You can find the explanation on the Pinna illusion here.
The important points useful to explain this illusion are the following: (i) the micropatterns have oriented low-frequency components, (ii) these engage low-level direction selective mechanisms, which (iii) are subject to the aperture problem. The implicit orientation polarity in the micropatterns (i.e., the low frequency luminance gradients) and not the black and white edges (i.e., the high-frequency components), is the basic attribute underlying this illusion. The notion of implicit orientation suggests that the illusion can be explained in terms of orthogonal biases (Grossberg, Mingolla & Viswanathan, 2001; Gurnsey & Pagé, 2006; Gurnsey et al., 2002; Mather, 2000; Pinna & Brelstaff, 2000; Pinna & Spillmann, 2005), on the basis of which the visual system produces an interpretation of image flow biased towards the strongest velocities perpendicular to the two-dimensional contours in the image. In Figure 10-left, the two bottom micropatterns show a blurred version of the two above. Under these conditions the high frequencies have been removed from the micropatterns. By translating the micropatterns to the right, they will most strongly stimulate neurons selective for the directions indicated by the white arrows. This bias can be considered to occur when the process of optical flow estimation is contaminated by spatiotemporal noise (Fermüler & Malm, 2004; Fermüller, Pless & Aloimonos, 2000; Weiss & Fleet, 2002; Weiss, Simoncelli & Adelson, 2002). More precisely, the interpretation of the motion effect depends on a step where image features such as lines, intersections of lines, black and white edges like those of Figure 1 and local image movement are derived.
These features contain many sources of noise or uncertainty that can cause bias. As a result, the locations of features are perceived erroneously and the appearance of the patterns is altered. Thus, the estimated flow vectors of Figure 1 are biased in the clockwise and in the counterclockwise directions as can be perceived in the outer and inner ring. The role of low-frequency luminance gradients is demonstrated by replacing the micropatterns with Gabor patches (Bayerl & Neumann, 2002; Gurnsey & Pagé, 2006; Gurnsey et al., 2002; Morgan, 2002). In this case, the strength of the illusion persists or is even enhanced (see Figure 10-right). Gurnsey et al. (2002) demonstrated that the strength of the illusion depends on the number of Gabor patches in the display, their wavelengths, and the orientation diﬀerence between adjacent micropatterns in the inner and outer rings. The illusion can be explained by the response of direction-selective neurons at the earliest cortical stage of visual processing, i.e., area V1. These neurons can signal the speed with which a line of its preferred orientation moves through its receptive field. This constraint may be considered as akin to the aperture effect (cf. Nakayama & Silverman, 1988) by which a moving straight line seen through an aperture can be perceived to move only along the direction of its normal. While this seems to explain how individual square elements receive a local illusory motion signal, the illusory rotational motion can be thought to be sensed by the higher cortical area such as MT (medium scale motion analysis, inhibition of opponent directions) and dorsal MST (MSTd – large scale motion analysis, directional decomposition) which collates all the signals provided by the local motion micropatterns. An FMRI study of the illusion showed activation of the motion specific complex hMT+ in addition to the V1/V2 areas to be involved in the perception of the illusion (Budnik et al., 2006).
Credit: Dr. Baingio Pinna, Dipartimento di Scienze dei Linguaggi, Università di Sassari, Italy