A Compact Representation of Random Phase and Gaussian Textures

In this paper, we are interested in the mathematical analysis of the micro-textures that have the property to be perceptually invariant under the randomization of the phases of their Fourier Transform. We propose a compact representation of these textures by considering a special instance of them: the one that has identically null phases, and we call it ''texton''. We show that this texton has many interesting properties, and in particular it is concentrated around the spatial origin. It appears to be a simple and useful tool for texture analysis and texture synthesis, and its definition can be extended to the case of color micro-textures

A Generative Model of Natural Texture Surrogates

Natural images can be viewed as patchworks of different textures, where the local image statistics is roughly stationary within a small neighborhood but otherwise varies from region to region. In order to model this variability, we first applied the parametric texture algorithm of Portilla and Simoncelli to image patches of 64X64 pixels in a large database of natural images such that each image patch is then described by 655 texture parameters which specify certain statistics, such as variances and covariances of wavelet coefficients or coefficient magnitudes within that patch. To model the statistics of these texture parameters, we then developed suitable nonlinear transformations of the parameters that allowed us to fit their joint statistics with a multivariate Gaussian distribution. We find that the first 200 principal components contain more than 99% of the variance and are sufficient to generate textures that are perceptually extremely close to those generated with all 655 components. We demonstrate the usefulness of the model in several ways: (1) We sample ensembles of texture patches that can be directly compared to samples of patches from the natural image database and can to a high degree reproduce their perceptual appearance. (2) We further developed an image compression algorithm which generates surprisingly accurate images at bit rates as low as 0.14 bits/pixel. Finally, (3) We demonstrate how our approach can be used for an efficient and objective evaluation of samples generated with probabilistic models of natural images.Comment: 34 pages, 9 figure

Phase Harmonic Correlations and Convolutional Neural Networks

A major issue in harmonic analysis is to capture the phase dependence of frequency representations, which carries important signal properties. It seems that convolutional neural networks have found a way. Over time-series and images, convolutional networks often learn a first layer of filters which are well localized in the frequency domain, with different phases. We show that a rectifier then acts as a filter on the phase of the resulting coefficients. It computes signal descriptors which are local in space, frequency and phase. The non-linear phase filter becomes a multiplicative operator over phase harmonics computed with a Fourier transform along the phase. We prove that it defines a bi-Lipschitz and invertible representation. The correlations of phase harmonics coefficients characterise coherent structures from their phase dependence across frequencies. For wavelet filters, we show numerically that signals having sparse wavelet coefficients can be recovered from few phase harmonic correlations, which provide a compressive representationComment: 26 pages, 8 figure

A survey of exemplar-based texture synthesis

Exemplar-based texture synthesis is the process of generating, from an input sample, new texture images of arbitrary size and which are perceptually equivalent to the sample. The two main approaches are statistics-based methods and patch re-arrangement methods. In the first class, a texture is characterized by a statistical signature; then, a random sampling conditioned to this signature produces genuinely different texture images. The second class boils down to a clever "copy-paste" procedure, which stitches together large regions of the sample. Hybrid methods try to combine ideas from both approaches to avoid their hurdles. The recent approaches using convolutional neural networks fit to this classification, some being statistical and others performing patch re-arrangement in the feature space. They produce impressive synthesis on various kinds of textures. Nevertheless, we found that most real textures are organized at multiple scales, with global structures revealed at coarse scales and highly varying details at finer ones. Thus, when confronted with large natural images of textures the results of state-of-the-art methods degrade rapidly, and the problem of modeling them remains wide open.Comment: v2: Added comments and typos fixes. New section added to describe FRAME. New method presented: CNNMR