Anisotropic drift-diffusion PDEs
In this project, we focus on a non-symmetric drift-diffusion equation, called osmosis for its analogies with the physical process, introduced by Weickert and collaborators in 2013. Compared to standard plain diffusion models, the osmosis model considers an additional drift term, making the process asymmetric and allowing non-constant steady-states.
Firstly, we propose an efficient numerical implementation of the osmosis equation, based on alternate directions and operator splitting techniques. We study their scale-space properties and show their efficiency in processing large Cultural Heritage images.
Secondly, the idea of using directional gradients for imaging applications is used for the generalisation of the osmosis equation to its anisotropic counter-part, which accommodate suitable directional information: this modification turns out to be useful to correct for the well-known blurring artefacts that the original osmosis model introduces when applied to the removal of (contant) shadow in images. Thus, anisotropic osmosis is applied to the shadow removal problem, where the directional interpolation on the shadow boundaries is based on the spectral decomposition of a refined Tensor Voting approach.
| Parisotto, Simone; Calatroni, Luca; Bugeau, Aurélie; Papadakis, Nicolas; Schönlieb, Carola-Bibiane: Variational Osmosis for Non-linear Image Fusion. In: IEEE Transactions on Image Processing, 2020. (Type: Journal Article | | )|
We propose a new variational model for non-linear image fusion. Our approach is based on the use of an osmosis energy term related to the one studied in Vogel et al.  and Weickert et al. . The minimization of the proposed non-convex energy realizes visually plausible image data fusion, invariant to multiplicative brightness changes. On the practical side, it requires minimal supervision and parameter tuning and can encode prior information on the structure of the images to be fused. For the numerical solution of the proposed model, we develop a primal-dual algorithm and we apply the resulting minimization scheme to solve multi-modal face fusion, color transfer and cultural heritage conservation problems. Visual and quantitative comparisons to state-of-the-art approaches prove the out-performance and the flexibility of our method.
| Parisotto, Simone; Calatroni, Luca; Caliari, Marco; Schönlieb, Carola-Bibiane; Weickert, Joachim: Anisotropic osmosis filtering for shadow removal in images. In: Inverse Problems, 35 (5), 2019. (Type: Journal Article | | )|
We present an anisotropic extension of the isotropic osmosis model that has been introduced by Weickert et al (2013 Energy Minimization Methods in Computer Vision and Pattern Recognition (Berlin: Springer)) for visual computing applications, and we adapt it specifically to shadow removal applications. We show that in the integrable setting, linear anisotropic osmosis minimises an energy that involves a suitable quadratic form which models local directional structures. In our shadow removal applications we estimate the local structure via a modified tensor voting approach (Moreno et al 2012 New Developments in the Visualization and Processing of Tensor Fields (Berlin: Springer)) and use this information within an anisotropic diffusion inpainting that resembles edge-enhancing anisotropic diffusion inpainting (Galić et al 2008 J. Math. Imaging Vis. 31 255–69; Weickert and Welk 2006 Visualization and Processing of Tensor Fields (Berlin: Springer)). Our numerical scheme combines the nonnegativity preserving stencil of Fehrenbach and Mirebeau (2014 J. Math. Imaging Vis. 49 123–47) with an exact time stepping based on highly accurate polynomial approximations of the matrix exponential. The resulting anisotropic model is tested on several synthetic and natural images corrupted by constant shadows. We show that it outperforms isotropic osmosis, since it does not suffer from blurring artefacts at the shadow boundaries.
| Parisotto, Simone; Calatroni, Luca; Daffara, Claudia: Digital Cultural Heritage Imaging via Osmosis Filtering. In: Mansouri, Alamin; Moataz, Abderrahim El; Nouboud, Fathallah; Mammass, Driss (Ed.): ICISP 2018: Image and Signal Processing, Lecture Notes in Computer Science, pp. 407-415, Springer, 2018. (Type: Inproceedings | | )|
In Cultural Heritage (CH) imaging, data acquired within different spectral regions are often used to inspect surface and sub-surface features. Due to the experimental setup, these images may suffer from intensity inhomogeneities, which may prevent conservators from distinguishing the physical properties of the object under restoration. Furthermore, in multi-modal imaging, the transfer of information between one modality to another is often used to integrate image contents.
In this paper, we apply the image osmosis model proposed in [4,10,12] to solve correct these problems arising when diagnostic CH imaging techniques based on reflectance, emission and fluorescence mode in the optical and thermal range are used. For an efficient computation, we use stable operator splitting techniques to solve the discretised model. We test our methods on real artwork datasets: the thermal measurements of the mural painting “Monocromo” by Leonardo Da Vinci, the UV-VIS-IR imaging of an ancient Russian icon and the Archimedes Palimpsest dataset.
| Parisotto, Simone; Calatroni, Luca; Daffara, Claudia: Mathematical osmosis imaging for multi-modal and multi-spectral applications in Cultural Heritage conservation. In: Image Processing for Art Investigation, Ghent, 2018. (Type: Inproceedings | | )|
In this work we present a dual-mode mid-infrared workflow , for detecting sub-superficial mural damages in frescoes artworks. Due to the large nature of frescoes, multiple thermal images are recorded. Thus, the experimental setup may introduce measurements errors, seen as inter-frame changes in the image contrast, after mosaicking. An approach to lowering errors is to post-process the mosaic  via osmosis partial differential equation (PDE) [12, 13], which preserves details, mass and balance the lights: efficient numerical study for osmosis on large images is proposed [2, 11], based on operator splitting . Our range of Cultural Heritage applications include the detection of sub-superficial voids in Monocromo (L. Da Vinci, Castello Sforzesco, Milan) , the light-balance for multi-spectral imaging and the data integration on the Archimedes Palimpsest .
| Calatroni, Luca; Estatico, Claudio; Garibaldi, Nicola; Parisotto, Simone: Alternating Direction Implicit (ADI) schemes for a PDE-based image osmosis model. In: Journal of Physics: Conference Series, IOP publishing, 2017. (Type: Inproceedings | | )|
We consider Alternating Direction Implicit (ADI) splitting schemes to compute efficiently the numerical solution of the PDE osmosis model considered by Weickert et al. in  for several imaging applications. The discretised scheme is shown to preserve analogous properties to the continuous model. The dimensional splitting strategy traduces numerically into the solution of simple tridiagonal systems for which standard matrix factorisation techniques can be used to improve upon the performance of classical implicit methods, even for large time steps. Applications to the shadow removal problem are presented.
| Parisotto, Simone: Anisotropic variational models and PDEs for inverse imaging problems. University of Cambridge, 2019. (Type: PhD Thesis | | )|
Supervisor: Prof Carola-Bibiane Schönlieb (University of Cambridge)
Co-supervisor: Prof Simon Masnou (Université Lyon 1)
In this thesis we study new anisotropic variational regularisers and partial differential equations (PDEs) for solving inverse imaging problems that arise in a variety of real-world applications.
Firstly, we introduce a new anisotropic higher-order total directional variation regulariser. We describe both the theoretical and the numerical details for its use within a variational formulation for solving inverse problems and give examples for the reconstruction of noisy images and videos, image zooming and the interpolation of scattered surface data.
Secondly, we focus on a non-symmetric drift-diffusion equation, called osmosis. We propose an efficient numerical implementation of the osmosis equation, based on alternate directions and operator splitting techniques. We study their scale-space properties and show their efficiency in processing large images. Moreover, we generalise the osmosis equation to accommodate suitable directional information: this modification turns out to be useful to correct for the well-known blurring artefacts the original osmosis model introduces when applied to shadow removal in images.
Last but not least, we explore applications of variational models and PDEs to cultural heritage conservation. We develop a new non-invasive technique that uses multi-modal imaging for detecting sub-superficial defects in fresco walls at sub-millimetre precision. We correct light-inhomogeneities in these imaging measurements that are due to measurement errors via osmosis filtering, in particular making use of the efficient computational schemes that we introduced before for dealing with the large-scale nature of these measurements.
Finally, we propose a semi-supervised workflow for the detection and inpainting of defects in damaged illuminated manuscripts.
Keywords: Total directional variation, anisotropic diffusion, osmosis filter, cultural heritage conservation, primal-dual hybrid gradient, dimensional splitting, inverse problems, image denoising, video denoising, image zooming, surface interpolation, digital elevation maps, shadow removal, thermal quasi-reflectography, non-destructive imaging, dual-mode mid-infrared imaging, inpainting, illuminated manuscripts.