= 0.97= 105

||A||||B||

g= 1.00= 90

||A||||B||

g

= 0.97= 75

||A||||B||

g

03

Curved Versus Straight DSGs Curved Versus Curved DSGs

Introduction

Conclusion

Visual Discontinuities In DSG StimuliResponse Curves (Characteristic)

A B C D

Response Curves (Predicted)

A B C D

period

phase

amplitude

A

BC

D

X-spacing

Y-spacing

Fixation (300 ms)

Mask (300 ms)

Blank (400 ms)

Stimulus (200 or 300 ms)

Response

= 1.0= 90

||A||||B||

g= 1.5= 90

||A||||B||

g= 2.0= 75

||A||||B||

g

= 1.0= 75

||A||||B||

g = 60g

= 90g

= 1.0||A||||B|| = 3.0||A||

||B||

f(x) = cos-1(1/(2*x))

02

Response Curves (Characteristic)

A B C D

Response Curves (Predicted)

A B C D

Dot-sampled Structured Grids From geometry to perceptual organizationDaniel Dunbar, Lars Strother, Michael Kubovy • University of Virginia, Department of Psychology

01

period

phase

amplitude

A

BC

D

X-spacing

Y-spacing

04

6a

0807

09

10

11

6b

5a

5b

Kubovy [1994] described the complete geometric space of dot lattices. This is shownin Fig. 2, along with samples of the stimuli found in each region of the space. Thisstimulus space defines the appropriate parameter ranges within which each dotlattice exhibits a unique organization. The geometry of a dot lattice can be describedby that of its basic parallelogram. Disregarding global orientation and scale, theparallelogram can be represented using only two parameters: an internal angle andthe ratio of the lengths of its sides. In Fig. 2 the basic parallelograms and thematching parameters are displayed in the lower right of each lattice display. Each dotcan be represented by an integer pair that specifies the repetitions along each axis.Thus we can view the entire stimulus as having a related structured grid, where eachcell of the grid is identical to the basic parallelogram.

When curvature is introduced into dot patterns by shifting the columns to lie along acurve, a single paral lelogram no longer suff ices to describe the entirestimulus.Rather, the space is composed of many distinct parallelograms (Fig. 3).However, the underlying grid is preserved and it is still possible to identify each dotwith an integer pair. This suggests that we can view the introduction of curvature intoa dot lattice as the replacement of one linear axis with a curvilinear one.

Here we identify two general types of DSGs as (1) ‘curved versusstraight’ (where one axis is defined by a curve and the other by astraight line) and (2) ‘curved versus curved’ (where both axis aredefined by a curves). We refer to the underlying curves used to defineeach axis as the characteristic curves of the DSG. And although thereis no inherent restriction upon these curves, some analyses requirethat they be twice-differentiable functions.

To complete the definition of a DSG we also must specify the relativespacing between dots along each axis, as in a dot lattice, and otherparameters to determine specifics of how the grid is sampled, such asthe dot density and the location at which the grid should be sampled.

A sinusoid is the simplest curve with complex behavior. A constructive diagram of thesinusoidal curved versus straight DSG is displayed in Fig. 4. By varying the location and scaleof the dot sampling it is possible to use this DSG to create stimuli that have simple (noinflection), symmetric, inflected, and repeated curvature. As seen in the figure, a DSG can alsobe manipulated by changing the properties of the characteristic sinusoid: period, phase, andamplitude.

However, DSGs are in no way limited to such simple curves. Fig. 5a shows a complex curvethat transitions from a simple slow varying sinusoid at the far left to a sophisticated noisefunction on the right. The three large panels in the figure demonstrate the effects of varying thespacing to emphasize perceptually the columns, rows, or to promote organizational competition(multi-stability) between the two. By selecting specific regions and zooming in on them, whilekeeping the dot density constant, we can gain a better understanding of the nature of thestimulus. The top four boxed figures of Fig. 5b display this; each corresponds roughly to theregion of the DSG above it.

The final four figures use partial line segments to display the four closest dot samples with thestimulus. These lines typically serve as an indication of what dots would be predicted by a PureDistance Model [Kubovy et al.,1998] to be percepually grouped into a particular contour. In eachof these displays, the predictions of the Pure Distance Model lie along the basic parallelogramabout each sample. This is, however, not always the case.

As the degree of curvature of the characteristic curves of the DSGincreases, so does the likelihood that the Pure Distance Model will predictgroupings that are not in accordance with the underlying characteristiccurve. In these cases visual discontinuities appear. Visual discontinuitiesare not inherently problematic although they tend to give the stimuli a morecomplex or confusing appearance.

In a typical stimulus without visual discontinuities, the contour that the PureDistance Model predicts to be perceived is identical to the underlyingcharacteristic curve of the DSG. This is especially valuable when anexperiment requires an observer to report a particular perceivedorganization. We would like to explicitly specify what that contour will be.

Fig. 6a and 6b display two stimuli within which the predicted responsecurves do not match the characteristic curves of the DSG. The responsescreens at the top of the diagram indicate the curves we would like topresent the subjects with; the response screens below indicate the curvesthat would be predicted visible. Fig. 7 shows the effect of increasingamplitude in causing visual discontinuity. Fig. 8 shows how the visualdiscontinuity occur as a transition between two alternate grouping regions.It is important that we understand the phenomena of visual discontinuities tobe able to control these situations.

Fortunately, Kubovy’s [1994] work on dot lattices defines the local behaviorof DSGs as well and visual discontinuities correspond to the areas withinthe lattice where the basic parallelogram would violate the dot lattice spaceequation shown in Fig. 2. By expressing this equation over the global DSGwe can compute the exact regions and parameter ranges at which visualdiscontinuities occur.

email: lars@virginia.edu

Curved versus straight DSGs are formed by replacing only one axis with a characteristiccurve. We can also replace both axes with curves to generate curved versus curved DSGs. Fig. 9is a constructive diagram of a curved versus curved DSG analogous to Fig. 4. Note that the onlydifference is that the linear axis before has been replaced by a sinusoid. Fig. 10 displays theresults of the equivalent modification to the complex curve of Fig. 5.

In a strict sense curved versus straight DSGs and dot lattices are a subset of curved versus curvedDSGs where curvature is zero along one or both axes. The concrete specification of a DSG isgiven by a two-dimensional vector equation that is sampled with integer coordinates. Forcing oneor both characteristic curves within this equation to linear functions gives rise to the curved versusstraight DSGs and dot lattices, respectively.

Although we only sample the DSG equation at integer coordinates for stimulus generation, thevector function itself is typically continuous. We use this fact to visualize the underlying gridstructure by sampling the function continuously along one axis while holding the other fixed at anintegral value, as well as to perform global analyses such as the discontinuity analysis describedpreviously.

Clearly the geometry of this newly defined stimulus is highly constrained. Nevertheless it allows fora considerable number of organizational scenarios which could be examined empirically using themethod developed by Kubovy and colleagues [1995, 1998; Fig. 11].

Kubovy, M., 1994. The Perceptual Organization of Dot Lattices, Psychonomic Bulletin & Review, 1(2): 182–190.Kubovy, M. and J. Wagemans, 1995. Grouping by Proximity and Multistability in Dot Lattices: A Quantitative Gestalt Theory, Psychological Science, 6(4): 225–234.Kubovy, M., A.O. Holcombe, and J. Wagemans, 1998. On the Lawfulness of Grouping by Proximity, Cognitive Psychology, 35(1): 71–98.

We have shown that a new family of dot stimuli, DSGs, can varied systematicallyto favor one perceptual organization over another. We have suggested that the methoddeveloped by Kubovy and colleagues [1995, 1998] for use with dot lattices can beextended to DSGs (Fig. 11). In fact, we have done this (see Abstract 0005).Preliminary results suggest that curvilinear contours are more salient than linear ones(in curved versus straight DSGs) when distances between dots along a straight contourare approximately equal to those between dots along the competing curvilinear one.Indeed, curvilinear organizations prevail even when the distance between those dots isgreater than those in the linear alternative.

In our current work we are investigating the hypothesis that contours with greaterinformation (based on curvature, inflection, discontinuities, etc.) captures visualattention more strongly than do than alternatives in the same stimulus that have lessinformation (based on the same standards). One might expect this given the visualsystems propensity to discover the 3D structure of objects and surfaces. That is, if weforce a competition between two alternative representations of the same surface(corresponding to the underlying structured grid) by using DSGs, one might predict thatthe contour representation (the outcome of perceptual organization) with moreinformation would be favored since it contains more structural information with regardto the corresponding surface.

For more information on DSGs experiments please attend the talk entitled,Beyond Grouping by Proximity in Regular Dot Patterns (L. Strother & M. Kubovy).

Edge detection and contour formation are fundamental to human vision. The mechanismsresponsible for these tasks are among the earliest involved in the various stages of perceptualorganization.The perceptual organization of dotted contours is a process by which dots aregrouped into larger structures. It is distinct from one that organizes contours consisting oforiented segments (Fig. 1) because these provide unambiguous local orientation information.Here we present a novel family of regular dot patterns with which to study perceptualorganization of dotted contours. We call these patterns dot-sampled structured grids (DSGs).

DSGs and other regular patterns can be generated via repetition or tiling, and can easily beformed from any dot-sampled curve by simply stacking the curve at fixed intervals. DSGsgenerated using dot-sampled lines (straight ones) are commonly called dot lattices [Kubovy,1994]. In fact, curvature can be easily introduced into a dot lattice simply by systematicallyshifting adjacent columns. We have extended the models and methods of Kubovy andcolleagues [1994, 1995] to develop a more extensive family of regular patterns that couldaccommodate curvillinear dot contours.