Iterative Methods at Lower Precision

Since numbers in the computer are represented with a fixed number of bits, loss of accuracy during calculation is unavoidable. At high precision where more bits (e.g. 64) are allocated to each number, round-off errors are typically small. On the other hand, calculating at lower precision, such as half (16 bits), has the advantage of being much faster. This research focuses on experimenting with arithmetic at different precision levels for large-scale inverse problems, which are represented by linear systems with ill-conditioned matrices. We modified the Conjugate Gradient Method for Least Squares (CGLS) and the Chebyshev Semi-Iterative Method (CS) with Tikhonov regularization to do arithmetic at lower precision using the MATLAB chop function, and we ran experiments on applications from image processing and compared their performance at different precision levels. We concluded that CGLS is a more stable algorithm, but overflows easily due to the computation of inner products, while CS is less likely to overflow but it has more erratic convergence behavior. When the noise level is high, CS outperforms CGLS by being able to run more iterations before overflow occurs; when the noise level is close to zero, CS appears to be more susceptible to accumulation of round-off errors

ME-PCN: Point Completion Conditioned on Mask Emptiness

Point completion refers to completing the missing geometries of an object from incomplete observations. Main-stream methods predict the missing shapes by decoding a global feature learned from the input point cloud, which often leads to deficient results in preserving topology consistency and surface details. In this work, we present ME-PCN, a point completion network that leverages `emptiness' in 3D shape space. Given a single depth scan, previous methods often encode the occupied partial shapes while ignoring the empty regions (e.g. holes) in depth maps. In contrast, we argue that these `emptiness' clues indicate shape boundaries that can be used to improve topology representation and detail granularity on surfaces. Specifically, our ME-PCN encodes both the occupied point cloud and the neighboring `empty points'. It estimates coarse-grained but complete and reasonable surface points in the first stage, followed by a refinement stage to produce fine-grained surface details. Comprehensive experiments verify that our ME-PCN presents better qualitative and quantitative performance against the state-of-the-art. Besides, we further prove that our `emptiness' design is lightweight and easy to embed in existing methods, which shows consistent effectiveness in improving the CD and EMD scores.Comment: Accepted to ICCV 2021; typos correcte