Tight frames of k-plane ridgelets and the problem of representing objects that are smooth away from d-dimensional singularities in Rn

For each pair (n, k) with 1 ≤ k < n, we construct a tight frame (ρλ : λ ∈ Λ) for L2 (Rn), which we call a frame of k-plane ridgelets. The intent is to efficiently represent functions that are smooth away from singularities along k-planes in Rn. We also develop tools to help decide whether k-plane ridgelets provide the desired efficient representation. We first construct a wavelet-like tight frame on the X-ray bundle χn,k—the fiber bundle having the Grassman manifold Gn,k of k-planes in Rn for base space, and for fibers the orthocomplements of those planes. This wavelet-like tight frame is the pushout to χn,k, via the smooth local coordinates of Gn,k, of an orthonormal basis of tensor Meyer wavelets on Euclidean space Rk(n−k) × Rn−k. We then use the X-ray isometry [Solmon, D. C. (1976) J. Math. Anal. Appl. 56, 61–83] to map this tight frame isometrically to a tight frame for L2(Rn)—the k-plane ridgelets. This construction makes analysis of a function f ∈ L2(Rn) by k-plane ridgelets identical to the analysis of the k-plane X-ray transform of f by an appropriate wavelet-like system for χn,k. As wavelets are typically effective at representing point singularities, it may be expected that these new systems will be effective at representing objects whose k-plane X-ray transform has a point singularity. Objects with discontinuities across hyperplanes are of this form, for k = n − 1.

Statistical analysis of shape through triangulation of landmarks: A study of sexual dimorphism in hominids

Two objects with homologous landmarks are said to be of the same shape if the configuration of landmarks of one object can be exactly matched with that of the other by translation, rotation/reflection, and scaling. In an earlier paper, the authors proposed statistical analysis of shape by considering logarithmic differences of all possible Euclidean distances between landmarks. Tests of significance for differences in the shape of objects and methods of discrimination between populations were developed with such data. In the present paper, the corresponding statistical methodology is developed by triangulation of the landmarks and by considering the angles as natural measurements of shape. This method is applied to the study of sexual dimorphism in hominids.

Fluorescence tomographic imaging in turbid media using early-arriving photons and Laplace transforms

We present a multichannel tomographic technique to detect fluorescent objects embedded in thick (6.4 cm) tissue-like turbid media using early-arriving photons. The experiments use picosecond laser pulses and a streak camera with single photon counting capability to provide short time resolution and high signal-to-noise ratio. The tomographic algorithm is based on the Laplace transform of an analytical diffusion approximation of the photon migration process and provides excellent agreement between the actual positions of the fluorescent objects and the experimental estimates. Submillimeter localization accuracy and 4- to 5-mm resolution are demonstrated. Moreover, objects can be accurately localized when fluorescence background is present. The results show the feasibility of using early-arriving photons to image fluorescent objects embedded in a turbid medium and its potential in clinical applications such as breast tumor detection.

Visual stimuli induce waves of electrical activity in turtle cortex

The computations involved in the processing of a visual scene invariably involve the interactions among neurons throughout all of visual cortex. One hypothesis is that the timing of neuronal activity, as well as the amplitude of activity, provides a means to encode features of objects. The experimental data from studies on cat [Gray, C. M., Konig, P., Engel, A. K. & Singer, W. (1989) Nature (London) 338, 334–337] support a view in which only synchronous (no phase lags) activity carries information about the visual scene. In contrast, theoretical studies suggest, on the one hand, the utility of multiple phases within a population of neurons as a means to encode independent visual features and, on the other hand, the likely existence of timing differences solely on the basis of network dynamics. Here we use widefield imaging in conjunction with voltage-sensitive dyes to record electrical activity from the virtually intact, unanesthetized turtle brain. Our data consist of single-trial measurements. We analyze our data in the frequency domain to isolate coherent events that lie in different frequency bands. Low frequency oscillations (<5 Hz) are seen in both ongoing activity and activity induced by visual stimuli. These oscillations propagate parallel to the afferent input. Higher frequency activity, with spectral peaks near 10 and 20 Hz, is seen solely in response to stimulation. This activity consists of plane waves and spiral-like waves, as well as more complex patterns. The plane waves have an average phase gradient of ≈π/2 radians/mm and propagate orthogonally to the low frequency waves. Our results show that large-scale differences in neuronal timing are present and persistent during visual processing.

Memory fields of neurons in the primate prefrontal cortex

Many prefrontal (PF) neurons convey information about both an object’s identity (what) and its location (where). To explore how they represent conjunctions of what and where, we explored the receptive fields of their mnemonic activity (i.e., their “memory fields”) by requiring monkeys to remember both an object and its location at many positions throughout a wide portion of central vision. Many PF neurons conveyed object information and had highly localized memory fields that emphasized the contralateral, but not necessarily foveal, visual field. These results indicate that PF neurons can simultaneously convey precise location and object information and thus may play a role in constructing a unified representation of a visual scene.

A complex ligase ribozyme evolved in vitro from a group I ribozyme domain

Like most proteins, complex RNA molecules often are modular objects made up of distinct structural and functional domains. The component domains of a protein can associate in alternative combinations to form molecules with different functions. These observations raise the possibility that complex RNAs also can be assembled from preexisting structural and functional domains. To test this hypothesis, an in vitro evolution procedure was used to isolate a previously undescribed class of complex ligase ribozymes, starting from a pool of 1016 different RNA molecules that contained a constant region derived from a large structural domain that occurs within self-splicing group I ribozymes. Attached to this constant region were three hypervariable regions, totaling 85 nucleotides, that gave rise to the catalytic motif within the evolved catalysts. The ligase ribozymes catalyze formation of a 3′,5′-phosphodiester linkage between adjacent template-bound oligonucleotides, one bearing a 3′ hydroxyl and the other a 5′ triphosphate. Ligation occurs in the context of a Watson–Crick duplex, with a catalytic rate of 0.26 min−1 under optimal conditions. The constant region is essential for catalytic activity and appears to retain the tertiary structure of the group I ribozyme. This work demonstrates that complex RNA molecules, like their protein counterparts, can share common structural domains while exhibiting distinct catalytic functions.

TectoRNA: modular assembly units for the construction of RNA nano-objects

Structural information on complex biological RNA molecules can be exploited to design tectoRNAs or artificial modular RNA units that can self-assemble through tertiary interactions thereby forming nanoscale RNA objects. The selective interactions of hairpin tetraloops with their receptors can be used to mediate tectoRNA assembly. Here we report on the modulation of the specificity and the strength of tectoRNA assembly (in the nanomolar to micromolar range) by variation of the length of the RNA subunits, the nature of their interacting motifs and the degree of flexibility of linker regions incorporated into the molecules. The association is also dependent on the concentration of magnesium. Monitoring of tectoRNA assembly by lead(II) cleavage protection indicates that some degree of structural flexibility is required for optimal binding. With tectoRNAs one can compare the binding affinities of different tertiary motifs and quantify the strength of individual interactions. Furthermore, in analogy to the synthons used in organic chemistry to synthesize more complex organic compounds, tectoRNAs form the basic assembly units for constructing complex RNA structures on the nanometer scale. Thus, tectoRNA provides a means for constructing molecular scaffoldings that organize functional modules in three-dimensional space for a wide range of applications.

Neural mechanisms for visual memory and their role in attention

Recent studies show that neuronal mechanisms for learning and memory both dynamically modulate and permanently alter the representations of visual stimuli in the adult monkey cortex. Three commonly observed neuronal effects in memory-demanding tasks are repetition suppression, enhancement, and delay activity. In repetition suppression, repeated experience with the same visual stimulus leads to both short- and long-term suppression of neuronal responses in subpopulations of visual neurons. Enhancement works in an opposite fashion, in that neuronal responses are enhanced for objects with learned behavioral relevance. Delay activity is found in tasks in which animals are required to actively hold specific information “on-line” for short periods. Repetition suppression appears to be an intrinsic property of visual cortical areas such as inferior temporal cortex and is thought to be important for perceptual learning and priming. By contrast, enhancement and delay activity may depend on feedback to temporal cortex from prefrontal cortex and are thought to be important for working memory. All of these mnemonic effects on neuronal responses bias the competitive interactions that take place between stimulus representations in the cortex when there is more than one stimulus in the visual field. As a result, memory will often determine the winner of these competitions and, thus, will determine which stimulus is attended.

On the role of selective attention in visual perception

What is the role of selective attention in visual perception? Before answering this question, it is necessary to differentiate between attentional mechanisms that influence the identification of a stimulus from those that operate after perception is complete. Cognitive neuroscience techniques are particularly well suited to making this distinction because they allow different attentional mechanisms to be isolated in terms of timing and/or neuroanatomy. The present article describes the use of these techniques in differentiating between perceptual and postperceptual attentional mechanisms and then proposes a specific role of attention in visual perception. Specifically, attention is proposed to resolve ambiguities in neural coding that arise when multiple objects are processed simultaneously. Evidence for this hypothesis is provided by two experiments showing that attention—as measured electrophysiologically—is allocated to visual search targets only under conditions that would be expected to lead to ambiguous neural coding.

Lightness constancy in primary visual cortex

When the illumination of a visual scene changes, the quantity of light reflected from objects is altered. Despite this, the perceived lightness of the objects generally remains constant. This perceptual lightness constancy is thought to be important behaviorally for object recognition. Here we show that interactions from outside the classical receptive fields of neurons in primary visual cortex modulate neural responses in a way that makes them immune to changes in illumination, as is perception. This finding is consistent with the hypothesis that the responses of neurons in primary visual cortex carry information about surface lightness in addition to information about form. It also suggests that lightness constancy, which is sometimes thought to involve “higher-level” processes, is manifest at the first stage of visual cortical processing.

Statistical analysis of shape of objects based on landmark data.

Two objects with homologous landmarks are said to be of the same shape if the configurations of landmarks of one object can be exactly matched with that of the other by translation, rotation/reflection, and scaling. The observations on an object are coordinates of its landmarks with reference to a set of orthogonal coordinate axes in an appropriate dimensional space. The origin, choice of units, and orientation of the coordinate axes with respect to an object may be different from object to object. In such a case, how do we quantify the shape of an object, find the mean and variation of shape in a population of objects, compare the mean shapes in two or more different populations, and discriminate between objects belonging to two or more different shape distributions. We develop some methods that are invariant to translation, rotation, and scaling of the observations on each object and thereby provide generalizations of multivariate methods for shape analysis.

Faithful representation of similarities among three-dimensional shapes in human vision.

Efficient and reliable classification of visual stimuli requires that their representations reside a low-dimensional and, therefore, computationally manageable feature space. We investigated the ability of the human visual system to derive such representations from the sensory input-a highly nontrivial task, given the million or so dimensions of the visual signal at its entry point to the cortex. In a series of experiments, subjects were presented with sets of parametrically defined shapes; the points in the common high-dimensional parameter space corresponding to the individual shapes formed regular planar (two-dimensional) patterns such as a triangle, a square, etc. We then used multidimensional scaling to arrange the shapes in planar configurations, dictated by their experimentally determined perceived similarities. The resulting configurations closely resembled the original arrangements of the stimuli in the parameter space. This achievement of the human visual system was replicated by a computational model derived from a theory of object representation in the brain, according to which similarities between objects, and not the geometry of each object, need to be faithfully represented.

Mnemonic neuronal activity in somatosensory cortex.

Single-unit activity was recorded from the hand areas of the somatosensory cortex of monkeys trained to perform a haptic delayed matching to sample task with objects of identical dimensions but different surface features. During the memory retention period of the task (delay), many units showed sustained firing frequency change, either excitation or inhibition. In some cases, firing during that period was significantly higher after one sample object than after another. These observations indicate the participation of somatosensory neurons not only in the perception but in the short-term memory of tactile stimuli. Neurons most directly implicated in tactile memory are (i) those with object-selective delay activity, (ii) those with nondifferential delay activity but without activity related to preparation for movement, and (iii) those with delay activity in the haptic-haptic delayed matching task but no such activity in a control visuo-haptic delayed matching task. The results indicate that cells in early stages of cortical somatosensory processing participate in haptic short-term memory.

Interisland and interarchipelago transfer of stone tools in prehistoric Polynesia.

Tracing interisland and interarchipelago movements of people and artifacts in prehistoric Polynesia has posed a challenge to archaeologists due to the lack of pottery and obsidian, two materials most readily used in studies of prehistoric trade or exchange. Here we report the application of nondestructive energy-dispersive x-ray fluorescence (EDXRF) analysis to the sourcing of Polynesian artifacts made from basalt, one of the most ubiquitous materials in Polynesian archaeological sites. We have compared excavated and surface-collected basalt adzes and adze flakes from two sites in Samoa (site AS-13-1) and the Cook Islands (site MAN-44), with source basalts from known prehistoric quarries in these archipelagoes. In both cases, we are able to demonstrate the importing of basalt adzes from Tutuila Island, a distance of 100 km to Ofu Island, and of 1600 km to Mangaia Island. These findings are of considerable significance for Polynesian prehistory, as they demonstrate the movement of objects not only between islands in the same group (where communities were culturally and linguistically related) but also between distant island groups. Further applications of EDXRF analysis should greatly aid archaeologists in their efforts to reconstruct ancient trade and exchange networks, not only in Polynesia but also in other regions where basalt was a major material for artifact production.

Numerical representations in primates.

Research has demonstrated that human infants and nonhuman primates have a rudimentary numerical system that enables them to count objects or events. More recently, however, studies using a preferential looking paradigm have suggested that preverbal human infants are capable of simple arithmetical operations, such as adding and subtracting a small number of visually presented objects. These findings implicate a relatively sophisticated representational system in the absence of language. To explore the evolutionary origins of this capacity, we present data from an experiment with wild rhesus monkeys (Macaca mulatta) that methodologically mirrors those conducted on human infants. Results suggest that rhesus monkeys detect additive and subtractive changes in the number of objects present in their visual field. Given the methodological and empirical similarities, it appears that nonhuman primates such as rhesus monkeys may also have access to arithmetical representations, although alternative explanations must be considered for both primate species.