Compressed Sensing

csCompressed sensing has gained a significant amount of attention since its introduction by Donoho, Candes, Tao and coworkes in 2006, see [Don06, CRT06]. These works show that it is possible to reconstruct discrete signals or images from a very limited number of random measurements. This is a significant improvement over Shannon's sampling theorem [Sha49], which states that stable reconstruction of signals is only possible when the measurements are performed at the so called Nyquist rate, which is twice the maximum frequency of the signal. In the recent years, compressed sensing has been proposed to speed up the measurement process in various medical imaging devices such as MRI, see [LDSP08]. Another prominent example is the single pixel camera proposed in [Bar08] to circumvent the use of expensive high resolution sensors in digital photography.

In our research, we develop compressed sensing techniques for tomographic image reconstruction problems and other imaging problems. The goal is to obtain high quality reconstructions from limited measurements and therefore speeding up the measurement process and to increase the robustness of the reconstruction process with respect to the noise. For example in [SKBBH15], we propose a novel compressed sensing scheme for speeding up data acquisition in photoacoustic tomography.

References:

[CRT06] Emmanuel Candes, Justin Romberg, and Terence Tao. Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Transactions on Information Theory 52, pages 489–509, 2006. 

[Bar08] Richard G Baraniuk. Single-pixel imaging via compressive sampling. IEEE Signal Processing Magazine, 2008. 

[Don06] David L Donoho. Compressed sensing. IEEE Transactions on Information Theory, 52(4):1289–1306, 2006. 

[LDSP08] Michael Lustig, David L Donoho, Juan M Santos, and John M Pauly. Compressed sensing MRI. Signal Processing Magazine, IEEE, 25(2):72–82, 2008. 

[Sha49] Claude E. Shannon. Communication in the presence of noise. Proc. I.R.E., 37, 1949. 

[SKBBH15] M. Sandbichler, F. Krahmer, T. Berer, P. Burgholzer, M. Haltmeier, A Novel Compressed Sensing Scheme for Photoacoustic Tomography, Submitted, 2015

 

Dictionary Learning

If you have read any compressed sensing or sparse approximation papers, you will most likely have run into the statement 'you have a signal y which is sparse in a given basis/dictionary'. The question is just who gave you that basis/dictionary :D? In dictionary learning the tasks is exactly this. You have a bunch of signals, let's say K of them stored in a matrix Y, and you want to find a dictionary D with N atoms, to approximate all your signals with S atoms. So you want to find a factorisation of the matrix Y into the dictionary D and coefficients X, with S nonzeros in every column, such that the error in E is small.

$$ Y=D*X + E, Y...(dxK), D....(dxN), X...(NxK), E...(dxK) $$

Then you can vary the question, how many atoms do I need for a given sparsity level and error, which sparsity level can I achieve, given the number of atoms and error? Also you can become more philosophic, how is the coherence of the atoms related to this, how can we characterise the usefulness of an atom?
As mentioned before these are all hard problems! Why? They are hard to treat theoretically because they are so non-linear and they are even hard to treat numerically as at some point you need to find sparse approximations to calculate the error, which is time consuming and depends on the algorithm you used.

 

Inverse Problems

inv pRoughly spoken, inverse problems are concerned with the determination of the cause of an observed or a desired effect. A typical exampe of an inverse problem is ultrasound imaging, whose application you probably have seen at a doctor. The goal is to estimate/image the so-called acoustic impedance of some interior region of the human body. For that purpose, ultrasound waves are sent into the patient and afterwards the pressure variations are measured outside of the patient. Because the wave propagation inside the patient depends on the acoustic impedance, it is possible to estimate the impedance from the measured pressure data. Ultrasound imaging has the advantage that it is very cheap, fast and mobile.

A drawback of pure ultrasound imaging, however, is that the acoustic impedance of many types of tumors is very similar to that of healthy tissue. An alternative, more recently developed imaging method offering improved contrast for cancer detection is photoacoustic tomography (PAT). The basic principle of PAT can be described as follows: The tissue of interest is illuminated with pulsed light, which becomes partially absorbed inside the tissue. As a consequence of the heating, pressure waves are generated within the tissue, which are measured outside the patient, for example by piezoelectric films or optical fibers. In our research, we work on various theoretical and practical aspects of PAT including image reconstruction using linear, planar or circular detectors, or taking acoustic wave dissipation into account. In our research, we also develop iterative and variational reconstruction methods that can be applied for solving more general types of inverse problems.

 

Computed Tomography

ctComputed tomography (CT) is one of the most important diagnostic tools in modern medicine. While the term computed tomography was initially reserved for x-ray based CT scanners, it nowadays covers various non-invasive imaging technologies, where mathematics plays a major role for obtaining diagnostic images. Examples include x-ray CT, single photon emission computed tomography (SPECT), positron emission tomography (PET), magnetic resonance tomography (MRT), ultrasound tomography, electrical impedance tomography, optical imaging, as well as photo- and thermoacoustic tomography (see, for example, [HalPer14, Her09, Kuc14, Nat01] for some general introductions).

A unifying element of all tomographic applications is that only indirect information about the quantity of interest (usually modelled as a function or some high dimensional parameter vector) can be collected when scanning the patient. Due to the modeling imperfections, measurement errors and statistical uncertainties, the data are additionally corrupted by deterministic or random noise. Such types of applications are most conveniently studied in the framework of inverse problems, where the reconstruction problem is formulated as a system of linear or non-linear equations in high or infinite dimensional spaces. In our research we work on various theoretical and practical for the solution of such tomographic image reconstruction problems. Examples include general iterative reconstruction methods (see, for example, [HKLS07, HLR09]), or special, problem adapted techniques for photoacoustic tomography (see, for example [FHR07, Hal14, Kow14]).

 

References:

[FHR07] Inversion of spherical means and the wave equation in even dimensions. D. Finch, M. Haltmeier, Rakesh SIAM J. Appl. Math. 68(2): 392-412, 2007.

[Hal14] Universal inversion formulas for recovering a function from spherical means. M. Haltmeier. SIAM J. Math. Anal. 46(1): 214–232, 2014. [arxiv.org]

[HalPer14] M. Haltmeier, S. Pereverzyev Jr. Introduction to the mathematics of computed tomography, Internat. Math. Nachrichten 226:29-51, 2014. [pdf pdf button]

[Her09] G. T. Herman. Fundamentals of computerized tomography, second edition. Springer, 2009.

[HKLS07] Kaczmarz methods for regularizing nonlinear ill-posed equations II: Applications M. Haltmeier, R. Kowar, A. Leitao, O. Scherzer. Inverse Probl. Imaging 1(3): 507-523, 2007.

[HLR09] M. Haltmeier, A. Leitao, E. Resmerita. On regularization methods of EM-Kaczmarz type. Inverse Problems 25(7): 075008, 2009.

[Kow14] On time reversal in photoacoustic tomography for tissue similar to water. R. Kowar SIAM J. Imaging Sciences 7(1): 509-527, 2014. [pdf pdf button] [arxiv.org]

[Kuc14] P. Kuchment. The Radon transform and medical imaging. SIAM, 2014.

[Nat01] F. Natterer. The Mathematics of Computerized Tomography. SIAM, 2001.

 

Learning Theory

ltSystems, functioning entities that take an input and give the output, arise in many scientific studies. Understanding how a specific system performs, that is given an input one aims to predict the output of a system. Especially, one looks for the prediction of the system output. For example, one is maybe interested in predicting the glucose concentration in blood at a given time of the day; or interested in predicting whether an individual customer would appreciate a specific product, such as a movie, book, song, etc. The interior structure of many systems is often unavailable, which makes it difficult to describe the internal mechanisms of the system that for a given input create the output. The only available information about the system is then the knowledge of certain input-output pairs. These pairs are obtained from the system observations, and are often called the empirical data.

In Machine Learning [Alp04, Bis06, Mit97], one is concerned with the design and development of algorithms, called (machine) learning algorithms, that allow computers (machines) to predict, or to make a decision about, the system output based on the empirical data from the system observations. The analysis of learning algorithms is done in the framework of (Computational) Learning Theories. One of such theories is the so called Statistical Learning Theory [PogSma03, Vap98]. According to this theory, the learning algorithm should construct a function, called an estimator, that approximates well the relationship between the system input and the system output. In our research, we construct learning algorithms using the so-called multi-penalty regularization (MPR) techniques, which are known in the field of Regularization of Inverse Problems. We also analyze the quality of the corresponding estimators. In particular, in [LPS12], we showed that MPR in comparison to the conventional approaches creates better extrapolating estimators. In [HNP14], we proposed and analyzed a new method based on the MPR approach for detecting relevant variables in the input-output relations. In the application of this method to the reconstruction of the gene regulatory networks, we obtained results that show a clear evidence of the competitiveness of the proposed method with respect to the previously known approaches.

References:

[Alp04] E. Alpaydin. Introduction to Machine Learning (Adaptive Computation and Machine Learning). MIT Press, 2004.

[Bis06] C. M. Bishop. Pattern Recognition and Machine Learning. Springer, 2006.

[HNP14] K. Hlavackova-Schindler, V. Naumova, S. Pereverzyev Jr. Multi-penalty regularization for detecting relevant variables. Applied Mathematics Preprint Nr. 11, University of Innsbruck, 2014, submitted. [pdf pdf button]

[LPS12] S. Lu, S. Pereverzyev Jr., S. Sampath. Multi-parameter regularization for construction of extrapolating estimators in statistical learning theory. In "Multiscale Signal Analysis and Modeling" (Eds.: X. Shen and A. I. Zayed). Springer Lecture Notes in Electrical Engineering, 2012.

[Mit97] T. M. Mitchell. Machine Learning. McGraw Hill, 1997.

[PogSma03] T. Poggio and S. Smale. The mathematics of learning: dealing with data. Notices Am. Math. Soc., 50(5):537–544, 2003.

[Kuc14] P. Kuchment. The Radon transform and medical imaging. SIAM, 2014.

[Nat01] F. Natterer. The Mathematics of Computerized Tomography. SIAM, 2001.

 

Photoacoustic Tomography

patPhotoacoustic tomography (PAT) is a recently developed medical imaging paradigm that combines the high spatial resolution of ultrasound imaging with the high contrast of optical imaging [Bea11, Kru99, XuWan06]. When a semitransparent sample is illuminated with a short pulse of electromagnetic energy near the visible range, then parts of the optical energy will be absorbed inside the sample, which causes a rapid, non-uniform increase of temperature. The increase of temperature yields a spatially varying thermoelastic expansion, which in turn induces an acoustic pressure wave. The acoustic pressure wave is measured outside of the object of interest, and mathematical algorithms are used to recover an image of the interior. PAT provides a good imaging contrast in soft tissues making it a very promising technique for detecting various types of early cancer, such as breast cancer or skin melanoma.

In our group, we develop and analyze various mathematical models and methods relevant for PAT. For example, in [Hal14, HP14, HP15] we derive theoretically exact reconstruction formulas for the cases that the imaging function imaging is supported inside spheres, ellipsoids, and other quadric hyper-surfaces. Mathematical models for PAT taking acoustic attenuation into account are developed and analyzed in [Kow14KS12]. In another direction of research, we develop models and algorithms for PAT using integrating detectors (see, for example, [PNHB09, ZHS09]).

References:

[Bea11] P. Beard. Biomedical photoacoustic imaging. Interface focus, 1(4): 602–631, 2011

[Hal14] M. Haltmeier. Universal inversion formulas for recovering a function from spherical means. SIAM J. Math. Anal., 46(1):214–232, 2014.

[HP14] M. Haltmeier and S. Pereverzyev Jr. The universal back-projection formula for spherical means and the wave equation on certain quadric hypersurfaces, 2014, submitted. [pdf pdf button]

[HP15] M. Haltmeier and S. Pereverzyev Jr. Recovering a function from circular means or wave data on the boundary of parabolic domains. SIAM J. Imaging Sci., 2015, to appear. [pdf pdf button]

[Kow14] R. Kowar. On time reversal in photoacoustic tomography for tissue similar to water. SIAM J. Imaging Sci. 7(1):509–527, 2014. [arxiv.org]

[Kru99] R. A. Kruger, K. K. Kopecky, A. M. Aisen, Reinecke D. R., G. A. Kruger, and W. L. Kiser. Thermoacoustic CT with radio waves: A medical imaging paradigm. Radiology 200(1): 275–278, 1999.

[KS12] R. Kowar and O. Scherzer. Attenuation models in photoacoustics. In Mathematical modeling in biomedical imaging II, pp. 85-130, 2012.

[PNHB09] G. Paltauf, R. Nuster, M. Haltmeier, P. Burgholzer. Photoacoustic tomography with integrating area and line detectors  Chapter 20 in L.-H. Wang (Editor): Photoacoustic imaging and spectroscopy, CRC Press, 2009

[XuWan06] M. Xu and L. V. Wang. Photoacoustic imaging in biomedicine. Rev. Sci. Instrum., 77(4): 041101 (22pp), 2006.

[ZHS09] G. Zangerl, M. Haltmeier, O. Scherzer. Exact series reconstruction in photoacoustic tomography with circular integrating detectors. Commun. Math. Sci. 7(3): 665-678, 2009 [arxiv.org]