Shearlet-based measures of entropy and complexity for two-dimensional patterns

New spatial entropy and complexity measures for two-dimensional patterns are proposed. The approach is based on the notion of disequilibrium and is built on statistics of directional multiscale coefficients of the fast finite shearlet transform. Shannon entropy and Jensen-Shannon divergence measures are employed. Both local and global spatial complexity and entropy estimates can be obtained, thus allowing for spatial mapping of complexity in inhomogeneous patterns. The algorithm is validated in numerical experiments with a gradually decaying periodic pattern and Ising surfaces near critical state. It is concluded that the proposed algorithm can be instrumental in describing a wide range of two-dimensional imaging data, textures, or surfaces, where an understanding of the level of order or randomness is desired.

Figure 1

Example of shearlet transform and shearlet coefficient power distribution. (a) Input image, a striped pattern with added Gaussian noise. (b) Shearlet coefficients $S_{j,k}(x,y)$ for scale and orientation with highest energy of $S_{j,k}$. (c) Corresponding shearlet $|{\widehat{\psi }}_{j,k}|$ in Fourier domain. (d) Corresponding shearlet ${\psi }_{j,k}$ in image domain. (e) Spectrum of normalized shearlet power at all scale and orientation combinations ${\sum }_{x,y}{S}_i^2(x,y)$.

Figure 2

Evolution of entropy and complexity in gradually randomized periodic patterns. (a) Input images with different probabilities of pixel value change shown on top; image size: $280\times 280$. (b) Entropy (blue) gradually rises with increase in probability of pixel value change, while complexity behavior is nonmonotonic (orange and red). Light-blue and orange lines, global estimate; dark-blue and red lines, “mean field” estimate. Dotted line, admissible complexity $C$ range for given entropy $H$. (c) Same as in (b) but plotted on the $H$–$C$ plane for “mean-field” estimates; pixel probability change is color coded. Region enclosed in the thin gray line, the admissible complexity range.

Figure 3

Ising surfaces and temperature dependence of complexity and entropy properties. (a) Examples of simulated Ising surfaces at different reduced temperature levels $T_r=T/T_c$ shown on top (image size: $256\times 256$, after $3\times {10}^4$ iterations; image intensities are renormalized to $0...1$ interval for better representation). (b) Dependence of entropy (blue) and complexity (orange and red) on reduced temperature. The critical temperature is characterized by a peak in complexity and a dip in entropy. Light-blue and orange lines, global estimate; dark-blue and red lines, “mean field” estimate. Dotted lines, admissible complexity $C$ range for given entropy $H$. Dotted lines, admissible complexity $C$ range for given entropy $H$.

Figure 4

Spatial mapping of local complexity and entropy and complexity-entropy spectra for Ising surfaces. (a) Ising surface at $T=T_c$ (left), local entropy (middle), and complexity (right). (b) Complexity-entropy spectra taken from local values and shown on the $H$–$C$ plane for Ising surfaces at different temperatures. Crosses, average, “mean field” values; stars, global estimates. Thin gray line encircles the admissible complexity $C$ range for each $H$ value. Thick gray line, point curve corresponding to power-law distributions.