Fig. 1

The oriented gradient operator G(x,y,z,θ,φ,r): At location (x,y,z), the local neighborhood is defined by a sphere with radius r. An equatorial plane \(\vec {n}\) (shaded green) along with its normal vector \(\vec {t}\) splits the sphere into two half spheres. The one above \(\vec {n}\) is shaded yellow and the one below is shaded blue. We histogram the intensity values of voxels that fall in the yellow and blue half spheres respectively. Finally, the local gradient magnitude is set to the χ² distance between the yellow and blue histograms in the direction \(\vec {t}\) at scale r