Samuli Siltanen

Saara Isoranta, Emilia L. K. Blåsten, Lílian Ferreira de Freitas et al.

We approach image denoising from a perception-driven perspective: how can we select the parameters that are best suited for human visual perception? We combine research methods in mathematics and psychology to develop a mathematical framework for measuring perceived similarity. We construct a sample set of differently denoised photographs by using the same base image as input data and by tuning the parameter value in a total variation denoising algorithm. A comparison test is conducted with human participants to survey perceived differences between the images. Analyzing the results with psychometric scaling provides us with a HaarPSI value to use as a threshold in discretizing parameter grids. As a result, we obtain psychometrically scaled, openly available image sets that are ready to use in further experiments in perception-driven imaging, as well as a framework for ensuing experiments involving comparison tests.

Martin Burger, Hendrik Dirks, Lena Frerking et al.

In this paper we study the reconstruction of moving object densities from undersampled dynamic X-ray tomography in two dimensions. A particular motivation of this study is to use realistic measurement protocols for practical applications, i.e. we do not assume to have a full Radon transform in each time step, but only projections in few angular directions. This restriction enforces a space-time reconstruction, which we perform by incorporating physical motion models and regularization of motion vectors in a variational framework. The methodology of optical flow, which is one of the most common methods to estimate motion between two images, is utilized to formulate a joint variational model for reconstruction and motion estimation. We provide a basic mathematical analysis of the forward model and the variational model for the image reconstruction. Moreover, we discuss the efficient numerical minimization based on alternating minimizations between images and motion vectors. A variety of results are presented for simulated and real measurement data with different sampling strategy. A key observation is that random sampling combined with our model allows reconstructions of similar amount of measurements and quality as a single static reconstruction.

Maarten de Hoop, Matti Lassas, Matteo Santacesaria et al.

A new computational method for reconstructing a potential from the Dirichlet-to-Neumann map at positive energy is developed. The method is based on D-bar techniques and it works in absence of exceptional points -- in particular, if the potential is small enough compared to the energy. Numerical tests reveal exceptional points for perturbed, radial potentials. Reconstructions for several potentials are computed using simulated Dirichlet-to-Neumann maps with and without added noise. The new reconstruction method is shown to work well for energy values between $10^{-5}$ and $5$, smaller values giving better results.

Andreas Hauptmann, Matteo Santacesaria, Samuli Siltanen

In Electrical Impedance Tomography (EIT) one wants to image the conductivity distribution of a body from current and voltage measurements carried out on its boundary. In this paper we consider the underlying mathematical model, the inverse conductivity problem, in two dimensions and under the realistic assumption that only a part of the boundary is accessible to measurements. In this framework our data are modeled as a partial Neumann-to-Dirichlet map (ND map). We compare this data to the full-boundary ND map and prove that the error depends linearly on the size of the missing part of the boundary. The same linear dependence is further proved for the difference of the reconstructed conductivities -- from partial and full boundary data. The reconstruction is based on a truncated and linearized D-bar method. Auxiliary results include an extrapolation method to obtain the full-boundary data from the measured one, an approximation of the complex geometrical optics solutions computed directly from the ND map as well as an approximate scattering transform for reconstructing the conductivity. Numerical verification of the convergence results and reconstructions are presented for simulated test cases.

Zenith Purisha, Sakari S. Karhula, Juuso Ketola et al.

X-ray tomography is a reliable tool for determining the inner structure of 3D object with penetrating X-rays. However, traditional reconstruction methods such as FDK require dense angular sampling in the data acquisition phase leading to long measurement times, especially in X-ray micro-tomography to obtain high resolution scans. Acquiring less data using greater angular steps is an obvious way for speeding up the process and avoiding the need to save huge data sets available memory. However, computing 3D reconstruction from such a sparsely sampled dataset is very sensitive to measurement noise and modelling errors. An automatic regularization method is proposed for robust reconstruction, based on enforcing sparsity in the three-dimensional shearlet transform domain. The inputs of the algorithm are the projection data and {\it a priori} known expected degree of sparsity, denoted $0<{\mathcal C}_{pr}\leq 1$. The number ${\mathcal C}_{pr}$ can be calibrated from a few dense-angle reconstructions and fixed. Human subchondral bone samples were tested and morphometric parameters of the bone reconstructions were then analyzed using standard metrics. The proposed method is shown to outperform the baseline algorithm (FDK) in the case of sparsely collected data. The number of X-ray projections can be reduced up to 10\% of the total amount while retaining the quality of the reconstruction images and of the morphometric paramaters.

Sarah Jane Hamilton, Samuli Siltanen

Electrical impedance tomography (EIT) is a non-invasive imaging method in which an unknown physical body is probed with electric currents applied on the boundary, and the internal conductivity distribution is recovered from the measured boundary voltage data. The reconstruction task is a nonlinear and ill-posed inverse problem, whose solution calls for special regularized algorithms, such as D-bar methods which are based on complex geometrical optics solutions (CGOs). In many applications of EIT, such as monitoring the heart and lungs of unconscious intensive care patients or locating the focus of an epileptic seizure, data acquisition on the entire boundary of the body is impractical, restricting the boundary area available for EIT measurements. An extension of the D-bar method to the case when data is collected only on a subset of the boundary is studied by computational simulation. The approach is based on solving a boundary integral equation for the traces of the CGOs using localized basis functions (Haar wavelets). The numerical evidence suggests that the D-bar method can be applied to partial-boundary data in dimension two and that the traces of the partial data CGOs approximate the full data CGO solutions on the available portion of the boundary, for the necessary small $k$ frequencies.

Samuli Siltanen, Janne P. Tamminen

The aim of electrical impedance tomography is to form an image of the conductivity distribution inside an unknown body using electric boundary measurements. The computation of the image from measurement data is a non-linear ill-posed inverse problem and calls for a special regularized algorithm. One such algorithm, the so-called D-bar method, is improved in this work by introducing new computational steps that remove the so far necessary requirement that the conductivity should be constant near the boundary. The numerical experiments presented suggest two conclusions. First, for most conductivities arising in medical imaging, it seems the previous approach of using a best possible constant near the boundary is sufficient. Second, for conductivities that have high contrast features at the boundary, the new approach produces reconstructions with smaller quantitative error and with better visual quality.

Janne P. Tamminen, Tanja Tarvainen, Samuli Siltanen

The D-bar method at negative energy is numerically implemented. Using the method we are able to numerically reconstruct potentials and investigate exceptional points at negative energy. Subsequently, applying the method to Diffusive Optical Tomography, a new way of reconstructing the diffusion coefficient from the associated Complex Geometrics Optics solution is suggested and numerically validated.

Heikki Haario, Aki Kallonen, Marko Laine et al.

A two-dimensional tomographic problem is studied. The target is assumed to be a homogeneous object bounded by a smooth curve. A Non Uniform Rational Basis Splines (NURBS) curve is used as computational representation of the boundary. This approach conveniently provides the result in a format readily compatible with computer-aided design (CAD) software. However, the linear tomography task becomes a nonlinear inverse problem due to the NURBS-based parameterization. Therefore, Bayesian inversion with Markov chain Monte Carlo (MCMC) sampling is used for calculating an estimate of the NURBS control points. The reconstruction method is tested with both simulated data and measured X-ray projection data. The proposed method recovers the shape and the attenuation coefficient significantly better than the baseline algorithm (optimally thresholded total variation regularization), but at the cost of heavier computation.

Chuyang Wu, Samuli Siltanen

Image reconstruction in X-ray tomography is an ill-posed inverse problem, particularly with limited available data. Regularization is thus essential, but its effectiveness hinges on the choice of a regularization parameter that balances data fidelity against a priori information. We present a novel method for automatic parameter selection based on the use of two distinct computational discretizations of the same problem. A feedback control algorithm dynamically adjusts the regularization strength, driving an iterative reconstruction toward the smallest parameter that yields sufficient similarity between reconstructions on the two grids. The effectiveness of the proposed approach is demonstrated using real tomographic data.

Markus Juvonen, Samuli Siltanen

We present a patch-based image reconstruction and animation method that uses existing image data to bring still images to life through motion. Image patches from curated datasets are grouped using k-means clustering and a new target image is reconstructed by matching and randomly sampling from these clusters. This approach emphasizes reinterpretation over replication, allowing the source and target domains to differ conceptually while sharing local structures.

Sara Sippola, Siiri Rautio, Andreas Hauptmann et al.

Electrical impedance tomography (EIT) is a non-invasive imaging method with diverse applications, including medical imaging and non-destructive testing. The inverse problem of reconstructing internal electrical conductivity from boundary measurements is nonlinear and highly ill-posed, making it difficult to solve accurately. In recent years, there has been growing interest in combining analytical methods with machine learning to solve inverse problems. In this paper, we propose a method for estimating the convex hull of inclusions from boundary measurements by combining the enclosure method proposed by Ikehata with neural networks. We demonstrate its performance using experimental data. Compared to the classical enclosure method with least squares fitting, the learned convex hull achieves superior performance on both simulated and experimental data.

Juan Pablo Agnelli, Fernando S. Moura, Siiri Rautio et al.

Electrical impedance tomography (EIT) is a non-invasive imaging method for recovering the internal conductivity of a physical body from electric boundary measurements. EIT combined with machine learning has shown promise for the classification of strokes. However, most previous works have used raw EIT voltage data as network inputs. We build upon a recent development which suggested the use of special noise-robust Virtual Hybrid Edge Detection (VHED) functions as network inputs, although that work used only highly simplified and mathematically ideal models. In this work we strengthen the case for the use of EIT, and VHED functions especially, for stroke classification. We design models with high detail and mathematical realism to test the use of VHED functions as inputs. Virtual patients are created using a physically detailed 2D head model which includes features known to create challenges in real-world imaging scenarios. Conductivity values are drawn from statistically realistic distributions, and phantoms are afflicted with either hemorrhagic or ischemic strokes of various shapes and sizes. Simulated noisy EIT electrode data, generated using the realistic Complete Electrode Model (CEM) as opposed to the mathematically ideal continuum model, is processed to obtain VHED functions. We compare the use of VHED functions as inputs against the alternative paradigm of using raw EIT voltages. Our results show that (i) stroke classification can be performed with high accuracy using 2D EIT data from physically detailed and mathematically realistic models, and (ii) in the presence of noise, VHED functions outperform raw data as network inputs.

Markus Juvonen, Samuli Siltanen, Fernando Silva de Moura

The photographic dataset collected for the Helsinki Deblur Challenge 2021 (HDC2021) contains pairs of images taken by two identical cameras of the same target but with different conditions. One camera is always in focus and produces sharp and low-noise images the other camera produces blurred and noisy images as it is gradually more and more out of focus and has a higher ISO setting. Even though the dataset was designed and captured with the HDC2021 in mind it can be used for any testing and benchmarking of image deblurring algorithms. The data is available here: https://doi.org/10.5281/zenodo.477228

Tatiana A. Bubba, Mathilde Galinier, Matti Lassas et al.

We propose a novel convolutional neural network (CNN), called $Ψ$DONet, designed for learning pseudodifferential operators ($Ψ$DOs) in the context of linear inverse problems. Our starting point is the Iterative Soft Thresholding Algorithm (ISTA), a well-known algorithm to solve sparsity-promoting minimization problems. We show that, under rather general assumptions on the forward operator, the unfolded iterations of ISTA can be interpreted as the successive layers of a CNN, which in turn provides fairly general network architectures that, for a specific choice of the parameters involved, allow to reproduce ISTA, or a perturbation of ISTA for which we can bound the coefficients of the filters. Our case study is the limited-angle X-ray transform and its application to limited-angle computed tomography (LA-CT). In particular, we prove that, in the case of LA-CT, the operations of upscaling, downscaling and convolution, which characterize our $Ψ$DONet and most deep learning schemes, can be exactly determined by combining the convolutional nature of the limited angle X-ray transform and basic properties defining an orthogonal wavelet system. We test two different implementations of $Ψ$DONet on simulated data from limited-angle geometry, generated from the ellipse data set. Both implementations provide equally good and noteworthy preliminary results, showing the potential of the approach we propose and paving the way to applying the same idea to other convolutional operators which are $Ψ$DOs or Fourier integral operators.

Zenith Purisha, Juho Rimpeläinen, Tatiana Bubba et al.

Tomographic reconstruction is an ill-posed inverse problem that calls for regularization. One possibility is to require sparsity of the unknown in an orthonormal wavelet basis. This in turn can be achieved by variational regularization where the penalty term is the sum of absolute values of wavelet coefficients. Daubechies, Defrise and De Mol (Comm. Pure Appl. Math. 57) showed that the minimizer of the variational regularization functional can be computed iteratively using a soft thresholding operation. Choosing the soft threshold parameter $μ>0$ is analogous to the notoriously difficult problem of picking the optimal regularization parameter in Tikhonov regularization. Here a novel automatic method is introduced for choosing $μ$, based on a control algorithm driving the sparsity of the reconstruction to an {\it a priori} known ratio of nonzero versus zero wavelet coefficients in the unknown function.

David Villacis, Santeri Kaupinmäki, Samuli Siltanen et al.

This is a photographic dataset collected for testing image processing algorithms. The idea is to have images that can exploit the properties of total variation, therefore a set of playing cards was distributed on the scene. The dataset is made available at www.fips.fi/photographic_dataset2.php

Teemu Helenius, Samuli Siltanen

This is a photographic dataset collected for testing image processing algorithms. The idea is to have sets of different but statistically similar images. In this work the images show randomly distributed peppercorns. The dataset is made available at www.fips.fi/photographic_dataset.php .