A Tree-based Dictionary Learning Framework

We propose a new outline for dictionary learning methods based on a hierarchical clustering of the training data. Through recursive application of a clustering method the data is organized into a binary partition tree representing a multiscale structure. The dictionary atoms are defined adaptively based on the data clusters in the partition tree. This approach can be interpreted as a generalization of the wavelet transform. The computational bottleneck of the procedure is then the chosen clustering method: when using K-means the method runs much faster than K-SVD. Thanks to the multiscale properties of the partition tree, our dictionary is structured: when using Orthogonal Matching Pursuit to reconstruct patches from a natural image, dictionary atoms corresponding to nodes being closer to the root node in the tree have a tendency to be used with greater coefficients.

A Region Based Easy Path Wavelet Transform For Sparse Image Representation

The Easy Path Wavelet Transform is an adaptive transform for bivariate functions (in particular natural images) which has been proposed in [1]. It provides a sparse representation by finding a path in the domain of the function leveraging the local correlations of the function values. It then applies a one dimensional wavelet transform to the obtained vector, decimates the points and iterates the procedure. The main drawback of such method is the need to store, for each level of the transform, the path which vectorizes the two dimensional data. Here we propose a variation on the method which consists of firstly applying a segmentation procedure to the function domain, partitioning it into regions where the variation in the function values is low; in a second step, inside each such region, a path is found in some deterministic way, i.e. not data-dependent. This circumvents the need to store the paths at each level, while still obtaining good quality lossy compression. This method is particularly well suited to encode a Region of Interest in the image with different quality than the rest of the image. [1] Gerlind Plonka. The easy path wavelet transform: A new adaptive wavelet transform for sparse representation of two-dimensional data. Multiscale Modeling & Simulation, 7(3):1474$-$1496, 2008.