An Extended Symbolic-Arithmetic Model for Teaching Double-Black Removal with Rotation in Red-Black Trees

Double-black (DB) nodes have no place in red-black (RB) trees. So when DB nodes are formed, they are immediately removed. The removal of DB nodes that cause rotation and recoloring of other connected nodes poses greater challenges in the teaching and learning of RB trees. To ease this difficulty, this paper extends our previous work on the symbolic arithmetic algebraic (SA) method for removing DB nodes. The SA operations that are given as, Red + Black = Black; Black - Black = Red; Black + Black = DB; and DB - Black = Black removes DB nodes and rebalances black heights in RB trees. By extension, this paper projects three SA mathematical equations, namely, general symbolic arithmetic rule; partial symbolic arithmetic rule1; and partial symbolic arithmetic rule2. The removal of a DB node ultimately affects black heights in RB trees. To balance black heights using the SA equations, all the RB tree cases, namely, LR, RL, LL, and RR, were considered in this work; and the position of the nodes connected directly or indirectly to the DB node was also tested. In this study, to balance a RB tree, the issues considered w.r.t. the different cases of the RB tree were i) whether a DB node has an inner, outer, or both inner and outer black nephews; or ii) whether a DB node has an inner, outer or both inner and outer red nephews. The nephews r and x in this work are the children of the sibling s to a DB, and further up the tree, the parent p of a DB is their grandparent g. Thus, r and x have indirect relationships to a DB at the point of formation of the DB node. The novelty of the SA equations is in their effectiveness in the removal of DB that involves rotation of nodes as well as the recoloring of nodes along any simple path so as to balance black heights in a tree.

Temporal compressive edge imaging enabled by a lensless diffuser camera

Lensless imagers based on diffusers or encoding masks enable high-dimensional imaging from a single shot measurement and have been applied in various applications. However, to further extract image information such as edge detection, conventional post-processing filtering operations are needed after the reconstruction of the original object images in the diffuser imaging systems. Here, we present the concept of a temporal compressive edge detection method based on a lensless diffuser camera, which can directly recover a time sequence of edge images of a moving object from a single-shot measurement, without further post-processing steps. Our approach provides higher image quality during edge detection, compared with the conventional post-processing method. We demonstrate the effectiveness of this approach by both numerical simulation and experiments. The proof-of-concept approach can be further developed with other image post-process operations or versatile computer vision assignments toward task-oriented intelligent lensless imaging systems.