Box Splines for Image Processing 
Investigators: Kunal Narayan Chaudhury 

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. 

Box splines are a multivariate extension of the polynomial Bsplines. 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 boxsplinebased 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 Bspline signal processing. Another interesting application is the reconstruction of color images from observations through a honeycomb filter.
The motivation is that several imageprocessing applications (e.g., denoising of biological images) call for spacevariant anisotropic filtering whereby the orientation and shape of the filters could be arbitrarily adapted to the local image characteristics. Due to their spacevariant nature, such filters cannot be implemented using Fourierbased 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 spaceinvariant anisotropic Gaussian filtering have been developed in the past, the spacevariant ones are subject of current research. We believe that one potential application of box splines is efficient scale and rotationadaptive 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 Gaussianlike filtering; the computational cost is O(1) per pixel, irrespective of the size/orientation of the smoothing kernel, but the filtering is spaceinvariant, meaning nonadaptive. We are also aware of two algorithms that can perform efficient scaleadaptive filtering, but which are not orientable. The first uses polynomial Bsplines 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. 

We specifically addressed the possibility of designing an efficient spacevariant elliptical filtering algorithm using box splines. In particular, these elliptical box splines filters were parametrized by a scalevector (corresponding to the widths of the constituent Bsplines) 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 Bsplines. 

Collaborations: Prof. Michael Unser, Maria Arrate Muñoz Barrutia (CIMA) 


Funding: Swiss National Science Foundation 


[2]  K.N. Chaudhury, Z. Püspöki, A. Muñoz Barrutia, D. Sage, M. Unser, "Fast Detection of Cells Using a Continuously Scalable MexicanHatLike Template," Proceedings of the Seventh IEEE International Symposium on Biomedical Imaging: From Nano to Macro (ISBI'10), Rotterdam, The Netherlands, April 1417, 2010, in press.


