In an earlier paper [J. Opt. Soc. Am. A 2, 1769 (1985)] a class of nonlinear image processing operators was introduced in which each photoreceptor creates a nonnegative point-spread function whose center height is proportional to its quantum catch and whose volume is constant, so that the local spatial-summation area varies inversely with the local quantum catch. These constant-volume (CV) operators are designed to maximize spatial resolution in the presence of photon noise. In the previous paper it was shown that when CV operators are applied to deterministic images, they produce a surprising range of effects that are reminiscent of human vision, including Mach bands and Weber's-law behavior. In this paper the consequences of applying CV operators to images containing Poisson noise are analyzed. It is shown that a fixed-parameter CV operator can duplicate the global qualitative properties of spatial vision for retinal illuminances ranging from absolute threshold to 1000 Td. Although there are fundamental obstacles to modeling the exact quantitative properties of human spatial vision by CV operators, these operators seem likely to be useful in machine vision.

The space-charge driven envelope instability can be of great danger in high intensity accelerators and was studied using a two-dimensional (2D) envelope model and three-dimensional (3D) macroparticle simulations before. In this paper, we study the instability for a bunched beam using a three-dimensional envelope model in a periodic solenoid and radio-frequency (rf) focusing channel and a periodic quadrupole and rf focusing channel. This study shows that when the transverse zero current phase advance is below 90°, the beam envelope can still become unstable if the longitudinal zero current phase advance is beyond 90°. For the transverse zero current phase advance beyond 90°, the instability stopband width becomes larger with the increase of the longitudinal focusing strength and even shows different structure from the 2D case when the longitudinal zero current phase advance is beyond 90°. Breaking the symmetry of two longitudinal focusing rf cavities and the symmetry between the horizontal focusing and the vertical focusing in the transverse plane in the periodic quadrupole and rf channel makes the instability stopband broader. This suggests that a more symmetric accelerator lattice design might help reduce the range of the envelope instability in parameter space.