Automated digital watermarking is an excellent technique for prevention of online digital data from theft and piracy. With current advances in internet, network and social media technologies, protecting online data theft and piracy to preserve brand ownership remains a top priority. A watermark is a pattern inserted into a digital image, audio or video file which identifies a file’s copyright information. Common types of signals to watermark are text, digital images, audio music clips and videos. Digital images are stored as arrays/matrices on computers which allows matrix-algebra based methods for image processing. We have applied mathematical modeling techniques using singular value decomposition (SVD) for digital watermarking of visual data. We wrote Python-based codes to digitize images and embedded copyright information into visual images using SVD based methods. We find that the embedded watermark is tamper resistant and the watermark could be retrieved from manipulated greyscale images subject to rotation and compression distortions. Our preliminary studies using greyscale images suggest that SVD based digital watermarking methods are robust and can be used to verify and authenticate data ownership. Digital watermarking can be used to prevent copyright infringement and data theft online. This project integrates advanced application of linear-algebra based mathematical methods with Python based programming to provide solutions to real-world problems of current interest.

Mathematical modeling and simulations of the gait and pose of mobile robots find important applications for mobile robot design for process automation, industrial applications, and deriving algorithms for walking styles. Here, we have used Webot robotic simulations and mathematical modeling methods to study, analyze and interpret the gait and pose of six-legged (hexapod) and four-legged (quadruped) robots that mimic dog-like movements. Hexapod and quadruped mobile robots are perfect for deployment for outer space exploration as these mobile robots can traverse uneven terrain and use artificial intelligence to plan their safe foothold positions to navigate their environment. We have analyzed the simulated gait and pose of the hexapod robot using rigid body inverse kinematics and symmetry analysis. The hexapod mantis insect-shaped robot motion in the simulations can be expressed as linear combinations of rigid translational and rotational motion. The hexapod robot moves using an alternating tripod-like gait wherein three legs move at a time while the other three remain stationary. The hexapod robot has greater dynamic stability for uneven terrain and can move more legs as compared to a quadruped robot. This research project serves as an elegant platform for applications of simulations and mathematical methods for studying the gait and stability of legged mobile robots that are relevant for the safety design of computer-controlled walking robots for radioactive waste management, space exploration, and for the design of mobile robots working in nuclear power stations and finding other industrial applications.