This paper reports psychophysical and modelling results concerning the contour-detection paradigm of D. J. Field, A. Hayes, and R. F. Hess (1993). We measured psychophysically the maximum tolerable contour curvature (path angle) as a function of contour length. We compared these data to the predictions of an association field (D. J. Field et al., 1993) model based on the relative positions and mutual orientations of nearby elements and to models that explicitly link adjacent elements into chains and characterize each chain by its sequence of contour bends. For every stimulus, a large set of chains is produced and the target identified as the chain with the lowest maximum bend. We tested two different types of linking process: isotropic (linking one element to any other nearby) and anisotropic (linking one element to any others nearby along the orientation of its axis). All of these models can account for our data. Moreover, we show that the pattern of results due to path angle is principally a product of the distribution of spurious contours in the randomly oriented background. Given that some of the models do not embody constraints of orientation relationships between linked elements, this finding shows the importance for early vision in deciding which local elements are to be associated.

^{2}. Optical imaging studies (e.g., Bonhoeffer & Grinvald, 1991) have revealed that primary visual cortex generates a retinotopic (local) representation of contour orientation. The local image structure that best drives V1 neurons is typically simple in spatial form: perhaps even as simple as to be spatially well approximated by local elongated structures varying in just orientation and spatial scale. If this is the case, then a set of templates (e.g., filters or receptive fields) can be used to create image descriptions of such local structure.

- an association field model which responds most strongly to elements surrounded by further contour with appropriate combinations of position and orientation;
- models that link each element with others solely on the basis of spatial separation and then select linked groups of elements with the lowest curvature; and
- models that link an element with others on the basis of spatial separation and the spatial direction of those others.

*where*elements are with respect to each other, and

*what*form their combination makes: a part of circle or something similar. The other two families only have selectivity for

*where*elements are with respect to each other, regardless of the resultant form. In the light of the computational need for generativity and to be able to recognize a wide range of forms, the second and third families of models might be computationally more desirable.

^{−2}.

*λ*(0.12°), Gaussian envelope of

*SD*

*λ*/2 (0.06°), and Michelson contrast of 50%.

*path angle*. For any one target the magnitude of the path angle was fixed, although its sign changed randomly after a minimum of three elements. The orientation of the element at each location was set to the orientation of the chord joining the elements on either side. This resulted in the paths having a small degree of departure from co-circularity.

*λ*(0.3°) from the center of any other element—until no more elements could be added. This results in a mean distance between adjacent elements of 3.3

*λ*. We will use this mean inter-element spacing as the basic unit of distance, 1Δ. The whole field was of size 38Δ and could accommodate about 1000 elements. Since background elements were added at random locations, the distribution of distances from one element center to those of its immediate neighbors is also random. The minimum distance is 0.75Δ; the mean distance is 1Δ and the maximum is 1.5Δ.

*maximum tolerable path angle*) was obtained from the fit and 95% confidence limits on this estimate were calculated using a bootstrap method.

*sigma_d*. The curvature function is also a Gaussian parameterized by a rate of orientation change

*sigma_c*.

*sigma_t*.

*x*–

*y*plane shows the “where” selectivity. A change in the distance parameter,

*sigma_d,*just changes the overall size of this selectivity. A change in the curvature parameter,

*sigma_c,*changes the extent to which this “where” selectivity extends away from the line

*y*= 0. The vertical dimension of the space corresponds to the “what” selectivity of the association field. At any one point in

*x*–

*y*space, the vertical line through the function gives the orientation selectivity at that point. A change in the orientation error parameter changes the degree of selectivity (thickness of the blades), setting a tolerance for the degree of misalignment of elements from a true circular contour.

*sigma_d, sigma_c,*and

*sigma_t,*were systematically varied. For any given set of values for the 3 model parameters, the proportion of targets correctly detected was obtained as a function of path angle. This function is not shown, but it exhibits the same decline in performance with path angle as the psychophysical data. From this function, the path angle supporting 75% correct performance was obtained—the

*maximum tolerable path angle*. The 3 parameters were all independently varied and for each combination of parameters, the maximum tolerable path angle was measured. This results in a 3-dimensional performance space. The top panel of Figure 4 shows an iso-surface in these dimensions that encloses all parameter combinations that are as good as the human data or better.

*sigma_d*= 1.0,

*sigma_c*= 35,

*sigma_t*= 7. At these values, the maximum tolerable path angle is found to be 45.6°, compared with a mean value of 32° for the observers. We note that the parameter values are broadly in line with the equivalent stimulus values.

*sigma_d*lead to very low performance because the association field does not stretch adequately from one element to the next. Larger values show a gradual decline in performance arising because larger distances include more elements, some of which will have spuriously good relationships. Small values for

*sigma_c*result in poor performance because this parameter sets a limit on the maximum bend that will be responded to. Interestingly, larger values show little if any decline in performance, suggesting that this parameter is not particularly important. Similarly there is very little consequence of variations in

*sigma_t*.

*m*there are 2

^{m}possible contours. Across each link between 2 successive elements in a chain, the implicit contour has 3 distinct directions: the direction of the first element, the direction of the link, and the direction of the second element. It follows that each link has 2 changes of contour direction (2 bends). Similarly at each element there are 2 bends, and the path angle is the sum of these bends.

*m*elements therefore comprises a sequence of 2

*m*− 1 directions (

*m*element directions plus

*m*− 1 link directions) and therefore 2

*m*− 2 changes of direction. We will represent the contour by a vector of 2

*m*− 2 bend values. For a given chain, it is possible to enumerate the vector of bend values for each of the possible contours (arising for different element directions), giving 2

^{m}vectors, each 2

*m*− 2 in length. The best contour for this chain is then the one with the lowest largest bend. The largest bend in this best contour through a chain will be called the chain bend. The example in Figure 6 shows the chain with the smallest chain bend in the field (marked in yellow) and the link in that chain with the largest bend (marked in red).

*Chain bend,*the largest bend in a chain, provides a heuristic for deciding which chain is the target path in all of the simulations that follow. The target is identified as the chain with the lowest chain bend.

- the optimum in Figure 13;
- the optimum threshold with the largest scale consistent with human performance;
- the optimum threshold with the smallest scale consistent with human performance; and
- the optimum scale and the largest threshold consistent with human performance.