TUT Inverse Problems

Research topics

From biology to cosmology, many of the topics we study are about surface reconstruction as an inverse problem, in which field we are an internationally leading group. In addition to deriving fundamental mathematical results on various types of inverse problems, we develop reconstruction procedures that converge efficiently and can handle large data sets. Our main application fields are space research, forest and environmental sciences, and remote sensing.

The following is a list of research topics our team is currently working on. Short descriptions are given for each topic and two types of keywords are also listed:

application
Descriptive keywords for the application.
theory
Mathematical concepts and techniques used to tackle the application.

For even more information see the list of publications and additional material.

Asteroid reconstruction

multiple data modalities asteroid shape asteroid spin
generalized projections regularization

We study the theory and practical solution procedures of inverse problems of generalized projections. An important application field is space research and solar system studies, where asteroids are modelled using multiple data modalities: their varying brightnesses (photometry), adaptive optics data, radar measurements, stellar occultations, thermal infrared radiation, interferometric projections, and flyby data from space missions. These sources sample the surface in various projective spaces, and the multiple relationships between the model parameters and data are utilized simultaneously. Mathematical results in this area include uniqueness and stability theorems and information content analyses for generalized projections, as well as a rigorous solution (maximum compatibility estimate) of the general inverse problem of combining multimodal data (data fusion).

An asteroid in space.

We are constructing a complete procedure for asteroid modelling using all available data sources, working in close collaboration with a number of international teams producing the data (large-scale sky surveys, large telescopes and space missions of ESO, ESA, etc.). The key data are photometry: this is the only way to model the shapes and rotation states of hundreds of thousands of asteroids and thus complete a comprehensive map of our solar system. Hundreds of asteroids can be imaged in more detail with the multimodal inversion. Most asteroid models in existence (several hundreds by now) have been constructed with our techniques.

For examples and more information on asteroid models, visit the Database of Asteroid Models from Inversion Techniques (DAMIT). You can also make your own contribution to asteroid research by joining the distributed computing project Asteroids@home.

Forest and tree research

reconstructed tree models Bayes forest soil models
least-squares fitting Bayes statistics optimization

Accurate tree models have a colossal range of applications. One of the pivotal research topics in environmental and climate research is the carbon cycle on both local and global scales. As the systems and their subsystems are very complex, all the related models require accurate estimates of biomass and its distribution in element size such as tree trunk/branch thickness. The same applies to forest inventory and the development of biological tree models in general. On the societal and economical fronts, proper inventory is necessary in bioenergy planning, timber volume measurement, estimation of timber quality (assessment of the best use for trees), carbon footprint budgeting, planning of harvesting, and forest fire prevention (fuel ladder models).

A reconstructed cylinder model of a maple tree.

Our goal is to develop a method that reconstructs a comprehensive, three-dimensional volume and surface model of a tree. Our model consists of the location, orientation and size of each reconstructed branch and the connections between branches. Therefore, at the moment, for example, the following properties are easily derived from our model: total or partial volumes, branch length and volume distributions, stem taper, lean and sweep, stem branch location and orientation. We are also working on adapting our modelling algorithm to work with pulled out tree stumps and root systems. Currently we use multidirection terrestrial laserscanning measurements as our data source, but we have plans to study the suitability of other measurements also.

At the moment we a working in close collaboration with numerous institutes both Finnish and international with research fields varying from mathematical modelling to environmental research, forest sciences, and remote sensing. Our Finnish partner organizations include the Remote sensing of the changing Earth research group at the Finnish Geospatial Research Institute (National Land Survey of Finland), the Finnish Environment Institute (SYKE) and the Natural Resources Institute Finland (LUKE).

Multidimensional reconstruction

dark matter sky surveys
Hamiltonian systems near-integrable systems

Visualization of a galaxy.

Surface reconstruction is a crucial inverse problem also in dimensions larger than three. Regular motion in Hamiltonian 3D systems is confined to 3-tori in six-dimensional phase space. The dynamics of a near-integrable system, such as our galaxy, can be described by perturbations of a foliated set of tori in 6D that defines the mass distribution and potential field of the system. Thus, the reconstruction of these surfaces from observations yields a well-defined model of the system, and especially the dark matter contained in it. We develop practical generally applicable procedures for torus construction, to be used in, e.g., the analysis of large-scale sky surveys.

Volumetric Low- and High-Frequency Inversion

biomedicine magnetoencephalography
volumetric inversion hierarchical Bayesian approach

We develop volumetric forward and inversion methodology for various technologically advanced applications including, the detection and stimulation of low-frequency (quasi-static) biopotential fields.

Recovered brain activity.

Characteristic to biopotential is, among other things, that the complexity of the biological tissue structures, such as the human brain, can be a major challenge that necessitates careful mathematical modeling. In some applications, the sparsity of the measurements and the large computational effort related to simulating and inverting full waveform data can set additional requirements for the imaging process.

Forward modeling and inversion in a volumetric domain: The figure shows brain activity recovered based on magnetoencephalography data with a divergence conforming source model and a hierarchical Bayesian approach.

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