1. Finding Semantic Structures in Image Hierarchies using Laplacian Graph Energy Yi-Zhe Song 1 , Pablo Arbelaez 2 , Peter Hall 1 , Chuan Li 1 , Anupriya Balikai 1 1: Computer Science, University of Bath. UK. 2: Computer Vision Group, University of California at Berkeley, USA. We provide a filter for levels in a segmentation hierarchy Our filter:  works on many hierarchies (Berkeley UCMs, topological mean shift, quad trees …)  reduces the number of levels by an order of magnitude  leads to zero or negligible loss in semantic content It is of value in applications (e.g. tracking) where the size of the representation impacts in efficiency. From N levels to n << N Our method uses (a modified form of) Laplacian graph energy as a filter. To develop intuition. Build a graph from empty by randomly adding edges. The red line shows standard LE, black line our modified form. The central minima occurs when there is a single connected component that is most like a polygon (fully connected, each node has two arcs). Polygonal graphs have lower energy than random graphs. Polygonal graphs tend to correspond to semantic structure. See paper for a fuller development. Our method: We look for minima in the modified graph energy as we build a graph from bottom-level to top-level. A graph, G, of m nodes and n arcs has Laplacian is L = D-A, where A is adjacency matrix and D is degree matrix;  I are the eigenvalues of L. Our modification is for disconnected graphs of K components. This means highly disconnected graphs have much more energy than normally. We conclude: LE penalizes complexity . no edges complete graph nearest polygon Results The method is shown on an example picture (left). The table on the right show Measures obtained by comparing Segmentations using the Berkeley Standard database. See the paper for more results, and an application in tracking.