ABSTRACT
Multi-resolution is an important property for any approximation algorithm. Whether one is compressing satellite images, trying to solve Partial Differential Equations (PDEs), or modeling some irregular function, there is broad interest in multi-resolution approximation. Spline approximations, wavelets and Finite Element Methods (FEM), are the most commonly used multiresolution algorithms. Qualitatively, multi-resolution refers to the ability of an approximation to represent macroscopic features as well as very fine structure features. The general strategy of all multi-resolution algorithms is similar; however, due to some additional properties specific to each algorithm, some algorithms are more suited for specific applications. For example, wavelets-based approximation methods are well suited for image processing and FEM are generally accepted as being effective for solving PDEs. These approximation methodologies are also shown to be compatible with a wide variety of disciplines, such as continuous function approximation, dynamic system modeling, time series prediction and image processing. The multiresolution properties of these algorithms have led to broadly useful approximation approaches that have good local approximation properties for any given input-output data. Multi-resolution approximation enables separation of concern between model complexity, accuracy and compression ratio. It also enables accurate characterization of noise and uncertainty in the data, leading to robust approximation models. The basis functions to construct local approximations at various granularity levels are usually chosen from a space of polynomial functions, trigonometric functions, splines, or radial basis functions. Ideally, one may prefer to choose these basis functions, based on prior knowledge about the problem, or based solely upon local approximability.
For example, if the local behavior is known to be highly oscillatory, harmonic functions with frequencies tuned to capture the actual system oscillations can be used in the basis set. Alternatively, these local models can also be constructed intelligently by studying local behavior of the input-output map by the use of methods like Principal Component Analysis or Fourier Decomposition [58-60]. Such special functions may be introduced either by themselves, or to supplement a previously existing approximation. While such freedom provides great flexibility and can immensely improve the approximability, it generally prevents the basis functions from constituting a conforming space; i.e., the inter-element continuity of the approximation is not ensured. Following traditional approaches, the specific multi-resolution approximation basis functions used to obtain different local approximations cannot be independent of each other without introducing discontinuity across the boundary of contiguous local regions. Hence, there is a need for rigorous methods to merge different independent local approximations to obtain a desired order, globally continuous, approximation.
