Geometric Transformation of Images 
Investigators: Stefan Horbelt, Maria Arrate Muñoz Barrutia 

Summary: We have designed a series of algorithms to implement geometric transformations of images in a way that minimizes the loss of information. The solutions are optimal in the leastsquares sense. 

Geometric transformations play an important role in biomedical image processing. Image translations, rotations, and/or scaling, are required for data visualization, reslicing of volumetric PET or MRI data sets, and image registration. The problem with most conventional interpolation procedures is that they produces noticeable artifacts (blocking, smoothing, and sometimes aliasing or ringing). Since image quality is a key concern, it is important to investigate techniques that result in less degradation.
The goal of this project is to develop new splinebased methods for implementing geometric transformations of images with the highestpossible quality. A special case of interest is the generation of a multiresolution representation of images (pyramids) for multiscale processing. Geometric methods are also very relevant for threedimensional data visualization, and for texture mapping. Our algorithms are designed to be optimal in the leastsquares sense, which is a principle that had not been used before in this particular context. 


Perspective texture mapping of a checkerboard pattern. (a) Point sampling of the source image; (b) new leastsquares solution. 
We have proposed a new, iterative texturemapping algorithm based on the idea of successive refinement. Our methods is optimal in a welldefined sense; it can deal with rather general (reversible) warping functions. Our new method compares favorably with the standard techniques in terms of image quality. It tends to produce sharper images while minimizing aliasing artifacts.
We have developed efficient algorithms for computing image pyramids (multiresolution approximation) that are optimal in the l_{p}norm. The case p = 1 was found to be of particular interest because it tends to simplify images while reducing ringing artifacts. 

Collaboration: Prof. Michael Unser 



[2]  S. Horbelt, P. Thévenaz, M. Unser, "Texture Mapping by Successive Refinement," Proceedings of the 2000 IEEE International Conference on Image Processing (ICIP'00), Vancouver BC, Canada, September 1013, 2000, vol. II, pp. 307310.

[3]  S. Horbelt, A. Muñoz Barrutia, T. Blu, M. Unser, "Spline Kernels for ContinuousSpace Image Processing," Proceedings of the TwentyFifth IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'00), Istanbul, Turkey, June 59, 2000, vol. IV, pp. 21912194.


