Summary

In this project, we investigate the possibility of filtering an image with elliptic windows of varying size, elongation, and orientation, at a fixed computational cost per pixel using box splines.

Introduction

Box splines are a multivariate extension of the polynomial B-splines. To date, there are only few applications of box splines in image processing and computer graphics. Noteworthy is the work of Richter who introduced two box-spline-based algorithms for tomographic reconstruction. Asahi et al. developed digital filtering algorithms for dealing with box splines, which are largely inspired by our earlier work on B-spline signal processing. Another interesting application is the reconstruction of color images from observations through a honeycomb filter.
The motivation is that several image-processing applications ( e.g. , denoising of biological images) call for space-variant anisotropic filtering whereby the orientation and shape of the filters could be arbitrarily adapted to the local image characteristics. Due to their space-variant nature, such filters cannot be implemented using Fourier-based methods; the only option is to filter the image locally with sampled Gaussian kernels, which proves to be extremely slow for wide kernels. Though fast recursive solutions for space-invariant anisotropic Gaussian filtering have been developed in the past, the space-variant ones are subject of current research. We believe that one potential application of box splines is efficient scale- and rotation-adaptive filtering. While we are not aware of anyone having used box splines for that purpose, there have been multiple efforts in the literature to develop fast directional filtering techniques. For instance, Smeulders proposed a fast recursive solution for directional Gaussian-like filtering; the computational cost is O (1) per pixel, irrespective of the size/orientation of the smoothing kernel, but the filtering is space-invariant, meaning non-adaptive. We are also aware of two algorithms that can perform efficient scale-adaptive filtering, but which are not orientable. The first uses polynomial B-splines and was developed by us for the fast computation of the Continuous Wavelet Transform. The second originated in Computer Graphics and essentially implements a rectangular smoothing using repeated integration. The challenge is to include directionality as well and to be able to orient and rescale the filters adaptively, depending on the local context.

Main Contribution

We specifically addressed the possibility of designing an efficient space-variant elliptical filtering algorithm using box splines. In particular, these elliptical box splines filters were parametrized by a scale-vector (corresponding to the widths of the constituent B-splines) that controlled their size, elongation, and orientation. This made the filter capable of adapting to local image features allowing for smoothing while preserving directional features (such as lines and edges) at the same time. Even more interesting were the facts that these box splines converged to a Gaussian as the order increases; and that we could arbitrarily steer these elliptical filters by suitable rescaling of the constituent B-splines.