Although visual systems are optimized to deal with the natural visual environment, our understanding of human *motion perception* is in large part based on the use of artificial stimuli. Here, we assessed observers' ability to estimate the direction of translating natural images and fractals by having them adjust the orientation of a subsequently viewed line. A system of interleaved staircases, driven by observers' direction estimates, ensured that stimuli were presented near one of 16 reference directions. The resulting error distributions (i.e., the differences between reported and true directions) reveal several anisotropies in global motion processing. First, observers' estimates are biased away from cardinal directions (*reference repulsion*). Second, the standard deviations of estimates show an “oblique effect” being ∼45% lower around cardinal directions. Third, errors around cardinal directions are more likely (∼22%) to approach zero than would be consistent with Gaussian-distributed errors, suggesting that motion processing minimizes the *number* as well as *magnitude* of errors. Fourth, errors are similar for natural scenes and fractals, indicating that observers do not use top-down information to improve performance. Finally, adaptation to unidirectional motion modifies observers' bias by amplifying existing repulsion (e.g., around cardinal directions). This bias change can improve direction discrimination but is not due to a reduction in variability.

*orientations*is less precise than discrimination of cardinal (horizontal or vertical) orientations (Appelle, 1972; Heeley & Timney, 1988). Similarly for motion, observers are more precise at fine direction discrimination (i.e., reporting whether a stimulus is clockwise or anticlockwise of a reference direction) for motions near the cardinal directions (0°, 90°, 180°, or 270°) than they are for oblique directions (Ball & Sekuler, 1979, 1980; Dakin, Mareschal, & Bex, 2005a, 2005b; Gros, Blake, & Hiris, 1998; Heeley & Buchanan-Smith, 1992; Krukowski, Pirog, Beutter, Brooks, & Stone, 2003). Interestingly, this oblique effect only applies to discrimination and not detection performance (at least in dot patterns; Gros et al., 1998). In plaids, it is pattern motion, and not component motion, that determines the oblique effect (Heeley & Buchanan-Smith, 1992). Generally, the oblique effect is thought to result from low-level tuning properties of orientation-sensitive neurons, with oblique orientations and motion directions being underrepresented relative to cardinal directions (Li, Peterson, & Freeman, 2003; McMahon & MacLeod, 2003). It has been suggested that this uneven distribution of neural sensitivities is due to the statistical properties of the natural environment, which exhibit a similar bias to vertical and horizontal (Essock, Haun, & Kim, 2009; Keil & Cristobal, 2000), and consequent underrepresentation of the obliques. However, an oblique effect for direction discrimination with natural images is yet to be established.

*local*in nature: it affects the precision with which the directions of individual elements are encoded but not how well they are integrated across space. The particular pattern of sensitivity loss associated with oblique motion, expressed in a polar format, resembled a fat “X”; low discrimination thresholds are observed only within a few degrees of the cardinal directions. The authors went on to compare this pattern of results with the local motion statistics of natural movies, using a video shot from the point of view of an individual walking through an urban environment. This revealed that there was substantially less energy at oblique than cardinal directions and that local energy profiles had broader directional bandwidths (i.e., standard deviations) away from the cardinals. This paper made several predictions, notably that the representation of global direction should be prone to the effects of anisotropies in the earlier motion coding stage. Specifically, distributions around cardinal directions should: first, have lower standard deviations, and second, be more leptokurtic.

^{1}

*reference repulsion*(Loffler & Orbach, 2001; Rauber & Treue, 1998) where it is assumed that observers use the cardinal directions (up, down, left and right) as implicit “reference” directions, even in the absence of an explicit reference boundary. Jazayeri and Movshon (2007) recently observed a systematic bias in observers' estimates of motion direction in a fine discrimination task. Their observations were well explained by a model of sensory decoding in which the most informative signals are those from neurons tuned away from the discrimination boundary; such signals are preferentially weighted in a fine discrimination task, but not in a coarse (up–down) task. Jazayeri and Movshon's (2007) model asserts that the direction channels used for motion perception will depend on the task. If this is the case, then so-called “reference repulsion” should not be seen in absolute judgments of direction, since the task is not a binary discrimination task. The “off-channel” neurons are only more informative if a decision must be made regarding the direction of motion with respect to a boundary (clockwise/anticlockwise of the boundary); the most informative neurons in an absolute direction judgment must be those tuned to the direction itself, since they are maximally responsive. On the other hand, it is possible that observers may compute absolute direction based on some internal representation of cardinal directions (up/down and left/right), in which case reference repulsion might still be seen for directions near the cardinals.

^{2}, respectively.

*π*ambiguity). This is consistent with scenes being dominated by texture, whereas images that are dominated by edge structure (Figure 2c, lower right) generate histograms whose orientation structure (both the mean and the range) is much more dependent on the direction of motion in which they are translated (Figure 2d, lower right). We took the covariance of these histograms as a simple index of directional ambiguity. The image eventually selected fell near the middle of the range of values computed and was also selected to contain both texture and edge information and to avoid the presence of specularities that can dominate the gray-level range in linearized natural scenes.

*Note that this procedure was used only to sample an informative range of directions around a given reference direction*. For a given reference direction, QUEST generated a motion cue value (which we assigned a random sign relative to the reference). In Figure 3a, that motion cue is positive (i.e., clockwise of the reference). The motion cue is presented to the observer who estimates its direction with an analog response (

*report*). In this case, their report is clockwise of the reference direction (computed using Equation 1 below), and this is classed a correct response that would lead QUEST to reduce cue size on the subsequent trial. In Figure 3b, a negative cue is presented, but the observer's report is clockwise (positive) of the reference direction and this is classed as an incorrect response (leading to an increase in cue size). In brief, observers' responses are correct only if they are the same “side” of the reference as the true/cued direction. On each trial, we record (a) the reference direction, (b) the cued direction, and (c) the reported direction. The difference between true and reported directions (

*θ*

_{error}) will form the basis of our analysis below.

*θ*

_{true}and a given estimated direction

*θ*

_{est}, we use Equation 1 to compute a signed directional error

*θ*

_{error}. In this paper, we will adhere to the convention that 0° indicates rightward horizontal motion, 90°, upward vertical motion, 180° leftward horizontal motion, etc. To remain consistent with this convention, positive errors are anticlockwise, and negative errors are clockwise, of the true direction.

*θ*

_{block}(0°, 22.5°, …, 337.5°) and data were derived from trials when

*θ*

_{true}fell in the corresponding range: 0° ± 11.25°, 22.5° ± 11.25°, …, 337.5°. Blocks contained between 75 and 311 samples (depending on observer and direction). For each block, we summarized the distribution of signed directional errors between the true and estimated directions (estimated as above) using the following statistics (Mardia & Jupp, 2000). The mean direction (

*θ*

_{block}for a given block. Thus when presented with vertical upward motion (90°) if observers systematically reported 95° (5° anticlockwise of the reference), then their bias would be +5°.

*σ*

_{ θ }). The mean resultant length

*V*≤ 1. Mardia and Jupp (2000) show that this measure can be transformed into a measure of circular standard deviation using

_{ p }and

_{ p }denote the sample mean direction and sample mean resultant length of

*pθ*

_{1}, …

*pθ*

_{ n }. Skewness quantifies the degree of asymmetry of a distribution—i.e., the difference in a distribution's slope either side of its peak—with positive and negative skews indicating that the slope is steeper clockwise or anticlockwise of the peak, respectively. Positive kurtosis indicates that a distribution is more “peaky” (leptokurtic) than a Gaussian distribution, and negative kurtosis that is “flatter/more uniform” (

*platykurtic*).

*reference repulsion*(Rauber & Treue, 1998): observers are likely to be using the cardinals as an internal “standard” for their judgments, and because the most informative channels for performing the task tend to be located slightly clockwise and anticlockwise of the cardinal reference, this can bias the appearance of the stimulus (Jazayeri & Movshon, 2007). Our results are not as clear as those from Rauber and Treue (superimposed in Figure 5a in orange), as our data indicate reliable repulsion only around 90° and 180°. However, we note that Rauber and Treue pooled across 11 subjects to gain their effect and that others have failed to find reliable reference repulsion effects at all (Wiese & Wenderoth, 2008).

*B*is the Beta function (evaluated using a numerical approximation in Matlab) and

*m*controls the kurtosis of the distribution (

*m*> 3/2). The parameter

*σ*and

*μ*control the mean and standard deviation of the distribution. An asymmetric prediction is derived by generating two distributions with a common mean and kurtosis (the first and second free parameters), but with different standard deviations (

*σ*

_{1}and

*σ*

_{2}, the third and fourth parameters), matching the peaks of the two distributions and joining the left component of one distribution with the right of the other, at the point of the mean. The final (fifth) parameter is an overall scaling factor. Note that these distributions are fit only for graphical purposes: all statistics of kurtosis, etc., reported below are computed from the raw data not from the parameters of the fit.

*toward*135°. There are two possible explanations for this. First, we sampled test directions much more coarsely (22.5° steps from 0 to 347.5°) in Experiment 1 compared to here (±15°, ±7.5°, ±3.75°, ±1.87°, and 0°), so that we may have simply missed the fine structure of the observers' bias around 135°. Second, the direction repulsion effects we see (in the pooled data) may in some sense be defined relative to the center of the range of directions presented. This makes sense if the observers are using this direction as an implicit reference relative to which they make their (absolute) judgments of direction. We return to this point in the General discussion section. An interesting difference with the data from Experiment 1 concerns the skew of the distributions. In Experiment 1, distributions of reported directions were fairly symmetrical (i.e., skew was on average 0) whereas here we see pronounced skew of some distributions (e.g., with a 94° test). The direction of this skew is such that the longer tails tend to point away from the true direction. We return to this point below when we compare quantitative estimates of skew across conditions.

*amplification*of underlying direction repulsion (i.e., present before adaptation). For example, the tendency to see directions that are slightly clockwise of 90° as further clockwise than is veridical is exaggerated by adaptation (and vice versa for anticlockwise directions). This effect is present but subtle for subject SCD but is pronounced for MB. The modulation of skew around 90° (with equal and opposite signs) observed in the unadapted condition is also seen after adaptation. Although the skew effect is not altered by adaptation, the replication of this subtle effect is striking.

*t*-tests of the bootstrapped estimates of the

*σ*parameter of the psychometric functions) in four of the five subjects (all but DMA1) and that adapting to 135° produces improvement in only one of the subjects (DMA2). The reliable improvement in discrimination following adaptation to 90° is interesting because it occurred even though the variability of direction judgments was not altered by adaptation. As the group averages in Figure 8 show, the primary effect of adaptation was to alter bias, not standard deviation. This simulation demonstrates that adaptation can lead to improved discrimination due to changes in bias.

*k*= 3.0,

*s*= 0.26; natural scene:

*k*= 3.0,

*s*= 0.27), whereas on the cardinal directions kurtosis rises to 4.3 (

*s*= 0.67) for fractals and 4.5 (

*s*= 0.35) for natural scenes. Our data also reveal that skewed distributions lie next to the cardinals (Figure 7, bottom of third column), which may be critical as flanking distributions play a key role in improving discrimination (Jazayeri & Movshon, 2007; Regan & Beverly, 1985). When the elevation in kurtosis on the cardinals (indicating “peaky” distributions) is combined with skewed flanking distributions (see Figure 9b), we consider there is ample scope for statistics other than simple variance to drive 2AFC performance lower than variance-only models would predict.