Perceptual Scaling
Vision can be posed as a statistical learning and inference problem. As an over-simplified account, let W be a description of the outside scene in terms of “what is where,” let I be the retina image, and let p(W, I) be the joint distribution ofW and I. Then visual learning is to learn p(W, I) from training data, and visual perception is to infer W from I based on p(W |I). There are two major schools on visual learning and perception. One school is operation-oriented and learns the inferential process defined by p(W |I) directly, often in the form of an explicit transformation W ≈ F (I). This scheme is mostly used in supervised learning, whereW is object category, and is given in training data. The other school is representation-oriented and learns the generative process p(W ) and p(I|W ) explicitly, then perception is to invert the generative process by maximizing or sampling p(W |I) ∝ p(W )p(I|W ). This scheme is Bayesian in nature, the prior distribution p(W ) may also be accounted for by a regularization term such as smoothness or sparsity. This scheme is often used in unsupervised learning where W is not available in training data. In the literature, there are a number of statistical theories proposed for vision. In representation-oriented school, Grenander (1993) and Mumford (1994) proposed pattern theory as a paradigm for vision (see also Geman and Geman, 1984; Amit, Grenander and Piccioni, 1991; Grenander and Miller, 1994; and S. Geman, Potter, and Chi, 2002, for important contributions that