This paper explores the direct motion estimation problem assuming that video-rate depth information is available, from either stereo cameras or other sensors. We use these depth measurements in the traditional linear brightness constraint equations, and we introduce a new depth constraint equation. As a result, estimation of certain types of motion, such as translation in depth and rotations out of the image plane, becomes more robust. We derive linear brightness and depth change constraint equations that govern the velocity field in 3-D for both perspective and orthographic camera projection models. These constraints are integrated jointly over image regions according to a rigid-body motion model, yielding a single linear system to robustly track 3D object pose. Results are shown for tracking the pose effaces in sequences of synthetic and real images.

Multipliers have been recently introduced by P. Balazs as operators for Bessel sequences and frames in Hilbert spaces. These are opera- tors that combine (frame-like) analysis, a multiplication with a fixed sequence ( called the symbol) and synthesis. One of the last extensions of frames is weighted and controlled frames that introduced by P.Balazs, J-P. Antoine and A. Grybos to improve the numerical efficiency of iterative algorithms for inverting the frame operator. Also g-frames are the most popular generalization of frames that include almost all of the frame extensions. In this manuscript the concept of the controlled g- frames will be defined and we will show that controlled g-frames are equivalent to g-frames and so the controlled operators C and C' can be used as preconditions in applications. Also the multiplier operator for this family of operators will be introduced and some of its properties will be shown.