We introduce graph wedgelets - a tool for data compression on graphs based on the representation of signals by piecewise constant functions on adaptively generated binary graph partitionings. The adaptivity of the partitionings, a key ingredient to obtain sparse representations of a graph signal, is realized in terms of recursive wedge splits adapted to the signal. For this, we transfer adaptive partitioning and compression techniques known for 2D images to general graph structures and develop discrete variants of continuous wedgelets and binary space partitionings. We prove that continuous results on best m-term approximation with geometric wavelets can be transferred to the discrete graph setting and show that our wedgelet representation of graph signals can be encoded and implemented in a simple way. Finally, we illustrate that this graph-based method can be applied for the compression of images as well.
翻译:我们引入图形 Wedgelets -- -- 图形中的数据压缩工具 -- -- 一种基于根据根据适应性生成的二进制图形分区的元素常量函数对信号的表示来对图形进行数据压缩的工具。 分区是获得图示信号的稀疏表示的一种关键成份, 其适应性通过循环式网格分割来实现, 适应性网格和压缩技术可适应于该信号。 为此, 我们将已知的2D图像的适应性分割和压缩技术传输到一般图形结构, 并开发连续 Wedgelets 和二进制空间分区的离散变体。 我们证明, 与几何波子的最佳中期近似结果可以传输到离散的图表设置, 并显示我们的图形信号的网格表示方式可以简单编码和实施。 最后, 我们说明, 基于图形的这个方法也可以用于图像的压缩。