The independent components of natural images are a set of linear filters which are optimized for statistical independence.
With such a set of filters images can be represented without loss of information. Intriguingly, the filter
shapes are localized, oriented, and bandpass, resembling important properties of V1 simple cell receptive fields.
Here we address the question of whether the independent components of natural images are also perceptually less
dependent than other image components. We compared the pixel basis, the ICA basis and the discrete cosine
basis by asking subjects to interactively predict missing pixels (for the pixel basis) or to predict the coefficients
of ICA and DCT basis functions in patches of natural images. Like Kersten (1987)1 we find the pixel basis to
be perceptually highly redundant but perhaps surprisingly, the ICA basis showed significantly higher perceptual
dependencies than the DCT basis. This shows a dissociation between statistical and perceptual dependence
measures.
There are two aspects to unsupervised learning of invariant representations of images: First, we can reduce the
dimensionality of the representation by finding an optimal trade-off between temporal stability and informativeness.
We show that the answer to this optimization problem is generally not unique so that there is still
considerable freedom in choosing a suitable basis. Which of the many optimal representations should be selected?
Here, we focus on this second aspect, and seek to find representations that are invariant under geometrical transformations
occuring in sequences of natural images. We utilize ideas of 'steerability' and Lie groups, which have
been developed in the context of filter design. In particular, we show how an anti-symmetric version of canonical
correlation analysis can be used to learn a full-rank image basis which is steerable with respect to rotations. We
provide a geometric interpretation of this algorithm by showing that it finds the two-dimensional eigensubspaces
of the average bivector. For data which exhibits a variety of transformations, we develop a bivector clustering
algorithm, which we use to learn a basis of generalized quadrature pairs (i.e. 'complex cells') from sequences of
natural images.
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