Operationally Optimal Vertex-Based Shape Coding

In this paper, we present a review of our work on rate-distortion based operationally optimal vertex-based lossy shape encoding techniques. We approximate the boundary of a given shape by a low order curve, such as a polygon, and consider the problem of nding the approximation which leads to the smallest distortion for a given number of bits. We also address the dual problem of nding the approximation which leads to the smallest bit rate for a given distortion. We consider two diierent classes of distortion measures. The rst class is based on the maximum operator (such as the maximum distance between a boundary and its approximation) and the second class is based on the summation operator (such as the total number of error pels between a boundary and its approximation). For the rst class, we derive a fast and operationally optimal scheme which is based on a shortest path algorithm for a weighted directed acyclic graph. For the second class, we propose a solution approach which is based on the Lagrange multiplier method, which uses the above mentioned shortest path algorithm. Examples illustrate the eeectiveness of the various techniques which are presented. 3 Figures a, b, c, and d show the nal result after the inner loop has been nished for v 1 , v 2 , v 3 and v 4 , for a maximum distance DM of one pel. The pels with a bold border indicate the original boundary and the other pels are the admissible control points, where the number of the associated boundary point is written inside it.. 4 The deenition of a state for a polygon and a second order B-spline curve. Figure (a) shows that for a polygon, knowing p k?1 makes the search for future control points independent from the control points already selected. Figure (b) shows that for a second order B-spline, knowing p k?2 and p k?1 makes the search for future control points independent from the 5 A second degree B-spline curve with 8 curve segments Q k and a double control point at the 6 Three diierent ways for deening the segment distortion for a polygon-based approximation of the original boundary. Figure (a) shows the maximum distance, Figure (b) the error area, and Figure (c) the maximum distance band concept.. 7 Three diierent ways for deening the segment distortion for a second order B-spline curve-based approximation of the original …