Robust coding has been proposed as a solution to the problem of minimizing decoding error in the presence of neural noise. Many real-world problems, however, have degradation in the input signal, not just in neural representations. This generalized problem is more relevant to biological sensory coding where internal noise arises from limited neural precision and external noise from distortion of sensory signal such as blurring and phototransduction noise. In this note, we show that the optimal linear encoder for this problem can be decomposed exactly into two serial processes that can be optimized separately. One is Wiener filtering, which optimally compensates for input degradation. The other is robust coding, which best uses the available representational capacity for signal transmission with a noisy population of linear neurons. We also present spectral analysis of the decomposition that characterizes how the reconstruction error is minimized under different input signal spectra, types and amounts of degradation, degrees of neural precision, and neural population sizes.

Neurons in the early stages of processing in the primate visual system efficiently encode natural scenes. In previous studies of the chromatic properties of natural images, the inputs were sampled on a regular array, with complete color information at every location. However, in the retina cone photoreceptors with different spectral sensitivities are arranged in a mosaic. We used an unsupervised neural network model to analyze the statistical structure of retinal cone mosaic responses to calibrated color natural images. The second-order statistical dependencies derived from the covariance matrix of the sensory signals were removed in the first stage of processing. These decorrelating filters were similar to type I receptive fields in parvo- or konio-cellular LGN in both spatial and chromatic characteristics. In the subsequent stage, the decorrelated signals were linearly transformed to make the output as statistically independent as possible, using independent component analysis. The independent component filters showed luminance selectivity with simple-cell-like receptive fields, or had strong color selectivity with large, often double-opponent, receptive fields, both of which were found in the primary visual cortex (V1). These results show that the “form” and “color” channels of the early visual system can be derived from the statistics of sensory signals.