On Properties of Semipositive Cones and Simplicial Cones

For a given nonsingular $n\times n$ matrix $A$, the cone $S_{A}=\{x:Ax\geq 0\}$ , and its subcone $K_A$ lying on the positive orthant, called as semipositive cone, are considered. If the interior of the semipositive cone $K_A$ is not empty, then $A$ is named as semipositive matrix. It is known that $K_A$ is a proper polyhedral cone. In this paper, it is proved that $S_{A}$ is a simplicial cone and properties of its extremals are analyzed. An one-one relation between simplicial cones and invertible matrices is established. For a proper cone $K$ in $\mathbb{R}^n$, $\pi(K)$ denotes the collection of $n\times n$ matrices that leave $K$ invariant. For a given minimally semipositive matrix (no column-deleted submatrix is semipositive) $A$, it is shown that the invariant cone $\pi(K_A)$ is a simplicial cone.

Misinterpreting Changes of Illuminant on Complex Mondrian Patterns

Ratios of cone excitations from different surfaces of the same coloured scene are almost invariant under illuminance changes, and might provide the cue by which the visual system discriminates illuminant from non-illuminant changes in coloured scenes. Previous work with pairs of surfaces showed that observers were able to detect small, naturally occurring, violations in these ratios (Nascimento and Foster, 1995 Perception24 Supplement, 60 – 61). In the present study, sensitivity to violations was assessed with more complex, Mondrian patterns. In a two-interval forced-choice experiment, two colour transformations of the same Mondrian pattern were compared by observers. The patterns comprised 7 × 7 coloured patches. Each patch was a simulation of a Munsell surface, and the whole pattern was illuminated by a Planckian illuminant of variable colour temperature. In one of the intervals only the colour temperature of the illuminant changed; in the other, the same colour-temperature change was made, but, in addition, the spectral reflectances of the surfaces were adjusted such that all cone ratios were exactly preserved for the three classes of cone. The task was to identify which of the intervals contained the pure illuminant change. Observers could reliably discriminate the intervals but systematically interpreted colour changes with invariant cone-ratios as being illuminant changes, with a probability that increased as the degree of violation of invariance increased. Performance depended mainly on long-wavelength-sensitive cones, less on medium-wavelength-sensitive cones, and little or not at all on short-wavelength-sensitive cones or luminance signals. Cone-excitation ratios, although sometimes unreliable, appear to be the dominant cue for deciding on the nature of colour changes.