X-radiography (X-ray imaging) is a widely used imaging technique in art investigation. It can provide information about the condition of a painting as well as insights into an artist's techniques and working methods, often revealing hidden information invisible to the naked eye. In this paper, we deal with the problem of separating mixed X-ray images originating from the radiography of double-sided paintings. Using the visible color images (RGB images) from each side of the painting, we propose a new Neural Network architecture, based upon 'connected' auto-encoders, designed to separate the mixed X-ray image into two simulated X-ray images corresponding to each side. In this proposed architecture, the convolutional auto encoders extract features from the RGB images. These features are then used to (1) reproduce both of the original RGB images, (2) reconstruct the hypothetical separated X-ray images, and (3) regenerate the mixed X-ray image. The algorithm operates in a totally self-supervised fashion without requiring a sample set that contains both the mixed X-ray images and the separated ones. The methodology was tested on images from the double-sided wing panels of the \textsl{Ghent Altarpiece}, painted in 1432 by the brothers Hubert and Jan van Eyck. These tests show that the proposed approach outperforms other state-of-the-art X-ray image separation methods for art investigation applications.

To help understand the underlying mechanisms of neural networks (NNs), several groups have, in recent years, studied the number of linear regions $\ell$ of piecewise linear functions generated by deep neural networks (DNN). In particular, they showed that $\ell$ can grow exponentially with the number of network parameters $p$, a property often used to explain the advantages of DNNs over shallow NNs in approximating complicated functions. Nonetheless, a simple dimension argument shows that DNNs cannot generate all piecewise linear functions with $\ell$ linear regions as soon as $\ell > p$. It is thus natural to seek to characterize specific families of functions with $\ell$ linear regions that can be constructed by DNNs. Iterated Function Systems (IFS) generate sequences of piecewise linear functions $F_k$ with a number of linear regions exponential in $k$. We show that, under mild assumptions, $F_k$ can be generated by a NN using only $\mathcal{O}(k)$ parameters. IFS are used extensively to generate, at low computational cost, natural-looking landscape textures in artificial images. They have also been proposed for compression of natural images, albeit with less commercial success. The surprisingly good performance of this fractal-based compression suggests that our visual system may lock in, to some extent, on self-similarities in images. The combination of this phenomenon with the capacity, demonstrated here, of DNNs to efficiently approximate IFS may contribute to the success of DNNs, particularly striking for image processing tasks, as well as suggest new algorithms for representing self similarities in images based on the DNN mechanism.

This paper proposes a novel non-oscillatory pattern (NOP) learning scheme for several oscillatory data analysis problems including signal decomposition, super-resolution, and signal sub-sampling. To the best of our knowledge, the proposed NOP is the first algorithm for these problems with fully non-stationary oscillatory data with close and crossover frequencies, and general oscillatory patterns. NOP is capable of handling complicated situations while existing algorithms fail; even in simple cases, e.g., stationary cases with trigonometric patterns, numerical examples show that NOP admits competitive or better performance in terms of accuracy and robustness than several state-of-the-art algorithms.

We address the uniform approximation of sums of ridge functions $\sum_{i=1}^m g_i(a_i\cdot x)$ on ${\mathbb R}^d$, representing the shallowest form of feed-forward neural network, from a small number of query samples, under mild smoothness assumptions on the functions $g_i$'s and near-orthogonality of the ridge directions $a_i$'s. The sample points are randomly generated and are universal, in the sense that the sampled queries on those points will allow the proposed recovery algorithms to perform a uniform approximation of any sum of ridge functions with high-probability. Our general approximation strategy is developed as a sequence of algorithms to perform individual sub-tasks. We first approximate the span of the ridge directions. Then we use a straightforward substitution, which reduces the dimensionality of the problem from $d$ to $m$. The core of the construction is then the approximation of ridge directions expressed in terms of rank-$1$ matrices $a_i \otimes a_i$, realized by formulating their individual identification as a suitable nonlinear program, maximizing the spectral norm of certain competitors constrained over the unit Frobenius sphere. The final step is then to approximate the functions $g_1,\dots,g_m$ by $\hat g_1,\dots,\hat g_m$. Higher order differentiation, as used in our construction, of sums of ridge functions or of their compositions, as in deeper neural network, yields a natural connection between neural network weight identification and tensor product decomposition identification. In the case of the shallowest feed-forward neural network, we show that second order differentiation and tensors of order two (i.e., matrices) suffice.

In this paper, computer-based techniques for stylistic analysis of paintings are applied to the five panels of the 14th century Peruzzi Altarpiece by Giotto di Bondone. Features are extracted by combining a dual-tree complex wavelet transform with a hidden Markov tree (HMT) model. Hierarchical clustering is used to identify stylistic keywords in image patches, and keyword frequencies are calculated for sub-images that each contains many patches. A generative hierarchical Bayesian model learns stylistic patterns of keywords; these patterns are then used to characterize the styles of the sub-images; this in turn, permits to discriminate between paintings. Results suggest that such unsupervised probabilistic topic models can be useful to distill characteristic elements of style.

Super-resolution methods form high-resolution images from low-resolution images. In this paper, we develop a new Bayesian nonparametric model for super-resolution. Our method uses a beta-Bernoulli process to learn a set of recurring visual patterns, called dictionary elements, from the data. Because it is nonparametric, the number of elements found is also determined from the data. We test the results on both benchmark and natural images, comparing with several other models from the research literature. We perform large-scale human evaluation experiments to assess the visual quality of the results. In a first implementation, we use Gibbs sampling to approximate the posterior. However, this algorithm is not feasible for large-scale data. To circumvent this, we then develop an online variational Bayes (VB) algorithm. This algorithm finds high quality dictionaries in a fraction of the time needed by the Gibbs sampler.

We introduce a useful tool for analyzing boosting algorithms called the ``smooth margin function,'' a differentiable approximation of the usual margin for boosting algorithms. We present two boosting algorithms based on this smooth margin, ``coordinate ascent boosting'' and ``approximate coordinate ascent boosting,'' which are similar to Freund and Schapire's AdaBoost algorithm and Breiman's arc-gv algorithm. We give convergence rates to the maximum margin solution for both of our algorithms and for arc-gv. We then study AdaBoost's convergence properties using the smooth margin function. We precisely bound the margin attained by AdaBoost when the edges of the weak classifiers fall within a specified range. This shows that a previous bound proved by R\"{a}tsch and Warmuth is exactly tight. Furthermore, we use the smooth margin to capture explicit properties of AdaBoost in cases where cyclic behavior occurs.

In order to understand AdaBoost's dynamics, especially its ability to maximize margins, we derive an associated simplified nonlinear iterated map and analyze its behavior in low-dimensional cases. We find stable cycles for these cases, which can explicitly be used to solve for Ada-Boost's output. By considering AdaBoost as a dynamical system, we are able to prove Rätsch and Warmuth's conjecture that AdaBoost may fail to converge to a maximal-margin combined classifier when given a'nonoptimal' weaklearning algorithm.

In order to understand AdaBoost's dynamics, especially its ability to maximize margins, we derive an associated simplified nonlinear iterated map and analyze its behavior in low-dimensional cases. We find stable cycles for these cases, which can explicitly be used to solve for Ada-Boost's output. By considering AdaBoost as a dynamical system, we are able to prove Rätsch and Warmuth's conjecture that AdaBoost may fail to converge to a maximal-margin combined classifier when given a'nonoptimal' weak learning algorithm.