Multiscale geometric analysis of three-dimensional data

Three-dimensional volumetric data are becoming increasingly available in a wide range of scientific and technical disciplines. With the right tools, we can expect such data to yield valuable insights about many important systems in our three-dimensional world.; This work presents new multiscale geometric tools for both analysis and synthesis of 3-D data that may be scattered or observed in voxel arrays. The data of interest is typically very noisy, and may contain one-dimensional structures such as line segments and filaments and/or two-dimensional structures such as plane patches and smooth surfaces. These tools mainly rely on two kinds of transforms. The 3-D Beamlet transform offers the collection of line integrals along a strategic multiscale set of line segments (Beamlets) running through the image at different orientations, positions, and lengths, while the 3-D Planelet transform computes a collection of plane integrals using a strategic multiscale set of plane patches (Planelets).; We present different strategies and algorithms for computing the Beamlet and Planelet transforms, direct evaluations as well as fast FFT-based methods. We compare the different algorithms with respect to their accuracy, speed, and cache memory usage. We also present backprojection and inversion procedures using iterative methods and preconditioning.; We present several basic applications for these tools, for example in finding faint structures buried in noisy data. We discuss in some detail the application of these tools for an important problem in astronomy, the analysis of 3-D Galaxy Catalogs, and show some promising results.