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In this paper, we explore how to decompose the global thermodynamic entropy of a network into components associated with its edges. Commencing from a statistical mechanical picture, in which the normalised Laplacian matrix plays the role of Hamiltonian operator, thermodynamic entropy can be calculated from partition function associated with different energy level occupation distributions arising from Maxwell–Boltzmann statistics. Using the spectral decomposition of the Laplacian, we show how to project the edge-entropy components so that the detailed distribution of entropy across the edges of a network can be achieved. We apply the resulting method to the brain functional connectivity networks using BOLD-fMRI data. The entropic measurement turns out to be an effective tool for the diagnosis of Alzheimer’s disease by finding the most salient functional connectivity features from the corresponding anatomical brain regions.

In this paper we show how to use blind source separation to estimate shape from polarised images from a fixed viewing and illumination direction. Our method does not require prior knowledge of the polariser angles nor the refractive index. We use weighted singular vector decomposition (SVD) to estimate the unknown parameters, where the weights are dictated by the physics of polarisation. In a two-step process, we firstly use least square fitting to obtain the polariser angles, and then estimate the refractive index using a mutual information criterion function. We show that the proposed method is capable of calculating robust shape information compared with alternative approaches relying on the same input information. The proposed method can be applied when using uncalibrated polarisation filters. It can also be used effectively when the subject moves during image capture.