normal for depth prediction
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Introduction
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Monocular depth prediction aims to predict distances between scene objects and the camera from a single monocular image. It is a critical task for understanding the 3D scene, such as recognizing a 3D object and parsing a 3D scene.

![Example results of ground truth (the first row), our method (the second row) and Hu [@Hu2018RevisitingSI] (the third row). By enforcing the geometric constraints of virtual normals, our reconstructed 3D point cloud can represent better shape of sofa (see the left part) and the recovered surface normal has much less errors (see green parts) even though the absolute relative error (rel) of our predicted depth is only slightly better than Hu ($0.108$ *vs.* $0.115$). []{data-label="fig:pcd normal dorn ours"}](./Fig/pcd_normal_dorn_ours){width="47.00000%"}

Although the monocular depth prediction is an ill-posed problem because many 3D scenes can be projected to the same 2D image, many deep convolutional neural networks (DCNN) based methods [@eigen2015predicting; @eigen2014depth; @fu2018deep; @guo2018learning; @laina2016deeper; @li2018deep; @saxena2009make3d] have achieved impressive results by using a large amount of labelled data, thus taking advantage of prior knowledge in labelled data to solve the ambiguity.

These methods typically formulate the optimization problem as either point-wise regression or classification. That is, 