On the reconstruction of planar lattice-convex sets from the covariogram

A finite subset $K$ of $\mathbb{Z}^d$ is said to be lattice-convex if $K$ is the intersection of $\mathbb{Z}^d$ with a convex set. The covariogram $g_K$ of $K\subseteq \mathbb{Z}^d$ is the function associating to each u \in \integer^d the cardinality of $K\cap (K+u)$. Daurat, G\'erard, and Nivat and independently Gardner, Gronchi, and Zong raised the problem on the reconstruction of lattice-convex sets $K$ from $g_K$. We provide a partial positive answer to this problem by showing that for $d=2$ and under mild extra assumptions, $g_K$ determines $K$ up to translations and reflections. As a complement to the theorem on reconstruction we also extend the known counterexamples (i.e., planar lattice-convex sets which are not reconstructible, up to translations and reflections) to an infinite family of counterexamples.Comment: accepted in Discrete and Computational Geometr

Preliminary evidence that both blue and red light can induce alertness at night

<p>Abstract</p> <p>Background</p> <p>A variety of studies have demonstrated that retinal light exposure can increase alertness at night. It is now well accepted that the circadian system is maximally sensitive to short-wavelength (blue) light and is quite insensitive to long-wavelength (red) light. Retinal exposures to blue light at night have been recently shown to impact alertness, implicating participation by the circadian system. The present experiment was conducted to look at the impact of both blue and red light at two different levels on nocturnal alertness. Visually effective but moderate levels of red light are ineffective for stimulating the circadian system. If it were shown that a moderate level of red light impacts alertness, it would have had to occur via a pathway other than through the circadian system.</p> <p>Methods</p> <p>Fourteen subjects participated in a within-subject two-night study, where each participant was exposed to four experimental lighting conditions. Each night each subject was presented a high (40 lx at the cornea) and a low (10 lx at the cornea) diffuse light exposure condition of the same spectrum (blue, λ_max= 470 nm, or red, λ_max= 630 nm). The presentation order of the light levels was counterbalanced across sessions for a given subject; light spectra were counterbalanced across subjects within sessions. Prior to each lighting condition, subjects remained in the dark (< 1 lx at the cornea) for 60 minutes. Electroencephalogram (EEG) measurements, electrocardiogram (ECG), psychomotor vigilance tests (PVT), self-reports of sleepiness, and saliva samples for melatonin assays were collected at the end of each dark and light periods.</p> <p>Results</p> <p>Exposures to red and to blue light resulted in increased beta and reduced alpha power relative to preceding dark conditions. Exposures to high, but not low, levels of red and of blue light significantly increased heart rate relative to the dark condition. Performance and sleepiness ratings were not strongly affected by the lighting conditions. Only the higher level of blue light resulted in a reduction in melatonin levels relative to the other lighting conditions.</p> <p>Conclusion</p> <p>These results support previous findings that alertness may be mediated by the circadian system, but it does not seem to be the only light-sensitive pathway that can affect alertness at night.</p

Changes in recognition memory over time: an ERP investigation into vocabulary learning

Although it seems intuitive to assume that recognition memory fades over time when information is not reinforced, some aspects of word learning may benefit from a period of consolidation. In the present study, event-related potentials (ERP) were used to examine changes in recognition memory responses to familiar and newly learned (novel) words over time. Native English speakers were taught novel words associated with English translations, and subsequently performed a Recognition Memory task in which they made old/new decisions in response to both words (trained word vs. untrained word), and novel words (trained novel word vs. untrained novel word). The Recognition task was performed 45 minutes after training (Day 1) and then repeated the following day (Day 2) with no additional training session in between. For familiar words, the late parietal old/new effect distinguished old from new items on both Day 1 and Day 2, although response to trained items was significantly weaker on Day 2. For novel words, the LPC again distinguished old from new items on both days, but the effect became significantly larger on Day 2. These data suggest that while recognition memory for familiar items may fade over time, recognition of novel items, conscious recollection in particular may benefit from a period of consolidation

Stability in Discrete Tomography:Linear Programming, Additivity and Convexity

The problem of reconstructing finite subsets of the integer lattice from X-rays has been studied in discrete mathematics and applied in several fields like image processing, data security, electron microscopy. In this paper we focus on the stability of the reconstruction problem for some lattice sets. First we show some theoretical bounds for additive sets, and a numerical experiment is made by using linear programming to deal with stability for convex sets

Reconstruction of Q-convex lattice sets

We present a class od lattice sets for which there are unique determination and a polynomial time reconstruction algorithm by X-rays in suitable directions. Moreover many reconstructions of different classes of lattice sets having convexity/connnectivity constrains can be seen as particular cases of the former case

Reconstruction of Discrete Sets From Two or More Projections in any Direction

We study the problem of reconstructing discrete sets satisfying properties of connectivity and convexity by projections taken along many directions. The members of the class we consider are called Q-convexes. We design a polynomial time reconstruction algorithm for this class

Determination of Q-convex bodies by X-rays

The class of Q-convex bodies is defined, and the uniqueness result proved by Gardner and McMullen in 1980 for planar convex bodies is extended to this new class

Reconstruction of Q-convex lattice sets

We study the reconstruction of special lattice sets from X-rays when some convexity constraints are imposed on the sets. Two aspects are relevant for a satisfactory reconstruction: the unique determination of the set by its X-rays and the existence of a polynomial-time algorithm reconstructing the set from its X-rays. For this purpose we present the notion of Q-convex lattice sets for which there are unique determination by X-rays in suitable directions, and a polynomial-time reconstruction algorithm. After discussing these results, we show that many reconstructions of sets with convexity and connectivity constraints can be seen as particular cases of the algorithm reconstructing Q-convex lattice sets