The most common scheme for estimating the orientation field of space-varying directional textures is based on a local nonlinear spatial averaging of the gradient field. This leads to locally biased orientations, especially in regions of nonlinearly distributed convergence, asymmetrically distributed curvature, or geometry superposition. In this paper, we propose an orientation estimation framework that is invariant toward the local geometry. Instead of applying the spatial averaging in the initial space, it is performed in a flattened space which is obtained from a parametric space transformation model deduced from the reconstruction of local hypersurfaces. The flattening can either be applied to the initial gradient field or to the intensity image, on which the flattened gradient field is computed. The process is iterated in order to define more accurate space transformations leading to refined orientation estimation. The computational efficiency is improved without noticeable loss of orientation accuracy by a patch-based approach. Experiments, both on synthetic as well as real-life fingerprint and fibrous material images, exhibit enhanced performance in comparison with conventional methods such as the structure tensor method.

Methods of seismic horizon reconstruction based on the solution of a partial derivative equation are generally robust to noise. However, they can only reconstruct horizons on rectangular domains when using fast iterative methods which is inconvenient in regions of uncertainty or irrelevant data in the seismic image. In this paper, we propose a method to reconstruct seismic horizons on any polygonal domain based on a succession of Schwarz-Christoffel transformations which conserve the angles locally. Since these are conformal transformations, the form of the equation is preserved and the solution on the polygonal domain corresponds to the solution on the rectangular domain using the grid data transformed from the rectangular into the polygonal domain.

Estimating the local orientations of a directional structure may prove to be complicated especially for complex geometries for which conventional methods, including those based on the gradient, are unsuitable. In this paper, we propose a recursive method of orientation estimation based on the reconstruction of hypersurfaces and the definition of a model of the structure in these so-called complex areas. The estimation proves to be more accurate when executed on the flattened data and driven to the initial referential. This process recursively leads to a more accurate orientation field estimation. When applied to synthetic data, the method reduces significantly the estimation bias. For real directional textures, the resulting orientation field coincides better with the perceptual orientation of the directional structures.