Coarse-to-Fine Image Reconstruction Based on Weighted Differential Features and Background Gauge Fields Bart Janssen, Remco Duits, and Luc Florack Eindhoven University of Technology, Dept. of Biomedical Engineering & Dept. of Mathematics and Computer Science {B.J.Janssen,R.Duits,L.M.J.Florack}@tue.nl Abstract. We propose an iterative approximate reconstruction method where we minimize the difference between reconstructions from subsets of multi scale measurements. To this end we interpret images not as scalar-valued functions but as sections through a fibered space. Informa- tion from previous reconstructions, which can be obtained at a coarser scale than the current one, is propagated by means of covariant deriva- tives on a vector bundle. The gauge field that is used to define the co- variant derivatives is defined by the previously reconstructed image. An advantage of using covariant derivatives in the variational formulation of the reconstruction method is that with the number of iterations the accuracy of the approximation increases. The presented reconstruction method allows for a reconstruction at a resolution of choice, which can also be used to speed up the approximation at a finer level. An appli- cation of our method to reconstruction from a sparse set of differential features of a scale space representation of an image allows for a weight- ing of the features based on the sensitivity of those features to noise. To demonstrate the method we apply it to the reconstruction from singular points of a scale space representation of an image. 1 Introduction Reconstruction from signal samples is a long standing problem in signal and im- age analysis [20]. We present a method for the approximation of a signal or im- age from its generalized samples, i.e. the samples are given on a non-equidistant grid and were obtained by means of spatially varying filters. Variational recon- struction of non-equidistant image samples has recently become of interest to the image compression community [9] where significant gains in reconstruction quality have been obtained by introducing anisotropic non-linear regularization strategies. In the scale space community a general interest in reconstruction from generalized samples has been there for quite some time [19, 18, 14, 12, 13]. We propose a method that produces an image that approximately satisfies all features. Features that are more robust to perturbations of the source image are given a higher weight, which steers the reconstruction method such that those X.-C. Tai et al. (Eds.): SSVM 2009, LNCS 5567, pp. 377–388, 2009. c  Springer-Verlag Berlin Heidelberg 2009