{"title": "Model selection and velocity estimation using novel priors for motion patterns", "book": "Advances in Neural Information Processing Systems", "page_first": 1793, "page_last": 1800, "abstract": "Psychophysical experiments show that humans are better at perceiving rotation and expansion than translation. These findings are inconsistent with standard models of motion integration which predict best performance for translation [6]. To explain this discrepancy, our theory formulates motion perception at two levels of inference: we first perform model selection between the competing models (e.g. translation, rotation, and expansion) and then estimate the velocity using the selected model. We define novel prior models for smooth rotation and expansion using techniques similar to those in the slow-and-smooth model [17] (e.g. Green functions of differential operators). The theory gives good agreement with the trends observed in human experiments.", "full_text": "Model selection and velocity estimation using novel\n\npriors for motion patterns\n\nShuang Wu\n\nDepartment of Statistics\n\nUCLA, Los Angeles, CA 90095\nshuangw@stat.ucla.edu\n\nHongjing Lu\n\nDepartment of Psychology\n\nUCLA, Los Angeles, CA 90095\n\nhongjing@ucla.edu\n\nAlan Yuille\n\nDepartment of Statistics\n\nUCLA\n\nLos Angeles, CA 90095\n\nyuille@stat.ucla.edu\n\nAbstract\n\nPsychophysical experiments show that humans are better at perceiving rotation\nand expansion than translation. These \ufb01ndings are inconsistent with standard\nmodels of motion integration which predict best performance for translation [6].\nTo explain this discrepancy, our theory formulates motion perception at two lev-\nels of inference: we \ufb01rst perform model selection between the competing models\n(e.g. translation, rotation, and expansion) and then estimate the velocity using the\nselected model. We de\ufb01ne novel prior models for smooth rotation and expansion\nusing techniques similar to those in the slow-and-smooth model [17] (e.g. Green\nfunctions of differential operators). The theory gives good agreement with the\ntrends observed in human experiments.\n\n1 Introduction\n\nAs an observer moves through the environment, the retinal image changes over time to create mul-\ntiple complex motion \ufb02ows, including translational, circular and radial motion. Human observers\nare able to process different motion patterns and infer ego motion and global structure of the world.\nHowever, the inherent ambiguity of local motion signals requires the visual system to employ an ef-\n\ufb01cient integration strategy to combine many local measurements in order to perceive global motion.\nPsychophysical experiments have identi\ufb01ed a variety of phenomena, such as motion capture and\nmotion cooperativity [11], which appear to be consequences of such integration. A number of com-\nputational Bayesian models have been proposed to explain these effects based on prior assumptions\nabout motion. In particular, it has been shown that a slow-and-smooth prior, and related models, can\nqualitatively account for a range of experimental results [17, 15, 16] and can quantitatively account\nfor others [7, 12].\nHowever, the integration strategy modeled by the slow-and-smooth prior may not generalize to more\ncomplex motion types, such as circular and radial motion, which are critically important for estimat-\ning ego motion. In this paper we are concerned with two questions. (1) What integration priors\nshould be used for a particular motion input? (2) How can local motion measurements be combined\nwith the proper priors to estimate motion \ufb02ow? Within the framework of Bayesian inference, the\nanswers to these two questions are respectively based on model selection and parameter estimation.\nIn the \ufb01eld of motion perception, most work has focused on the second question, using parame-\nter estimation to estimate motion \ufb02ow. However, Stocker and Simoncelli [13] recently proposed a\nconditioned Bayesian model in which strong biases in precise motion direction estimates arise as a\nconsequence of a preceding decision about a particular hypothesis (left vs. right motion).\n\n\fThe goal of this paper is to provide a computational explanation for both of the above questions\nusing Bayesian inference. To address the \ufb01rst question, we develop new prior models for smooth\nrotation and expansion motion. To address the second, we propose that the human visual system has\navailable multiple models of motion integration appropriate for different motion patterns. The visual\nsystem decides the best integration strategy based upon the perceived motion information, and this\nchoice in turn affects the estimation of motion \ufb02ow.\nIn this paper, we \ufb01rst present a computational theory in section (3) that includes three different in-\ntegration strategies, all derived within the same framework. We test this theory in sections (4,5) by\ncomparing its predictions with human performance in psychophysical experiments, in which sub-\njects were asked to discriminate motion direction in translational, rotational, and expanding stimuli.\nWe employ two commonly used stimuli, random dot patterns and moving gratings, to show that the\nmodel can apply to a variety of inputs.\n\n2 Background\n\nThere is an enormous literature on visual motion phenomena and there is only room to summarize\nthe work most relevant to this paper. Our computational model relates most closely to work [17, 15,\n7] that formulates motion perception as Bayesian inference with a prior probability biasing towards\nslow-and-smooth motion. But psychophysical [4, 8, 1, 6], physiological [14, 3] and fMRI data [9]\nsuggests that humans are sensitive to a variety of motion patterns including translation, rotation, and\nexpansion. In particular, Lee et al [6] demonstrated that human performance on discrimination tasks\nfor translation, rotation, and expansion motion was inconsistent with the predictions of the slow-and-\nsmooth theory (our simulations independently verify this result). Instead, we propose that human\nmotion perception is performed at two levels of inference: (i) model selection, and (ii) estimating\nthe velocity with the selected model. The concept of model selection has been described in the\nliterature, see [5], but has only recently been applied to model motion phenomena [13]. Our new\nmotion models for rotation and expansion are formulated very similarly to the original slow-and-\nsmooth model [17] and similar mathematical analysis [2] is used to obtain the forms of the solutions\nin terms of Greens functions of the differential operators used in the priors.\n\n3 Model Formulation\n\n3.1 Bayesian Framework\n\nWe formulate motion perception as a problem of Bayesian inference with two parts. The \ufb01rst part\nselects a model that best explains the observed motion pattern. The second part estimates motion\nproperties using the selected model.\nThe velocity \ufb01eld {(cid:126)v} is estimated from velocity measurements {(cid:126)u} at discrete positions {(cid:126)ri, i =\n1, . . . N} by maximizing\n\np({(cid:126)v}|{(cid:126)u}, M) = p({(cid:126)u}|{(cid:126)v})p({(cid:126)v}|M)\n\n,\n\np({(cid:126)u}|M)\n\nThe prior\n\np({(cid:126)v}|M) = exp(\u2212E({(cid:126)v}|M)/T ),\n\ndiffers for different models M and is discussed in section 3.2.\nThe likelihood function\n\np({(cid:126)u}|{(cid:126)v}) = exp(\u2212E({(cid:126)u}|{(cid:126)v})/T )\ndepends on the measurement process and is discussed in section 3.3.\nThe best model that explains measurement {(cid:126)u} is chosen by maximizing the model evidence\n\n(cid:90)\n\np({(cid:126)u}|M) =\n\np({(cid:126)u}|{(cid:126)v})p({(cid:126)v}|M)d{(cid:126)v}\n\nwhich is equivalent to maximizing the posterior probability of the model M (assuming uniform prior\non the models):\n\nM\u2217 = arg max\n\nM\n\nP (M|{(cid:126)u}) = arg max\n\nM\n\nP ({(cid:126)u}|M)P (M)\n\nP ({(cid:126)u})\n\n= arg max\nM\n\nP ({(cid:126)u}|M).\n\n(5)\n\n(1)\n\n(2)\n\n(3)\n\n(4)\n\n\f3.2 The Priors\n\nWe de\ufb01ne three priors corresponding to the three different types of motion \u2013 translation, rotation, and\nexpansion. For each motion type, we encourage slowness and smoothness. The prior for translation\nis very similar to the slow-and-smooth prior [17] except we drop the higher-order derivative terms\nand introduce an extra parameter (to ensure that all three models have similar degrees of freedom).\nWe de\ufb01ne the priors by their energy functions E({(cid:126)v}|M), see equation (2). We label the models by\nM \u2208 {t, r, e}, where t, r, e denote translation, rotation, and expansion respectively. (We note that\nthe prior for expansion will also account for contraction).\n\nE({(cid:126)v}|M = t) =\n\n\u03bb(|(cid:126)v|2 + \u00b5|\u2207(cid:126)v|2 + \u03b7|\u22072(cid:126)v|2)d(cid:126)r\n\n(6)\n\n1. slow-and-smooth-translation:\n\n(cid:90)\n\n2. slow-and-smooth-rotation:\n\n3. slow-and-smooth-expansion:\n\n(cid:90)\n(cid:90)\n\nE({(cid:126)v}|M = r) =\n\n\u03bb{|(cid:126)v|2 + \u00b5[( \u2202vx\n\u2202x\n\n)2 + ( \u2202vy\n\u2202y\n\n)2 + ( \u2202vx\n\u2202y\n\n+ \u2202vy\n\u2202x\n\n)2] + \u03b7|\u22072(cid:126)v|2}d(cid:126)r (7)\n\nE({(cid:126)v}|M = e) =\n\n\u03bb{|(cid:126)v|2 + \u00b5[( \u2202vx\n\u2202y\n\n)2 + ( \u2202vy\n\u2202x\n\n)2 + ( \u2202vx\n\u2202x\n\n\u2212 \u2202vy\n\u2202y\n\n)2] + \u03b7|\u22072(cid:126)v|2}d(cid:126)r (8)\n\n{vx = \u2212\u03c9(y \u2212 y0), vy = \u03c9(x \u2212 x0)},{vx = e(x \u2212 x0), vy = e(y \u2212 y0)\n\nThese models are motivated as follows. The |(cid:126)v|2 and |\u22072(cid:126)v|2 bias towards slowness and smoothness\nand are common to all models. The \ufb01rst derivative term gives the differences among the models.\nThe translation model prefers constant translation motion with (cid:126)v constant, since \u2207(cid:126)v = 0 for this\ntype of motion. The rotation model prefers rigid rotation and expansion, respectively, of ideal form\n(9)\nwhere (x0, y0) are the (unknown) centers, \u03c9 is the angular speed and e is the expansion rate. These\nforms of motion are preferred by the two models since, for the \ufb01rst type of motion (rotation) we have\n{ \u2202vx\n\u2202y = 0} (independent of (x0, y0) and \u03c9). Similarly, the second type of\n\u2202y + \u2202vy\n\u2202x = 0}\nmotion is preferred by the expansion (or contraction) model since { \u2202vx\n(again independent of (x0, y0) and e).\nThe translation model is similar to the \ufb01rst three terms of the slow-and-smooth energy function\n[17] but with a restriction on the set of parameters. Formally \u03bb(|(cid:126)v|2 + \u03c32\n8 |\u22072(cid:126)v|2)d(cid:126)r\nm!2m|Dm(cid:126)v|2d(cid:126)r. Our computer simulations showed that the translation model performs\n\n\u2248 \u03bb(cid:80)\u221e\n\n\u2202x = 0, \u2202vx\n\n\u2202x = \u2202vy\n\n\u2202y = 0, \u2202vx\n\n\u2202y = \u2202vy\n\n2 |\u2207(cid:126)v|2 + \u03c34\n\n\u2202x \u2212 \u2202vy\n\n\u03c32m\n\nm=0\n\nsimilar to the slow-and-smooth model.\n\n3.3 The Likelihood Functions\n\nThe likelihood function differs for the two classes of stimuli we examined: (i) For the moving dot\nstimuli, as used in [4], there is enough information to estimate the local velocity (cid:126)u; (ii) For the\ngratings stimuli [10], there is only enough information to estimate one component of the velocity\n\ufb01eld.\nFor the dot stimuli, the energy term in the likelihood function is set to be\n\nE({(cid:126)u|(cid:126)v}) =\n\n|(cid:126)v((cid:126)ri) \u2212 (cid:126)u((cid:126)ri)|2\n\nN(cid:88)\n\ni=1\n\nN(cid:88)\n\nFor the gratings stimuli, see 2, the likelihood function uses the energy function\n\nEn({(cid:126)u}|{(cid:126)v}) =\n\n|(cid:126)v((cid:126)ri) \u00b7 \u02c6(cid:126)u((cid:126)ri) \u2212 |(cid:126)u((cid:126)ri)||2\n\nwhere \u02c6(cid:126)u((cid:126)ri) is the unit vector in the direction of (cid:126)u((cid:126)ri) and normally it is the direction of local image\ngradient.\n\ni=1\n\n(10)\n\n(11)\n\n\f3.4 MAP estimator of velocities\n\nThe MAP estimate of the velocities for each model is obtained by solving\n\n(cid:126)v\u2217 = arg max\n\np({(cid:126)v}|{(cid:126)u}, M) = arg min\n\n{E({(cid:126)u|(cid:126)v}) + E({(cid:126)v}|M)}\n\n(cid:126)v\n\n(cid:126)v\n\n(12)\n\nFor the slow-and-smooth model [17], it was shown using regularization analysis [2] that this solution\ncan be expressed in terms of a linear combination of the Green function G of the differential operator\nwhich imposes the slow-and-smoothness constraint (the precise form of this constraint was chosen\nso that G was a Gaussian).\nWe can obtain similar results for the three types of models M \u2208 {t, r, e} we have introduced in this\npaper. The main difference is that the models require two vector valued Green functions (cid:126)GM\n1 =\n(GM\n1y. These\nvector-valued Green functions are required to perform the coupling between the different velocity\ncomponent required for rotation and expansion, see \ufb01gure (1). For the translation model there is no\ncoupling required and so GM\n\n2y), with the constraint that GM\n\n1y) and (cid:126)GM\n\n2y and GM\n\n2 = (GM\n\n2x = GM\n\n1x = GM\n\n1x, GM\n\n2x, GM\n\n2x = GM\n\n1y = 0.\n\n1x\n\n, GM =e\n\nFigure 1: The vector-valued Green function (cid:126)G = (G1, G2).\nGM =t\n, GM =r\n1x\nright: GM =t\nare similar for all models, GM =t\nity components), and GM =r\nof rotation and expansion. Recall that GM\n\nleft-to-right:\nleft-to\nfor translation, rotation, and expansion models. Observe that the GM\n1x\nvanishes for the translation model (i.e. no coupling between veloc-\n2x\nboth have two peaks which correspond to the two directions\nand GM =e\n1y = GM\n\nfor the translation, rotation and expansion models. Bottom panel:\n2x\n\n2x and GM\n\n2y = GM\n1x.\n\nTop panel,\n\n1x\n, GM =r\n\n, GM =e\n\n2x\n\n2x\n\n2x\n\n2x\n\nThe estimated velocity for the M model is of the form:\n\n(cid:126)v((cid:126)r) =\n\n[\u03b1i (cid:126)GM\n\n1 ((cid:126)r \u2212 (cid:126)ri) + \u03b2i (cid:126)GM\n\n2 ((cid:126)r \u2212 (cid:126)ri)],\n\n(13)\n\nN(cid:88)\n\ni=1\n\nN(cid:88)\n\nFor the dot stimuli, the {\u03b1},{\u03b2} are obtained by solving the linear equations:\n\nj=1\n\n[\u03b1j (cid:126)GM\n\n1 ((cid:126)ri \u2212 (cid:126)rj) + \u03b2j (cid:126)GM\n\n2 ((cid:126)ri \u2212 (cid:126)rj)] + \u03b1i(cid:126)e1 + \u03b2i(cid:126)e2 = (cid:126)u(ri), i = 1, . . . N,\n\n(14)\nwhere (cid:126)e1, (cid:126)e2 denote the (orthogonal) coordinate axes. If we express the {\u03b1},{\u03b2} as two N-dim\nvectors A and B, the {ux} and {uy} as vectors U = (Ux, Uy)T , and de\ufb01ne N \u00d7 N matrices\n2y((cid:126)ri \u2212 (cid:126)rj) re-\n1x((cid:126)ri \u2212 (cid:126)rj), GM\n1x, gM\ngM\nspectively, then we can express these linear equations as:\n\n2y to have components GM\n\n2x, gM\n\n1y , gM\n\n(cid:18) gM\n(cid:18) \u02dcgM\n\n1x + I\ngM\n1y\n\n1x + I\n\u02dcgM\n1y\n\n(cid:19)(cid:18) A\n(cid:19)(cid:18) A\n\n1y((cid:126)ri \u2212 (cid:126)rj), GM\n2x((cid:126)ri \u2212 (cid:126)rj), GM\n(cid:19)\n(cid:19)\n(cid:19)\n(cid:19)\n\n(cid:18) Ux\n(cid:18) Ux\n\nUy\n\n=\n\nB\n\n=\n\nB\n\nUy\n\ngM\n2x\n2y + I\ngM\n\n\u02dcgM\n2x\n\u02dcgM\n2y + I\n\n(15)\n\n(16)\n\nSimilarly for the gratings stimuli,\n\n\f1x is the matrix with components \u02dcGM\n\n1x((cid:126)ri \u2212 (cid:126)rj) = [ (cid:126)GM\n\n1 ((cid:126)ri \u2212 (cid:126)rj) \u00b7 \u02c6(cid:126)u(ri)]\u02c6(cid:126)ux(ri), and\n\nin which \u02dcgM\nsimilarly for \u02dcgM\n\n1y, \u02dcgM\n\n2x and \u02dcgM\n2y.\n\n3.5 Model Selection\nWe re-express model evidence p({(cid:126)u}|M) in terms of (A, B):\n\n(cid:90)\n\np({(cid:126)u}|M) =\n\np({(cid:126)u}|A, B, M)p(A, B)dAdB\n\n(17)\n\n(cid:18) gM\n\n(cid:19)\n\nWe introduce new notation in the form of 2N \u00d7 2N matrices: gM =\n\u02dcgM .\nThe model evidence for the dot stimuli can be computed analytically (exploiting properties of multi-\ndimensional Gaussians) to obtain:\n\n, similarly for\n\ngM\n2x\ngM\n2y\n\n1x\ngM\n1y\n\np({(cid:126)u}|M) =\n\nexp[\u2212 1\nT\n\n(U T U \u2212 U T\n\ngM\n\ngM + I\n\nU)]\n\n(18)\n\nSimilarly, for the gratings stimuli we obtain:\n\np({(cid:126)u}|M) =\n\n(U T U \u2212 U T \u02dcgM \u02dc\u03a3\u22121(\u02dcgM )T U)]\n\n(19)\n\n1\n\n(\u03c0T )N(cid:112)det(gM + I)\n(cid:112)det(gM )\n(cid:113)\n\n1\n\ndet(\u02dc\u03a3)\n\n(\u03c0T )N\n\nexp[\u2212 1\nT\n\nwhere \u02dc\u03a3 = (\u02dcgM )T \u02dcgM + gM .\n\n4 Results on random dot motion\n\nWe \ufb01rst investigate motion perception with the moving dots stimuli used by Freeman and Harris\n[4], as shown in \ufb01gure (2). The stimuli consist of 128 moving dots in a random spatial pattern.\nAll the dots have the same speed in all three motion patterns, including translation, rotation and\nexpansion. Our simulations \ufb01rst select the correct model for each stimulus and then estimate the\nspeed threshold of detection for each type of motion. The parameter values used are \u03bb = 0.001,\n\u00b5 = 12.5, \u03b7 = 78.125 and T = 0.0054.\n\nFigure 2: Moving random dot stimuli. Left panel: translation; middle panel: rotation; right panel:\nexpansion.\n\n4.1 Model selection\n\nModel selection results are shown in \ufb01gure (3). As speed increases in the range of 0.05 to 0.1, model\nevidence decreases for all models. This is due to slowness term in all model priors. Nevertheless the\ncorrect model is always selected over the entire range of speed, and for all 3 type of motion stimuli.\n\n\u22123\u22122\u221210123\u22122.5\u22122\u22121.5\u22121\u22120.500.511.522.5\u22122.5\u22122\u22121.5\u22121\u22120.500.511.522.5\u22122.5\u22122\u22121.5\u22121\u22120.500.511.522.5\u22123\u22122\u221210123\u22123\u22122\u221210123\fFigure 3: Model selection results with random dot motion. Plots the log probability of the model as\na function of speed for each type of stimuli. left: translation stimuli; middle: rotation stimuli; right:\nexpansion stimuli. Green curves with cross are from translation model. Red curves with circles are\nfrom rotation model. Blue curves with squares are from expansion model.\n\n4.2 Speed threshold of Detection\n\nAs reported in [4], humans have lower speed threshold in detecting rotation/expansion than trans-\nlation motion. The experiment is formulated as a model selection task with an additional \u201cstatic\u201d\nmotion prior. The \u201cstatic\u201d motion prior is modeled as a translation prior with \u00b5 = 0 and \u03bb sig-\nni\ufb01cantly large to emphasize slowness. In the simulation, \u03bb = 0.3 for this \u201cstatic\u201d model, while\n\u03bb = 0.001 for all other models.\nAt low speed, the \u201cstatic\u201d model is favored due to its stronger bias towards slowness, as stimulus\nspeed increases, it loses its advantage to other models. The speed thresholds of detection for different\nmotion patterns can be seen from the model evidence plots in \ufb01gure (4), and they are lower for\nrotation/expansion than translation. The threshold values are about 0.05 for rotation and expansion\nand 0.1 for translation. This is consistent with experimental result in [4].\n\nFigure 4: Speed threshold of detection. Upper left panel: model evidence plot for translation stimuli.\nUpper right panel: model evidence plot for rotation stimuli. Lower left panel: model eviddence plot\nfor expansion stimuli. Lower right panel: bar graph of speed thresholds.\n\n0.050.060.070.080.090.1482.75482.8482.85482.9speedlog(P(u)) rotation modelexpansion modeltranslation model0.050.060.070.080.090.1477478479480481482speedlog(P(u)) rotation modelexpansion modeltranslation model0.050.060.070.080.090.1477478479480481482speedlog(P(u)) rotation modelexpansion modeltranslation model0.10.1020.1040.1060.1080.11478480482484486488Speedlog(P(u)) rotation modelexpansion modeltranslation modelstatic model0.05020.05040.05060.0508480.2480.4480.6480.8481481.2481.4481.6Speedlog(P(u)) rotation modelexpansion modeltranslation modelstatic model0.05020.05040.05060.0508480480.5481481.5Speedlog(P(u)) rotation modelexpansion modeltranslation modelstatic modeltranslationrotationexpansion00.020.040.060.080.10.12Speed threshold\f5 Results on randomly oriented gratings\n\n5.1 Stimuli\n\nWhen randomly oriented grating elements drift behind apertures, the perceived direction of motion\nis heavily biased by the orientation of the gratings, as well as by the shape and contrast of the aper-\ntures. Recently, Nishida and his colleagues developed a novel global motion stimulus consisting of\na number of gratings elements, each with randomly assigned orientation [10]. A coherent motion\nis perceived when the drifting velocities of all elements are consistent with a given velocity. Ex-\namples of the stimuli used in these psychophysical experiments are shown in left side of \ufb01gure (6).\nThe stimuli consisted of 728 gratings (drifting sine-wave gratings windowed by stationary Gaus-\nsians). The orientations of the gratings were randomly assigned, and their drifting velocities were\ndetermined by a speci\ufb01ed global motion \ufb02ow pattern. The motions of signal grating elements were\nconsistent with global motion, but the motions of noise grating elements were randomized. The\ntask was to identify the global motion direction as one of two alternatives: left/right for translation,\nclockwise/counterclockwise for rotation, and inward/outward for expansion. Motion sensitivity was\nmeasured by the coherence threshold, de\ufb01ned as the proportion of signal elements that yielded a\nperformance level of 75% correct.\nSimilar stimuli with 328 gratings were generated to test our computational models. The input for\nthe models is the velocity component perpendicular to the assigned orientation for each grating, as\nillustrated in the upper two panels of \ufb01gure (5).\n\nFigure 5: Randomly-oriented grating stimuli and estimated motion \ufb02ow. Upper left panel: rotation\nstimulus (with 75% coherence ratio). Upper right panel: expansion stimulus (with 75% coherence\nratio). Lower left panel: motion \ufb02ow estimated from stimulus in \ufb01rst panel with rotation model.\nLower right panel: motion \ufb02ow estimated from stimulus in second panel with expansion model.\n\n5.2 Result\n\nThe results of psychophysical experiments (middle panel of \ufb01gure 6) showed worse performance\nfor perceiving translation than rotation/expansion motion [6]. Clearly, as shown in the third panel\nof the same \ufb01gure, the model performs best for rotation and expansion, and is worst for translation.\nThis \ufb01nding agrees with human performance in psychophysical experiments.\n\n6 Conclusion\n\nHumans motion sensitivities depend on the motion patterns (translation/rotation/expansion). We\npropose a computational model in which different prior motions compete to \ufb01t the data by levels\n\n\u221215\u221210\u22125051015\u221215\u221210\u22125051015\u221215\u221210\u22125051015\u221215\u221210\u22125051015\u221215\u221210\u22125051015\u221215\u221210\u22125051015\u221215\u221210\u22125051015\u221215\u221210\u22125051015\fFigure 6: Stimulus and results. Left panel: illustration of grating stimulus. Blue arrows indicate the\ndrifting velocity of each grating. Middle panel: human coherence thresholds for different motion\nstimuli. Right panel: Model prediction of coherence thresholds which are consistent with human\ntrends.\n\nof inference. This analysis involves formulating two new prior models for rotation and expansion\nmodel and deriving their properties. This competitive prior approach gives good \ufb01ts to the empirical\ndata and accounts for the dominant trends reported in [4, 6].\nOur current work aims to extend these \ufb01ndings to a range of different motions (e.g. af\ufb01ne motion)\nand to use increasingly naturalistic appearance/intensity models. It is also important to determine\nif motion patterns to which humans are sensitive correspond to those appearing regularly in natural\nmotion sequences.\n\nReferences\n[1] J.F. Barraza and N.M. Grzywacz. Measurement of angular velocity in the perception of rotation. Vision Research, 42.2002.\n[2] J. Duchon. Lecture Notes in Math. 571, (eds Schempp, W. and Zeller, K.) 85-100. Springer-Verlag, Berlin, 1979.\n[3] C. J. Duffy, and R. H. Wurtz. Sensitivity of MST neurons to optic \ufb02ow stimuli. I. A continuum of response selectivity to large \ufb01eld\n\nstimuli. 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Global motion with multiple Gabors - A tool to investigate motion integration\n\nacross orientation and space. VSS 2006.\n\n[11] R. Sekuler, S.N.J. Watamaniuk and R. Blake. Perception of Visual Motion. In Steven\u2019s Handbook of Experimental Psychology. Third\n\nedition. H. Pashler, series editor. S. Yantis, volume editor. J. Wiley Publishers. New York. 2002.\n\n[12] A.A. Stocker and E.P. Simoncelli. Noise characteristics and prior expectations in human visual speed perception Nature Neuroscience,\n\nvol. 9(4), pp. 578\u2013585, Apr 2006.\n\n[13] A.A. Stocker, and E. Simoncelli. A Bayesian model of conditioned perception. Proceedings of Neural Information Processing Systems.\n\n2007.\n\n[14] K. Tanaka, Y. Fukada, and H. Saito. Underlying mechanisms of the response speci\ufb01city of expansion/contraction and rotation cells in the\n\ndorsal part of the MST area of the macaque monkey. Journal of Neurophysiology. 62, 642-656. 1989.\n\n[15] Y. Weiss, and E.H. Adelson. Slow and smooth: A Bayesian theory for the combination of local motion signals in human vision Technical\n\nReport 1624. Massachusetts Institute of Technology. 1998.\n\n[16] Y. Weiss, E.P. Simoncelli, and E.H. Adelson. Motion illusions as optimal percepts. Nature Neuroscience, 5, 598-604. 2002.\n[17] A.L. Yuille and N.M. Grzywacz. A computational theory for the perception of coherent visual motion. Nature, 333,71-74. 1988.\n\ntranslationrotationexpansion00.10.20.30.40.5HumanCoherence Ratio Thresholdtranslationrotationexpansion00.050.10.150.20.25ModelCoherence Ratio Threshold\f", "award": [], "sourceid": 977, "authors": [{"given_name": "Shuang", "family_name": "Wu", "institution": null}, {"given_name": "Hongjing", "family_name": "Lu", "institution": null}, {"given_name": "Alan", "family_name": "Yuille", "institution": null}]}