An optimal polygonal boundary encoding scheme in the rate distortion sense

Abstract

In this paper, we present fast and efficient methods for the lossy encoding of object boundaries that are given as eight-connect chain codes. We approximate the boundary by a polygon, and consider the problem of finding the polygon which leads to the smallest distortion for a given number of bits. We also address the dual problem of finding the polygon which leads to the smallest bit rate for a given distortion. We consider two different classes of distortion measures. The first class is based on the maximum operator and the second class is based on the summation operator. For the first class, we derive a fast and optimal scheme that is based on a shortest path algorithm for a weighted directed acyclic graph. For the second class we propose a solution approach that is based on the Lagrange multiplier method, which uses the above-mentioned shortest path algorithm. Since the Lagrange multiplier method can only find solutions on the convex hull of the operational rate distortion function, we also propose a tree-pruning-based algorithm that can find all the optimal solutions. Finally, we present results of the proposed schemes using objects from the Miss America sequence.

Data reduction of large vector graphics

“…The processing time is To sum up, the time complexity of the reduced search algorithm for multiple-object min-e problem is defined by the first step, and it is between O(N) and O(N 2 ). This is better than the O(N 3 ) of the full search, and the O (N 2 logN) of the method proposed in [41].…”

“…Two approaches have been proposed in [41] for the problem. The first approach is based on the Lagrangian multipliers method, which uses the DP algorithm for the shortest path in a directed acyclic graph.…”

“…In the case of L ¥ error measure, the problem reduces to the single-object min-# problem as the optimization can be solved for every object independently [41]. In the case of additive error measures (L 1 , L 2 , etc.…”

“…In the case of additive error measures (L 1 , L 2 , etc. ), on the other hand, the problem is not trivial [41]. Fortunately, in practical applications we mostly have to deal with error measure L ¥ in the case of min-# problem.…”

“…The optimal solution have been introduced by Schuster and Katsaggelos [41] but the algorithm has time complexity of O(N 2 )-O(N 3 ) depending on the number of segments. This can be suitable for the encoding of object contours for MPEG-4 standard [42] but it can be too slow in the case of large vector maps.…”