Colinear Line Research Articles

Origin Point Must Represent Critical Line as Location for Nontrivial Zeros of Riemann Zeta Function, and Set Prime Gaps With Subsets Odd Primes Are Arbitrarily Large in Number

We prove the Countably Infinite Subsets of odd primes have cardinality of Arbitrarily Large in Number. This is achieved by demonstrating the asymptotic law of distribution of prime numbers that involves natural logarithm function to be applicable to all these subsets of odd primes derived from every even Prime gaps which are again Arbitrarily Large in Number. We prove the location for Countably Infinite Set of nontrivial zeros computed using Dirichlet eta function (proxy function for Riemann zeta function) is the unique critical line. This is achieved by plotting the infinitely many colinear lines of Riemann zeta function using both critical line and non-critical lines.

Long-range spatial integration across contrast signs: a probabilistic mechanism?

It is widely assumed that the integration of orientation contrast across spatial gaps within the long-range regime is not selective to the contrast sign of the individual stimuli. Probabilistic models of perceptual integration, however, suggest that long-range spatial integration should be, if not selective, at least sensitive to local contrast signs. To clarify this issue, we tested predictions of a model based on conditional probabilistic weights of identical and opposite contrast signs in a simple spatial configuration of two co-linear lines. Contrast detection thresholds of the target line presented either by itself (control condition) or simultaneously with the co-linear inducer (test condition) were measured. The contrast sign of targets and inducers was varied so that all four possible combinations of signs were produced in the test conditions: (1) dark target with dark inducer, (2) dark target with bright inducer, (3) bright target with bright inducer and (4) bright target with dark inducer. The contrast intensity (Weber ratio) of dark and bright inducers was identical. The coaxial distance between target and inducer was constant in each of two experiments, testing for two distances that corresponded to an angular separation within the long-range domain of spatial integration as defined previously. It is found that targets and inducers with identical contrast signs produce significantly stronger facilitating effects on detection than stimuli with opposite signs. The data closely match predictions consistent with those of a probabilistic model of line contrast integration across spatial gaps and contrast signs within the long-range regime.

Two-frequency analysis of interactions elicited by Vernier stimuli.

In five subjects, we measured visual evoked potentials (VEPs) elicited by Vernier targets in which the contrast of the two components of the stimuli were modulated by sinusoids at distinct frequencies fl and f2. This approach allows for the extraction of VEP signatures of spatial interactions, namely, responses at intermodulation frequencies n1f1 + n2f2, without the need to introduce motion into the stimulus. The most prominent interactions were at the sum frequency f1 + f2, and, for frequency pairs that were sufficiently separated, the difference frequency f1- f2. These responses had a systematic dependence on the temporal parameters of the stimulus, corresponding to an effective latency of 145 to 165 ms. Fourth-order interactions were also detected, particularly at the frequencies 2f1 +/- 2f2. These VEP signatures of interaction were similar to interactions seen for colinear line segments separated by a gap. Thus, for Vernier stimuli devoid of motion, VEP signatures of interaction are readily detected but are not specific to hyperacuity displacements. The distribution of interactions across harmonic orders is consistent with local rectification preceding the spatial interactions. Their effective latencies and dependence on spatial parameters are consistent with interactions within V1 receptive fields or mediated by horizontal connections between cells with a similar orientation tuning within V1.

Do alignment thresholds define a critical boundary in long-range detection facilitation with co-linear lines?

Thresholds for line contrast detection (experiment 2) were measured with a two-alternative temporal forced-choice procedure as a function of the spatial position of a vertical target line with regard to two co-linear context lines. The different spatial positions of the target line corresponded to values near the position discrimination threshold (experiment 1) reflecting the just detectable lateral offset, or non-co-linearity, between the context lines which were vertically separated by about 100 minutes of visual arc. Target and context lines were vertically separated by about 30 minutes of arc, had equal contrast polarity in one case, and opposite contrast polarity in the other. Strong line contrast detection facilitation is found at perceptually co-linear target locations. This facilitation decreases noticeably at a horizontal target offset that corresponds to the alignment threshold measured with the context lines. The effects are independent of the relative contrast polarity of target and context and, as shown in a third experiment, also independent of both the relative length or number of lines, and the magnitude of their absolute co-axial separation. This independence seems to hold, provided individual line length and co-axial distance between lines are larger than what appears to be the lower limit of the long-range spatial domain for orientation or contour integration (i.e. 20 minutes of arc), as determined by previous studies. The findings reported here suggest that alignment thresholds are likely to define a critical lateral boundary in long-range detection facilitation with co-linear lines. They support models of contour integration based on interactions between neural mechanisms that integrate local signals of contrast, orientation, and relative position or end-to-end alignment. Such mechanisms may help to explain the formation of representations of virtual contours and object contours in human perception.

THE DISTURBANCE OF A UNIFORM HEAT FLOW BY TWO LINE CRACKS IN AN INFINITE ANISOTROPIC THERMOELASTIC SOLID

Integral transform techniques are used to determine the stress intensity factors of a pair of colinear line cracks that disturb a uniform heat flow in an anisotropic thermoelastic solid that is under uniaxial tension. A solution is given for the case in which the tension is large enough to ensure that the cracks open fully.

Two effects of context on the presence/absence of connecting segments.

Stimulus A consisted of two proximal brackets with colinear lines separated by small gaps. Stimulus B was a square-like rectangle produced by inserting connecting segments into the gaps. Stimulus A was frequently represented as B (i.e., as closed). Following Pomerantz and Pristach (1989) and Treisman and Paterson (1984), perhaps A produced a closure emergent feature that was salient but otherwise independent of other features. However, adding the same two vertical lines (context) to both A and B produced a contrasting division outcome, even though the putative closure emergent feature was an element of the A+context stimulus, and even though it matched a physical feature of the B+context stimulus. Therefore, this emergent feature did not produce the closure. Two additional experiments indicated that the two context lines made the two connecting segments more visible--a context-produced increase in visibility occurred that is not comparable to other known evidence of perceptual improvement. This greater visibility also indicates that the division was not due to the two context lines inhibiting the perception of the two connecting segments, and the closure was not due to good continuation.

Improvements and systolic implementation of the hough transformation for straight line detection

Hough Transformation (HT) is an efficient method to detect straight lines in digital pictures. In the conventional HT, pixel contiguity is not taken into account, and this leads to the following drawbacks: (1) actual length of line segments cannot be computed; (2) colinear line segments cannot be distinguished; and (3) very often, false lines are detected and short lines go undetected. This paper proposes a modified Hough Transformation which performs contiguity check in a simple and efficient way. A systolic architecture that implements this modified transform is presented. The systolic array takes the bit-map of the binary picture as input and processes one row/column of pixels concurrently. The area-time complexity of the proposed architecture is shown to be superior to the conventional sequential algorithm. Preliminary simulation results are presented.

Evoked potentials elicited by brief vernier offsets: Estimating vernier thresholds and properties of the neural substrate

The characteristics of the neural generator producing an evoked potential in response to the brief presentation of a vernier offset was investigated in three experiments. In the first study, averaged evoked potentials (EPs) recorded in response to a single vernier offset stimulus consisting of a horizontal line which changed from colinearity to noncolinearity for 100 msec every 1.5 sec were compared to responses elicited by other vernier configurations consisting of: (1) stimuli with multiple offsets; (2) stimuli presented in different orientations; (3) targets with different offset features; and (4) with the simple displacement of a colinear line. The results showed that a single vernier offset elicited a robust response if the offset was located in the central zone (1°) of the target. Other features of the target configuration were unimportant. Displacement of a colinear line over the same range without an offset evoked little, if any, response. In the second study, EPs were recorded in response to a single offset target which varied in magnitude from 21 to 82 sec of visual angle on different trials. The latency and amplitude of the EP response varied systematically with the amplitude of the vernier offset. Plots of EP amplitude against log of the offset magnitude were linear over the range of offsets employed. Straight lines fitted to the data and extrapolated to zero amplitude provided estimates of vernier threshold. These estimates agreed closely with psychophysical measures taken with the same targets and confirm the initial observations by Levi et al. (1983). In the third experiment, irrelevant contours were added to the vernier target in various spatial and temporal configurations. The addition of stationary, contiguous contours to the vernier target reduced the amplitude of the EP response when the contours were within 4–8 min of the offset, producing progressively less EP attenuation with increasing distance from the offset. However, brief presentation of the irrelevant contour (e.g. a single line passing through the offset) with an onset asynchrony relative to the vernier offset stimulus appropriate to assure simultaneity of the line-elicited EP and the offset-elicited EP yielded an enhanced response, i.e. to two responses added algebraically. The long latency of the offset evoked response and the summation results of the EPs generated by an offset and by a briefly presented linear contour suggests independence of the neural generators producing the response to these two targets.

Binocular adaptation that cannot be measured monocularly.

The tilt aftereffect (TAE) is used to demonstrate the existence of a purely binocular process in human vision. A purely binocular process is a process that can be activated only by matched inputs to the two eyes. It is insensitive to monocular stimulation. The TAE can be produced by exposure to a bipartite field with the top tilted to the left of vertical, the bottom to the right. After such adaptation a pair of colinear lines appears bent in the opposite direction. A cyclopean random-dot stereogram can be used as an adapting stimulus. It produces a 2 degree TAE when a cyclopean test stimulus is used and a 1 degree TAE when a binocular but noncyclopean test stimulus is used. If the same noncyclopean pattern is viewed monocularly, no TAE is measurable. The TAE does not transfer from cyclopean adaptation to monocular testing. Apparently, cyclopean stimuli activate a 'purely binocular process' that cannot be activated by either eye alone.

Application of signal detection theory to perception of differences in line length

This paper is an attempt to specify the equinormal model for application to different tasks which are currently used in psychophysics (i.e. phenomenal report, detection, and 2AFC). The same set of stimuli (pairs of horizontal colinear lines) is presented in procedures which differ by the task, the probability of the signal, and the succession of the stimuli of the set. This approach leads to the specification of the sensorial and the decisional aspects of the model. The data indicate that (1) the sensitivity index d' to differences in line length is the same function of the physical stimulus independently of paradigm, as far as the same task is concerned. However, and according to the predictions, its value becomes larger by a factor of 2 in a two alternative forced choice task; (2) d' increases with stimulus exposure time; (3) latency of responses generally decreases with increasing d', and are dissimilar for correct and incorrect responses; (4) when the temporal succession of the stimuli is random the subject's criterion, x c , is not chosen to optimize the percentage correct, which would require a dependance of criterion upon stimulus amplitude and prior probabilities.

Vernier offset produced by rotary target motion

Two separated colinear lines appear displaced from colinearity when either the target or the subject’s head is rotated in a frontal plane. The direction of perceived offset is reversed for opposite directions of rotation. The present experiments prove that the effect depends on some property of the visual system that is responsive to stimulus motion per se and is not manifested in the response to stationary targets. Two mechanisms which may be responsible for the rotation-contingent effect are considered: (1) An induction mechanism based on the dynamics of induced tilt or of figural aftereffect displacement. (2) A mechanism based on variation of visual latency with stimulus energy/time.