A Survey on Non-rigid 3D Shape Analysis <s> Random 3D model synthesis <s> A circuit for controlling a display panel identifying malfunctions in an engine generator receives a plurality of electrical signals from the engine generator, each of which identifies a particular trouble. The electrical signal may be produced by closing a switch. It is caused to operate a latch that lights a light associated with the particular malfunction. Indications of other malfunctions are suppressed until the circuit is reset. A manual reset tests all lights and then leaves them off ready to respond. A power-up reset does not test lights but leaves all lights off ready to respond. The circuit is rendered especially appropriate for military use by hardening against radiation and against pulses of electromagnetic interference. <s> BIB001 </s> A Survey on Non-rigid 3D Shape Analysis <s> Random 3D model synthesis <s> Recent developments in elastic shape analysis (ESA) are motivated by the fact that it provides comprehensive frameworks for simultaneous registration, deformation, and comparison of shapes. These methods achieve computational efficiency using certain square-root representations that transform invariant elastic metrics into Euclidean metrics, allowing for applications of standard algorithms and statistical tools. For analyzing shapes of embeddings of $\mathbb{S}^2$ in $\mathbb{R}^3$, Jermyn et al. introduced square-root normal fields (SRNFs) that transformed an elastic metric, with desirable invariant properties, into the $\mathbb{L}^2$ metric. These SRNFs are essentially surface normals scaled by square-roots of infinitesimal area elements. A critical need in shape analysis is to invert solutions (deformations, averages, modes of variations, etc) computed in the SRNF space, back to the original surface space for visualizations and inferences. Due to the lack of theory for understanding SRNFs maps and their inverses, we take a numerical approach and derive an efficient multiresolution algorithm, based on solving an optimization problem in the surface space, that estimates surfaces corresponding to given SRNFs. This solution is found effective, even for complex shapes, e.g. human bodies and animals, that undergo significant deformations including bending and stretching. Specifically, we use this inversion for computing elastic shape deformations, transferring deformations, summarizing shapes, and for finding modes of variability in a given collection, while simultaneously registering the surfaces. We demonstrate the proposed algorithms using a statistical analysis of human body shapes, classification of generic surfaces and analysis of brain structures. <s> BIB002
Finally, we consider the problem of synthesizing arbitrary 3D models that are similar to, but note exactly the same as, a given collection of 3D models. We consider a collection of 398 human shapes BIB001 composed of multiple subjects in different poses. The first subject has 35 different poses including a neutral one. All other subjects have a neutral pose and a few other poses. We first compute the SRNF representations of these surfaces, perform statistical analysis in the SRNF shape space S q , using standard linear statistics such as Principal Component Analysis, and finally map the results to the surface space, F, using the proposed SRNF inversion algorithm. Formally, ifq is the mean and V i , i = 1, · · · , k are the leading eigenvectors in the space of SRNFs, then any arbitrary SRNF map q can be written as: where λ i are real coefficients. An arbitrary shape can then be synthesized by finding a surface f such that its SRNF is q. This can be efficiently computed using the SRNF inversion procedure described in Section 4.2.2 and detailed in BIB002 . Figure 8 -(a) shows the mean shape and the first five modes of variation. Figure 9 , on the other hand, shows 10 randomly syntehsized human body shapes. Observe that the statistical model can generate shapes of arbitrary subjects in arbitrary poses.