achieved using sparsity-based multi-illumination imaging with very few frames, whereby the spatio-temporal super-resolution becomes possible.

Onsite Non-Line-of-Sight Imaging via Online Calibrations

There has been an increasing interest in deploying non-line-of-sight (NLOS) imaging systems for recovering objects behind an obstacle. Existing solutions generally pre-calibrate the system before scanning the hidden objects. Onsite adjustments of the occluder, object and scanning pattern require re-calibration. We present an online calibration technique that directly decouples the acquired transients at onsite scanning into the LOS and hidden components. We use the former to directly (re)calibrate the system upon changes of scene/obstacle configurations, scanning regions, and scanning patterns whereas the latter for hidden object recovery via spatial, frequency or learning based techniques. Our technique avoids using auxiliary calibration apparatus such as mirrors or checkerboards and supports both laboratory validations and real-world deployments.

Unsupervised Domain Adaptive Person Re-Identification via Human Learning Imitation

Unsupervised domain adaptive person re-identification has received significant attention due to its high practical value. In past years, by following the clustering and finetuning paradigm, researchers propose to utilize the teacher-student framework in their methods to decrease the domain gap between different person re-identification datasets. Inspired by recent teacher-student framework based methods, which try to mimic the human learning process either by making the student directly copy behavior from the teacher or selecting reliable learning materials, we propose to conduct further exploration to imitate the human learning process from different aspects, \textit{i.e.}, adaptively updating learning materials, selectively imitating teacher behaviors, and analyzing learning materials structures. The explored three components, collaborate together to constitute a new method for unsupervised domain adaptive person re-identification, which is called Human Learning Imitation framework. The experimental results on three benchmark datasets demonstrate the efficacy of our proposed method.