{"title": "Human and Ideal Observers for Detecting Image Curves", "book": "Advances in Neural Information Processing Systems", "page_first": 1459, "page_last": 1466, "abstract": "", "full_text": "Human and Ideal Observers for Detecting Image\n\nCurves\n\nAlan Yuille\n\nDepartment of Statistics & Psychology\nUniversity of California Los Angeles\n\nLos Angeles CA\n\nyuille@stat.ucla.edu\n\nFang Fang\n\nPsychology, University of Minnesota\n\nMinneapolis MN 55455\n\nfang0057@tc.umn.edu\n\nPaul Schrater\n\nDaniel Kersten\n\nPsychology, University of Minnesota\n\nPsychology, University of Minnesota\n\nMinneapolis MN 55455\nschrater@umn.edu\n\nMinneapolis MN 55455\nkersten@umn.edu\n\nAbstract\n\nThis paper compares the ability of human observers to detect target im-\nage curves with that of an ideal observer. The target curves are sam-\npled from a generative model which speci\ufb01es (probabilistically) the ge-\nometry and local intensity properties of the curve. The ideal observer\nperforms Bayesian inference on the generative model using MAP esti-\nmation. Varying the probability model for the curve geometry enables us\ninvestigate whether human performance is best for target curves that obey\nspeci\ufb01c shape statistics, in particular those observed on natural shapes.\nExperiments are performed with data on both rectangular and hexagonal\nlattices. Our results show that human observers\u2019 performance approaches\nthat of the ideal observer and are, in general, closest to the ideal for con-\nditions where the target curve tends to be straight or similar to natural\nstatistics on curves. This suggests a bias of human observers towards\nstraight curves and natural statistics.\n\n1 Introduction\n\nDetecting curves in images is a fundamental visual task which requires combining local\nintensity cues with prior knowledge about the probable shape of the curve. Curves with\nstrong intensity edges are easy to detect, but those with weak intensity edges can only be\nfound if we have strong prior knowledge of the shape, see \ufb01gure (1) But, to the best of\nour knowledge, there have been no experimental studies which test the ability of human\nobservers to perform curve detection for semi-realistic stimuli with locally ambiguous in-\ntensity cues or to explore how the dif\ufb01culty of the task varies with the geometry of the\ncurve.\n\nThis paper formulates curve detection as Bayesian inference. Following Geman and Je-\ndynak [6] we de\ufb01ne probability distributions PG(.) for the shape geometry of the target\ncurve and Pon(.), Po\ufb00 (.) for the intensity on and off the curve. Sampling this model gives\nus semi-realistic images de\ufb01ned on either rectangular or hexagonal grids. The human ob-\n\n\f(cid:65)(cid:108)(cid:109)(cid:111)(cid:115)(cid:116)(cid:32)(cid:105)(cid:109)(cid:112)(cid:111)(cid:115)(cid:115)(cid:105)(cid:98)(cid:108)(cid:101)\n\n(cid:73)(cid:110)(cid:116)(cid:101)(cid:114)(cid:109)(cid:101)(cid:100)(cid:105)(cid:97)(cid:116)(cid:101)\n\n(cid:69)(cid:97)(cid:115)(cid:121)\n\nFigure 1: It is plausible that the human visual system is adapted to the shape statistics of curves\nand paths in images like these. Left panel illustrates the trade-off between the reliability of intensity\nmeasurements and priors on curve geometry. The tent is easy to detect because of the large intensity\ndifference between it and the background, so little prior knowledge about its shape is required. But\ndetecting the goat (above the tent) is harder and seems to require prior knowledge about its shape.\nCentre panel illustrates the experimental task of tracing a curve (or road) in clutter. Right panel shows\nthat the \ufb01rst order shape statistics from 49 object images (one datapoint per image) are clustered\nround P (straight) = 0.64 (with P (lef t) = 0.18 and P (right) = 0.18) for both rectangular and\nhexagonal lattices, see [1].\n\nserver\u2019s task is to detect the target curve and to report it by tracking it with the (computer)\nmouse. Human performance is compared with that of an ideal observer which computes\nthe target curve using Bayesian inference (implemented by a dynamic programming algo-\nrithm). The ideal observer gives a benchmark against which human performance can be\nmeasured.\nBy varying the probability distributions PG, Pon.Po\ufb00 we can explore the ability of the\nhuman visual system to detect curves under a variety of conditions. For example, we can\nvary PG and determine what changes in Pon.Po\ufb00 are required to maintain a pre-speci\ufb01ed\nlevel of detection performance.\n\nIn particular, we can investigate how human performance depends on the geometrical dis-\ntribution PG of the curves. It is plausible that the human visual system has adapted to the\nstatistics of the natural world, see \ufb01gure (1), and in particular to the geometry of salient\ncurves. Our measurements of natural image curves, see \ufb01gure (1), and studies by [16],\n[10], [5] and [2], show distributions for shape statistics similar to those found for image\nintensities statistics [11, 9, 13]. We therefore investigate whether human performance ap-\nproaches that of the ideal when the probability distributions PG is similar to that for curves\nin natural images.\n\nThis investigation requires specifying performance measures to determine how close hu-\nman performance is to the ideal (so that we can quantify whether humans do better or\nworse relative to the ideal for different shape distributions PG). We use two measures of\nperformance. The \ufb01rst is an effective order parameter motivated by the order parameter\ntheory for curve detection [14], [15] which shows that the detectability of target curves, by\nan ideal observer, depends only on an order parameter K which is a function of the prob-\nability distributions characterizing the problem. The second measure computes the value\nof the posterior distribution for the curves detected by the human and the ideal and takes\nthe logarithm of their ratio. (For theoretical reasons this is expected to give a performance\nmeasure similar to the effective order parameter).\n\nThe experiments are performed by human observers who are required to trace the target\ncurve in the image. We simulated the images \ufb01rst on a rectangle grid and then on a hexag-\nonal grid to test the generality of the results. In these experiments we varied the probability\ndistributions of the geometry PG and the distribution Pon of the intensity on the target curve\nto allow us to explore a range of different conditions (we kept the distribution Po\ufb00 \ufb01xed).\nIn section (2) we brie\ufb02y review previous psychophysical studies on edge detection. Sec-\n\n\f0.25\n\n0.2\n\n0.15\n\n0.1\n\n0.05\n\ny\nt\ni\nl\ni\n\nb\na\nb\no\nr\nP\n\nP\n\nPon\n\n0\n\n0\n\n4\n\n8\n\nIntensity\n\n12\n\n16\n\nFigure 2: Left panel: the tree structure superimposed on the lattice. Centre panel: a pyramid struc-\nture used in the simulations on the rectangular grid. Right panel: Typical distributions of Pon, Po\ufb00\n\ntion (3) describes our probabilistic model and speci\ufb01es the ideal observer. In section (4), we\ndescribe the order parameter theory and de\ufb01ne two performance measures. Sections (5,6)\ndescribe experimental results on rectangular and hexagonal grids respectively in terms of\nour two performance measures.\n\n2 Previous Work\n\nPrevious psychophysical studies have shown conditions for which the human visual sys-\ntem is able to effectively group contour fragments when embedded in an array of distract-\ning fragments [3, 8]. Most of these studies have focused on the geometrical aspects of\nthe grouping process. For example, it is known that the degree to which a target contour\n\u201cpops out\u201d depends on the degree of similarity of the orientation of neighboring fragments\n(typically gabor patches) [3], and that global closure facilitates grouping [8].\n\nRecently, several researchers have shown that psychophysical performance for contour\ngrouping may be understood in terms of the statistical properties of natural contours [12, 5].\nFor example, Geisler [5] has shown that human contour detection for line segments can be\nquantitatively predicted from a local grouping rule derived from measurements of local\nedge statistics.\n\nHowever, apart from studies that manipulate the contrast of gabor patch tokens [4], there\nhas been little work on how intensity and contour geometry information is combined by the\nvisual system under conditions that begin to approximate those of natural contours. In this\npaper we attempt to \ufb01ll this gap by using stimuli sampled from a generative model which\nenables us to quantitatively characterize the shape and intensity information available for\ndetecting curves and compare human performance with that of an ideal detector.\n\n3 The Probabilistic Model for Data Generation\n\nWe now describe our model in detail. Following [6], we formulate target curve detection as\ntree search, see \ufb01gure (2), through a Q-nary tree. The starting point and initial direction is\nspeci\ufb01ed and there are QN possible distinct paths down the tree. A target curve hypothesis\nconsists of a set of connected straight-line segments called segments. We can represent a\npath by a sequence of moves {ti} on the tree. Each move ti belongs to an alphabet {a\u00b5} of\nsize Q. For example, the simplest case sets Q = 3 with an alphabet a1, a2, a3 correspond-\ning to the decisions: (i) a1 \u2013 go straight (0 degrees), (ii) a2 \u2013 go left (-5 degrees), or (iii)\na3 \u2013 go right (+ 5 degrees). This determines a path x1, . . . , xN in the image lattice where\nxi, xi+1 indicate the start and end points of the ith segment. The relationship between the\ntwo representations is given by xi+1 = xi + w(xi \u2212 xi\u22121, ti), where w(xi \u2212 xi\u22121, ti) is\na vector of approximately \ufb01xed magnitude (choosen to ensure that the segment ends on a\npixel) and whose direction depends on the angle of the move ti relative to the direction of\nthe previous segment xi \u2212 xi\u22121. In this paper we restrict Q = 3.\n\n\u0001\n\u0001\n\fWe put a prior probability on the geometry of paths down the tree. This is of form\nP ({ti}) = (cid:81)N\ni=1 P (ti). We will always require that the probabilities to go left or right\nare equal and hence we can specify the distribution by the probability P (straight) that the\ncurve goes straight. Our analysis of image curve statistics suggests that P (straight) =\n0.64 for natural images, see \ufb01gure (1).\nWe specify the probability models Pon, Pof f for the image intensity on and off to be\nof Poisson form de\ufb01ned over the range (1, ..., 16), see \ufb01gure (2). This reduced range\nmeans that the distributions are expressed as Pon(I = n) = (1/Kon)e\u2212\u03bbon\u03bbn\non/n! and\nof f /n!, where Kon, Kof f are normalization factors. We\nPof f (I = n) = (1/Kof f )e\u2212\u03bbof f \u03bbn\n\ufb01x \u03bbof f = 8.0 and will vary \u03bbon. The quantity \u03bbon \u2212 \u03bbof f is a measure of the local in-\ntensity contrast of the target contour and so we informally refer to it as the signal-to-noise\nratio (SNR).\n\nThe Ideal Observer estimates the target curve trajectory by MAP estimation (which we\ncompute using dynamic programming). As described in [6], MAP estimation corresponds\nto \ufb01nding the path {ti} with \ufb01lter measurements {yi} which maximizes the (scaled) log-\nlikelihood ratio, or reward function,\n\nr({ti}, {yi}) =\n\n1\nN\n\n{log P (Y |X) + log P (X) \u2212\n\nN\n\n(cid:88)\n\ni=1\n\nlog U (ti)}\n\n=\n\n1\nN\n\nN\n\n(cid:88)\n\ni=1\n\nlog{Pon(yi)/Pof f (yi)} +\n\n1\nN\n\nN\n\n(cid:88)\n\ni=1\n\nlog{PG(ti)/U (ti)},\n\n(1)\n\ni=1 log U (ti) =\n\nwhere U (.) is the uniform distribution (i.e. U (t) = 1/3 \u2200t) and so (cid:80)N\n\u2212N log 3 which is a constant. The length of the curve is N = 32 in our experiments.\nWe implement this model on both rectangular and hexagonal lattices (the hexagonal latt-\ntices equate for contrast at borders, and are visually more realistic). The tree representation\nused by Geman and Jedynak must be modi\ufb01ed when we map onto these lattices. For a\nrectangular lattice, the easiest way to do this involves de\ufb01ning a pyramid where paths start\nat the apex and the only allowable \u201cmoves\u201d are: (i) one step down, (ii) one step down\nand one step left, and (iii) one step down and one step right. This can be represented by\nxi+1 = xi+w(ti) where ti \u2208 {\u22121, 0, 1} and w(\u22121) = \u2212(cid:126)i\u2212(cid:126)j, w(0) = \u2212(cid:126)j, w(1) = +(cid:126)i\u2212(cid:126)j\n(where (cid:126)i,(cid:126)j are the x, y directions on the lattice).\nA similar procedure is used on the hexagonal lattice. But for certain geometry probabil-\nities we observed that the sampled curves had \u201cclumping\u201d where the path consists of a\nlarge number of zig-zags. This was sometimes confusing to the human observers. So we\nimplemented a higher-order Markov model which explicitly forbade zig-zags. We show\nexperimental results for both the Clumping and No-Cluming models.\n\nTo obtain computer simulations of target curves in background clutter we proceed in two\nstages. In the \ufb01rst stage, we stochastically sample from the distribution PG(t) to produce\na target curve in the pyramid (starting at the apex and moving downwards). In the second\nstage, we must sample from the likelihood function to generate the image. So if a pixel\nx is on or off the target curve (which we generated in the \ufb01rst stage) then we sample the\nintensity I(x) from the distribution Pon(I) or Pof f (I) respectively.\n\n4 Order Parameters and Performance Measures\n\nYuille et al [14],[15] analyzed the Geman and Jedynak model [6] to determine how the\nability to detect the target curve depended on the geometry Pg and the intensity properties\nPon.Po\ufb00 . The analysis showed that the ability to detect the target curve behaves as e\u2212KN,\n\n\fwhere N is the length of the curve and K is an order parameter. The larger the value of K\nthen the easier it is to detect the curve.\nThe order parameter is given by K = D(Pon||Pof f ) + D(PG||U ) \u2212 log Q [15], where U is\nthe uniform distribution. If K > 0 then detecting the target curve is possible but if K < 0\nthen it becomes impossible to \ufb01nd it (informally, it becomes like looking for a needle in a\nhaystack).\n\nThe order parameter illustrates the trade-off between shape and intensity cues and deter-\nmines which types of curves are easiest to detect by an ideal observer. The intensity cues\nare quanti\ufb01ed by D(Pon||Pof f ) and the shape cues by D(PG||U ). The easiest curves to\ndetect are those which are straight lines (i.e. D(PG||U ) takes its largest possible value).\nThe hardest curves to detect are those for which the geometry is most random. The stronger\nthe intensity cues (i.e. the bigger D(Pon||Pof f )) then, of course, the easier the detection\nbecomes.\n\nSo when comparing human performance to ideal observers we have to take into account\nthat some types of curves are inherent easier to detect (i.e. thay have larger K). Human\nobservers are good at detecting straight line curves but so are ideal obervers. We need\nperformance measures to quantify the relative effectiveness of human and ideal observers.\nOtherwise, we will not be able to conclude that human observers are biased towards partic-\nular curve shapes (such as those occuring in natural images).\n\nWe now de\ufb01ne two performance measures to quantify the relative effectivenes of human\nand ideal observers. Our \ufb01rst measure is based on the hypothesis that human observers\nhave an \u201ceffective order parameter\u201d. In other words, their performance on the target curve\ntracking task behaves like e\u2212N KH where KH is an effective order parameter which differ-\nence from the true order parameter K might re\ufb02ect a human bias towards straight lines or\necological shape priors. We estimate the effective order parameters by \ufb01xing PG, Po\ufb00 and\nadjusting Pon until the observers achieve a \ufb01xed performance level of at most 5 errors on\non for the ideal and human observers\na path of length 32. This gives distributions P I\nrespectively. Then we set KH = K \u2212 D(P H\non are\nthe distributions used by the human and the ideal (respectively) to achieve similar perfor-\nmance.\n\non||Pof f ), where P H\n\non||Pof f ) + D(P I\n\non, P H\n\non, P I\n\nOur \ufb01rst performance measure is the difference \u2206K = D(P H\nbetween the effective and the true order parameters.\n\non||Pof f ) \u2212 D(P I\n\non||Pof f )\n\nBut order parameter analysis should be regarded with caution for the curve detection task\nused in our experiments. The experimental criterion that the target path be found with 5\nor less errors, see section (5), was not included in the theoretical analysis [14],[15]. Also\nsome small corrections need to be made to the order parameters due to the nature of the\nrectangular grid, see [15] for computer calculations of the size of these corrections. These\ntwo effects \u2013 the error criterion and the grid correction \u2013 means that the order parameters\nare only approximate for these experimental conditions.\n\nThis motivates a second performance measure where we calculate the value of the posterior\nprobability (proportional to the exponential of r in equation (1)) for the curve detected by\nthe human and the ideal observer (for identical distributions PG, Pon, Po\ufb00). We measure\nthe logarithm of the ratio of these values. (A theoretical relationship can be shown between\nthese two measures).\n\n5 Experimental Results on Rectangular Grid\n\nTo assess human performance on the road tracking task, we \ufb01rst had a set of 7 observers \ufb01nd\nthe target curve in a tree de\ufb01ned by a rectangular grid \ufb01gure (3)A. The observer tracked the\n\n\fStimulus\n\nIdeal\n\nHuman\n\nStimulus\n\nIdeal\n\nHuman\n\nA\n\nRectangular grid\n\nB\n\nC\n\n\u0002\u0001\n\n\t\u000b\n\n\u000f\u000e\n\n\u0005 \u000e\n\n\u001b\u0001\n\n\u001c\u0002\u0006\n\n\t\u000b\n\n\u001e\u000e\n\n\u0005 \u000e\n\nFigure 3: A. Rectangular Grid Stimulus (Left), Example Path: Ideal (Center), Example Path: Human\n(Right). B & C. Hexagonal Grid Stimulus (Left), Example Path: Ideal (Center), Example Path: Hu-\nman (Right). Panel C shows an example of a path with higher order constraints to prevent \u201cclumping\u201d.\nThere were a number of other differences between the rectangular and hexagonal grid psychophysics,\nincluding rectangle samples were slightly smaller than the hexgaons, and feedback was presented to\nthe observers without (rectangular) or with background (hexagonal), and the lowestp(straight) was\n0.0 for rectangular and0.1 for hexagonal grids.\ncontour by starting at the far left corner and making a series of 32 key presses that moved\nthe observer\u2019s tracked contour either left, right, or straight at each key press. Each contour\nestimate was scored by counting the number of positions the observer\u2019s contour was off the\ntrue path. Each observer had a training period in which the observer was shown examples\nof contours produced from the four different geometry distributions and practiced tracing\nin noise.\n\nDuring an experimental session, the geometry distribution was \ufb01xed at one the four possi-\nble values and observers were told which geometry distribution was being used to generate\nthe contours. The parameter \u03bbon of Pon was varied using an adaptive procedure until the\nhuman observer managed to repeatedly detect the target curve with at most \ufb01ve misclassi-\n\ufb01ed pixels. This gave a threshold of \u03bbon \u2212 \u03bbof f for each probability distribution de\ufb01ned\nby P (straight). This threshold could be compared to that of the Ideal Observer (obtained\nby using dynamic programming to estimate the ideal, also allowing for up to \ufb01ve errors).\nThe process was repeated several times for the four geometry distribution conditions.\n\nThe thresholds for 7 observers and the ideal observer are shown in \ufb01gure 4. These thresh-\nolds can be used to calculate our \ufb01rst performance measure (\u2206K ) and determine how\neffectively observers are using the available image information at each P (straight).\nThe results are illustrated in \ufb01gure (4)B where the human data was averaged over seven\nsubjects. They show that humans perform best for curves with P (straight) = 0.66 which\nis closest to the natural priors, see \ufb01gure (1). Conversely, \u2206K is biggest for the curves with\nP (straight) = 0.0, which is the condition that differs most from the natural statistics.\nWe next compute our second performance measure (for which Pon, Pof f , PG are the same\nfor the ideal and the human observer). The average difference of this performance measure\nfor the each geometry distribution is an alternative way how well observers are using the in-\ntensity information as a function of geometry, with a zero difference indicating optimal use\nof the information. The results are shown in \ufb01gure (4)C. Notice that the best performance\nis achieved with P (straight) = 0.9.\nObserve that the two performance measures give different answers for this experiment.\nWe conclude that our results are consistent either with a bias to ecological statistics or to\nstraight lines. But the rectangular lattice\n\n6 Experiments on Hexagonal Lattices\n\nIn these experiments we used a hexagonal lattice because, for the human observers, the\ncontrast at the edges corresponding to a left, straight, or right move is the same (in contrast\nto the rectangular grid, in which left and right moves only share a corner). We also use the\nsame values of Pon, Po\ufb00 , P (straight) for the humans and the ideal.\n\n\u0003\n\u0004\n\u0005\n\u0006\n\u0007\n\b\n\f\n\u0010\n\u0007\n\u0011\n\u0012\n\u0013\n\u0014\n\u0004\n\u0010\n\u0005\n\u0015\n\u0013\n\u0016\n\u0017\n\u0018\n\u0019\n\u001a\n\u001a\n\u0003\n\u0004\n\u0005\n\u0006\n\u0007\n\b\n\u001d\n\f\n\u0010\n\u0007\n\u0011\n\u0012\n\u0013\n\u0014\n\u0004\n\u0010\n\u0005\n\u0015\n\u0013\n\u0016\n\u0017\n\u0018\n\u0019\n\u001a\n\u001a\n\fFigure 4: A-C. Psychophysical results on rectangular grid. A. Threshold \u03bbon \u2212 \u03bbof f plotted against\nP (straight). The top seven curves are the results of the seven subjects. The bottom curve is for\nthe ideal observer. B. The difference between human and ideal K order parameters. C. The average\nreward difference between ideal and human observers. D-I shows psychophyscial results on a hexag-\nonal grid. D-F are for the Clumping condition, and G-I for the No Clumping condition for which\nhigh order statistics prevented sharp turns that result in \u201cclumps\u201d.\n\nWe performed experiments on the hexagonal lattice under four different probabilities for\nthe geometry. These were speci\ufb01ed by P (straight) = 0.10, 0.33, 0.66, 0.90 (in other\nwords, the straightest curves will be sampled when P (straight) = 0.90 and the least\nstraight from P (straight) = 0.10). For reasons described previously, we did the experi-\nment in two conditions. (1) allowing zig-zags \u201cClumping\u201d, (2) forbidding zig-zags \u201cNo-\nClumping\u201d. We show examples of the stimuli, the ideal results (indicated by dotted path),\nand the human results (indicated by dotted path) for the Clumping amd No-Clumping cases\nin \ufb01gure (4B & C), respectively.\n\nThe threshold SNR results for Clumping and No Clumping are summarized in \ufb01gures (4D\n& G. The average \u2206K = Khuman \u2212 Kideal results for Clumping and No Clumping are\nsummarized in \ufb01gure (4E & H). The average reward difference, \u2206r = rideal \u2212 rhuman,\nresults for Clumping and No Clumping are summarized in \ufb01gure (4F & I).\n\nBoth performance measures give consistent results for the Clumping data suggesting that\nhumans are best when detecting the straightest lines (P (straight) = 0.9). But the situation\nis more complicated for the No Clumping case where human observers show preferences\nfor P (straight) = 0.9 or P (straight) = 0.66.\n\n\f7 Summary and Conclusions\n\nThe results of our experiments suggest that humans are most effective at detecting curves\nwhich are straight or which obey ecological statistics. But further experiments are needed\nto clarify this. Our two performance measures were not always consistent, particularly for\nthe rectangular grid (we are analyzing this discrepency theoretically). The \ufb01rst measure\nsuggested a bias towards ecological statistics on the rectangular grid and for No Clumping\nstimuli on the hexagonal grid. The second measure showed a bias towards curves with\nP (straight) = 0.9 on the rectangular and hexagonal grids.\nTo our knowledge, this is the \ufb01rst experiment which tests the performance of human ob-\nservers for detecting target curves by comparison with that of an ideal observer with am-\nbiguous intensity data. Our novel experimental design and stimuli may cause artifacts due\nto the rectangular and hexagonal grids. Further experiments may need to \u201dquantize\u201d curves\nmore carefully and reduce the effect of the grids.\n\nFurther experiments performed on a larger number of subjects may be able to isolated\nmore precisely the strategy that human observers employ. Do they, for example, make use\nof a speci\ufb01c geometry prior based on empirical edge statistics [16], [10]. If so, this might\naccount for the bias towards straigthness and natural priors observed in the experiments\nreported here.\n\nAcknowledgments\n\nSupported by NIH RO1 EY11507-001, EY02587, EY12691 and, EY013875-01A1, NSF\nSBR-9631682, 0240148.\n\nReferences\n[1] Brady, M. J. (1999). Psychophysical investigations of incomplete forms and forms with back-\n\nground. Ph. D., University of Minnesota.\n\n[2] Elder J.H. and Goldberg R.M.. Ecological Statistics of Gestalt Laws for the Perceptual Organi-\n\nzation of Contours. Journal of Vision, 2, 324-353. 2002.\n\n[3] Field, D. J., Hayes, A., & Hess, R. F. Contour integration by the human visual system: evidence\n\nfor a local \u201cassociation \ufb01eld\u201d. Vision Res, 33, (2), 173-93. 1993.\n\n[4] Field, D. J., Hayes, A., & Hess, R. F. The roles of polarity and symmetry in the perceptual\n\ngrouping of contour fragments. Spat Vis, 13, (1), 51-66.2000.\n\n[5] Geisler W.S. , Perry J.S. , Super B.J. and Gallogly D.P. . Edge co-occurrence in natural images\n\npredicts contour grouping performance. Vision Res, 41, (6), 711-24. 2001.\n\n[6] Geman D. and Jedynak B. . \u201cAn active testing model for tracking roads from satellite images\u201d.\n\nIEEE Trans. Pattern Anal. Mach. Intell., 18, 1-14, 1996.\n\n[7] Hess, R., & Field, D. . Integration of contours: new insights. Trends Cogn Sci, 3, (12), 480-\n\n486.1999.\n\n[8] Kovacs, I., & Julesz, B. A closed curve is much more than an incomplete one: effect of closure\n\nin \ufb01gure-ground segmentation. Proc Natl Acad Sci U S A, 90, (16), 7495-7. 1993.\n\n[9] Lee A.B., Huang J.G., and Mumford D.B., \u201cRandom collage model for natural images\u201d, Int\u2019l J.\n\nof Computer Vision, Oct. 2000.\n\n[10] Ren X. and Malik J. . \u201cA Probabilistic Multi-scale Model for Contour Completion Based on\n\nImage Statistics\u201d. In Proceedings ECCV. 2002\n\n[11] Ruderman D.L. and Bialek W. , \u201cStatistics of natural images: scaling in the woods\u201d, Phy. Rev.\n\nLetter, 73:814-817, 1994.\n\n[12] Sigman, M., Cecchi, G. A., Gilbert, C. D., & Magnasco, M. O. . On a common circle: natural\n\nscenes and Gestalt rules. Proc Natl Acad Sci U S A, 98, (4), 1935-40. 2001.\n\n[13] Wainwright M.J. and Simoncelli E.P., \u201cScale mixtures of Gaussian and the statistics of natural\n\nimages\u201d, NIPS, 855-861, 2000.\n\n[14] Yuille A.L. and Coughlan J.M. . \u201cFundamental Limits of Bayesian Inference: Order Parameters\n\nand Phase Transitions for Road Tracking\u201d . IEEE PAMI. Vol. 22. No. 2. February. 2000.\n\n[15] Yuille A.L. , Coughlan J.M., Wu Y-N. and Zhu S.C. . \u201cOrder Parameters for Minimax Entropy\n\nDistributions: When does high level knowledge help?\u201d IJCV. 41(1/2), pp 9-33. 2001.\n\n[16] Zhu S.C. . \u201cEmbedding Gestalt Laws in Markov Random Fields \u2013 A theory for shape modeling\n\nand perceptual organization\u201d. IEEE PAMI, Vol. 21, No.11, pp1170-1187, Nov, 1999.\n\n\f", "award": [], "sourceid": 2479, "authors": [{"given_name": "Fang", "family_name": "Fang", "institution": null}, {"given_name": "Daniel", "family_name": "Kersten", "institution": null}, {"given_name": "Paul", "family_name": "Schrater", "institution": null}, {"given_name": "Alan", "family_name": "Yuille", "institution": null}]}