This paper proposes a further improved technique on the stochastic functional approach for randomly rough surface scattering. The original improved technique has been established in the previous paper [Waves in Random and Complex Media, vol.19, no.2, pp.181-215, 2009] as a novel numerical-analytical method for a Wiener analysis. By deriving modified hierarchy equations based on the diagonal approximation solution of random wavefields for a TM plane wave incidence or even for a TE plane wave incidence under large roughness, large slope or low grazing incidence, such a further improved technique can provide a large reduction of required computational resources, in comparison with the original improved technique. This paper shows that numerical solutions satisfy the optical theorem with very good accuracy, by using small computational resources.

When a monochromatic electromagnetic plane wave is incident on an infinitely extending surface with the translation invariance property, a curious phenomenon often takes place at a low grazing angle of incidence, at which the total wave field vanishes and a dark shadow appears. This paper looks for physical and mathematical reasons why such a shadow occurs. Three cases are considered: wave reflection by a flat interface between two media, diffraction by a periodic surface, and scattering from a homogeneous random surface. Then, it is found that, when a translation invariant surface does not support guided waves (eigen functions) propagating with real propagation constants, such the shadow always takes place, because the primary excitation disappears at a low grazing angle of incidence. At the same time, a shadow form of solution is proposed. Further, several open problems are given for future works.

The joint histogram of second order scale space differential invariants of natural images (including textures) is typically clustered about parabolic surface patches, whereas symmetrical elliptical patches (local convexities or concavities) are very rare and symmetrical hyperbolical patches also occur less frequently than parabolic patches. We trace the origin of this striking effect in the context of Gaussian random noise. For this case one may derive the joint histogram of curvedness and shape index analytically. The empirical observations are fully corroborated. In deriving these results we introduce a polar coordinate system in the space of second order scale space derivatives that turns out to be particularly useful in the study of the statistics of local curvature properties. The empirical observations apply also to non-Gaussian noise (e.g., Brownian noise) as well as to photographs of natural scenes. We discuss general arguments that help explain these observations.