In the arts and sciences, as well as in our daily lives, symmetry has made a profound and lasting impact. Likewise, a computational treatment of symmetry and group theory (the ultimate mathematical formalization of symmetry) has the potential to play an important role in computational sciences. Though the term computational symmetry was formally defined a decade ago by the first author, referring to algorithmic treatment of symmetries, seeking symmetry from digital data has been attempted for over four decades. Computational symmetry on real world data turns out to be challenging enough that, after decades of effort, a fully automated symmetry–savvy system remains elusive for real world applications. The recent resurging interests in computational symmetry for computer vision and computer graphics applications have shown promising results. Recognizing the fundamental relevance and potential power that computational symmetry affords, we offer this survey to the computer vision and computer graphics communities. This survey provides a succinct summary of the relevant mathematical theory, a historic perspective of some important symmetry-related ideas, a partial yet timely report on the state of the arts symmetry detection algorithms along with its first quantitative benchmark, a diverse set of real world applications, suggestions for future directions and a comprehensive reference list.

Gait is an attractive biometric for vision-based human identification. Previous work on existing public data sets has shown that shape cues yield improved recognition rates compared to pure motion cues. However, shape cues are fragile to gross appearance variations of an individual, for example, walking while carrying a ball or a backpack. We introduce a novel, spatiotemporal shape variation-based frieze pattern (SVB frieze pattern) representation for gait, which captures motion information over time. The SVB frieze pattern represents normalized frame difference over gait cycles. Rows/columns of the vertical/horizontal SVB frieze pattern contain motion variation information augmented by key frame information with body shape. A temporal symmetry map of gait patterns is also constructed and combined with vertical/horizontal SVB frieze patterns for measuring the dissimilarity between gait sequences. Experimental results show that our algorithm improves gait recognition performance on sequences with and without gross differences in silhouette shape. We demonstrate superior performance of this computational framework over previous algorithms using shape cues alone on both CMU MoBo and UoS HumanID gait databases.

Quantitative evaluation of human stability using foot pressure/force measurement hardware and motion capture (mocap) technology is expensive, time consuming, and restricted to the laboratory. We propose a novel image-based method to estimate three key components for stability computation: Center of Mass (CoM), Base of Support (BoS), and Center of Pressure (CoP). Furthermore, we quantitatively validate our image-based methods for computing two classic stability measures, CoMtoCoP and CoMtoBoS distances, against values generated directly from laboratory-based sensor output (ground truth) using a publicly available, multi-modality (mocap, foot pressure, two-view videos), ten-subject human motion dataset. Using Leave One Subject Out (LOSO) cross-validation, experimental results show: 1) our image-based CoM estimation method (CoMNet) consistently outperforms state-of-the-art inertial sensor-based CoM estimation techniques; 2) stability computed by our image-based method combined with insole foot pressure sensor data produces consistent, strong, and statistically significant correlation with ground truth stability measures (CoMtoCoP r = 0.79 p < 0.001, CoMtoBoS r = 0.75 p < 0.001); 3) our fully image-based estimation of stability produces consistent, positive, and statistically significant correlation on the two stability metrics (CoMtoCoP r = 0.31 p < 0.001, CoMtoBoS r = 0.22 p < 0.043). Our study provides promising quantitative evidence for the feasibility of image-based stability evaluation in natural environments.

Pose stability analysis is the key to understanding locomotion and control of body equilibrium, with applications in numerous fields such as kinesiology, medicine, and robotics. In biomechanics, Center of Pressure (CoP) is used in studies of human postural control and gait. We propose and validate a novel approach to learn CoP from pose of a human body to aid stability analysis. More specifically, we propose an end-to-end deep learning architecture to regress foot pressure heatmaps, and hence the CoP locations, from 2D human pose derived from video. We have collected a set of long (5min +) choreographed Taiji (Tai Chi) sequences of multiple subjects with synchronized foot pressure and video data. The derived human pose data and corresponding foot pressure maps are used jointly in training a convolutional neural network with residual architecture, named PressNET. Cross-subject validation results show promising performance of PressNET, significantly outperforming the baseline method of K-Nearest Neighbors. Furthermore, we demonstrate that our computation of center of pressure (CoP) from PressNET is not only significantly more accurate than those obtained from the baseline approach but also meets the expectations of corresponding lab-based measurements of stability studies in kinesiology.

Quantitative evaluation of human stability using foot pressure/force measurement hardware and motion capture (mocap) technology is expensive, time consuming, and restricted to the laboratory. We propose a novel image-based method to estimate three key components for stability computation: Center of Mass (CoM), Base of Support (BoS), and Center of Pressure (CoP). Furthermore, we quantitatively validate our image-based methods for computing two classic stability measures, CoMtoCoP and CoMtoBoS distances, against values generated directly from laboratory-based sensor output (ground truth) using a publicly available, multi-modality (mocap, foot pressure, two-view videos), ten-subject human motion dataset. Using Leave One Subject Out (LOSO) cross-validation, experimental results show: 1) our image-based CoM estimation method (CoMNet) consistently outperforms state-of-the-art inertial sensor-based CoM estimation techniques; 2) stability computed by our image-based method combined with insole foot pressure sensor data produces consistent, strong, and statistically significant correlation with ground truth stability measures (CoMtoCoP r = 0.79 p < 0.001, CoMtoBoS r = 0.75 p < 0.001); 3) our fully image-based estimation of stability produces consistent, positive, and statistically significant correlation on the two stability metrics (CoMtoCoP r = 0.31 p < 0.001, CoMtoBoS r = 0.22 p < 0.043). Our study provides promising quantitative evidence for the feasibility of image-based stability evaluation in natural environments.