Conditionals in Homomorphic Encryption and Machine Learning Applications

Homomorphic encryption aims at allowing computations on encrypted data without decryption other than that of the final result. This could provide an elegant solution to the issue of privacy preservation in data-based applications, such as those using machine learning, but several open issues hamper this plan. In this work we assess the possibility for homomorphic encryption to fully implement its program without relying on other techniques, such as multiparty computation (SMPC), which may be impossible in many use cases (for instance due to the high level of communication required). We proceed in two steps: i) on the basis of the structured program theorem (Bohm-Jacopini theorem) we identify the relevant minimal set of operations homomorphic encryption must be able to perform to implement any algorithm; and ii) we analyse the possibility to solve -- and propose an implementation for -- the most fundamentally relevant issue as it emerges from our analysis, that is, the implementation of conditionals (requiring comparison and selection/jump operations). We show how this issue clashes with the fundamental requirements of homomorphic encryption and could represent a drawback for its use as a complete solution for privacy preservation in data-based applications, in particular machine learning ones. Our approach for comparisons is novel and entirely embedded in homomorphic encryption, while previous studies relied on other techniques, such as SMPC, demanding high level of communication among parties, and decryption of intermediate results from data-owners. Our protocol is also provably safe (sharing the same safety as the homomorphic encryption schemes), differently from other techniques such as Order-Preserving/Revealing-Encryption (OPE/ORE).Comment: 14 pages, 1 figure, corrected typos, added introductory pedagogical section on polynomial approximatio

The near shift-invariance of the dual-tree complex wavelet transform revisited

The dual-tree complex wavelet transform (DTCWT) is an enhancement of the conventional discrete wavelet transform (DWT) due to a higher degree of shift-invariance and a greater directional selectivity, finding its applications in signal and image processing. This paper presents a quantitative proof of the superiority of the DTCWT over the DWT in case of modulated wavelets.Comment: 15 page

Symmetric units satisfying a group identity

AbstractLet K be an infinite field of characteristic p≠2, G a locally finite group and KG its group algebra. Let φ:KG→KG denote the K-linear extension of an involution φ defined on G. In this paper we prove, under some assumptions, that if the set of φ-symmetric units of KG satisfies a group identity then KG satisfies a polynomial identity. Moreover, in case the prime radical of KG is nilpotent we characterize the groups for which the φ-symmetric units satisfy a group identity

Free groups and subgroups of finite index in the unit group of an integral group ring

In this article we construct free groups and subgroups of finite index in the unit group of the integral group ring of a finite non-abelian group G for which every non-linear irreducible complex representation is of degree 2 and with commutator subgroup G0 a central elementary abelian 2-group.Research partially supported by the Onderzoeksraad of Vrije Universiteit Brussel, Fonds voor Wetenschappelijk Onderzoek (Belgium) and Bilateral Scientific and Technological Cooperation BWS 05/07 (Flanders-POland). Postdoctoraal Onderzoeker van het Fonds voor Wetenschappelijk Onderzoek- Vlaanderen. Research partially supported by the Fundación Séneca of Murcia and D.G.I. of Spain

Spatiogram features to characterize pearls in paintings

Objective characterization of jewels in paintings, especially pearls, has been a long lasting challenge for art historians. The way an artist painted pearls reflects his ability to observing nature and his knowledge of contemporary optical theory. Moreover, the painterly execution may also be considered as an individual characteristic useful in distinguishing hands. In this work, we propose a set of image analysis techniques to analyze and measure spatial characteristics of the digital images of pearls, all relying on the so called spatiogram image representation. Our experimental results demonstrate good correlation between the new metrics and the visually observed image features, and also capture the degree of realism of the visual appearance in the painting. In that sense, these results set the basis in creating a practical tool for art historical attribution and give strong motivation for further investigations in this direction

Digital image processing of the Ghent altarpiece : supporting the painting's study and conservation treatment

In this article, we show progress in certain image processing techniques that can support the physical restoration of the painting, its art-historical analysis, or both. We show how analysis of the crack patterns could indicate possible areas of overpaint, which may be of great value for the physical restoration campaign, after further validation. Next, we explore how digital image inpainting can serve as a simulation for the restoration of paint losses. Finally, we explore how the statistical analysis of the relatively simple and frequently recurring objects (such as pearls in this masterpiece) may characterize the consistency of the painter’s style and thereby aid both art-historical interpretation and physical restoration campaign

Virtual restoration of the Ghent altarpiece using crack detection and inpainting

In this paper, we present a new method for virtual restoration of digitized paintings, with the special focus on the Ghent Altarpiece (1432), one of Belgium's greatest masterpieces. The goal of the work is to remove cracks from the digitized painting thereby approximating how the painting looked like before ageing for nearly 600 years and aiding art historical and palaeographical analysis. For crack detection, we employ a multiscale morphological approach, which can cope with greatly varying thickness of the cracks as well as with their varying intensities (from dark to the light ones). Due to the content of the painting (with extremely many fine details) and complex type of cracks (including inconsistent whitish clouds around them), the available inpainting methods do not provide satisfactory results on many parts of the painting. We show that patch-based methods outperform pixel-based ones, but leaving still much room for improvements in this application. We propose a new method for candidate patch selection, which can be combined with different patch-based inpainting methods to improve their performance in crack removal. The results demonstrate improved performance, with less artefacts and better preserved fine details

Total Variation Reconstruction From Quasi-Random Samples

Abstract-Pseudo-random numbers are often used for generating incoherent uniformly distributed sample distributions. However randomness is a sufficient -not necessary -condition to ensure incoherence. If one wants to reconstruct an image from few samples, choosing a globally optimized set of evenly distributed points could capture the visual content more efficiently. This work compares classical random sampling with a simple construction based on properties of the fractional Golden ratio sequence and the Hilbert space filling curve. Images are then reconstructed using a total variation prior. Results show improvements in terms of peak signal to noise ratio over pseudo-random sampling

Continuous Ultrasound Speckle Tracking with Gaussian Mixtures

Speckle tracking echocardiography (STE) is now widely used for measuring strain, deformations, and motion in cardiology. STE involves three successive steps: acquisition of individual frames, speckle detection, and image registration using speckles as landmarks. This work proposes to avoid explicit detection and registration by representing dynamic ultrasound images as sparse collections of moving Gaussian elements in the continuous joint space-time space. Individual speckles or local clusters of speckles are approximated by a single multivariate Gaussian kernel with associated linear trajectory over a short time span. A hierarchical tree-structured model is fitted to sampled input data such that predicted image estimates can be retrieved by regression after reconstruction, allowing a (bias-variance) trade-off between model complexity and image resolution. The inverse image reconstruction problem is solved with an online Bayesian statistical estimation algorithm. Experiments on clinical data could estimate subtle sub-pixel accurate motion that is difficult to capture with frame-to-frame elastic image registration techniques