On the existence of optimal multi-valued decoders and their accuracy bounds for undersampled inverse problems

Undersampled inverse problems occur everywhere in the sciences including medical imaging, radar, astronomy etc., yielding underdetermined linear or non-linear reconstruction problems. There are now a myriad of techniques to design decoders that can tackle such problems, ranging from optimization based approaches, such as compressed sensing, to deep learning (DL), and variants in between the two techniques. The variety of methods begs for a unifying approach to determine the existence of optimal decoders and fundamental accuracy bounds, in order to facilitate a theoretical and empirical understanding of the performance of existing and future methods. Such a theory must allow for both single-valued and multi-valued decoders, as underdetermined inverse problems typically have multiple solutions. Indeed, multi-valued decoders arise due to non-uniqueness of minimizers in optimisation problems, such as in compressed sensing, and for DL based decoders in generative adversarial models, such as diffusion models and ensemble models. In this work we provide a framework for assessing the lowest possible reconstruction accuracy in terms of worst- and average-case errors. The universal bounds bounds only depend on the measurement model $F$, the model class $\mathcal{M}_1 \subseteq \mathcal{X}$ and the noise model $\mathcal{E}$. For linear $F$ these bounds depend on its kernel, and in the non-linear case the concept of kernel is generalized for undersampled settings. Additionally, we provide multi-valued variational solutions that obtain the lowest possible reconstruction error

Medial Sural Perforator "Nerve through Flap": Anatomical Study and Clinical Application

Background: Nerve recovery after a complex trauma is affected by many factors and a poorly vascularized bed is often the cause of failure and perineural scar. Many techniques have been devised to avoid this problem and the possibility to transfer a nerve with a surrounding viable sliding tissue could help in this purpose; Methods: We performed an anatomic study on 8 injected specimens to investigate the possibility to raise a medial sural artery perforator (MSAP) flap including the sural nerve within its vascularized sheath; Results: In anatomic specimens, a visible direct nerve vascularization was present in 57% of legs (8 out of 14). In 43% a vascular network was visible in the fascia layer. There were no vascular anomalies. In one patient the MSAP flap was raised including the sural nerve with its proximal tibial and peroneal components within the deep sheath. The tibial and peroneal component of the sural nerve were anastomized independently with the common digital nerve of 4th and 5th fingers and with the collateral nerve for the ulnar aspect of the 5th. After 9 months, the patient showed an improving nerve function both clinically and electromyographically without any problem due to nerve adherence; Conclusions: Given the still debated advantage of a vascularized nerve graft versus a non-vascularized one, this flap could be useful in those cases of composite wounds with nerve lesions acting as a "nerve through flap", in order to reduce nerve adherence with a viable surrounding gliding tissue

Cardiac magnetic resonance predictors of left ventricular remodelling following acute ST elevation myocardial infarction: The VavirimS study

Left ventricular (LV) remodelling (REM) ensuing after ST-elevation myocardial infarction (STEMI), has typically been studied by echocardiography, which has limitations, or cardiac magnetic resonance (CMR) in early phase that may overestimate infarct size (IS) due to tissue edema and stunning. This prospective, multicenter study investigated LV-REM performing CMR in the subacute phase, and 6 months after STEMI

Compressed Sensing Photoacoustic Tomography Reduces to Compressed Sensing for Undersampled Fourier Measurements

Photoacoustic tomography (PAT) is an emerging imaging modality that aims at measuring the high-contrast optical properties of tissues by means of high-resolution ultrasonic measurements. The interaction between these two types of waves is based on the thermoacoustic effect. In recent years, many works have investigated the applicability of compressed sensing to PAT in order to reduce measuring times while maintaining a high reconstruction quality. However, in most cases, theoretical guarantees are missing. In this work, we show that in many measurement setups of practical interest, compressed sensing PAT reduces to compressed sensing for undersampled Fourier measurements. This is achieved by applying known reconstruction formulae in the case of the free-space model for wave propagation, and by applying the theories of Riesz bases and nonuniform Fourier series in the case of the bounded domain model. Extensive numerical simulations illustrate and validate the approach

Microstructural Features in Corroded Celtic Iron Age Sword Blades

Archaeological artefacts made from iron and steel are often of critical importance for archaeometallurgical studies, which aim to understand the process of manufacturing, as the nearly complete alloy mineralization does not allow for any type of metallographic interpretation. In this study, three Iron Age sword blades dated from the second century BC (LaTe`ne B2/D1) found in the archaeological site of Tintignac (Commune de Naves, Corre`ze, France), were investigated. A multianalytical approach was employed to acquire a complete range of data from the partially or totally corroded objects. Analyses were carried out with the use of light optical microscopy, micro Raman spectroscopy, and scanning electron microscopy equipped with energy-dispersive x-ray spectroscopy (EDXS). Remnants of metallographic features\u2014ghost microstructure\u2014in the corrosion layers of the blades were observed, allowing for a partial reconstruction of the manufacturing process

On accuracy and existence of approximate decoders for ill-posed inverse problems

Based on work by Cohen, Damen and Devore \and Bourrier et. al., we propose a framework that highlights the importance of knowing the measurement model $F$ and model class $\mathcal{M}_1$, for solving ill-posed (non-)linear inverse problems. Previous work has assumed that the measurement model is injective on the model class $\mathcal{M}_1$ and we obviate the need for this assumption. We establish fundamental upper and lower bounds on the reconstruction accuracy of an inverse problem in terms of the kernel size. The key definition introduced in this work, the kernel size of an inverse problem, only requires the measurement model $F$ and model class $\mathcal{M}_1$ to be computed. Thus, it is applicable in deep learning (DL) based settings where $\mathcal{M}_1$ can be an arbitrary data set

Micro-Raman spectroscopy for the characterization of artistic patinas produced on copper-based alloys

In order to contribute to the improvement of restoration and conservation sciences of metallic artefacts belonging to Cultural Heritage, a wide research has been started on the study of artificial (artistic) and natural (corrosion or alteration) patinas of copper-based alloys. Micro-Raman spectroscopy (mRS) coupled with scanning electron microscopy (SEM) equipped with energy Dispersive X-Ray Spectroscopy (EDXS) were applied for the characterization of a number of patinas made on metallic substrate reproducing the typical copper-based alloys used for statuary. A non alloyed copper sheet has been also used to simulate the architectural cover typically used after a specific cold spray coating. To reach these goals, a number of patinas were experimentally produced in our laboratory using the torch technique and reactive solutions based on water as a solvent containing, respectively, copper nitrate, iron nitrate and potassium sulfide (\u2018liver of sulfur\u2019). The only exception is a sample reproducing the architectural cover of a recent building nearly completely covered with copper sheets coated with copper salts. All the produced patinas were aged in a salt spray chamber and then studied with mRS, SEM and EDXS in order to understand their evolution between the two known terms and therefore the reactions involved in all patinas. mRS allowedus todetect asmain constituents of the \u2018artistic\u2019 patinas, before and after ageing: copper oxides (Cuprite and Tenorite), copper nitrates (Rouaite and Gerhardtite), copper chlorides (Botallackite, Atacamite and Paratacamite), a copper sulfate (Posnjakite), iron oxides (Magnetite and Hematite) and an iron hydroxide (Limonite)