Correlated Percolation, Fractal Structures, and Scale-Invariant Distribution of Clusters in Natural Images

Natural images are scale invariant with structures at all length scales.We formulated a geometric view of scale invariance in natural images using percolation theory, which describes the behavior of connected clusters on graphs.We map images to the percolation model by defining clusters on a binary representation for images. We show that critical percolating structures emerge in natural images and study their scaling properties by identifying fractal dimensions and exponents for the scale-invariant distributions of clusters. This formulation leads to a method for identifying clusters in images from underlying structures as a starting point for image segmentation.

[1]  Saeed Saremi,et al.  Hierarchical model of natural images and the origin of scale invariance , 2013, Proceedings of the National Academy of Sciences.

[2]  K. Wilson.  Problems in Physics with many Scales of Length , 1979 .

[3]  S. Redner,et al.  Introduction To Percolation Theory , 2018 .

[4]  J. V. van Hateren,et al.  Independent component filters of natural images compared with simple cells in primary visual cortex , 1998, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[5]  William Bialek,et al.  Statistics of Natural Images: Scaling in the Woods , 1993, NIPS.

[6]  Jitendra Malik,et al.  A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics , 2001, Proceedings Eighth IEEE International Conference on Computer Vision. ICCV 2001.

[7]  Eero P. Simoncelli,et al.  Natural image statistics and neural representation. , 2001, Annual review of neuroscience.

[8]  M. Newman,et al.  Efficient Monte Carlo algorithm and high-precision results for percolation. , 2000, Physical review letters.

[9]  David J. Schwab,et al.  An exact mapping between the Variational Renormalization Group and Deep Learning , 2014, ArXiv.

[10]  Daniel L. Ruderman,et al.  Origins of scaling in natural images , 1996, Vision Research.

[11]  Terrence J. Sejnowski,et al.  On Criticality in High-Dimensional Data , 2014, Neural Computation.

[12]  W. Bialek,et al.  Statistical thermodynamics of natural images. , 2008, Physical review letters.

[13]  David J. Field,et al.  How Close Are We to Understanding V1? , 2005, Neural Computation.

[14]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[15]  Jitendra Malik,et al.  Normalized cuts and image segmentation , 1997, Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition.