# Spatial Proximity and Similarity

## Abstract and Keywords

The author describes the quantification of grouping by proximity using a perturbation approach applied to briefly seen arrays of dots distributed in space or space-time. This work revealed a law of grouping by proximity (the pure distance law): the log-odds of seeing alternative groupings is a linear function of the ratios of their interdot distances, independent of the configuration of the array of dots. Surprisingly, curved organizations are more salient than rectilinear ones. Studies of grouping in space-time revealed that perception of apparent motion is inconsistent with any notion of spatiotemporal distance. When grouping principles (such as proximity and similarity) are conjoined (i.e., more than one grouping principle is applied to the same stimulus), they sometimes affect perception additively. When they do not operate additively, they interact to produce a new emergent property.

Keywords: Spatial grouping, temporal grouping, dot lattice, similarity, curvature, apparent motion, indifference curve, emergent property, perturbation method, attraction function, additivity

Highlight

The quantification of grouping by proximity in space and time provides valuable and counterintuitive insights into perceptual emergent properties.

Proximity

The proximity principle is an empirical law that was first applied to the perception of static scenes (Hochberg & Silverstein, 1956; Kubovy, Holcombe, & Wagemans, 1998; Kubovy & van den Berg, 2008; Oyama, 1961): the closer the elements of a scene are to one another, the more likely it is that they will appear to be parts of the same object.

When Wertheimer (1923) first discussed grouping by proximity, he illustrated the text with examples such as Figure 1. About this figure he wrote,

In allen besprochenen Fällen zeigt sich ein … einfaches Prinzip. Die Form der Gruppen der Punkte mit kleinem Abstand ist die natürlich resultierende, die Form der Gruppen der Punkte mit großem Abstand entsteht nicht oder schwerer, [unsichere,] nur künstlich, und ist labiler. In vorläufiger Formulierung: Die Zusammengefaßtheit resultiert—ceteris paribus—

im Sinn des kleinen Abstandes. (Faktor der Nähe)^{1}(§6, p. 308, the interpolation in square brackets is from §3, p. 307)

His *ceteris paribus* hedge meant that this effect of proximity could be relied on only if either

1. No other grouping factors were present.

2. Other grouping factors were held constant. This was the case when he pitted grouping by proximity against grouping by similarity, as shown in Figure 2 (Wertheimer, 1923, p. 309).

Even though his examples were not multistable, Wertheimer was aware of the competition between groupings within the same figure:

Auch … wenn … der Ausfall … nicht eindeutig ist, kann durch verschiedene Methoden oft noch festgestellt werden, daß für die eine Form die stärkere Tendenz vorliegt… .

^{2}(p. 307)

## Quantifying Grouping by Proximity

Over the years, the Gestalt psychologists have been indicted for muddled thinking (Marr, 1982; Spearman, 1925; 1926). Attempts to apply information theory to the elucidation and quantification of perceptual organization (Garner, 1962; 1970; 1974) have left a conceptual legacy but did not lead to quantification and the formulation of perceptual laws.

### The Perturbation Approach

*Perturbation theory* is used when a problem has no exact solution or when the exact solution is too complex to be useful (Berglund, 2001). Suppose that *ε* is a parameter relevant to the problem and that an exact solution exists when *ε* = 0. Perturbation theory allows us to solve the problem by perturbing *ε* so that it is non-zero, but still small.

We developed an analogous *empirical perturbation approach* to the measurement of grouping strength. We made three assumptions:

1. When a stimulus is at a perceptual equilibrium between several grouping tendencies, the groupings are equiprobable.

2. Small perturbations of the stimulus away from equilibrium result in smooth deviations from equiprobability.

3. The magnitude of the effect of these perturbations on the deviation of the response probabilities from equiprobability is a measure of the strength of grouping.

### Dot Lattices

To implement this approach, I developed regular arrays of dots called *dot lattices* (Kubovy, 1994), an example of which is Figure 3. Each dot of the lattice is surrounded by eight neighbors at four different distances from it, shown by the four red arrows in the figure and labeled by lowercase bold letters, ** a**, …,

**(to which—for notational convenience—we add**

*d**v*to denote a generic vector, where

**∈ {**

*v***,**

*b***,**

*c***}).**

*d*We can rearrange these distances into what is called the *basic parallelogram* (BP) of the dot lattice. Its two sides are ** a** and

**; their magnitudes (or lengths) are $\Vert a\Vert $ and $\Vert b\Vert $. By convention, $\Vert b\Vert $**

*b**≥*$\Vert a\Vert $. Thus, $\Vert a\Vert $ and $\Vert b\Vert $ are the two shortest distances among the dots. (The two other distances are $\Vert c\Vert $ and $\Vert d\Vert $, the diagonals of BP.) The acute angle between

**and**

*a***is**

*b**γ*.

The entire dot lattice can be shown at different orientations. By convention, this orientation is represented by the deviation of ** a** from the horizontal and is denoted by

*θ*. We randomize

*θ*in our experiments to overcome the orientation biases that inevitably affect grouping: we note that observers prefer vertical and horizontal groupings to oblique ones.

^{3}

If we are not interested in the scale of a lattice, dot lattices can be located in a two-dimensional space with dimensions $\Vert b\Vert $/$\Vert a\Vert $ and *γ* (Figure 4), in which we can identify six different types of lattices, characterized by their symmetry properties.

When we use dot lattices, we hold the length $\Vert a\Vert $ constant and manipulate $\Vert b\Vert $ (and *θ*) from trial to trial. Changes in the length of $\Vert b\Vert $ result in changes in the lengths $\Vert c\Vert $ and $\Vert d\Vert $. (Manipulating *γ* also causes the lengths $\Vert c\Vert $ and $\Vert d\Vert $ to vary.)

Dot lattices can be seen organized into strips along ** a**,

**,**

*b***, or**

*c***. If $\Vert b\Vert $/$\Vert a\Vert $ $\underset{\u02dc}{<}$ 1.5, the lattice is multistable**

*d*^{4}; the perceived organizations are in competition and they can appear to change (or can be willed to change). To preclude such changes in experiments, we show dot lattices for 300 ms or less. Nevertheless, it is still multistable: the same stimulus is seen differently on different presentations.

The first attempt to use dot lattices to measure the strength of grouping by proximity is attributed to Oyama (1961), who used rectangular dot lattices at fixed orientation. He recorded the amount of time subjects reported seeing the competing vertical and horizontal groupings. He found that the ratio of the time they saw the vertical and the horizontal organizations is a power function of the ratio of the vertical and horizontal distances.

## Grouping by Proximity Is Lawful in Visible Static Displays

Kubovy and Wagemans (1995) and Kubovy et al. (1998) manipulated $\Vert b\Vert $/$\Vert a\Vert $ and *γ* in briefly exposed dot lattices and asked observers to report their organization. Each trial of this type of experiment (Figure 5) is a sequence consisting of (a) a fixation point, (b) a dot lattice, (c) a dynamic mask (three random dot patterns containing the same dots as the stimulus), and (d) a response screen consisting of four response icons—gray circles traversed by a white line in the same orientation as one of four possible organizations of the lattice. The observers’ task is to report the dominant orientation along which the dots were grouped by clicking on one of these icons. The screen then goes dark until the next trial begins.

Figure 6 shows schematic data for such an experiment. We denote the four possible responses by lowercase italic letters, *a, …, d*, and a generic response by *v* (where *v* ∈ {*b, c, d*}). The *x*-axis of this figure is $\Vert \nu \Vert $/$\Vert a\Vert $, and the *y*-axis is log[*p(v)/p(a*)] (i.e., the log-odds of responding *v* rather than *a*). The figure shows the results for two dot lattices, denoted *lattice 1* and *lattice 2* (whose $\Vert b\Vert $/$\Vert a\Vert $ and *γ* values are shown in the inset). We first consider the *b* responses. Recalling that in *lattice 1*, $\Vert b\Vert $/$\Vert a\Vert $ = 1.1 and in *lattice 2*, $\Vert b\Vert $/$\Vert a\Vert $ = 1.2, we mark their locations on the $\Vert \nu \Vert $/$\Vert a\Vert $ axis. The frequency of *b* responses relative to the frequency of *a* responses for each lattice is represented by blue data points, which show the corresponding values of log[*p(b)/p(a*)].

We then consider the *c* responses. In *lattice 1*, $\Vert c\Vert $/$\Vert a\Vert $ = 1.3; in *lattice 2*, $\Vert c\Vert $/$\Vert a\Vert $ = 1.39. The brown data points show the corresponding values of log[*p(c)/p(a*)]. Turning to the *d* responses, the purple data points show the corresponding values of log[*p(d)/p(a*)]. Finally, there is one point for which we don’t need data: when $\Vert b\Vert $/$\Vert a\Vert $ = 1, it is inevitable that log[*p(b)/p(a*)] = 0 (the black data point) because *p(b*) = *p(a*) when $\Vert b\Vert $ = $\Vert a\Vert $.^{5}

All these data points fall on a straight line, known as the *attraction function*, which shows that grouping by proximity follows a *pure distance law*. This means that grouping by proximity is determined by the three distance ratios $\Vert b\Vert $/$\Vert a\Vert $, $\Vert c\Vert $/$\Vert a\Vert $, and $\Vert d\Vert $/$\Vert a\Vert $, and that it is unaffected by the symmetries of the lattice (as described by Kubovy, 1994). As Kubovy et al. (1998) show, this means that the organization of a dot lattice can be modeled as if it were a collection of *unorganized dots in an isotropic Cartesian space*.

## The Surprising Dominance of Curvature in Grouping by Proximity

Most of the recent research on grouping analyzed the competition between rectilinear organizations. There was no point in investigating the competition between a rectilinear and a curvilinear organization because there are so many good reasons to believe that rectilinear organizations would be stronger. For example, models of good continuation—in dot contours (Fantoni & Gerbino, 2003; Feldman, 2001; Feldman & Singh, 2005; Pizlo, Rosenfeld, & Weiss, 1997; Smits & Vos, 1986; Uttal, Bunnell, & Corwin., 1970) or in arrays of oriented visual elements (e.g., Field, Hayes, & Hess, 1993)—had shown that rectilinear organizations are more readily detected than are curvilinear ones. Nonetheless, Strother and Kubovy (2006) produced an unexpected result: *curvilinear perceptual organizations are stronger than rectilinear ones*.

For this work, Strother and Kubovy (2006; 2012) created dot-sampled structured grids (DSGs). Figure 7 shows the main parameters of a DSG with two sets of curvilinear gridlines. The distances between dots on the higher curvature lines were $\Vert {h}_{i}\Vert $, and the distances along the lower curvature lines were $\Vert {l}_{i}\Vert $. Because these distances deviate slightly from uniformity, we will use the ratio of the average distances, $\overline{\Vert l\Vert}/\overline{\Vert h\Vert}$, to characterize proximity. The curvature along ** h** is

*κ*

_{h}

_{,}and the curvature along

**is**

*l**κ*

_{l}(where

*κ*

_{h}

*≥ κ*

_{l}and if

*κ*

_{l}= 0 the

**lines are straight).**

*l*Strother and Kubovy (2006) studied DSGs in which one organization was linear and the other curvilinear (Figure 8a). They left a question unresolved: is the linear organization at a disadvantage, or is it low curvature? Strother and Kubovy (2012) answered the question: they showed that in a competition between higher and lower curvature, higher curvature wins.

Figure 9 gives a schematic summary of the Strother and Kubovy (2012) results. The abscissa represents the ratio of the distance between dots on the lower curvature paths and the higher curvature paths, $\overline{\Vert l\Vert}/\overline{\Vert h\Vert}$. The ordinate represents the log-odds of choosing the low curvature organization rather than the high curvature organization, log[*p(l)/p(h*)].

Here, just as had been established with dot lattices (Figure 6), log[*p(l)/p(h*)] varies linearly with $\overline{\Vert l\Vert}/\overline{\Vert h\Vert}$. Since the data are linear in log-odds space, the odds of seeing the low curvature organization is an exponentially decreasing function of the relative distances between dots in the two organizations.

The upper line is the attraction function for dot lattices (i.e., competing rectilinear organizations), as in Figure 3. The next three lines represent the three pairings of high, moderate, and low curvature (Figures 8b-d). Thus, we know that the Strother and Kubovy (2006) result is not due to an advantage of curvilinearity over linearity, but rather to an advantage of high curvature over low curvature. We revisit these data later, in the later section “Proximity and Curvature Are Not Additive.”

How to account for these results? Strother and Kubovy (2012) argued that this phenomenon is an instance of the reinforcement of grouping by nonaccidentality, by what Barlow (1986, p. 89) called “suspicious coincidences.” To begin, they assumed that accidentality and nonaccidentality are complementary: *p*(accidentality) = 1*–p*(nonaccidentality). They then asked how probable is the null hypothesis that a collection of contours (a pattern) is the outcome of *n* independent curve processes (i.e., a coincidence)? Following Feldman (2001), they assume that the probability of a single contour *c, p*_{c}, decreases as a function of the contour’s curvature. Furthermore, also following Feldman (1997; 2001; 2004), they assume that the probability of *n* contours of curvature *κ* being parallel (P_{n}) by accident (NP) is

Suppose that the curvatures of contours *c*_{1} and *c*_{2} are *κ*_{1} and *κ*_{2}. If *κ*_{1} *> κ*_{2}, then *p*_{c}_{1} *< p*_{c}_{2}. Therefore, *p*(P_{n}*|*NP, *κ*_{1}) *< p*(P_{n}*|*NP, *κ*_{2}), meaning that the null hypothesis of accidentality is lower for *κ*_{1} than for *κ*_{2}. Therefore the visual system has more reason to notice that the high-curvature parallel curves are real (i.e., nonaccidental) than the low-curvature parallel curves.

## Grouping by Proximity May Not Occur in Visible Space-Time Displays

### Tradeoff of Distance Components

The proximity principle implies the *tradeoff of distance components*. To explain this concept, we consider a dot lattice in which $\Vert a\Vert =\Vert b\Vert $ (Figure 10a); as we have seen, this implies that *p(a*) = *p(b*).

In Figure 10b, we show the vectors in a Cartesian plane with coordinates *x* and *y*. The projections of ** a** and

**onto the**

*b**x-*axis are

*X*

_{a}

*X*

_{b}and onto the

*y*-axis are

*Y*

_{a}and

*Y*

_{b}. There are two ways to visualize a transformation that will turn

**into**

*a***and**

*b***into**

*b***: (a) a clockwise rotation of**

*a***by**

*b**γ*and a concurrent counterclockwise rotation of

**, also by**

*a**γ*; or (b) a tradeoff between the lengths $\Vert {X}_{a}\Vert $ and $\Vert {X}_{b}\Vert $ and a concurrent tradeoff between $\Vert {Y}_{a}\Vert $ and $\Vert {Y}_{b}\Vert $. The latter is called the

*tradeoff of distance components*(Gepshtein et al., 2011).

We now ask the same question about space-time. Suppose one of the two dimensions in Figure 10b is time. To preserve an equality of distances in space-time, the spatial and temporal components of spatiotemporal distance must trade off, just as they did in space: an increment or decrement in the spatial distance between elements must be accompanied by a decrement or an increment in temporal distance.

### Resolving Two Views of Motion Perception

A tradeoff was found by Burt and Sperling (1981): the longer they made the spatial gap between dots, the more they had to shorten the temporal interval between dots for apparent motion to be seen. In contrast, however, according to Korte’s Third Law of Motion (Korte, 1915; also Koffka, 1935 [1963]), the larger the spatial gap between alternating lights, the slower the rate at which they need to be flashed in alternation for apparent motion to be seen. Koffka (1935 [1963], p. 293) himself found this result counterintuitive:

[W]hen Korte and I discovered it, I was surprised …: if one separates the two successively exposed objects more and more, either spatially or temporally, one makes their unification more and more difficult. Therefore increase of distance should be compensated by decrease of time interval, and

vice versa.

Why did Korte’s law puzzle psychologists while the result of Burt and Sperling does not appear surprising? It is probably because space-time coupling contradicts the widespread intuition of distance: the fact that to preserve distance its components must trade off (Gepshtein, Tyukin, & Kubovy, 2011).

In an attempt to resolve these inconsistent results, Gepshtein and Kubovy (2007) devised the following procedure. Three short-lived dots, O, a, and b, appear and disappear sequentially at three locations in space (Figure 11a). Nothing prevents us from seeing apparent motion O *→* a or O *→* b. (The distance $\Vert \text{ab}\Vert $ is too long for a *→* b.) We call the O *→* a motion *m*_{a} and the O *→* b motion *m*_{b}. Each of these has a temporal and a spatial component. For *m*_{a}, they are (*T*_{a}, *S*_{a}), and for *m*_{b} they are (*T*_{b}, *S*_{b}).

This allows us to represent each motion as a point in a plot of distances (Figure 11b). The spatial distance $\Vert \text{Ob}\Vert $ is *S*_{b}. In the figure, it is represented by the interval between ❶ and ❷ connected by the thick double-headed arrow.

As we have seen, we have two spatial distances (*S*_{a}, *S*_{b}) and two temporal intervals (*T*_{a}, *T*_{b}). Let us constrain the relation between the temporal distances to be *T*_{a} = 2*T*_{b} (11a and 11b and ). Furthermore, let us hold *S*_{a}, *T*_{a}, and *T*_{b} constant, so that only *S*_{b} can vary. We can vary *S*_{b} until we find a value ${S}_{\text{b}}={S}_{\text{b}}^{*}$ for which an observer is equally likely to report *m*_{a} and *m*_{b}: *p*(*m*_{b}) = *p*(*m*_{a}). In light of the previous literature, we pit two hypotheses against each other:

•

*Space-time tradeoff*$({S}_{\text{b}}^{*}<{S}_{\text{a}})$, which supports the proximity principle in space-time (because*T*_{b}*> T*_{a}). In Figure 11b, this result is represented by outcome ❶ where the line connecting the conditions of equilibrium has a negative slope.•

*Space-time coupling*$({S}_{\text{b}}^{*}>{S}_{\text{a}})$, where the proximity principle is not applicable. In Figure 11b, this result is represented by outcome ❷ where the line connecting the conditions of equilibrium has a positive slope.

Gepshtein and Kubovy (2007) did not hold *T*_{a} constant as in the explanation we gave a moment ago. Instead, they manipulated *S*_{b} and *T*_{a} (and, since *T*_{a} = 2*T*_{b}, perforce *T*_{b} changed as well). They held *S*_{a} constant. In Figure 11b, the manipulation of *S*_{b} is represented by changes in the length of the thick double-headed arrow (i.e., the distance between ❶ and ❷). Likewise, the manipulation of *T*_{a} results in a movement of *m*_{a} to the left or right. This imposes a double-distance left or right movement of *m*_{b}.

Gepshtein and Kubovy’s manipulation of *T*_{a} and *S*_{b} is shown in the lower left panel of Figure 12. Consider the blue triangle on the lower right of this panel. Below it, you read that the distance covered by *S*_{b} was 3°. To the extreme right of the figure, you read that the time it was allotted to cover that distance was 0.027 s. (Thus, symbols of the same color shared a value of *T*_{a}, and symbols of the same shape connected by lines shared a value of *S*_{b}.) If *S*_{b} = 3^{◦} and *T*_{a} = 0.027s, the speed of motion is *S*_{b} */T*_{a} = 111.1°*/*s. This value corresponds to a position on the abscissa of the upper left panel.

Using psychophysical methods, Gepshtein and Kubovy, for each of the 20 (*S*_{b}, *T*_{a}) pairs in the bottom left panel, obtained a value of *S*_{b} for which *p*(*m*_{b}) = *p*(*m*_{a}). They denoted this value ${S}_{\text{b}}^{*}$ and plotted the results as a function of the response variable ${r}_{13}^{*}={S}_{\text{b}}^{*}/{S}_{\text{a}}$(shown in the upper left panel of Figure 12). As we said earlier, when ${S}_{\text{b}}^{*}<{S}_{\text{a}}({r}_{13}^{*}<1)$, we have *space-time tradeoff*, shown as the darker area in the upper left panel of Figure 12. But when ${S}_{\text{b}}^{*}>{S}_{\text{a}}({r}_{13}^{*}>1)$, we have *space-time coupling*, shown as the lighter area in the upper left panel of Figure 12.

Noticing that the data in panel A looked like a hyperbolic branch, Gepshtein and Kubovy replotted the data in panel B as a function of the reciprocal of motion speed; that is, slowness (*T*_{a} */S*_{b}) revealing a simple relation between ${r}_{13}^{*}$ and motion slowness.

Because the functions in panels A and B cross the boundary ${r}_{13}^{*}=1$, both tradeoff and coupling occur, depending on the speed (or slowness) of the motion. Tradeoff occurs at low speeds (i.e., at small spatial and large temporal distances), but as the speed is increased (i.e., toward large spatial and small temporal distances), eventually we observe coupling.

We conclude that a concept of proximity in space-time is a special case that occurs when motion is slow. But when it is fast, the notion of proximity in space-time is refuted. A mnemonic might be drawn from physics: Newtonian laws of motion work when speeds of motion are far from the speed of light (i.e., motion is slow) but fail when the speed of motion approaches the speed of light. The overall conclusion is that grouping by proximity is not the rule in dynamic displays.

Conjoined Grouping Principles

Emergent properties are thought to be the product of nonlinear systems (Hopfield, 1982), and the essence of Gestalt is emergence (Lehar, 2003). It is thus reasonable to characterize many Gestalt phenomena as the output of nonlinear mechanisms that generate an emergent whole from a collection of elements (Stadler & Kruse, 1995; Vetter, 1995). It would appear to follow that when two grouping principles operate at the same time on the same stimulus (we say that they are *conjoined*), their joint effect would differ from the sum of their individual effects.

## Grouping by Proximity and Similarity Are Additive

To study the effect of grouping by proximity conjoined with grouping by similarity, we must use two kinds of elements (called *motifs*) in the lattice, as did Wertheimer (1923), our Figure 1. We call them *dimotif lattices* (Figure 13) in which one of the translations transforms one motif into the other (see Grünbaum & Shepard, 1987, pp. 215, 247–248).

Once we have determined how grouping varies as a function of relative distance, we can determine the effect of conjoined grouping principles. In order to characterize the relation between two grouping principles, we can use *grouping indifference curves,* analogous to what microeconomists call *indifference curves* (Krantz, Luce, Suppes, & Tversky, 1971): imagine a consumer who would be equally satisfied with a market basket consisting of 1 lb of meat and 4 lbs of potatoes and another consisting of 2 lbs of meat and 1 lb of potatoes. In such a case, the $\u3008\text{meat},\text{potato}\u3009$ pairs $\u30081,4\u3009$ and $\u30082,1\u3009$ are said to lie on an indifference curve.

In grouping indifference curves (Figure 14), one axis (here the ordinate) represents the strength of grouping by proximity, due to the $\Vert b\Vert /\Vert a\Vert $ ratio, and the other (the abscissa) represents the strength of grouping by similarity, due to *δ*.

To do this, Kubovy and van den Berg (2008) obtained the results shown schematically in Figure 15a. The effect of proximity is on the *x*-axis, as in Figure 6. The parameter, *δ,* is the difference in lightness between the dots in the dimotif lattice. When *δ >* 0, the coloring favors ** a**, as in Figure 13a. When

*δ <*0, the coloring favors

**, as in Figure 13b.**

*b*The data of Figure 15a translate into the family of grouping indifference curves depicted in Figure 15b. It follows from these results that when grouping by proximity and similarity are conjoined, they affect the outcome independently, as summarized in Figure 16.

In the cue combination literature (which studies, for example, how cues such as stereo, motion parallax, texture, etc. combine to produce the perception of depth; see, e.g., Landy, Maloney, Johnsten, & Young, 1995; Young, Landy, & Maloney, 1993; Yuille & Bülthoff, 1996) this is called *weak fusion* or *weak coupling*. These results are in stark contrast with what one would expect from Gestalt phenomena: that the conjoining of Gestalts would itself be a Gestalt. In the language of cue integration, we would expect *strong fusion* or *strong coupling*: the modules interact and are not combined linearly.

## Certain Grouping Principles May Not Act Additively

### Proximity and Curvature Are Not Additive

In the earlier section “The Surprising Dominance of Curvature in Grouping by Proximity,” we reviewed the discovery that in grouping by proximity, organizations with high curvature are more salient than organizations with lower curvature. We glossed over the question of additivity, which we pick up now.

In their experiments, Strother and Kubovy (2006; 2012) conjoined the grouping principles of curvature and proximity. As mentioned earlier, Strother and Kubovy (2006) were surprised to find that grouping along curved paths was stronger that grouping along rectilinear paths. In order to further understand this phenomenon, Strother and Kubovy (2012) pitted curved paths against each other. They manipulated $\overline{\Vert l\Vert}/\overline{\Vert h\Vert}$and relative curvature to vary the strength of this competition. Using procedures described earlier, they showed that, all other things being equal, the organization with the higher curvature is the stronger.

Let us return to Figure 9. It depicts the functions for the curved DSGs with higher slopes than the attraction function. (It also depicts the functions for the curved DSGs as if they had, but we do not have enough data to tell us whether there is a variation in slope among them.)

A greater slope means that when the DSGs are curved, observers are more sensitive to changes in relative distance. Why this happens is an open question.

We can speculate on the source of this failure of additivity. Suppose that additivity means that the conjoined grouping principles do not produce a new emergent property. For example, when we conjoin proximity and similarity, we still get a dot lattice organized in parallel strips, just as we do with grouping by proximity. In contrast, perhaps nonadditivity implies that the conjoined grouping principle produced a new emergent property. Indeed, when we see the curved DSGs (such as in Figure 8b), the pattern ceases to be planar and suggests dots on a sphere. This sphericity can be thought of as an emergent property.

### Another Example of Nonadditivity and Emergence

A similar nonadditivity was observed by Gepshtein and Kubovy (2000) in dynamic scenes. Suppose that in perceiving dynamic scenes, vision takes a series of snapshots. When successive snapshots are different, motion may be seen. The snapshot metaphor raises two questions: (1) *The spatial grouping problem:* how does the visual system put together elements within each snapshot to form objects? (2) *The temporal grouping problem:* when the snapshots are different, how does the visual system know which element in one snapshot corresponds to which element in the next? If spatial and temporal grouping are independent, then their effects are additive. Gepshtein and Kubovy showed that they are not: the two grouping mechanisms are not separable. They used spatiotemporal dot lattices, which allowed them to independently manipulated the strength of spatial and temporal groupings. Their results refuted the independence of spatial and temporal grouping: even though they did not vary the spatial configuration of the stimulus, the perception of spatial grouping was affected by manipulations of the temporal configuration of the stimulus.

Conclusion

We have reviewed two lines of work: (1) the quantification of grouping by proximity and the limits of applicability of the concept of proximity; and (2) the quantification of the conjoint effects of multiple grouping principles. Here, we have shown that sometimes their effects are additive and sometimes they are not, and we have speculated why this may be the case.

## Envoi

It often takes observers hours of tedious work to generate the amount of data required to reach the conclusions we have reviewed. This creates an extraordinary opportunity for a plastic visual system to develop ad hoc tools to cope with the tasks imposed on it. As a result, we do not know—and may never know—whether we are studying the system as it operates when it is not being obsessively observed. This concern does not of course apply only to research on grouping and should also be a source of worry for other fields of perception that rely on psychophysical methods.

But not all is lost. The more we obtain quantitative laws using different methods, and the more diverse converging operations we perform to confirm them, the more likely that they will fit into a coherent theoretical framework that can allay such concerns. But by “theoretical framework” I do not mean what passes for theory in psychology. Psychology has played fast and loose with this notion. In psychology, a theory is akin to what might be called a conjecture in other sciences. A classic example of a theory is the (quantitative) ideal gas law, which states the relationship among pressure, volume, and temperature of a fixed mass of gas. It is explained by kinetic molecular theory.

It goes without saying that the work described here does not begin to approach the rigor and generality of laws in physics.

Acknowledgments

Portions of this chapter were reprinted with permission from the following sources:

• Kubovy, M. & Van den Berg, M. (2008). The whole is equal to the sum of its parts: A probabilistic model of grouping by proximity and similarity in regular patterns.

*Psychological Review, 115*(1), 131–154. doi: 10.1037/0033-295X.115.1.131• Gepshtein, S., Tyukin, I., & Kubovy, M. (2011). A failure of the proximity principle in the perception of motion.

*Humana.Mente: Journal of Philosophical Studies, 17*, 21–34.• Wagemans, J., Elder, J. H., Kubovy, M., Palmer, S. E., Peterson, M. A., Singh, M., & von der Heydt, R. (2012). A century of Gestalt psychology in visual perception. I. Perceptual grouping and figure-ground organization.

*Psychological Bulletin, 138*(6), 1172–1217. doi: 10.1037/a0029333

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## Notes:

(^{1})
All these cases reveal a … simple principle. The groupings of points separated by small distances emerge naturally, whereas the groupings of dots separated by large distances are harder to see, [uncertain,] artificial, and unstable. Hence our provisional statement: The whole—ceteris paribus—is determined by the smaller distance. (Factor of proximity.)

(^{2})
Even … when … the outcome … is not unambiguous, there are ways to determine which grouping is stronger… .

(^{3})
Martin van den Berg collected (as yet unpublished) data showing that no one prefers oblique orientations over vertical or horizontal. His data also show individual differences: some prefer vertical over all other orientations, some prefer horizontal.

(^{4})
If $\Vert b\Vert /\Vert a\Vert $ 1.5, the likelihood of seeing organizations along b, c, or d, is too low to be estimated.

(^{5})
In practice, this may not be the case because log[*p(b*)/*p(a*)] is a random variable.