Mutual Information Loss in Pyramidal Image Processing

Gaussian and Laplacian pyramids have long been important for image analysis and compression. More recently, multiresolution pyramids have become an important component of machine learning and deep learning for image analysis and image recognition. Constructing Gaussian and Laplacian pyramids consists of a series of filtering, decimation, and differencing operations, and the quality indicator is usually mean squared reconstruction error in comparison to the original image. We present a new characterization of the information loss in a Gaussian pyramid in terms of the change in mutual information. More specifically, we show that one half the log ratio of entropy powers between two stages in a Gaussian pyramid is equal to the difference in mutual information between these two stages. We show that this relationship holds for a wide variety of probability distributions and present several examples of analyzing Gaussian and Laplacian pyramids for different images.