Poirson and colleagues (JOSA A, 7, 783-789, 1990) presented a simple and compelling argument to show that one cannot determine the sensitivity of mechanisms that underlie discrimination performance if thresholds lie on an ellipsoid (and they almost always do, at least within measurement error). In brief, the ellipsoidal model assumes that stimuli p and q are n-dimensional (n=3 for color vision, but is typically far larger for spatial visual stimuli), and that discrimination performance is based on their difference d=p-q as processed by m mechanisms. If the mechanism sensitivities to each dimension are summarized in an mXn matrix M, the ellipsoidal model predicts that performance is a function of |Md|=|RMd| for any orthonormal (e.g., rotation) matrix R. That is, any such rotation applied to the mechanisms results in an alternative model that makes the same predictions. On the other hand, Chubb, Landy & Econopouly (Vis Res, 44, 3223-3232, 2004) claimed to have isolated and measured the sensitivity of an individual texture-sensitive mechanism: the blackshot mechanism. How can this be? The method used by Chubb and colleagues requires one to use a high-dimensional space of stimuli and to find a relatively high-dimensional subspace of stimuli in which almost all dimensions of that subspace trade off linearly in controlling discrimination performance. If these conditions hold, one can be practically certain that a single mechanism determines discrimination performance within that subspace. And, by trading off stimulus power in that subspace with power in other directions, one can measure the complete sensitivity function of the isolated mechanism. Robustness of the method to small violations of its assumptions will be discussed.