José É Lezama

Rafael Grompone

Von Gioi

Jean-Michel Morel

Gregory Randall

Rafael Grompone Von Gioi

Point Alignment Detection

Keywords: point alignment detection, clustering, a contrario methods, Poisson point process

In spite of many interesting attempts, the problem of automatically finding alignments in a 2D set of points seems to be still open. This paper tries to explain why it is so difficult, and to elaborate a solution. We show that a correct alignment detection depends on not less than four interlaced criteria, namely the amount of masking in texture, the relative bilateral local density of the alignment, its internal regularity, and finally a redundancy reduction step. Extending tools of the a contrario detection theory, we show that all of these detection criteria can be naturally embedded in a single probabilistic a contrario model with a single user parameter, the number of false alarms. Our contribution to the a contrario theory is the use of sophisticated conditional events on random point sets, for which expectation we nevertheless find easy bounds. By these bounds the mathematical consistency of our detection model receives a simple proof. Our final algorithm also includes a new formulation of the exclusion principle in Gestalt theory to avoid redundant detections. Aiming at reproducibility, a source code and an online demo open to any data point set are provided. The method is carefully compared to three state-of-the-art algorithms and counterexamples to the final methods are also explained.

INTRODUCTION

We will consider the problem of finding collinear subsets within a planar set of points. This problem arises in many contexts of data analysis: Alignments are among the simplest structures observable in a point set. They constitute a classic example in statistical shape analysis [START_REF] Small | Techniques of shape analysis on sets of points[END_REF]. Alignment detection is relevant in geology, where the alignment of features, for example earthquake epicenters, reflects underlying faults and joints [START_REF] Lutz | An analysis of the orientations of largescale crustal structures: A statistical approach based on areal distributions of pointlike features[END_REF], [START_REF] Hall | Locating lines among scattered points[END_REF], [START_REF] Hammer | New statistical methods for detecting point alignments[END_REF]. In archaeology, geometric configurations of post holes, in particular alignments, often reveal the disposition of buildings even in presence of overlaps from different time periods [START_REF] Small | Techniques of shape analysis on sets of points[END_REF], [START_REF] Litton | Computer analysis of post hole distributions[END_REF], [START_REF] Broadbent | Simulating the Ley Hunter[END_REF]. The computer vision applications include the detection of Fig. 1. Exactly the same set of aligned dots is present in the three images, but it is only perceived as such in the first one. The second one is a classic "masking by texture" case and the third shows a "masking by structure", often called "Gestalt conflict". grids [START_REF] Dubská | Real-time precise detection of regular grids and matrix codes[END_REF], calibration patterns [START_REF] De La Escalera | Automatic chessboard detection for intrinsic and extrinsic camera parameter calibration[END_REF] or vanishing points [START_REF] Havel | Vanishing points in pointto-line mappings and other line parameterizations[END_REF], [53], and the interpretation of high resolution remote sensing images [START_REF] Vanegas | Detection of aligned objects for high resolution image understanding[END_REF].

Dot patterns are often used in the study of visual perception. Several psychophysical studies led by Uttal have investigated the effect of direction, quantity and spacing in dot alignment perception [START_REF] Uttal | On the detectability of straight lines in visual noise: An extension of French's paradigm into the millisecond domain[END_REF], [START_REF]The effect of deviations from linearity on the detection of dotted line patterns[END_REF], [START_REF]An Autocorrelation Theory of Form Detection[END_REF], [START_REF]The Perception of Dotted Forms[END_REF]. The detection of collinear dots in noise was the target of other studies attempting to assess quantitatively the masking effect of the background noise [START_REF] Smits | The perception of a dotted line in noise: a model of good continuation and some experimental results[END_REF], [START_REF] Kiryati | On the perception of dotted lines[END_REF], [START_REF] Tripathy | Detecting collinear dots in noise[END_REF], [START_REF] Mussap | Amblyopic deficits in detecting a dotted line in noise[END_REF]. A recent work by Preiss analyzes various perceptual tasks on dot patterns from a psychophysical and computational perspective [START_REF] Preiss | A Theoretical and Computational Investigation into Aspects of Human Visual Perception: Proximity and Transformations in Pattern Detection and Discrimination[END_REF]. An interesting computational approach to detect gestalts in dot patterns is presented in [START_REF] Ahuja | Extraction of early perceptual structure in dot patterns: Integrating region, boundary, and component Gestalt[END_REF], although the study is limited to very regularly sampled patterns. The work described here was initiated in the context of a psychophysical research; here we concentrate, however, on the general problem of detection of point alignments in noise.

While it may seem that point alignments are simple structures, Fig. 1 shows how complex an alignment event can be. From a purely factual point of view, the same alignment is present in the three figures. However, it is only perceived as such by most viewers in the first one. The second and the third figures illustrate two occurrences of the masking phenomenon discovered by gestaltists [START_REF] Kanizsa | Vedere e pensare[END_REF]: the masking by texture, which occurs when a geometric structure is surrounded by a clutter of randomly distributed similar objects or distractors, and the masking by structure, which happens when the structure is masked by other perceptually more relevant structures, a phenomenon also called perceptual conflict by gestaltists [START_REF] Metzger | Gesetze des Sehens[END_REF], [START_REF] Metzger | Laws of Seeing[END_REF], [START_REF] Kanizsa | Grammatica del vedere[END_REF]. The magic disappearance of the alignment in the second and third figures can be accounted for in two very different ways. As for the first one, a probabilistic a contrario model [START_REF] Desolneux | From Gestalt Theory to Image Analysis, a Probabilistic Approach[END_REF] is relevant and can lead to a quantitative prediction. As for the second disappearance, it requires the intervention of another more powerful grouping law, the good continuation [START_REF] Kanizsa | Organization in Vision[END_REF] in the perceptual conflict.

These examples show that a mathematical definition of point alignment perception is required before even starting to discuss how to detect them. A purely geometric-physical description is clearly not sufficient to account for the masking phenomenon. Indeed, an objective observer making use of a ruler would be able to state the existence of the very same alignment at the same precision on all three figures. But this statement would contradict our perception, as it would contradict any reasonable computational (definition and) theory of alignment detection.

This experiment also shows that alignment detection is highly dependent on the context of the alignment. It is therefore a complex question, and must be decided by building mathematical definitions and detection algorithms, and confronting them to human perception. As the patterns of Fig. 1 already suggest, simple computational definitions with increasing complexity will nevertheless find perceptual counterexamples.

A classic approach to this problem uses the Hough transform [START_REF] Hough | Method and Means for Recognizing Complex Patterns[END_REF], [START_REF] Duda | Use of the Hough transformation to detect lines and curves in pictures[END_REF], first used for the detection of subatomic particles in bubble chamber pictures [START_REF] Hough | Machine analysis of bubble chamber pictures[END_REF]. To compute the Hough transform, each point votes in a parameter space for the lines that pass through it. After accumulation of the votes of all points, the lines that correspond to local maxima in the parameter space are selected as detections. Several variations of the basic method were proposed; in particular, the methods proposed in [START_REF] Thrift | Approximating point-set images by line segments using a variation of the Hough transform[END_REF], [START_REF] Kiryati | On navigating between friends and foes[END_REF], [START_REF] Kiryati | What's in a set of points[END_REF] are robust to errors in the point positions. When the Hough transform is applied to a random set of points, it will still find a local maximum which does not correspond to a significant collinear subset. A threshold on the number of votes is usually imposed to cope with this problem. Even if the Hough transform methods provide successful solutions in many applications [START_REF] Wadge | Quantitative methods for detecting aligned points: An application to the volcanic vents of the Michoacan-Guanajuato volcanic field[END_REF], a sound criterion for setting this threshold is missing.

Other approaches use point clustering methods especially adapted to clusters of linear shape [START_REF] Murtagh | Fitting straight lines to point patterns[END_REF]. Using a particular distance between points and clusters [START_REF] Davé | Use of the adaptive fuzzy clustering algorithm to detect lines in digital images[END_REF], general clustering algorithms can be used to detect collinear subsets [START_REF] Frigui | A robust competitive clustering algorithm with applications in computer vision[END_REF], [START_REF] Figueiredo | Unsupervised learning of finite mixture models[END_REF]. The same problem can be approached using a parametric model fitting [START_REF] Danuser | Parametric model fitting: From inlier characterization to outlier detection[END_REF].

We are particularly interested in methods that provide an evaluation of the statistical significance of the detected aligned structures. An example in astronomy may illustrate the importance of such evaluation: In 1980 the discovery of several very precise alignments of quasars in the sky raised the question of a theory explaining this presence [START_REF] Arp | Peculiar configurations of quasars in two adjacent areas of the sky[END_REF]. These alignments, however, were later dismissed by a statistical analysis, first by simulations [START_REF] Edmunds | Random alignment of quasars[END_REF] and then analytically [START_REF] Zuiderwijk | Alignment of randomly distributed objects[END_REF], showing that alignments of such precision could easily occur just by chance.

The expected number of events where k among n random points are to be found in some rectangle of a given shape was already computed in 1950 using a Poisson random model [START_REF] Mack | The expected number of aggregates in a random distribution of n points[END_REF]. This could be the origin of the strip method for defining alignments as a large number of points covered by a thin rectangle (the thinner, the more precise). The same random model was used in [START_REF] Broadbent | Simulating the Ley Hunter[END_REF], now explicitly used for detecting point alignments. But the alignment was defined differently: three points are considered aligned when the triangle formed by them is flat enough. Alignments of more points are evaluated by all the possible triangles observed among the points. Various theoretical results about the flat triangles methods are described in [START_REF] Kendall | Alignments in twodimensional random sets of points[END_REF], where Poisson as well as Gaussian distributions are considered in different domain shapes.

Since then, many different algorithms have been proposed, most of them variations of the strip method. Monte Carlo simulations of random points provide the estimate of the significance in [START_REF] Zhang | Structural control of igneous complexes and kimberlites: a new statistical method[END_REF] while a binomial model is used in [START_REF] Amorese | A point pattern analysis of the distribution of earthquakes in Normandy (France)[END_REF]. A set of heuristics are added in [START_REF] Arcasoya | Comprehensive strip based lineament detection method (COSBALID) from point-like features: a GIS approach[END_REF]. The method in [START_REF] Hall | Locating lines among scattered points[END_REF] also applies the strip method with a Poisson model, but the density is estimated locally. A refined statistical test, including angular statistics, is proposed in [START_REF] Hammer | New statistical methods for detecting point alignments[END_REF].

Here we develop a method deriving from the a contrario methodology proposed by Desolneux, Moisan and Morel [START_REF] Desolneux | Meaningful alignments[END_REF], [START_REF] Desolneux | From Gestalt Theory to Image Analysis, a Probabilistic Approach[END_REF]. It is a mathematical formalization of the so-called non-accidentalness principle proposed for perception [START_REF] Witkin | On the role of structure in vision[END_REF], [START_REF] Albert | Genericity in spatial vision[END_REF], [START_REF] Wagemans | Perceptual use of nonaccidental properties[END_REF] (sometimes called Helmholtz principle). In a nutshell, an observed structure is relevant if it would rarely occur by chance. The result is a statistical framework that provides estimates of significance similar in spirit to the methods mentioned before.

As a simple example to introduce the methodology, Desolneux, Moisan and Morel showed how to handle point alignment detection using a simple strip method with a Poisson model [START_REF] Desolneux | From Gestalt Theory to Image Analysis, a Probabilistic Approach[END_REF]Sect. 3.2]. We shall nevertheless show that this initial method (and the similar ones we mentioned above) is far from sufficient for detecting perceptual alignments. To cope with obvious objections and counterexamples, we shall prove that three new features are necessary to cope with the variety of alignments. We shall show that a reliable algorithm requires: a) a local Poisson density estimation, b) an evaluation of the regularity of the spacing of the points in the alignment, and c) a criterion to select the best interpretation among redundant detections.

The rest of this article is organized as follows:

BASIC POINT ALIGNMENT DETECTOR

The key point in a statistical detection method is how to define accidental detections. This requires a stochastic model H 0 , the so-called a contrario model, characterizing unstructured or random data in which the sought structure could only be observed by chance. This basic idea of comparing the detection to the expected number of detections in a random model in an a contrario model is inevitable [START_REF] Broadbent | Simulating the Ley Hunter[END_REF], [START_REF] Kendall | Alignments in twodimensional random sets of points[END_REF], [START_REF] Hammer | New statistical methods for detecting point alignments[END_REF].

Consider a set of N points defined in a domain D with total area S D , see Fig. 2. We are interested in detecting groups of points that are well aligned. A reasonable a contrario hypothesis H 0 for this problem is to suppose that the N points are the result of a random process where points are independent and uniformly distributed in the domain. The question is then to evaluate whether the presence of aligned points contradicts the a contrario model or not.

Given an observed set of N points x = {x i } i=1...N and a rectangle r (a candidate for alignment), we will denote by k(r, x) the number of those points observed inside r. The decision of whether to keep this candidate or not is based on two principles: a good candidate should be non-accidental, and any equivalent or better candidate should be kept as well. The degree of non-accidentalness of an observed rectangle r can be measured by how small the probability P k(r, X) ≥ k(r, x) is, where X denotes a random set of N points following H 0 . In the same vein, a rectangle r ′ will be considered at least as good as r given the observation

x, if P k(r ′ , X) ≥ k(r ′ , x) ≤ P k(r, X) ≥ k(r, x) .
The question is now to control the expected number of accidental detections. Given that N tests candidates will be tested, the expected number of rectangles which are as good as r under H 0 is, following [START_REF] Desolneux | From Gestalt Theory to Image Analysis, a Probabilistic Approach[END_REF], less than

N tests • P k(r, X) ≥ k(r, x) . (1)
The H 0 stochastic model fixes the probability law of the random number of points in the rectangle, k(r, X).

The discrete nature of this law implies that (1) is not actually the expected value but an upper bound of it [START_REF] Desolneux | From Gestalt Theory to Image Analysis, a Probabilistic Approach[END_REF], [START_REF] Grompone Von Gioi | On computational Gestalt detection thresholds[END_REF]. Let us now analyze the two factors in [START_REF] Ahuja | Extraction of early perceptual structure in dot patterns: Integrating region, boundary, and component Gestalt[END_REF].

The a contrario model H 0 assumes that the N points are i.i.d. with uniform density on the domain. (H 0 is a planar Poisson process [START_REF] Miles | On the homogeneous planar Poisson point process[END_REF].) Under the a contrario hypothesis H 0 , the probability that one point falls into the rectangle r is p = Sr S D , where S r is the area of the rectangle and S D the area of the domain. As a consequence of the independence of the random points, k(r, X) follows a binomial distribution. Thus, the probability term P k(r, X) ≥ k(r, x) is given by

P k(r, X) ≥ k(r, x) = B N, k(r, x), p (2)
where B(n, k, p) is the tail of the binomial distribution

B(n, k, p) = n j=k n j p j (1 -p) n-j . (3)
The number of tests N tests corresponds to the total number of rectangles that could contain an alignment, which in turn is related to the number of pairs of points defining such rectangles. With a set of N points this gives N (N -1)

2

different pairs of points. The set of rectangle widths to be tested must be specified a priori as well. In the a contrario approach, a compromise must be found between the number of tests and the precision of the structures that are being sought for. The larger the number of tests, the lower the statistical relevance of detections. However, if the set of tests is chosen wisely, structures fitting accurately the tests will have a very low probability of occurrence under H 0 and will therefore be more significant.

For a particular problem, one may have reasons to restrict the shape of rectangles. Nevertheless, this inquiry is not aimed at any particular application. Thus, we will rely on the following general criteria. An alignment should be an elongated structure, so a minimal ratio between the length and the width of the rectangle must be fixed. Then, a fixed number of widths must be tested, decreasing geometrically the maximal width. (The choice of a geometric series is justified by the obvious scale invariance of the detection problem.) Our implementation uses a length/width max ratio of 10 and a geometric series of 8 width values with a factor 1/ √ 2. The total number of widths to be tested will be denoted by W . Then the total number of tested rectangles is

N tests = N (N -1) 2 W. (4)
The fundamental quantity of an a contrario approach is the Number of False Alarms (NFA) associated with a rectangle r and a set of points x,

NFA 1 (r, x) = N tests • P k(r, X) ≥ k(r, x) (5)
= N (N -1) 2 W • B N, k(r, x), p .
This quantity gives a precise meaning to Eq. ( 1). It will be interpreted as a bound of the expected number of rectangles containing enough points to be as rare as r under H 0 . When the NFA associated with a rectangle is large, this means that such an event is to be expected under the a contrario model and therefore is not relevant. On the other hand, when the NFA is small, the event is rare and probably meaningful.

A rarity threshold ε must nevertheless be fixed for each application. Rectangles with NFA 1 (r, x) ≤ ε will be called ε-meaningful rectangles [START_REF] Desolneux | From Gestalt Theory to Image Analysis, a Probabilistic Approach[END_REF], constituting the detection result of the algorithm. We will refer to this method as Algorithm 1.

Theorem 1 ([10]

).

E r∈R 1 NFA1(r,X)≤ε ≤ ε
where E is the expectation operator, 1 is the indicator function, R is the set of test rectangles, and X is a random set of points under H 0 .

The theorem states that the average number of εmeaningful rectangles under the a contrario model H 0 is bounded by ε. Thus, the number of detections in noise is controlled by ε and it can be made as small as desired. In other words, this detector satisfies the non-accidentalness principle.

As shown in [START_REF] Desolneux | From Gestalt Theory to Image Analysis, a Probabilistic Approach[END_REF], the detection result is not very sensitive to the value of ε. Following Desolneux, Moisan, and Morel [START_REF] Desolneux | Meaningful alignments[END_REF], [START_REF] Desolneux | From Gestalt Theory to Image Analysis, a Probabilistic Approach[END_REF], we shall therefore fix ε = 1 for our experiments. This corresponds to accepting on average at most one false detection per data set in the a contrario model.

Figure 3 shows the results of the basic algorithm in two simple cases. The results are as expected: the visible alignment in the first example is detected, while no detection is produced in the second. Actually, the points in the first example are also present in the second one, but the addition of random points masks the alignment to our perception. The first example produces many redundant detections; this issue will be addressed in Sect. 6.

LOCAL DENSITY ESTIMATION

The basic point alignment detector of section 2 takes as a contrario assumption a uniform point density . Each of the detected rectangles certainly has a non-accidental excess of points in the rectangles with respect to the global density, but this is definitely not what we are looking for. This example shows that we are actually interested in nonaccidental events with an excess of points conditioned to the observation of a local density (which may well be lower or higher than the global density). Such local density estimations for the random point models have been used in [START_REF] Amorese | A point pattern analysis of the distribution of earthquakes in Normandy (France)[END_REF], [START_REF] Hall | Locating lines among scattered points[END_REF], [START_REF] Hammer | New statistical methods for detecting point alignments[END_REF]. In the interpretation proposed here, a more sophisticated definition of the alignment event should not measure the nonaccidentalness by an unusually small probability, but by an unusually small conditional probability. The local density is estimated by counting the points in a rectangular local window, with the same length as the alignment and a given width. To account for the scale invariance of the detection, the width When the rectangle to be tested lies near the border of the domain, the local window may be partly outside it, where no point information is available, leading to a wrong density estimation. This also happens when the rectangle covers the diagonal of the domain. In accordance with perception theory and the good continuation principle, a sound solution is obtained by creating a mirror-symmetric extension of the point set outside the domain. The extended point set is only used for the local density estimation, but not for selecting candidates, and its points are never counted as part of an alignment.

Let R be the local window surrounding the alignment r, as shown in Fig. 5(a). The probability of one point in R falling in r is p = Sr S R where S r and S R are the areas of r and R respectively. The degree of nonaccidentalness of an observation will be measured by the probability that a rectangle has a higher density than its surroundings, conditioned by the observation of the surrounding density. The NFA for the new detector is accordingly defined as

NFA 2 (r, R, x) = N tests • P k(r, X) ≥ k(r, x) n(R, X) = n(R, x) = N (N -1) 2 W L • B n(R, x), k(r, x), p , (6)
where n(R, x) is the number of points observed in x inside R. We will call this method Algorithm 2. The next theorem shows that the method controls the number of false detections in H 0 ; its proof is very similar to the proof of the more general Th. 4 and will therefore be omitted.

Theorem 2. E   r∈R R∈R ′ (r) 1 NFA2(r,R,X)≤ε   ≤ ε
where E is the expectation operator, 1 is the indicator function, R is the set of rectangles considered, R ′ (r) is the set of surrounding local windows for each rectangle r, and X is a random set of points under H 0 .

LATERAL DENSITY ESTIMATION

While the local density estimation can provide a more adjusted background model, it can also introduce new problems such as a "border effect", as shown in Fig. 4 (c). Indeed, the density estimation is lower on the border of the left half of the image than inside it. Thus, the previous algorithm (Algorithm 2) detects alignments on the border with non-accidental, meaningful excess with respect to the local density.

In order to avoid this effect, the more sophisticated Algorithm 3 used in Fig. 4 (d) takes, as a conservative estimation of the background density, the maximum of the densities measured on both sides of the alignment. In short, to be detected, an alignment must show a higher point density than in both regions immediately on its left and right. This local alignment detector is therefore similar to a classic second order Gabor filter where an elongated excitatory region is surrounded by two inhibitory regions. The local density estimation is calculated as illustrated in Fig. 5 (b): The local window is divided in three parts. R 1 is the rectangle formed by the area of the local window on the left of the alignment. R 3 is the area of the local window on the right of the alignment, and R 2 is the rectangle which forms the candidate alignment. Note that the length of the local window is the same as the alignment and that we can consider any arbitrary orientation for it. Next, the algorithm counts the numbers of points M 1 , M 2 , and M 3 in R 1 , R 2 and R 3 , respectively, and defines the conservative estimate of the local number of points as

n * (R, x) = 2 max(M 1 , M 3 ) + M 2 . ( 7
)
We then define the NFA of the event "the density in R 2 has a significant excess with respect to the density estimated in R" by

NFA 3 (r, R, x) = N tests • P k(r, X) ≥ k(r, x) n(R, X) = n * (R, x) = N (N -1) 2 W L • B n * (R, x), k(r, x), p . (8)
Indeed, conditioned to the fact that we assume n(R, X) = n * (R, x) under the model H 0 , the n * (R, x) points in R are still uniformly and independently distributed. We call this method Algorithm 3. The following theorem shows that the method controls the number of false detections in H 0 . Again, its proof is omitted because it is very similar to the proof of Th. 4.

Theorem 3. E   r∈R R∈R ′ (r) 1 NFA3(r,R,X)≤ε   ≤ ε
where E is the expectation operator, 1 is the indicator function, R is the set of rectangles considered, R ′ (r) is the set of surrounding local windows for each rectangle r, and X is a random set of points under H 0 .

ALIGNMENT REGULARITY

We proceed incrementally toward a more complex algorithm, by showing that each new more sophisticated version of an alignment detection is still not sufficient to cope with all features of this problem.

There is indeed still an objection to Algorithm 3: One can stir wrong detections by introducing small point clusters as shown in Fig. 6 (a). The detected alignment in Fig. 6 (b) seems clearly wrong. It is nevertheless explainable in the setting of Algorithm 3: there is indeed a meaningful point density excess inside the red rectangle. But this excess is caused by the clusters, not by what could be termed an alignment. While the algorithm counted every point, human perception seems to group the small clusters into a single entity, and to count them only once. This unwanted result is a consequence of the fact that Algorithm 3 is searching for elongated clusters of higher density without any cluster regularity requirement. As suggested in other studies [START_REF] Preiss | A Theoretical and Computational Investigation into Aspects of Human Visual Perception: Proximity and Transformations in Pattern Detection and Discrimination[END_REF], [START_REF] Tripathy | Detecting collinear dots in noise[END_REF], [START_REF]The effect of deviations from linearity on the detection of dotted line patterns[END_REF], the density is not the only property that makes an alignment perceptually meaningful; another characteristic to consider is the uniform spacing or regularity of the points in it, which the gestaltists call the law of constant spacing. This observation forces to a still more sophisticated version of the alignment detector. To cope with both issues (avoiding small clusters and favoring regular spacing) a more advanced version of the alignment detector divides each candidate rectangle into equal boxes. Instead of counting the total number of points, the algorithm counts the number of boxes that are occupied by at least one point. We call them occupied boxes. In this way, the minimal NFA is attained when the points are perfectly distributed along the alignment. In addition, a concentrated cluster in the alignment has no more influence on the alignment detection than a single point in the same position. We want to estimate the expected number of occupied boxes in the background model H 0 . The probability of one point falling in one of the boxes is p 0 = S B S L , where S B and S L are the areas of the boxes and the local window respectively. Then, the probability of having one box occupied by at least one of the n * (R, x) points (i.e., of an occupied box) is

p 1 (R, c) = 1 -(1 -p 0 ) n * (R,x) . (9)
We will denote by b(r, c, x) the observed number of occupied boxes in the rectangle r when divided into c boxes. Finally, the probability of having at least b(r, c, x) of the c boxes occupied is

B c, b(r, c, x), p 1 (R, c) . (10)
A set C of different values are tried for the number of boxes c into which the rectangle is divided, and the one producing the lowest NFA is taken. Thus, the number of tests must be multiplied by its cardinality

N tests = N (N -1) 2 W LC = N (N -1) 2 W L √ N .
The NFA of the new event definition is then

NFA 4 (r, R, c, x) = N tests • P b(r, c, X) ≥ b(r, c, x) n(R, X) = n * (R, x) = N (N -1) 2 W LC • B c, b(r, c, x), p 1 (R, c) . ( 11
)
Fig. 6 (c) shows an example of the resulting algorithm and we will show some more after discussing the masking problem, in section 6. Algorithm 4 presents the pseudo-code for this final refined version of the alignment detector.

Theorem 4. E   r∈R R∈R ′ (r) c∈C 1 NFA4(r,R,c,X)≤ε   ≤ ε
where E is the expectation operator, 1 is the indicator function, R is the set of rectangles considered, R ′ (r) is the set of surrounding local windows for each rectangle r, C is the set of number of boxes tested, and X is a random set of points under H 0 .

Proof:

We define b(r, R, c, M ) as b(r, R, c, M ) = min β ∈ N, P b(r, c, X) ≥ β n(R, X) = M ≤ ε N (N -1) 2 W LC .
R determines the domain of the local window and M the number of points in it. The probabilistic model inside R, conditioned to the fact that the number of observed points is M , is still uniform and independent, and the conditional law of the number of points inside any subset of R follows a binomial law.

Then, NFA 4 (r, R, c, X) ≤ ε is equivalent to b(r, c, X) ≥ b(r, R, c, M ) when M = n * (R, X). Now, E   r∈R R∈R ′ (r) c∈C 1 NFA4(r,R,c,X)≤ε   = r∈R R∈R ′ (r) c∈C P NFA 4 (r, R, c, X) ≤ ε = r∈R R∈R ′ (r) c∈C 2N M =0 P NFA 4 (r, R, c, X) ≤ ε n(R, X) = M • P n(R, X) = M = r∈R R∈R ′ (r) c∈C 2N M =0 P b(r, c, X) ≥ b(r, R, c, M ) n(R, X) = M • P n(R, X) = M . ( 12
)
Algorithm 4: Point alignment detector with boxes input : A set x of N points [W = 8, L = 8, ε = 1] output: A list out of point alignments

for i = 1 to N and j = 1 to i -1 do l ← distance(x i , x j ) w ← l/10 for 1 to W do r ← rect(x i , x j , w) w L ← l for 1 to L do R 1 ← local-window-left(x i , x j , w L ) R 3 ← local-window-right(x i , x j , w L ) for c ∈ C do Compute NFA 4 (r, R, c, x) [Eq. 11] if NFA 4 (r, R, c, x) ≤ ε then out ← r end end w L ← w L / √ 2 end w ← w/ √ 2 end end
Note that, because of the maximum density estimation n * (R, x), the estimated number of points inside a rectangle can theoretically be as large as 2N , and thus the range for M . By definition of b(r, R, c, M ) we know that

P b(r, c, X) ≥ b(r, R, c, M ) n(R, X) = M ≤ ε N (N -1) 2
W LC ,

and using #R = N (N -1) 2 W , #R ′ (r) = L, #C = C and 2N M =0 P n(R, X) = M = 1 we get E   r∈R R∈R ′ (r) c∈C 1 NFA4(r,R,c,X)≤ε   ≤ r∈R R∈R ′ (r) c∈C ε N (N -1) 2 W LC 2N M =0 P n(R, X) = M = ε,
which concludes the proof.

REDUNDANCY

As was observed in Fig. 3, all the described alignment detectors may produce redundant detections.

In a very meaningful alignment many smaller or larger rectangles overlapping the main alignment are also meaningful. This redundancy phenomenon can involve points that belong to the real alignment as well as background points near the alignment, as illustrated in Fig. 7. In such cases, humans usually perceive only one alignment. The question is how to detect this best rectangle, both explaining and masking the redundant detections.

A simple model for this masking process was proposed by Desolneux et al. [START_REF] Desolneux | From Gestalt Theory to Image Analysis, a Probabilistic Approach[END_REF] under the name of "exclusion principle". The main idea is that each basic element (a point in our case) cannot contribute to more than one perceived group or gestalt. This leads to a masking algorithm proceeding as follows: The most meaningful observed structure (the one with smallest NFA) is kept as a valid detection. Then, all the basic elements (the points) that were part of that validated group are assigned to it and the remaining candidate structures cannot use them anymore. The NFA of the remaining candidates is re-computed without counting the excluded elements. In that way, redundant structures lose most of their supporting elements and are no longer meaningful. On the other hand, a candidate that corresponds to a different structure keeps most or all of its supporting basic elements and remains meaningful. The most meaningful candidate among the remaining ones is then validated and the process is iterated until there are no more meaningful candidates.

This formulation of the masking process often leads to good results, removing redundant detections while keeping the good ones. But it may also lead to unsatisfactory results as illustrated in Fig. 8. The problem arises when various valid alignments have many elements in common. As one alignment is evaluated after the other, it may happen that all of its elements have been removed, even if the alignment is in fact not redundant with any of the other ones. In the example of Fig. 8, individual horizontal and vertical alignments are not redundant, but if all the vertical ones have been detected first, the remaining horizontal ones will be (incorrectly) masked. This example shows a fundamental flaw of the exclusion principle: it is not sound to impose that a basic element belongs to a single perceptually valid structure. There must be a global explanation of the organization of the basic elements in visible structures which is at the same time coherent with each individual structure (eliminating local redundancy) and with the general explanation of the scene in such a way that some basic elements can participate of several structures without contradiction. The solution seems to be in a sort of relaxation of the exclusion principle. The following definitions sketch a possible solution.

Definition 1 (Building Elements). We call building element any atomic component that can be a constituent element of several structures. An example of building elements are points that can be recursively grouped in alignments.

Definition 2 (Masking Principle). A meaningful structure B will be said "masked by a structure A" if B is no longer meaningful when evaluated without counting its building elements belonging to A. In such a situation, the structure B is not retained as detected.

In short, a meaningful structure will be detected if it is not masked by any other detected structure. The difference with the former exclusion principle is that here a structure can only be masked by another individual structure and not by the union of several structures. A procedural way to attain this result is to validate alignments one by one, starting by the one with smallest NFA. Before accepting a new alignment, it is checked that it is not masked by any one of the previously detected alignments.

Figures 13 and14 show some point alignment detection results when combining Algorithm 4 with the masking principle. The results obtained in these examples are as expected and this masking procedure was applied to all experiments below. For simplicity, we shall still refer to it as "Algorithm 4".

COMPARATIVE EXPERIMENTS

We will first compare the proposed algorithm with two methods described in the literature designed for point alignment detection [START_REF] Hall | Locating lines among scattered points[END_REF], [START_REF] Hammer | New statistical methods for detecting point alignments[END_REF]. These two methods were selected because they were introduced recently and include statistical significance tests. As we shall see, these methods depend critically on user parameters; we will describe their result for three data sets, commenting on the results while varying the parameters. Next, we will show the results, for the same data, of a well-known clustering method [START_REF] Figueiredo | Unsupervised learning of finite mixture models[END_REF], which, as our algorithm, requires no parameter tuning. Finally, the results of our method for these and other point sets will be shown and commented. More experiments can be performed by the reader using the freely available online demo and source code. 2 The method introduced in 2006 by Hall, Tajvidi and Malin [START_REF] Hall | Locating lines among scattered points[END_REF] makes measurements similar to Algorithm 3: the alignment is evaluated as a thin strip and two lateral rectangles are used to estimate the point density; the set of candidates and the statistical test are different. The results presented here were computed using our own implementation of the method; we reproduced the experiments in the original publication to verify the correctness of our code.

The statistical test of the method by Hall et al. is designed to reject alignments in a uniform Poisson random point model. The method works well when the intensity of the point process is high. Indeed, the authors showed that the method approaches optimality as the intensity increases [START_REF] Hall | Locating lines among scattered points[END_REF]. The method is less efficient when the density of points is low relative to the size of the operator; in such conditions the sampling is not suitable, the point density estimation is poor, and the statistical test is not able to reject 2. http://bit.ly/point alignments random configurations. The test depends on two parameters, u and v, to be set manually. The first parameter controls the statistical level. In extreme cases of wrong density estimation (e.g. no point is observed in the local window), the statistical test fails. The second parameter, v, is a threshold imposed on the number of points in the strip. It is necessary to cope with cases of density undersampling. The method assumes that the domain is the unit square and tests candidates centered in an n × n grid, at regular orientations with an angle step θ; three parameters define the strip: the length b, the strip width c, and the local window width a. Thus seven parameters must be provided by the user: n, θ, a, b, c, u, and v.

Figure 9 shows two detection results in a set of 100 points generated according to a Poisson model. The first result (left) is using the same parameters as in [START_REF] Hall | Locating lines among scattered points[END_REF]Sect. 3.1]; in accordance with the results of the original article, in these conditions is observed an average of 3 detections per data set (2 in the example shown). However, when the shape parameters are modified, the statistical test is no longer able to control the number of false detections, see Fig. 9 (right). In the second experiment, the number of candidates is larger, (the grid is 20 × 20 instead of 10 × 10), and the density estimation is worse because the candidates are half as long (thus the local window is half as big). Under the new conditions, the same u and v values lead to 47 detections in the same point set. This experiment shows the need to set manually the statistical significance parameters (u, v) to produce reliable results. This behavior was also observed for sets of 1000 random points.

We will see now how the method handles data sets that do contain point alignments. Figure 10(a) shows a set of 186 points; perceptually one can see three alignments, three clusters, and random points. We first adjusted the statistical test parameters u and v so as to obtain very few detections with the same number of random points. The candidate shape parameters (a, b, c) where then adjusted to obtain the best result, see Fig. 10(b). As one can see, the three alignments were detected. Nevertheless, all detections only partially cover the perceived alignment; this is of course due to the selected length (b = 0.4), but longer candidates produced less complete results. Also, there is some redundancy in the detections; no redundancy reduction step is included in Hall et al.'s method. Finally, one can observe that one of the clusters led to a false detection. When the shape parameters are changed to less optimal values, Figs. 10(c) and (d), we obtains less useful results: some alignments or parts of them are missing, and many spurious detections were produced. Some are due indeed to deviations from the random model as is the case of the clusters, and this shows the need for a more complex event definition. Others, as in Fig. 10(d), reveal a failure of the statistical test. This experiment shows that the candidate shape parameters must be carefully adjusted to produce good results.

The second point alignment detection method we used for comparison was introduced by Hammer in 2009 [START_REF] Hammer | New statistical methods for detecting point alignments[END_REF]. This method requires an alignment length parameter (defined as a radius). Each point of the input set defines a candidate. The distribution of angles from the center point to each of the points inside the radius is evaluated. When the circularuniform distribution is rejected, using a Rayleigh test, the candidate produces a detection. The last decision requires a significance level α.

The experiments presented here were done using the author implementation of the algorithm, included in the software package PAST [START_REF] Hammer | PAST: Paleontological statistics software package for education and data analysis[END_REF]. Figure 11 shows the results for the three data sets considered in this comparison and for four parameter sets (no detection was produced for l = 0.6 and α = 0.00001). As one can see in the first row, the default significance level used in PAST is not satisfactory: it produces many detections on random sets of points (left and middle). Using this significance level one gets some of the expected alignments in the data set containing alignments (right). But, as in the previous method, redundancy is observed and many false detections, mainly caused by the presence of the clusters. When the significance level is changed to α = 0.00001 (second row), the number of false detections in noise is reduced significantly; unfortunately, the true alignments also disappear, leaving only two detections due to a cluster. Using a longer alignment length (third row) produced no better results: false detections in random points and unexpected detections in the structured data set. Increasing the significance level for long alignments led to no detection.

As discussed in the introduction, general clustering methods can provide point alignments when a criterion is added to select elongated clusters. We will show results of this approach using the well-known algorithm by Figueiredo and Jain of 2002 [START_REF] Figueiredo | Unsupervised learning of finite mixture models[END_REF]. This algorithm adds an important aspect to our comparison: like our algorithm it is unsupervised. Figure 12 shows the results for the same point sets used before. Being a randomized algorithm, different results are obtained at each run; the best results obtained in our tests are presented. The method was used in its standard form, fitting Gaussian mixtures. Each ellipse in the figure corresponds to a Gaussian cluster. The results obtained for the structured point sets are surprisingly good: each one of the perceived alignments and clusters is well represented. The middle result on random point sets is far less satisfactory as it includes many elongated clusters interpretable as alignment detections.

Figure 13 shows the results of the proposed algorithm for the same data sets. As one can see, no detection is produced in the random points, and the three alignments were found. Two of the detections are however shorter than expected and the top vertex of the "A" is missing. Note how the method correctly handled the redundant detections. Some further results of our method, with increasingly difficulty, are shown in Figs. 14 and15. The first four are correctly solved. Notice how the very low relative density alignment in Fig. 14(d) was correctly detected. Figure 15 shows some failures and requires a longer comment because they show the limitations of our algorithm. All the alignments in Fig. 15(a) were found; however, the redundancy reduction step did not select the candidates best covering the alignments from a global gestaltic viewpoint. The set of points in Fig. 15(b) is the same already shown in Fig. 1; the alignment found by the algorithm is correct, but as discussed before, does not corresponds to the most common interpretation by a human observer. A natural way of handling this problem would be to detect the "curves" by good continuation and then forcing a global interpretation by methods similar to our masking methodology, that would discard the detected alignment as "masked" by the curves. In Fig. 15(c) the presence of a large cluster masks an alignment: the large number of points causes a large number of tests, raising the detection threshold and imposing a very restrictive test on the alignments. The seven points alignment is detected but the six points one is missing. Handling this example probably implies a round cluster detector but also a recursive approach: once a cluster is detected and removed, our current algorithm would easily detect both alignments. Finally, two of the structures in Fig. 15(d) are slightly curved, rendering them inaccurate as alignments; they are therefore not detected. Again, the detection of good continuation may provide a solution. In short, no alignment detection algorithm can be fully satisfactory per se; it requires the interaction (and conflicts) with other feature detectors, namely cluster and curve detectors.

CONCLUSION

In this work we have presented a series of algorithms with growing efficiency for detecting alignments of points in a point pattern. The two key aspects of the alignment have been shown to be its local density and its regularity. Our final method combines both criteria into a single coherent detection. We have also introduced a new procedure to resolve the problem of redundant detections. As future work, the same research methodology can be used to address the detection of related structures, such as the Good Continuation or curves of points. This will in particular be necessary to resolve the still unsolved gestalt conflict between good continuation and alignment pointed out in Fig. 1.

Fig. 2 . 2 W

 22 Fig. 2. A schematic representation of the evaluated rectangle. In a domain with N points, there are N (N -1) 2 W possible rectangles. In this example, N = 47 and k(r, x) = 8 among them are inside the rectangle r.

Fig. 3 .

 3 Fig. 3. Results from the basic point alignment detector (Algorithm 1). (a) and (c) are the input data, and (b) and (d) are the corresponding results. Each detection is represented by a rectangle. In (b) the algorithm correctly detects the obvious alignment. Notice that multiple and redundant rectangles were detected; this issue will be dealt with in Sect. 6. The data set (c) contains the same set of points in (a) plus added noise points. The aligned points are still present but hardly perceptible. The algorithm handles correctly this masking phenomenon and produces no detection.

Fig. 4 .

 4 Fig. 4. Local vs. global density estimation. (a) The set of points. (b) Alignments found using global density estimation (Algorithm 1). The many detected rectangles indeed have a high point density compared to the average image density used as background model. (c) Alignments found using local density estimation (Algorithm 2). The local density is in that case lower on the border, hence the deceptive detection. (d) No alignment is found when the local density is estimated by the maximum density on both sides of the alignment (Algorithm 3).

Fig. 5 .

 5 Fig. 5. (a) The point density is estimated in the local window R surrounding the alignment r. (b) In a refined version of the algorithm, the density of points is measured on each side of the evaluated rectangle. The maximum of the densities in R 1 and R 3 is taken as an estimation of the point density in both R 1 and R 3 .

Fig. 6 .

 6 Fig. 6. Counting occupied boxes to avoid false detections due to the presence of clusters. The dot pattern shown in image (a) presents two point clusters but no alignment. However, Algorithm 3 finds a thin rectangle with a high point density, hence a false detection, as shown in (b). Algorithm 4 divides the rectangle into boxes and counts the occupied ones, avoiding this misleading cluster effect, as seen in (c), where the occupied boxes are marked in red and no alignment is actually detected.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Redundant detections. Left: point pattern. Center: all significant alignments found by the refined point alignment detector (Algorithm 4) described in Sect. 5. The color represents the relative NFA value, where red is the most significant (smallest NFA value) and blue the least (highest NFA value). Right: Result of the masking process.

Fig. 9 .

 9 Fig. 9. Result of Hall, Tajvidi, Malin 2006 [18] on a set of 100 uniform and independent random points. Each detection is represented by a thin rectangle, surrounded by the local window. Left: The same parameters as in [18, Sect. 3.1] were used: 10 × 10 grid, 5 degree angle step, a = 0.1, b = 0.6, c = 0.01, u = 6 v = 2. As in the original publication, about 3 alignments were detected (2 in this example). Right: Result with a slightly different candidate set: 20 × 20 grid and b = 0.3, producing 47 detections. A similar behavior is observed with sets of 1000 random points.

Fig. 10 . 1 l 1 Fig. 11 .

 101111 Fig. 10. Result of Hall, Tajvidi, Malin 2006 [18]. Each detection is represented by a thin rectangle, surrounded by the local window. In the following results a 20 × 20 candidate center grid was used, a 5 degree angle step, u = 7 and v = 3. (a) Input set of 186 points, containing 3 alignments, 3 clusters, and random points. (b) Result of the method for a = 0.2, b = 0.4, and c = 0.02. (c) Result of the method for a = 0.3, b = 0.6, and c = 0.02. (d) Result of the method for a = 0.05, b = 0.6, and c = 0.005.

Fig. 12 .

 12 Fig. 12. Unsupervised clustering of the sets of points in Figs. 9, 10, 11 obtained by the Figueiredo and Jain method [15]. Each ellipse represents a detected cluster.

Fig. 13 .Fig. 14 .Fig. 15 .

 131415 Fig. 13. Result of the proposed algorithm for the sets of points in Figs. 9, 10, 11 and 12. Each detection is represented by a thin rectangle divided into boxes, and surrounded by the local window.


Namkje Koudenburg
email: n.koudenburg@rug.nl

Tom Postmes

Ernestine H Gordijn

Disrupting the flow: How brief silences in group conversations affect social needs

Keywords: Conversational flow, rejection, social needs, social validation, perceived consensus

We all know the awkward feeling when a conversation is disrupted by a brief silence. This papers studies why such moments can be unsettling. We suggest that silences are particularly disturbing if they disrupt the conversational flow. In two experiments we examined the effects of a single brief instance of silence on social needs, perceived consensus, emotions, and rejection. Study 1 demonstrated that fluent conversations are associated with feelings of belonging, self-esteem, and social validation. If a brief silence disrupts this fluency, negative emotions and feelings of rejection increase. Study 2 replicated these effects in a more realistic setting, and showed that the effects of a brief silence are considerable despite participants' unawareness of the silence. Together, results show that conversational flow induces a sense of belonging and positive self-esteem. Moreover, this research suggests an implicit route to social validation, in which consensus is inferred from fluent group conversation.

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Disrupting the flow 3

Disrupting the flow: How brief silences in group conversations affect social needs.

Sometimes there is a brief pause in a conversation, just after we have said something. We experience prolonged silences as deadly or ear-splitting. But even brief silences are unsettling. Why is this so? We suggest that silences threaten social needs. This hypothesis is derived from research in pragmatics, showing that disfluency in conversations may signal conflict, and the ostracism literature, which demonstrates that being ignored harms social needs.

Conversational Flow and Social Needs

Conversations are more than mere exchanges of information: The social dynamic of a conversation can be compared to other cooperative social activities. When dancing together, the coordinated movements of two partners may arouse a variety of positive emotions (Haidt, Seder, & Kesebir, 2008). A fluent conversation, although different in many respects, shares these characteristics of close coordination and predictability, because of the harmonious exchange of information through smooth turn-taking [START_REF] Chapple | Culture and biological man: Explorations in behavioral anthropology[END_REF]. Another similarity is that this experience of conversational flow is associated with a pleasant state of contentment [START_REF] Burgoon | Interpersonal Adaptation: Dyadic Interaction Patterns[END_REF].

The positive experience of conversational flow may serve four different social needs. First, the pragmatics literature demonstrates that numerous interactions with a partner increase conversational flow and interpersonal bonding (Rabinowitz, 2008).

Furthermore, people synchronize their behaviors in interactions (Marsh, Richardson, Baron, & Schmidt, 2006), which increases feelings of entitativity and rapport [START_REF] Bernieri | Interactional synchrony and rapport: Measuring synchrony in displays devoid of sound and facial affect[END_REF][START_REF] Bernieri | Interactional synchrony and rapport: Measuring synchrony in displays devoid of sound and facial affect[END_REF]Lakens, 2010). This suggests that conversational flow could increase people's sense of belonging.

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Disrupting the flow 4

Fluency generally indicates a positive state of affairs and is thereby related to positive affect (Winkielman, Schwarz, Fazendeiro, & Reber, 2003). Similarly, synchronous interaction induces a positive state [START_REF] Cappella | On Defining Conversational Coordination and Rapport[END_REF] and decreases the chance of a breakdown or "awkward silence" [START_REF] Burgoon | The unspoken dialogue: An introduction to nonverbal communication[END_REF], suggesting that fluency gives interaction partners a sense of control over the communication.

Accordingly, we expect fluent conversations to serve two different needs: the need for self-esteem and the need for control.

Finally, research shows that talking in unison or completing each other's talk induces a sense of consensus [START_REF] Edwards | Joint remembering; constructing an account of shared experience through conversational discourse[END_REF]. This is probably a largely unconscious process in which the absence of a need for systematic information processing leads to the heuristic inference of consensus-harmony as a proxy for agreement. Moreover, Smith and Postmes (in press) show that consensual group interaction produces a sense of social validation. Thus, we predict that conversational flow could increase perceived consensus and social validation.

Taken together, we expect that conversational flow implicitly fosters feelings of belonging, social validation, control, and self-esteem. However, conversational flow can be disrupted by a brief silence. As we expect conversational flow to satisfy social needs, we expect that disrupting it will threaten these needs.

Disrupting conversational flow

Research on Italian melodrama suggests that silences are often used to signal noncompliance or confrontation, and are also known as disaffiliative disfluency (Piazza, 2006). Although this research confirms that disruptions of conversational flow can undermine feelings of social cohesion, the question remains why they may threaten social

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Disrupting the flow 5 needs and signal negative events such as conflict? One explanation stems from the ostracism literature.

The ostracism literature suggests that silence can be a way of socially excluding people, and that this negatively affects emotions and social needs (e.g., Williams & Zadro, 2001;[START_REF] Beest | When inclusion costs and ostracism pays, ostracism still hurts[END_REF]2001). The present studies do not investigate this so-called silent treatment: in our studies nobody is actually excluded. However, the notion that people are, due to the evolutionary importance of group membership, highly sensitive to perceiving exclusion is relevant to our research. This may explain why people are highly sensitive even to very minor disruptions in conversational flow. That is, conversational silences negatively affect emotions and threaten needs because they could signal social rejection.

The present research

Two studies examined the psychological effects of conversational flow by comparing it to conversations that were briefly interrupted by a silence. We tested two hypotheses. First, conversational flow is associated with positive feelings of belonging, control, self-esteem, social validation, and perceived consensus. Second, disruptions instigate feelings of rejection and negatively affect emotions.

Study 1

Participants read a scenario in which a fluent conversation was either disrupted by a silent moment (disrupted flow condition) or was not disrupted (flow condition). The silence occurred after a speaker (with whom the participant was asked to identify) made a mildly controversial statement. The statement was chosen such that participants could feel validated in their opinions by the positive feelings induced by the conversational

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Disrupting the flow 6 flow. In contrast, if there was no conversational flow, there could be legitimate doubts about the level of consensus in the group, reducing validation and increasing feelings of rejection.

Method

Participants and design. Participants (102 undergraduates, 57 female) were randomly assigned to one of two conditions (flow vs. disrupted flow).

Procedure. Participants where instructed to imagine being the narrator when reading a scenario. In the scenario, the narrator had a conversation with two fellow students, in which he or she made a mildly controversial statement, (i.e., "I think obese people should pay for two seats in the bus"). In the flow condition one of the fellow students smoothly continued the conversation on the previous topic making no further reference regarding this statement. In the disrupted flow condition the fellow student resumed the conversation as in the flow condition, but after a brief silence had been described (i.e., "Briefly, it remains silent. Suzanne stirs her coffee").

Dependent measures. After reading the scenario, participants' emotions during the conversation were assessed by asking whether they felt distressed, afraid, angry, and hurt (negative emotions, Cronbach's alpha =.78), along with 10 filler emotions (1= not at all, 7= completely). Belonging, control, and self-esteem were assessed by means of the 15-item Need Threat Scale (NTS; [START_REF] Beest | When inclusion costs and ostracism pays, ostracism still hurts[END_REF]), all alphas>.81). Additionally, five items measured social validation ("I had the feeling my opinions were validated", alpha =. pay for an extra seat in the bus". Needs, rejection, and perceived consensus items were rated on a scale from 1 (strongly disagree) to 7 (strongly agree).

Results and discussion

Rejection and emotions. Means are reported in Results show that flowing conversations are associated with higher feelings of belonging, control, self-esteem, social validation, and perceived consensus, than conversations that are disrupted by a silent moment. This confirms that conversational flow serves different social needs. Furthermore, disrupted conversations increase negative emotions and feelings of rejection, resembling ostracism experiences (e.g., Williams, 2001). This indicates that a brief disruption of conversational flow is interpreted as rejection, even when nobody is factually excluded from the conversation.

Study 2

Study 2 was designed as a replication in a more realistic setting. Because conversational flow was expected to be a pleasant state, we included a measure for positive emotions. Furthermore, Study 2 aimed to disrupt the conversational flow more subtly than Study 1. To this end, participants imagined being a student in a videotaped

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Disrupting the flow 8 conversation in which a silent moment occurred (disrupted flow condition) or not (flow condition). The duration of silence was chosen so that it was unlikely to be noticed consciously. In order to assess whether conversational flow or disrupted flow were stronger contributors to the effects, we added a baserate condition, in which participants received equivalent information about the group discussion, but saw no video.

Method

Participants and design. Sixty undergraduates (51 female) were randomly assigned to one of three conditions (flow vs. disrupted flow vs. baserate).

Procedure. Participants were seated behind personal computers in individual cubicles. In the flow and the disrupted flow condition, participants were instructed to watch a 6-minute video of three female students who knew each other superficially having a conversation about relationships. Participants were instructed to imagine being one of the conversation partners (named Linda). After four minutes of ongoing conversation Linda said: "Recently, I heard about a teacher having sex with students, I think that this should not be allowed. Such a teacher should be fired immediately". In the flow condition the other group members smoothly continued the conversation on a topic not directly related to Linda's statement. The conversation continued approximately one more minute with no further reference to Linda's statement. In the disrupted flow condition, the statement was followed by four seconds of silence, after which the conversation continued similar to the flow condition. In the baserate condition participants received the same information as given in the video (a script and photos of the discussants), but no information about the fluency of the conversation. This assessed participants' baseline assumptions about the emotions and

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Disrupting the flow 9 needs triggered by the conversation, irrespective of the actual flow. Afterwards, all participants filled out a number of questionnaires.

The duration of silence in the video was pilot-tested, as the appropriate time varies between interactions [START_REF] Burgoon | The unspoken dialogue: An introduction to nonverbal communication[END_REF]. A pilot (n = 40) showed the video with edited silences of either 2.5, 4, or 6 seconds. Four seconds appeared to be an optimal duration of silence to ensure that participants did not notice the silence consciously, still perceived the conversation as natural, but nevertheless felt that the conversation was significantly less pleasant.

Dependent measures. Negative emotions, the four needs, perceived consensus, and rejection were examined as in Study 1. Three positive emotions were added to the emotions scale (i.e., happy, strong, & independent, alpha =.69).

Participants estimated the duration in seconds between Linda's statement and the other group member's response. Finally, to check whether participants noticed the video edit, they were asked whether they had seen something remarkable in the video.

Results and discussion

Manipulation Checks. Estimated time passing before the group members responded to Linda's statement did not differ between conditions (M disrupted flow = 3.58, SD disrupted flow = 2.17, M flow = 3.28, SD flow = 1.41, F(1,37)=.25, ns), implying that participants did not consciously detect the silence. Four participants reported that they thought the video was edited. They were removed from the analyses.

Rejection and Emotions. All means are reported in

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Disrupting the flow 10 η²=.25. In the disrupted flow condition participants reported more rejection, more negative emotions and less positive emotions than in the flow and baserate conditions (which were not different from each other).

Need Threats and Consensus. Condition significantly influenced participants' feelings of belonging, F(2,57)= 6.03, p<.01, η² =.18; self-esteem, F(2,57)= 8.78, p<.001, η² =.24; social validation, F(2,57)= 6.39, p<.01, η² =.18; and perceived consensus, F(2,57)= 5.65, p<.01, η² =.17. Higher need satisfaction and consensus were reported in the flow condition and baserate condition than in the disrupted flow condition. 1 Control feelings were not influenced by condition (F<1).

Study 2 shows that conversational flow is associated with positive emotions, feelings of belonging, self-esteem, social validation, and perceived consensus. The effects in the baserate condition resemble those in the flow condition, suggesting that conversational flow may be the standard in conversations: Without any information about the fluency, people assume that there is flow. However, a mere four-seconds silence (in a six-minute video clip) suffices to disrupt the conversational flow and make one feel distressed, afraid, hurt, and rejected. These effects occur despite participants' unawareness of the short, single silence.

General Discussion

The present research shows that conversational flow is associated with positive These findings extend previous research in several respects. Our finding that fluent conversations induce higher levels of belonging than disrupted conversations adds to research by Lakens (2010), showing that synchronically moving people are perceived as a group. We show that groups that converse harmoniously make people feel they belong. Moreover, we found that conversational flow relates to higher self-esteem, which is compatible with the suggestion that fluency signals a positive state of affairs (Winkielman et al., 2003). However, our studies revealed no relation between conversational flow and feelings of control. Possibly, this is because conversational flow to some extent implies a relinquishing of control and allowing oneself to be led by others. The present research uses a scenario study and a videotaped conversation to assess the psychological impact of conversational flow. Previous research on ostracism has confirmed that the consequences of observation are broadly similar to the actual experience itself (Wesselmann, Bagg, & Williams, 2009). The effects of observation and actual experience are especially similar when participants take the target's perspective, as was the case in our experiments. Nevertheless, it is likely that stronger effects may be observed in real-life situations: our methodology would appear to be a conservative test.

The current studies reveal that conversational flow serves social needs and maintains perceived consensus. As such, this is one of the first studies showing that conversational characteristics are major contributors to social psychological processes such as social validation and belonging. We think that comprehension of social behavior will benefit from investigating the role of conversational characteristics in such processes.

1. Consensus was correlated with rejection (r=-.35), belonging (r=.43), social validation (r=.30), and self-esteem (r=.33). Consensus did not mediate the effects of flow on the other dependent variables.

2. We would like to thank Daniël Lakens for his useful feedback on a previous version of this paper.

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Disrupting the flow 17

  85) and five items measured rejection (alpha =.90, Gaertner & Iuzzini, 2005). Participants indicated their perceived consensus in the group by rating their agreement with the statement: "the group members agreed with one another on whether obese people should A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT Disrupting the flow 7

  emotions and a heightened sense of belonging, self-esteem, social validation, and consensus. Disrupting the flow by a brief silence produces feelings of rejection and negative emotions. Study 2 also shows that the high levels of need satisfaction in the flow condition resembled baserate levels, produced by mere reading of the script. Presumably, to be fluent and therefore experience disruptions as relatively harmful.

Further, complementary to

  explicit processes of social validation (e.g., Festinger 1954), our research suggests a more implicit route to social validation. People do not always actively search for opinions of others, but can also validate their opinions by deriving a general feeling of consensus from fluent conversations. In the case of disfluency, for instance instigated by a silent moment, validating opinions becomes less of an automatic event: people may attend more closely to what is actually being said by others.Finally, the present research reveals that although people do not consciously notice brief silences, they are influenced by conversational disfluency in a way quite similar to ostracism experiences (e.g.,Williams, 2001). That is, people report feeling more rejected and experience more negative emotions when a conversation is disrupted , rather than when it flows. Thus, disrupted flow can implicitly elicit feelings of rejection, confirming human sensitivity to social exclusion cues.

Table 1

 1

	. Participants in the

reported more belonging, F(1,100)= 8.09, p<.01, η² =.08, more self-esteem, F(1,100)= 18.85, p<.001, η² =.16, more social validation, F(1,100)= 16.67, p<.001, η² =.14, and higher levels of perceived consensus, F(1,100)= 15.98, p<.001, η² =.14. Control feelings were not significantly influenced by the silence, F(1,100)= 1.09, ns.

Table 2

 2

	. There was a

emotions, F(2,57)= 9.70, p=.001, η² =.23., and positive emotions, F(2,57)= 9.58, p<.001,

Table 1 .

 1 Mean effects of flow in Study 1 (SDs in brackets). Within each row, means with different subscripts differ significantly at p < .01, means with no subscripts do not differ significantly.

	Variable	Flow (n = 50)	Disrupted flow (n = 51)
	Rejection	2.11 a (1.01)	3.07 b (1.10)
	Negative Emotions	2.06 a (.92)	2.93 b (1.17)
	Belonging	5.12 a (.85)	4.56 b (1.11)
	Control	5.04 (1.01)	4.82 (1.08)
	Self-Esteem	5.58 a (1.06)	4.59 b (1.23)
	Validation	4.85 a (.88)	4.01 b (1.17)
	Perceived Consensus	3.40 a (1.99)	2.12 b (1.17)
	Note:

Table 2 .

 2 Mean effects of flow in Study 2 (SDs in brackets). Within each row, means with different subscripts differ significantly at p < .05, means with no subscripts do not differ significantly.

	Variable	Baserate (n = 23)	Flow (n = 18)	Disrupted flow (n = 19)
	Rejection	1.83 a (.85)	1.90 a (1.04)	2.80 b (1.18)
	Negative Emotions	2.01 a (.96)	2.23 a (.94)	3.32 b (1.26)
	Positive Emotions	4.97 a (.63)	4.96 a (.77)	3.96 b (1.05)
	Belonging	5.90 a (.72)	5.66 a (.78)	4.89 b (1.30)
	Control	4.90 (.92)	4.76 (1.26)	4.39 (1.09)
	Self-Esteem	5.70 a (1.03)	5.53 a (.89)	4.42 b (1.18)
	Validation	5.17 a (.86)	4.86 a (.85)	4.18 b (1.00)
	Perceived Consensus	4.70 a (1.33)	4.28 a (1.32)	3.32 b (1.38)
	Note:

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ACCEPTED MANUSCRIPT

Disrupting the flow 14 Festinger, L. (1954). A theory of social comparison processes. Human Relations, 7, 117-140. doi: 10.1177


Joachim Hüffmeier
email: hueffmeier@uni-muenster.de

Guido Hertel

When the whole is more than the sum of its parts: Motivation gains in the wild

Keywords: motivation gains in groups, group potential, motivation in groups, performance in groups

Whereas negative effects of groups on individual motivation have been reported for many years, recent research has begun to show when and why working in a group can produce motivation gains compared to individual work. So far, this evidence has been limited to laboratory settings and rather simple tasks. Using data from swimming competitions at the 2008 Olympics, evidence is presented that motivation gains in groups also occur in field settings with more complex tasks.

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When the Whole is More Than the Sum of Its Parts: Motivation Gains in the Wild "Relays are exciting for me, I love team sports (.…)

That"s why I have always done well in relays in the past."

(Jason Leszak, anchor of the gold medal winning US 4 x 100 meter freestyle relay at the 2008 Olympics, after the victory)

People are often quite enthusiastic about working in groups. However, is working in a group indeed more motivating than working individually? 1 This question was among the first to be addressed by scientific psychology (e.g., [START_REF] Triplett | The dynamogenic factors in pacemaking and competition[END_REF]. Indeed, some early studies (e.g., [START_REF] Köhler | Kraftleistungen bei Einzel-und Gruppenarbeit [Physical performance in individual and group work[END_REF][START_REF] Moede | Der Wetteifer, seine Struktur und sein Ausmaß. Ein Beitrag zur experimentellen Gruppenpsychologie [The rivalry, its structure, and its range. A contribution to experimental group psychology[END_REF] suggested that groups can trigger higher motivation than individual work, illustrating what later has been defined as "motivation gains in groups".

While these early studies have been long neglected, The swimmers were asked successively to imagine starting from the four positions in an important relay competition. For each starting position, they indicated the perceived indispensability of their individual contribution for the relay group ("How well can a bad performance on your part be compensated by the other swimmers of your relay?"). The item was rated on a 7-point scale ranging from 0 (very well) to 6 (not at all).

Results

A one-factorial repeated-measures ANOVA on participants' indispensability ratings with position (first vs. second vs. third vs. fourth swimmer in the relay) as within-subject factor revealed the expected significant linear contrast, F(1, 28) = 11.85, p < .001, η² = .30.

Corresponding with our theorizing, participants perceived their individual contributions to the relay' performance to be increasingly less compensable with later serial position (M = 2.71, For each swimmer, we collected swimming times from both types of competition (i.e., individual and relay) from the website http://www.swimrankings.net/. We focused our analysis on the data from the semi-finals to obtain a reasonable sample-size. 2 If a swimmer did not advance to the semi-finals in the individual competition, we included her/his individual performance time from the first heats. 3 We corrected both performance times for the swimmer's respective reaction time by subtracting the time the athlete spent on the starting block after the starting signal (also retrieved from the webpage above). This was done to control for differences in the starting procedure in individual and relay swim competitions.

SD
Please note, however, that previous research did not find any differences between individual and relay competition after a swimming distance of ten meters [START_REF] Mclean | Addition of an approach to a swimming relay start[END_REF]. Thus, faster swimming times for relay swimmers are unlikely to be merely due to differences in the starting procedure.

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The corrected times were z-transformed separately for the 100 and 200 meter competitions. Complying with the required non-interdependence of data in the statistical analyses below, we excluded two data sets of swimmers who performed both in the 100 and 200 meter freestyle competitions. The swimming distance (100 vs. 200 meters) did not cause main or interaction effects so that we accumulated the data across swimming distance.

Study Design and Statistical Analysis. The analysis followed a 2 (individual vs. relay competition) x 4 (position: first vs. second vs. third vs. fourth swimmer in the relay) x 2 (gender: female vs. male) design with the first factor as a within-subject factor.

Results

An insignificant Box's M-Test resulted, F(18, 1113,43) = .74, ns., which indicates similarity of the variance-covariance matrices between groups. The 2 (individual vs. relay) x 4

(position) x 2 (gender) ANOVA on the reaction-time corrected and standardized swimming times revealed a significant main effect of the first factor, F(1,56) = 11.46, p ≤ .001, η² = .17 (see Table 1, for means and standard deviations), which corroborates Hypothesis 1 by reflecting an overall process gain in the relay compared to the individual competitions.

Moreover, the analysis revealed a significant main effect for position in the relay, F(3,56) = 4.28, p < .01, η² = .19, identifying the fourth swimmers as strongest swimmers on average.

More interestingly, the main effects were qualified by a significant interaction, F(3,56) = 8.11, p < .01, η² = .30. Consistent with Hypothesis 2, the performance advantage of the relay increased with later relay positions (see Figure 1). While the a priori contrast between

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individual and group performance conditions revealed slight and non-significant process losses for the starting swimmers in the relay, t(26) = -1.64, p = .11, the following swimmers showed significant process gains in the relay compared to the individual performance times.

Second, third, and last swimmers swam significantly faster in the relay than in the individual competition, t(14) = 2.60, p ≤ .01, t(4) = 2.23, p < .05, and t(16) = 4.24, p ≤ .001, respectively (for the raw swimming scores, see Table 2). Further confirming Hypothesis 2, process gains

shown by swimmers at the second and third position (M = .38, SD = .51) were less pronounced than process gains shown by the anchor swimmers (M = .71, SD = .69), t(35) = 1.68, p ≤ .05, all tests one-tailed. Finally, a marginally significant gender main effect occurred, F(1,56) = 2.78, p = .10, η² = .05, indicating that male swimmers tended to swim faster than female swimmers. However, gender showed no interaction effects with the other factors.

General Discussion

While the pilot study showed that perceived indispensability for the relay group increases with later positions in a relay, the main study reveals performance data that nicely correspond with the assumed motivational processes. Indeed, the data of the main study demonstrate significant process gains in four-person groups compared to individual performance, using a complex and meaningful task in a field setting. Consistent with our hypotheses, these process gains seem to result from two different sources, i.e. intergroup competition and social indispensability. As expected in Hypothesis 1, a significant main effect of the individual vs. relay factor suggests that performing as part of a relay team generally increases effort and (average) swimming speed compared to individual competitions.

Moreover, the results support Hypothesis 2 predicting that motivation in the relays increases with later serial positions due to higher perceived indispensability for the team. Indeed, while the relays' starting swimmers swam as quickly in the relay as in the individual competition, the three later swimmers showed significant process gains.

As a consequence, intergroup competition per se was obviously not sufficient to elicit additional effort when compensation by other team members was viable. This finding extends previous research on intergroup competition (e.g., [START_REF] Erev | Constructive intergroup competition as a solution to the free rider problem: A field experiment[END_REF][START_REF] Tauer | The effects of cooperation and competition on intrinsic motivation and performance[END_REF] Alternative explanations for our findings such as beneficial conditions in the swimming pool for later swimmers, fatigue or habituation to the Olympic atmosphere in the relay competitions are rather unlikely to explain our findings. Water waves caused by

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previous swimmers in a pool should worsen rather than improve swimming times. Moreover, the chronological order of the relay and individual competitions was opposed for the 100 and the 200 meter freestyle semi-finals, excluding order effects as plausible alternative explanation. The observed moderation effect of serial order within the relays also speaks against fatigue or habituation as alternative explanations for our results.

On the other hand, this research of course has several limitations. First of all, the sample size--at least for athletes filling the relay's third spot--is small. The limited sample size for this position is due to the specifics of international swimming championships such as the Olympics: The (relatively) weaker swimmers on a certain distance from one nation, who usually swim at the third relay position, cannot qualify for the individual competition as often as strong swimmers, but they can still be crucial members of their country's relays. One important aspect of our research makes us, however, confident that we observed valid findings even with this small sample size at the relay's third position: Our data pattern for the relay's third position is consistent across both reported studies and it accords exactly with our hypothesis. Nevertheless, the results for the third position have to be interpreted with caution and replications with data from other championships and larger samples are desirable.

Second, it is possible that the specific effect of this research is restricted to very tight competitions like the Olympics with relays performing at the same performance level. The perceived indispensability of the fourth swimmer might be quite different when the differences between the relays are more pronounced and the final outcome is clear even before the fourth swimmer starts. Third, we used performance data instead of subjective ratings as indicator of effort, similar to most other research on motivation in groups (e.g., [START_REF] Weber | Motivation gains of inferior group members: A metaanalytical review[END_REF]. Although subjective ratings can be informative in many ways (cf. [START_REF] Hüffmeier | How to motivate member in sports teams: A field investigation of motivation gains and motivation losses[END_REF], performance data are probably more valid measures of effort because they are less likely to be biased by limited accessibility, social desirability, etc.

Nevertheless, the interpretation of performance differences in terms of motivation is a

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conjecture that should be further validated using different measures of motivation. To address these issues, future replications of the present research might combine behavior-related data (i.e., performance) with subjective ratings, allowing also to explicitly assessing perceived indispensability as mediator of the observed effects. Moreover, biological (lactate) and physiological data (heart rate) might cover additional aspects of motivational processes due to group performance.

Our research has important implications for the management of teams in different contexts. We could show that teams can actually generate significant motivation gains outside the scientific laboratory and exceed when the appropriate conditions are in place. In particular, perceived indispensability for the group seems to be a central process that is not restricted to conjunctive task structures as shown in earlier work [START_REF] Weber | Motivation gains of inferior group members: A metaanalytical review[END_REF] but also works in additive group tasks affecting both weaker and stronger group members.

A Footnotes 1 Motivation generally refers to both effort and direction of behavior (e.g., [START_REF] Geen | Human motivation: A social psychological approach[END_REF]. In this research, we focus on motivation as effort, reflected in the intensity and persistence of behavior. To accord with existing terminological standards, however, we use the labels motivation gains and effort increases in groups interchangeably.

2 Focusing on the data from the finals would have minimized the sample size and allowed solely for an overall assessment of motivation gains. Analyses involving athletes' position in the relay would not have been possible.

3 The inclusion of these swimmers (N = 33) in our sample seems warranted because it cannot be assumed that athletes who practice for many years to have the chance to qualify for the Olympics intentionally show an inferior performance in this individual competition.

  repeatedly demonstrated motivation gains in groups when persons perceive their contribution as highly instrumental for the group's success

  = 1.57; M = 2.78, SD = 1.31; M = 3.09, SD = 1.25; M = 4.22, SD = 1.69; for the first to fourth position). More specifically, compensation at the fourth position was perceived as significantly less possible than at the second and third position (M = 2.93, SD = 1.22), t(28) = -3.88, p ≤ .001.

  by suggesting that motivation gains during intergroup competition are not only a consequence of support by ingroup members or social identification, but can also be driven by perceived indispensability of individual contributions to the team outcome (cf.Wittchen et al., 2010). Thus, the present research is highly compatible with previous work showing indispensability concerns as trigger of motivation gains in groups in other settings (e.g.,[START_REF] Hertel | Motivation gains in performance groups: Paradigmatic and theoretical developments on the Köhler effect[END_REF][START_REF] Weber | Motivation gains of inferior group members: A metaanalytical review[END_REF]. In fact, intergroup competition and high indispensability are often connected in previous research (e.g., a basketball player performing as a team representative) so that the relative impact of these two sources of motivation gains remains an open question.A number of additional aspects of this research are noteworthy. This research is the first to show that the group's net effort, indicated in the performance data of the relay teams, was significantly higher than the combined effort . Second, motivation gains at the Olympics are especially remarkable because athletes should be already maximally motivated in their individual competitions Therefore, the opportunities for additional improvement should be rather small compared to other competitions (e.g., in training or in scientific laboratory studies) as well as compared to other instances of group work (e.g., in business organizations). Third, compared to earlier research on the indispensability effect (e.g.,[START_REF] Weber | Motivation gains of inferior group members: A metaanalytical review[END_REF], the motivation gains found in the present research occurred regardless of gender and relative capability of individual group members.

  .,Schlereth, D., & Hertel, G. (2007). Indispensability effects in a sequential task: Motivation gains during anonymous cooperation on the Internet. International Journal of Internet Science, 2, 12-27. Wittchen, M., van Dick, R., & Hertel, G. (2010). Motivation gains in groups during intergroup competition: A review of underlying processes and development of an integrative model. Manuscript submitted for publication.

Figure 1 .

 1 Figure 1. Reaction-time corrected and standardized swimming times as a function of

Table 1 .

 1 Reaction-time Corrected and Standardized (z-transformed) Means and Standard Deviations of Swimming Times and Reaction-time Corrected Raw Swimming Scores as a Function of Individual vs. Relay Competition and Position in the Relay (N = 64).

		Position in the	N	M standardized	SD standardized	M raw swimming
		relay		swimming times	swimming times	scores
		1	27	-.07	.96	78.19
	Individual	2	15	.47	.77	87.30
		3	5	.92	1.12	87.73
		4	17	-.14	.82	87.40
		1	27	.11	.99	78.38
	Relay	2	15	.09	.90	86.92
		3	5	.57	1.17	87.39
		4	17	-.84	.85	86.66
	Note. Smaller numbers indicate faster swimming times.

Acknowledgements

The authors thank Sebastian Fischer, Jens Kanthak and Jörg Mikoleit for their help in the data collection process. This research has been supported by a grant from the Deutsche Forschungsgemeinschaft (He 2745/8-3) to the second author.

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