Depth information is important for autonomous systems to perceive environments and estimate their own state. Traditional depth estimation methods, like structure from motion and stereo vision matching, are built on feature correspondences of multiple viewpoints. Meanwhile, the predicted depth maps are sparse. Inferring depth information from a single image (monocular depth estimation) is an ill-posed problem. With the rapid development of deep neural networks, monocular depth estimation based on deep learning has been widely studied recently and achieved promising performance in accuracy. Meanwhile, dense depth maps are estimated from single images by deep neural networks in an end-to-end manner. In order to improve the accuracy of depth estimation, different kinds of network frameworks, loss functions and training strategies are proposed subsequently. Therefore, we survey the current monocular depth estimation methods based on deep learning in this review. Initially, we conclude several widely used datasets and evaluation indicators in deep learning-based depth estimation. Furthermore, we review some representative existing methods according to different training manners: supervised, unsupervised and semi-supervised. Finally, we discuss the challenges and provide some ideas for future researches in monocular depth estimation.

The fractional Laplacian (− ) γ /2 commutes with the primary coordination transformations in the Euclidean space R d : dilation, translation and rotation, and has tight link to splines, fractals and stable Levy processes. For 0 < γ < d, its inverse is the classical Riesz potential I γ which is dilationinvariant and translation-invariant. In this work, we investigate the functional properties (continuity, decay and invertibility) of an extended class of differential operators that share those invariance properties. In particular, we extend the definition of the classical Riesz potential I γ to any noninteger number γ larger than d and show that it is the unique left-inverse of the fractional Laplacian (− ) γ /2 which is dilation-invariant and translationinvariant. We observe that, for any 1 ≤ p ≤ ∞ and γ ≥ d(1 − 1/ p), there exists a Schwartz function f such that I γ f is not p-integrable. We then introduce the new unique left-inverse I γ,p of the fractional Laplacian (− ) γ /2 with the property that I γ,p is dilation-invariant (but not translation-invariant) and that I γ,p f is p-integrable for any Schwartz function f . We finally apply that linear operator I γ,p with p = 1 to solve the stochastic partial differential equation (− ) γ /2 = w with white Poisson noise as its driving term w.

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In his 1932 paper, Carleman proposed a linearization method to transform a given finite-dimensional nonlinear system with analytic right-hand into an equivalent infinitedimensional linear system with (usually) unbounded operators. Finite truncation of the transformed system has been used to study dynamic properties, learning, and control of nonlinear systems. One of the remaining outstanding problems in this context is quantifying the effectiveness of such finitely truncated models. Intuitively, one expects that as the truncation length increases, the truncated system will approximate the original nonlinear more effectively. In this paper, we provide explicit error bounds and prove that the trajectory of the truncated system stays close to that of the original nonlinear system over a quantifiable time interval. This is particularly important in applications, such as Model Predictive Control, to choose proper truncation lengths for a given sampling period and employ the resulting truncated system as a good approximation of the nonlinear system. Several examples are discussed to support our theoretical estimates.