**The motion of a 1D image feature, such as a line, seen through a small aperture, or the small receptive field of a neural motion sensor, is underconstrained, and it is not possible to derive the true motion direction from a single local measurement. This is referred to as the aperture problem. How the visual system solves the aperture problem is a fundamental question in visual motion research. In the estimation of motion vectors through integration of ambiguous local motion measurements at different positions, conventional theories assume that the object motion is a rigid translation, with motion signals sharing a common motion vector within the spatial region over which the aperture problem is solved. However, this strategy fails for global rotation. Here we show that the human visual system can estimate global rotation directly through spatial pooling of locally ambiguous measurements, without an intervening step that computes local motion vectors. We designed a novel ambiguous global flow stimulus, which is globally as well as locally ambiguous. The global ambiguity implies that the stimulus is simultaneously consistent with both a global rigid translation and an infinite number of global rigid rotations. By the standard view, the motion should always be seen as a global translation, but it appears to shift from translation to rotation as observers shift fixation. This finding indicates that the visual system can estimate local vectors using a global rotation constraint, and suggests that local motion ambiguity may not be resolved until consistencies with multiple global motion patterns are assessed.**

*s*, according to the relationship

*x*is the current horizontal position of the trackball-controlled cursor (from 1 to 1024 pixels) and

*h*is the horizontal midpoint of the screen. This allowed subjects to vary the speed from 1/3 to 3 times the veridical speed of the array. The exponential relationship between

*x*and

*s*ensures that a shift of a given number of pixels will always produce the same proportional change in speed. This relationship was chosen as speed discrimination has been shown to be consistent with Weber's law, i.e., discrimination threshold is proportional to the absolute speed (Snowden & Braddick, 1991). The vertical axis controlled the curvature of the velocity field by varying the distance,

*d*, of the center of rotation from the center of the display (note the center of rotation was always on the horizontal meridian), i.e. where

*v*is the vertical midpoint of the display and

*y*is the current vertical position, measured in Gabor patch sized units (i.e., 1 unit = 1.18 cm). The denominator goes to zero when

*v*=

*y*and in this case the distance to center was set at 10,000 (i.e., a nominal distance of 118 m), which approximates a translation (a translation can be considered to be identical to a rotation about some notional point “at infinity”). Subjects could therefore adjust the speed and curvature of the plaid stimulus, while freely switching between plaids and Gabors. Subjects viewed the stimulus for as long as they wished; and when they were satisfied the two motions appeared similar, a button press recorded the coordinates of the cursor and initiated the next trial. The cursor was not displayed on screen at any time during the experiment. Three fixation points were used: the center of the display, 13° to the left and 13° to the right of center. These three fixation points were randomly interleaved over trials. There were 120 trials per block (20 repetitions of three fixation conditions for each of two direction conditions: upwards and downwards). No fixed limit was placed on trial duration, but each block lasted approximately 30 minutes.

*is*solved in MT but with the acceptance that the IOC solution (often termed the Pattern Direction) is only one of many possible solutions if we allow the local velocity to vary over space, as occurs with rotation. We note that many (>40%) neurons in MT do not appear to signal either the local (Component Direction) or global (Pattern Direction, i.e., IOC) motion (Majaj et al., 2007; Movshon et al., 1985). The population of candidate solutions to the aperture problem would then be examined by neurons in MST for consistency with global complex flows such as rotation.

*Nature*, 300 (5892), 523–525.

*Journal of Neurophysiology*, 107 (12), 3493–3508, doi:10.1152/jn.00821.2011.

*Vision Research*, 39 (13), 2229–2238.

*Nature Reviews Neuroscience*, 9 (9), 686–695, doi:10.1038/nrn2472.

*Vision Research*, 37 (5), 645–648.

*The visual neurosciences*(Vol. 2, pp. 1260–1283. Cambridge MA: MIT Press.

*Journal of Neuroscience*, 15 (7 Pt. 2), 5192–5208.

*Vision Research*, 44 (16), 1941–1950, doi:10.1016/j.visres.2004.03.016.

*Vision Research*, 32 (1), 81–87.

*Journal of Neuroscience*, 14, 54–67.

*Current Biology*, 21 (23), 2023–2028, doi:10.1016/j.cub.2011.10.049.

*Proceedings of the Royal Society of London Series B: Biological Sciences*, 221 (1223), 189–220.

*Vision Research*, 48 (19), 1940–1948, doi:10.1016/j.visres.2008.06.015.

*Journal of Neuroscience*, 28 (51), 13889–13906, doi:10.1523/JNEUROSCI.1946-08.2008.

*Frontiers in Computational Neuroscience*, 7, 146, doi:10.3389/fncom.2013.00146.earch

*Vision Research*, 26, 1099–1109.

*Vision Research*, 26 (1), 161–179.

*The senses: A comprehensive reference*. (Vol. 2, pp. 133–154. Oxford: Elsevier Inc.

*Nature Neuroscience*, 4 (7), 745–751.

*Vision Research*, 32 (2), 263–273.

*Visual Neuroscience*, 16 (4), 755–770.

*Journal of Neuroscience*, 27 (2), 366–370, doi:10.1523/Jneurosci.3183-06.2007.

*Perception*, 14 (2), 163–166.

*Vision Research*, 32 (6), 1015–1031.

*Pattern recognition mechanisms*(pp. 117–151 ). New York: Springer.

*Vision Research*, 33 (15), 2091–2107.

*The senses: A comprehensive reference*( Vol. 2, pp. 189–218 ). Cambridge, MA: Academic Press.

*Neuron*, 39 (4), 671–680.

*Vision Research*, 35 (4), 463–476.

*Vision Research*, 31 (5), 907–914.

*Vision Research*, 38 (20), 3069–3082.

*Journal of Neuroscience*, 6 (1), 134–144.

*Vision Research*, 46 (22), 3881–3885, doi:10.1016/j.visres.2006.06.010.

*Psychologische Forschung*, 20, 325–380.

*Technical report AI memo 1624*. Cambridge, MA: MIT AI Lab.

*Nature Neuroscience*, 5 (6), 598–604, doi:10.1038/Nn858.

*Vision Research*, 34, 1835–1842.

*Vision Research*, 32, 135–147.

*Nature*, 333 (6168), 71–74.

*Proof: The concentric dynamic Gabor array is consistent with multiple global motion solutions including a translation and rotation about some point (excluding the central point) along a line orthogonal to the specified translation through the center of the display*.

_{1},x

_{2}).

**U**= (0,

_{T}*T*). Then the 1D motion of each Gabor is given by projecting

**U**onto

_{T}**X**: where · denotes the dot product and × denotes multiplication by a scalar. Now consider a rotation centered on some arbitrary point on the

*x*axis, i.e.,

**P**= (

_{c}*x*, 0). Then the underlying 2D motion at each point is

_{c}**U**=

_{R}*R*(

*x*

_{2},−(

*x*

_{1}–

*x*)), where

_{c}*R*is the angular momentum. Projecting this onto

**X**we can again find the 1D motion in the normal component direction that is consistent with this rotation.

*Rx*=

_{c}*T*. So for a fixed translation speed,

*T*, and for every point on the

*x*axis, (

*x*) (except for the case where

_{c}, 0*x*), there is a speed of rotation,

_{c}= 0*R*, around that point such that the concentric Gabor array is entirely consistent, i.e., an infinite number of underlying global motions will produce the exact same local velocities in this Gabor array.

*Proof: The global translation solution is always slower than the average speed of any global rotation solution for our stimulus*.

*x*

_{1},

*x*

_{2}) the local motion of a field rotating about some arbitrary point (

*x*,0) is given by

_{c}**U**=

_{R}*R*(

*x*

_{2},–(

*x*

_{1}−

*x*)) and similarly the translation motion is given by

_{c}**U**= (0,

_{T}*T*), and, as we have already seen in Appendix A,

*Rx*=

_{c}*T*. Now if we take a circle of points, centered on the center of our array and of radius

*a*, then (x

_{1},x

_{2}) = (

*a*cos

*θ*,

*a*sin

*θ*) for

*θ*∈ [0,2

*π*]. The speed of the rotating field at these points can be found by,

*f*(

*θ*) over

*θ*∈ [0,2

*π*] and divide by 2

*π*. However, the integral of

*f*(

*θ*) the has no closed form. But we note that the square root in Equation 3 refers to the positive root (speed is always positive) so the whole is always positive. Now, if we first assume that

*x*≥

_{c}*a*then

*R*(

*x*–

_{c}*a*cos

*θ*) ≥ 0 and, so (

*f*(

*θ*))

^{2}≥ (

*R*(

*x*−

_{c}*a*cos

*θ*))

^{2}and because they are both positive this implies that

*f*(

*θ*) ≥

*R*(

*x*−

_{c}*a*cos

*θ*). We can easily find the mean of the right hand side of this inequality to put a lower bound on the mean of

*f*(

*θ*), i.e.

*x*<

_{c}*a*then we can swap

*a*and

*x*in Equation 4, noting that now

_{c}*R*(

*a*−

*x*cos

_{c}*θ*) > 0, and similarly show that

*f*(

*θ*) ≥

*R*(

*a*–

*x*cos

_{c}*θ*), and the mean of this is

*Ra*>

*Rx*=

_{c}*T;*so again the average speed of rotation is greater than translation. This is true for all values of

*x*and for any circle centred on the centre of our array, so it is certainly true for our annular arrangement. For completeness we note that the inequality in Equation (4) becomes an equality (for all values of

_{c}*θ*) if and only if

*a*= 0, i.e., the singular point at the centre of the array.