We propose a deep learning-based method for object detection in UAV-borne thermal images that have the capability of observing scenes in both day and night. Compared with visible images, thermal images have lower requirements for illumination conditions, but they typically have blurred edges and low contrast. Using a boundary-aware salient object detection network, we extract the saliency maps of the thermal images to improve the distinguishability. Thermal images are augmented with the corresponding saliency maps through channel replacement and pixel-level weighted fusion methods. Considering the limited computing power of UAV platforms, a lightweight combinational neural network ComNet is used as the core object detection method. The YOLOv3 model trained on the original images is used as a benchmark and compared with the proposed method. In the experiments, we analyze the detection performances of the ComNet models with different image fusion schemes. The experimental results show that the average precisions (APs) for pedestrian and vehicle detection have been improved by 2%~5% compared with the benchmark without saliency map fusion and MobileNetv2. The detection speed is increased by over 50%, while the model size is reduced by 58%. The results demonstrate that the proposed method provides a compromise model, which has application potential in UAV-borne detection tasks.

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Divergence functions are the non-symmetric “distance” on the manifold, Μθ, of parametric probability density functions over a measure space, ( Χ,μ). Classical information geometry prescribes, on Μθ: (i) a Riemannian metric given by the Fisher information; (ii) a pair of dual connections (giving rise to the family of α-connections) that preserve the metric under parallel transport by their joint actions; and (iii) a family of divergence functions ( α-divergence) defined on Μθ x？ Μθ, which induce the metric and the dual connections. Here, we construct an extension of this differential geometric structure from Μθ (that of parametric probability density functions) to the manifold, Μ, of non-parametric functions on X, removing the positivity and normalization constraints. The generalized Fisher information and α-connections on M are induced by an α-parameterized family of divergence functions, reflecting the fundamental convex inequality associated with any smooth and strictly convex function. The infinite-dimensional manifold, M, has zero curvature for all these α-connections; hence, the generally non-zero curvature of M can be interpreted as arising from an embedding of Μθ into Μ. Furthermore, when a parametric model (after a monotonic scaling) forms an affine submanifold, its natural and expectation parameters form biorthogonal coordinates, and such a submanifold is dually flat for α？= ± 1, generalizing the results of Amari’s α-embedding. The present analysis illuminates two different types of duality in information geometry, one concerning the referential status of a point (measurable function) expressed in the divergence function (“referential duality”) and the other concerning its representation under an arbitrary monotone scaling (“representational duality”).

The transformed $l_1$ penalty (TL1) functions are a one parameter family of bilinear transformations composed with the absolute value function. When acting on vectors, the TL1 penalty interpolates $l_0$ and $l_1$ similar to $l_p$ norm, where $p$ is in $(0,1)$. In our companion paper, we showed that TL1 is a robust sparsity promoting penalty in compressed sensing (CS) problems for a broad range of incoherent and coherent sensing matrices. Here we develop an explicit fixed point representation for the TL1 regularized minimization problem. The TL1 thresholding functions are in closed form for all parameter values. In contrast, the $l_p$ thresholding functions ($p$ is in $[0,1]$) are in closed form only for $p=0;1;1=2;2=3$, known as hard, soft, half, and 2/3 thresholding respectively. The TL1 threshold values differ in subcritical (supercritical) parameter regime where the TL1 threshold functions are continuous (discontinuous) similar to soft-thresholding (half-thresholding) functions. We propose TL1 iterative thresholding algorithms and compare them with hard and half thresholding algorithms in CS test problems. For both incoherent and coherent sensing matrices, a proposed TL1 iterative thresholding algorithm with adaptive subcritical and supercritical thresholds (TL1IT-s1 for short), consistently performs the best in sparse signal recovery with and without measurement noise.

In this paper, we formulate the deep residual network (ResNet) as a control problem of transport equation. In ResNet, the transport equation is solved along the characteristics. Based on this observation, deep neural network is closely related to the control problem of PDEs on manifold. We propose several models based on transport equation, Hamilton-Jacobi equation and Fokker-Planck equation. The discretization of these PDEs on point cloud is also discussed.

In this paper, we propose a novel low dimensional manifold model (LDMM) and apply it to some image processing problems. LDMM is based on the fact that the patch manifolds of many natural images have low dimensional structure. Based on this fact, the dimension of the patch manifold is used as a regularization to recover the image. The key step in LDMM is to solve a Laplace-Beltrami equation over a point cloud which is solved by the point integral method. The point integral method enforces the sample point constraints correctly and gives better results than the standard graph Laplacian. Numerical simulations in image denoising, inpainting and super-resolution problems show that LDMM is a powerful method in image processing.

A weighted and convex regularized nuclear norm model is introduced to construct a rank constrained linear transform on feature vectors of deep neural networks. The feature vectors of each class are modeled by a subspace, and the linear transform aims to enlarge the pairwise angles of the subspaces. The weight and convex regularization resolve the rank degeneracy of the linear transform. The model is computed by a dierence of convex function algorithm whose descent and convergence properties are analyzed. Numerical experiments are carried out in convolutional neural networks on CAFFE platform for 10 class handwritten digit images (MNIST) and small object color images (CIFAR-10) in the public domain. The transformed feature vectors improve the accuracy of the network in the regime of low dimensional features subsequent to principal component analysis. The feature transform is independent of the network structure, and can be applied without retraining the feature extraction layers of the network.

Let $q=p^{e}$ be a prime power and $\ell$ be an integer with $0\leq \ell\leq e-1$. The $\ell$-Galois hull of classical linear codes is a generalization of the Euclidean hull and Hermitian hull. We provide a necessary and sufficient condition under which a codeword of a GRS code or an extended GRS code belongs to its $\ell$-Galois dual code, generalizing both the Euclidean case and Hermitian case in the literature. By using four different tools: 1) the norm mapping from $\mathbb{F}_{q}^{\ast}$ to $\mathbb{F}_{p^{\ell}}^{\ast}$; 2) the direct product of two cyclic subgroups; 3) the coset decomposition of a cyclic group; 4) an additive subgroup of $\mathbb{F}_{q}$ and its cosets, we construct eleven families of $q$-ary MDS codes with $\ell$-Galois hulls of arbitrary dimensions, and give the related eleven families of $[[n,k,d;c]]_{q}$ entanglement-assisted quantum error-correcting codes (EAQECCs) with relatively large minimum distance in the sense that $2d=n-k+2+c$. We show that developing the theory of $\ell$-Galois hulls of $q$-ary MDS codes in this paper enables us to obtain new $q$-ary EAQECCs with different kinds of length sets via different $\ell$, where $2\ell\mid e$.

Inspired by the graph Laplacian and the point integral method, we introduce a novel weighted graph Laplacian method to compute a smooth interpolation function on a point cloud in high dimensional space. The numerical results in semi-supervised learning and image inpainting show that the weighted graph Laplacian is a reliable and efficient interpolation method. In addition, it is easy to implement and faster than graph Laplacian.

We demonstrate how path integrals often used in problems of theoretical physics can be adapted to provide a machinery for performing Bayesian inference in function spaces. Such inference comes about naturally in the study of inverse problems of recovering continuous (infinite dimensional) coefficient functions from ordinary or partial differential equations, a problem which is typically ill-posed. Regularization of these problems using L2 function spaces (Tikhonov regularization) is equivalent to Bayesian probabilistic inference, using a Gaussian prior. The Bayesian interpretation of inverse problem regularization is useful since it allows one to quantify and characterize error and degree of precision in the solution of inverse problems, as well as examine assumptions made in solving the problem—namely whether the subjective choice of regularization is compatible with prior knowledge. Using path-integral formalism, Bayesian inference can be explored through various perturbative techniques, such as the semiclassical approximation, which we use in this manuscript. Perturbative path-integral approaches, while offering alternatives to computational approaches like Markov-Chain-Monte-Carlo (MCMC), also provide natural starting points for MCMC methods that can be used to refine approximations. In this manuscript, we illustrate a path-integral formulation for inverse problems and demonstrate it on an inverse problem in membrane biophysics as well as inverse problems in potential theories involving the Poisson equation.