Since 3D printing is quite easy and inexpensive nowadays (can be done even in public libraries often), I used a the free software Autodesk 123D to design some 3D printable prototypes for static and dynamic CT. I am here sharing both the ready-made STL file (ready to be printed) and the 123D-project file, in case someone wants to do some personal edits. Anything can be freely use, just please quote the author and the source.

If you are aware of other open data repositories for CT or would like to share suggestions, feel free to comment below. I will update this post and the shared repositories in the future (last update: Feb 22nd, 2016).

Static CT open data

[real data] Tomographic data of a walnut: open dataset from FIPS, authors are indicated at the webpage.

[real data] 3D printable simple phantom prototype: to test contrast agents, geometry preservation of reconstruction method, how different attenuation values are reconstructed. Please quote the author (Paola Elefante) and the link to this post as a source.

[real data] 3D printable blood vessels prototype: to test a realistic static geometry of blood vessels splitting in capillaries. Please quote the author (Paola Elefante) and the link to this post as a source.

Dynamic CT open data

[real data] DirLab repository: a open data repository, mostly for image registration researchers.

[real data] 3D printable dynamic blood vessels prototype:  to test a realistic dynamic simulation of blood or fluid flowing. In the featured picture you can spot an old version of this prototype. I made some major edits in the design, but I still did not test it. Please quote the author (Paola Elefante) and the link to this post as a source.

[simulated data] 2D dynamic "Y-phantom": a binary phantom where meaningful topological changes happen, good for interface detection methods (level-set, etc.).

L'articolo Open Data: CT datasets and prototypes sembra essere il primo su Paola Elefante.

Since January 1st, 2016, the Helsinki Institute of Physics has a new appointed director, Prof. Paula Eerola from University of Helsinki. Paula is the 4th director of the Institute since its start of operations in 1996 and the first woman to be appointed. She has a rich research background in particle physics and an academic career which made her travel to Switzerland and Sweden, before coming back to her home university in Helsinki in 2008.

Congratulations for the great achievement! Can you tell me something more about HIP?

Thank you. The Helsinki Institute of Physics is a joint research institute operated by five Finnish universities: University of Helsinki, Aalto University, University of Jyväskylä, Tampere University of Technology and Lappeenranta University of Technology. Its board consists of representatives of the five universities and a scientific advisory board. Formally the Institute belongs to the Faculty of Science in University of Helsinki.

What are you main responsibilities as director?

I am a sort of CEO of the Institute. I supervise scientific operations, take care of personnel issues, make budget plans, and operate under the national mandate HIP was granted to manage all Finnish research at CERN. Soon we will take part also to FAIR research projects, a new center for subatomic research currently under construction in Germany. HIP also helps the Finnish “CERN co-operation high school network”, which allows high school (“lukio”) classes to visit the CERN facilities. Our researchers act as lecturers and guides during the visits. About 80% of all Finnish high schools take part in this educational project, which involves a long preparation and several school subjects: physics, of course, but also English language - since the visit is in English - and Finnish language class. The students usually write an article in the local Finnish newspaper about the trip. I once asked to first-year physics students in Helsinki how many of them took part in such a programme and many raised their hands. I think this kind of inspiration is very important for younger people, to understand what they want to do.

Can you disclose some cool physics HIP does at CERN?

HIP is involved in experiments at the Large Hadron Collider (LHC), which is the biggest and highest energy particle accelerator in the world. The experiments at the LHC we are contributing to are called CMS, ALICE and TOTEM. CMS is one of the two experiments which discovered the Higgs Boson. I used to be leader of the Finnish team of CMS and becoming director of HIP felt like a natural continuation of this path.

Do you have any career advice for aspiring or young scientists?

I think the basis of everything is your own interest and motivation. Do not hesitate and calculate too much which job you will end up doing. The academic career is not deterministic, you need strong faith and you have to accept uncertain conditions.

One obstacle in particular in Finland is that people tend to work too much alone, too afraid of asking questions or discussing their work. They tend to go home, make their calculations alone and come the day after with an answer. However, science doesn’t work like this, it requires constant interaction, not working in a sort of “vacuum”, it asks for cooperation and feedback. If you truly collaborate with someone, the final entity will be greater than the sum of two single parts. I think this scientist ideology should be revised. I advice not to be afraid to ask or to look stupid.

What is your perspective on women in physics? Any advice?

I think it is still harder for women physicists to be considered in a non-biased way, compared to male colleagues. I have been member of Nordic Women in Physics (NorWiP) for many years and I even took part in a focused training for women in leadership at Lund University, in Sweden. Sometimes younger people don’t acknowledge the issue until it hits them hard. At the same time, we need not to make girls depressed. They simply need to be aware and alert, so that if something happens, they can react. I have made my career as a single parent, it often has been hard to plan and organise everything. External baby-sitting help has been mandatory.

Did you have to travel or live abroad a lot?

Yes, I lived for six years in Geneva. Then, I was a researcher and later a professor in Sweden, at Lund University. My son was three when I was offered the first position in Sweden. It was a hard decision to take, moving to a new place without any social network.

Thank you again to Paula for sharing her story and point of view, and, again, congratulations!

Picture: Linda Tammisto.

L'articolo Paula Eerola, coordinating Finnish physics research at CERN sembra essere il primo su Paola Elefante.

Let's start from defining what an interface is. I could not find a rigorous definition, but the concept is very intuitive. It is a "boundary" which clearly splits the space in two subsets ("inside" and "outside"). You can imagine a closed (even self-intersecting) curve on the plane, for instance. Or the surface of a ball or a torus in . From now on, let's work with planar interfaces, for better visual intuition. However, everything I will discuss here can be extended to any . Now, imagine we are working with a dynamic interface, meaning that our closed curve, for instance, changes in time.

Rigorously speaking, we are given an initial curve, and a velocity field which we assume is normal to the curve at any instant . We would like to determine and parametrise the evolution of the curve, that is . One intuitive idea is the following: let's choose some ordered points on our curve (Fig. A), let's follow their evolution ( will tell us where they are going) and let's complete the curve between any subsequent points and by interpolation. However, it may happen that our curve will split under the action of and Fig. B shows how our method would fail, because we told our algorithm to connect with  and  with .

How could we explain to our algorithms when the curve splits or merges? It's hard, especially since we are searching for a general method. This is where level set methods come to the rescue.

The idea is very intuitive: what if we would add one extra-dimension (time) and "record" the evolution with a surface? For instance, if our curve is a disc expanding, one candidate surface could be a truncated cone. If our disc would evolve in a "8-shape" and then split, one candidate surface would be some sort of 3-dimensional "Y". In other words, we are looking for a function such that:

Here I denote the "inside" region at the time by . At any time, the zero level set of will detect the interface. In addition, its sign will detect the inside and outside regions. Now, observe that from the previous equation:

By applying the chain rule, we get . We assumed that our velocity field was orthogonal to the interface at any instant. In other words,

.

Hence, we can write the following evolution equation:

.

This, in addition to the given initial condition , will define and consequently the interface at any time. Suddenly we are in front of a PDE problem, for which there are many well-developed theoretical and numerical tools. Also, this approach handles perfectly topological changes, such as splitting and merging. Plus, it makes it really easy to compute geometric quantities as the curvature of the interface (simply differentiate ).

This new framework was introduced in 1987 by Stanley Osher and James Sethian. Since then, it has been a thriving topic of research: just know that their seminal paper to date has been cited more than 11 500 times! Level set methods have been applied to an incredible variety of problems and settings: medical imaging, computer vision, image denoising, active contour segmentation, scattering, obstacle detection, and more. It has been widely explored both theoretically and numerically. One its richest areas of application is computer graphics. One of Osher's students, Ron Fedkiw, now full professor at Stanford, won an Academy Scientific and Technical Award in 2008. Fedkiw is a consultant for Industrial Light and Magic, a big name in the special effects industry. He worked on blockbusters as Terminator III, Star Wars Episode III, the Pirates of the Caribbean's saga and some Harry Potter movies. Level set methods are widely used in fluid, fire, hair simulations in animation movies. Think of water, with all his splashes (=topological changes): this framework works very well. One drawback is that this approach does not conserve some physical quantities as the volume. However, there are nowadays many tricks to work around this. For instance, there are hybrid methods that mix level set and volume tracking methods or sometime rendering techniques that fill up for the missing physical properties. You can see many animations at the PhysBAM project page.

If you got curios, I include a selection of references:

Osher – Paragios, “Geometric Level Set Methods in Imaging, Vision and Graphics”, Springer 2003.

Osher – Fedkiw, “Level Set Methods and Dynamic Implicit Surfaces”, Springer 2003.

Links:

http://step.polymtl.ca/~rv101/levelset/ explanations

http://www.ams.org/notices/201005/rtx100500614p.pdf

http://physbam.stanford.edu/~fedkiw/papers/stanford2003-04.pdf

L'articolo Mathematicians Go Hollywood sembra essere il primo su Paola Elefante.

When the first tests on the static case were ran, Kolehmainen, Lassas and Siltanen noticed that the reconstruction was not good, but the level set function itself resembled the infinite precision data. Hence, they decided to use a new cut-off function:

that is the identity function, with a non-negativity constraint. In my own simulations, I approximated the latter by a map:

Numerical results were slightly better and the corresponding objective functional was Frechet differentiable (not only Gateaux differentiable, as before).

Recently Niemi et al. proved that is equivalent to the non-negativity constraint Tikhonov functional. Hence, they generalized it to higher orders. For instance, the functional of order 2 to minimize is:

In this case, existence of a global minimizer was proved.

The first simulation Esa Niemi ran, was on the (2+1)D phantom shown in part 2. The intensity value of the medium is constantly 1, while outside it we have a background constant at value 0. At each time frame, measurements were collected around a full-angle, from only 7 equally distant directions. In the following chosen time frames (“sections” of the 3D surface) you can see the outcome.

The first column depict the infinite precision data, that is the simulated body. In the second column the same sections are reconstructed through Filtered Back Projection, that is the method currently used by industrial machineries. FBP does not work with undersampled data, as you can see. In the third and fourth column you can compare the reconstructions by the level set method I explained, respectively by the order 1 and the order 2 functionals. In the last column, I show how a classical regularization method works in this case, namely Tikhonov regularization. Our new method, with the order 2 functional, works much better, as you can see by the approximation errors shown in each frame.

The second step Esa faced was testing on real data. To reproduce the same measurement setting of the simulation, he created a stop-motion animation. He put some sugar cubes and measured around a full-angle. Then he added one or a couple of sugar cubes and measured again, and so on. The new sugar cubes represented the dynamic change (sudden, in this case) in the data. Sugar cubes are also a good choice because they have corners, which the simulated data was missing. The results can be seen in the following pictures (I selected only three time frames).

The first column shows a fine reconstruction, done by FBP, using many projection angles. From the second column on, only 10 projections were used. Our method is compared with another classical reconstruction method, as Total Variation is. Again, the outcome is very promising: of course in this case you cannot compute an approximation error but you can compare visually with ground truth.

There is still an extensive investigation to carry on. Personally, one of my next goals is to make the codes work in a more realistic measurement setting, namely helicoidal acquisition. I would like to sample the data while the dynamic change happens. To this purpose, I designed the following prototype, inspired by the potential application of angiography.

The top part of the model has the practical purpose of collecting the viscous contrast agent and buy some time for it while we start the measurement procedure. The relevant part of the model are the “veins” that would be (slowly) filled up while we rotate the sample and acquire the data. This will be the next (2+1)D real data I will test on. Currently I am experimenting to find the right contrast agent together with my colleague Alexander Meaney. In the meantime I am experimenting with simulated data with promising results.

This is the current state of my project. Personally, I find it to be a perfect mix of theoretical aspects, computer simulations and great potential for applications. I also hope this will make me get in touch with professionals of other areas. For instance, it would be nice to get suggestions for testing data, or measurement settings. So… feel free to comment and share your view.

L'articolo 4D tomography: walkthrough of my project - part 3 sembra essere il primo su Paola Elefante.

Level set methods

A level set method is an elaborate, yet geometrically intuitive, framework to deal with a dynamic front. Imagine the problem of a 2D object changing in time. For instance, let's say we have a disk that stays still for a while, then expands in a "eigth shape" and then splits into disks that keep moving. After a while, a smaller disk originates from one of the previous two. In a situation like this, we would witness a topological change that is quite hard to parametrize (*). The intuitive idea behind level set methods is to model such situation in 3D, including time as a spatial dimension. The dynamic 2D object will then "build" a continuous surface. You can observe the case I depicted in the following video (**).

https://youtu.be/VtOpVH7pwrI

On the left, you can observe the 2D dynamic object changing in time. On the right, the level set surface is built accordingly.

Level set methods were originally developed in the 1980s by mathematicians Stanley Osher and James Sethian. The motivating application was (still is) computer graphics, where problems like the one I described above are frequent, for instance, in reproducing animation of fluids, where topological changes are routine.

Video from Dongsoo Han Youtube channel. See also this video about Disney animation.

As Osher put it, "when a catastrophe in the movies should look realistic, Hollywood calls for the mathematicians".

Our model

Level set methods were applied to several inverse problems and you can learn more about it from this nice survey (2004). In this case, we model the X-ray attenuation (that is the unknown we want to recover) as , where is a cut-off function we choose (I will explain how in the next post) and is the minimizer of the following Tikhonov-like functional:

(1)

For someone who works in iterative reconstruction algorithms, this looks familiar (***). The main difference is the presence of the function . Here is the regularization parameter, that has the task to balance the two norms. Now, through Gateaux differentiation (§), one can see that solving this minimization problem is equivalent to finding the limit solution of the evolution equation:

(2)

In this sense, this is a level set method, since equation (2) recalls an evolution equation of a level set method. Anyway, I approach the numerical solution of the problem by the formulation (1) and apply gradient descent methods.

In the next post I will explain who is and how we choose it. Also, I will show some published results to present a comparison with well-known methods in the case of undersampled data.

(*) If you had the instinct of running away at "topological change", don't panic. In simpler words, the trouble is at the instant when the disk splits in two. Such geometric change is tricky.

(**) The phantom was created by postgrad student Esa Niemi, the video was assembled by master student Topias Rusanen. Please mention the authors if you embed the video somewhere.

(***) For those who do not, this is a classical regularization problem formulation.

(§) For details, see Niemi et al. and Kolehmainen et al..

L'articolo 4D tomography: walkthrough of my project - part 2 sembra essere il primo su Paola Elefante.

The project

When I started, I basically took up the good work of soon-to-be-doctor Esa Niemi. Esa studied a novel tomography algorithm based on a level set method in the case of a dynamic 2D object. Such approach had been already investigated in the paper by Kolehmainen, Lassas and Siltanen in the static 2D case. My aim is to expand the algorithm to the dynamic 3D cases and to include non-trivial acquisition geometries.

Why dynamic tomography?

The motivation behind this project is strong and our team is definitely not the only one working on these issues. In our case, we are mostly - but not limited to - interested in biomedical applications. One powerful example of potential applications is angiography. In the featured image of this post, you can see a 2D radiography of a hand where a contrast agent has been injected. Angiography represents a fundamental non-invasive diagnostic and treatment tool in medicine.

https://youtu.be/jEfHnwEi2n4

In the video above you can observe a contrast agent injected into some heart's blood vessels, while dynamic CT allows to monitor what happens. Coronary angiography can be useful to detect obstructions or ruptures. During the treatment procedure known as angioplasty, it is fundamental for the physician to monitor the evolution of the operation. To date, coronary angiography is available only in the dynamic 2D case, meaning that it is possible to observe only a section of the heart. It would be extremely useful for a doctor to have a sense of the missing spatial dimension.

Another interesting biomedical application of dynamic CT is radiation therapy. During radiation therapy, cancerous cells are hit by ionizing radiation. If a tumour is placed along moving organs (i.e. lungs, etc.), the radiation flow would miss it for a portion of time and irradiate healthy tissue. As I mentioned in a previous post, radiation can contribute to cancer, so you want to tune the radiation dose down.

Dynamic tomography could allow to synchronise a radiation therapy machinery with the real movement of the tumour, thus reducing useless and potentially damaging radiation.

Then we come to the other attribute: sparse. Sparse measurement is synonym of undersampling, meaning that one tries to get the best he can with few data. Few measured data means lower X-ray dose in tomography. To date, industrial machineries mostly reconstruct measured data through the Filtered Back Projection algorithm (FBP). FBP usually guarantees good image quality but asks for a lot of sampled data (*). Iterative methods - that is what we use and research - reconstruct images with less quality (anyway good enough) but with definitely fewer data (even one tenth!). This idea motivates our testing of a novel algorithm, in the hope of massively reduce a patient irradiation.

If the radiation is minimised, CT can be safely prescribed as a prevention examination to monitor some cases. Also, this would mean less sensors and detectors (= less money) and less time (if we succeed to beat FBP computationally speaking).

Here is my/our motivation so far. Next I'll explain what level set method and how we apply it in the dynamic tomography case. To next time!

(*) I here promise I'll take the time to develop in a post what FBP is and show some comparisons with other reconstruction methods, with fewer projections.

Featured image comes from Wikipedia.

L'articolo 4D tomography: walkthrough of my project - part 1 sembra essere il primo su Paola Elefante.

Sofia's family

Sofia was born under the family name of Korvin-Krukovskaya in Moscow in 1850. Her family background surely had a strong influence on the development of her personality and aspirations. Her mother was a woman from aristocratic family, 20 years younger than her husband. He used to treat her as a child, being probably rights, since she did not care much for her daughters. She was an aspiring actress who could never realise her dream. This made her a weak figure in Sofia’s early life and maybe also her failure in life was a warning. On the other hand, her father was an educated men, with high military rank, and in early childhood Sofia grew naturally close to him. Sofia was the second born, when her parents were hoping for a son. Her older sister Anna was the incarnation of the ideal girl, while Sofia had a more intense and rebel personality, darker hair and carnation. This created a natural bonding between Anna and her mother, and as a consequence Sofia grew closer to her father.

Despite such strong differences, her sister Anna, “Aniuta”, was never an enemy. When she grew older she became more independent and rebel, and opposed with strength their father. She became an important role model to Sofia. Anna was lively and active in the local cultural literary scene and quickly abandoned all frivolousness she learned from her mother: she was so strong and fascinating, that she even got a marriage proposal from Dostoevsky (which she rejected)! Sofia was looking up to Anna, but wanted to find her own way and perhaps this was one of the reasons that led her to science and mathematics. Also, two of her uncles were amateur scientists and she could listen to their conversations when they were visiting. In her memoirs, Sofia tells about one unused room in the family estate of Palibino (in the current Belarus) that had no wallpaper, and was temporarily covered with some lectures notes in calculus, which she spent hours looking at, when very young.

When teenagers, Sofia and Anna were craving for freedom. As women, they were not allowed to travel to Europe to study without permission of their father or their husband. At the time, it was frequent for girls to persuade boys into fictitious marriages to gain freedom. Therefore around 1868 (Sofia being about 18), she therefore proposed to Vladimir Kovalevsky, a scientific writer. Her father opposed to the marriage, but Sofia found a trick: she sent around some letters to his friends declining some regular invitations, telling her fiancé and her were too busy preparing the wedding and could not attend. Her father was faced with the choice of admitting she was rebelling to his authority in front of his close friends, or playing along, and chose the latter.

Studying in Europe

The fake marriage was the plan for the two sisters to freely travel to Europe. Anna told her parents the couple would have chaperoned her but secretly traveled to Paris alone. Sofia and her husband started living in Heidelberg (Germany). Here, after a long struggle with the administration, she was allowed to follow university courses when the teachers would grant permission. She studied under Bunsen, Kirchhoff, Helmholtz, P. du Bois-Reymond (integral equations and Fourier series) and Koenigsberger (elliptic functions, differential equations).

The newlyweds soon started having serious money problems, that would lead Vladimir to commit suicide eventually. In 1870 Sofia travelled to Berlin, hoping that Weierstrass, one of the best mathematicians at the time, would grant her admission to the university. Because of university rules, he could not admit her to his lectures, but he was so impressed by her talent that despite having little time, he started giving her private lessons.

In 1871, Sofia lived a life-changing experience. At the time France was losing in the war against Prussia declared by Napoleon. People in Paris got agitated and tried to establish an independent government of the capital (La Commune). Anna and her lover were fighting as supporters of La Commune and Paris was surrounded by the Prussian army. In the spring of that year, Vladimir and Sofia rushed there, masking it as a study trip of Vladimir, to save them. Anna was captured but managed to escape, while her lover was saved by her parents who travelled there and intervened by using their good name and bribes; in exchange they required Anna and her lover to legally marry (in front of them!). All the family was miraculously safe at the end.

When Sofia came back, a turning point in the relationship with Weierstrass happened. She revealed him the secret behind her marriage. Weierstrass grew humanly closer to Sofia’s struggling, how she was prevented from attending courses and graduating, despite of her talent. From that moment, he became Sofia's first champion.

During the time span of 18 months (1873-1874), Sofia wrote three dissertations, two of which were considered excellent by Weierstrass. He recommended her work to the more liberal University of Goettingen (Germany) and she was finally awarded the doctoral degree, summa cum laude. She was the first woman in modern Europe to receive such a title in mathematics.

The Cauchy-Kovalevskaya theorem and the majorant method

One of her dissertations contained important material, not only for the final result, but also for the method used to obtain it. Let us give a simple version of the Cauchy-Kovalevskaya theorem:

Let be a function analytic near . Then there exist a neighbourhood of in and a unique solution to the initial value problem:

(1)

that is analytic in .

There are more general versions of this theorem where you have higher order derivatives, several analytic given functions instead of the only and several variables. The result is valid also for complex analyticity.

I will here give a simple explanation of the majorant method that Kovalevskaya used to prove it, following the example of Cauchy before her. Given the system, by using the chain rule of differentiation:

...

In general, the -derivative of can be written as a polynomial with non-negative integer coefficients depending on . If you imagine the following computations, there is no way a minus sign can show up. In symbols, we write that:

When evaluating those derivatives at , we can eliminate the dependence on the previous derivatives by noticing that:

( depends only on ) ...

If you iterate and eliminate the dependence on step by step, you can conclude that:

(*)

This shows that if such analytic solution exists, it must be unique, because its Maclaurin coefficients depend only on the given function . Now, let's recall we assumed that was analytic near . This means that its Taylor series near converges:

for near

We can surely find some numbers such that:

The term is the -th Mclaurin coefficient of the analytic function .

Consider the twin initial value problem:

(2)

Such problem has the following explicit solution (it's an exercise to verify): , that converges in an open neighbourhood of .

By previous calculations, that are valid also now, we already know that . Since the polynomials have non-negative coefficients, we can majorize as follows:

(3)

We now stand at this point. We know that (1) has an analytic solution that we can build it uniquely by (*). The convergence in an open disc near is now guaranteed by formula (3) and the local analyticity of . There is one thing left to prove, that is being a solution to (1).

Since are analytic, their composition is and has the following expansion:

(4)

but by construction, . If you write the Taylor series of (exercise) you can conclude it coincides with (4) and this concludes the proof.

Back to Russia and afterwards

After graduating, Sofia was exhausted and the family moved back to Russia, hoping to find jobs in the academia (Vladimir graduated as well during their stay in Europe). In 1875, after 7 years of platonic relationship, her marriage turned into a real one. In 1878 their first and only daughter Sofia “Fufa” was born. Shortly afterwards, Sofia’s mother died and Sofia and her husband lost an important investment that was supposed to grant them security, while finding a job was so hard. Despite their titles, they were fighting against unemployment. Sofia was discriminated as a woman and told she could teach to elementary schoolgirls at most. She was also struggling with her new family responsibilities and with the feeling of the waste of effort to gain a high education. Anyway, following Weierstrass' advice, she managed to send a paper to the 6^th Congress of Mathematicians and Physicians. This was a lucky move, because one of the readers was the Swedish mathematician Mittag-Leffler. Mittag-Leffler at the time was a professor at University of Helsinki (1877-1881). He became Sofia’s second champion and tried very hard to get a position for her in Helsinki. Ironically, she did not have a chance, not because she was a woman, but because she was Russian. At the time, Finland was struggling to affirm her own cultural independence after centuries of domination by Sweden and Russia. In 1881 Mittag-Leffler was appointed professor at the University of Stockholm, a more modern university than Uppsala and Lund, and some hope for Sofia to get a position got back.

In the meantime, Sofia’s marriage was facing a deep crisis: Vladimir was depressed for the previous financial loss and was acting irrationally, even wasting job opportunities. Sofia started planning life for herself and her child, and moved to Berlin. In 1883 Vladimir commited suicide and Sofia reacted by letting herself starve to coma. Luckily her friends took good care of her and despite her family tragedy, shortly afterwards that she finally was appointed in Stockholm, even thought not with a paid nor permanent position. In 1884 she gave her first lecture. She was not paid by university but from her students, but since she was a very good lecturer, this was not a problem. In 1886 Sofia went to Russia to assist her severely sick Anna, who died at the end of 1887.

One year later, despite struggling with her personal life tragic events, she won a prestigious prize for her work on the rotation dynamics of a top with a fixed point.

Before her result, only the case where the center of gravity and the fixed point were the same, and the case were such 2 points were on the symmetry axis were known. The jury even doubled the prize because of originality and important of her results. This achievement granted her to become full professor in Stockholm in 1889. Unfortunately she could not enjoy the position for long: in 1891, after returning from a brief holiday, she ended up under a pouring rain. This resulted in a flu that degenerated to pneumonia and led her quickly to death.

After Sofia

Sofia left us a great inheritance, not only in terms of mathematical results, but also being a pioneer of women in science and mathematics. Even though much progress in terms of gender equality has occurred, we are still far from good.

A report of the Department of Mathematics and Statistics of University of Helsinki (2008), shows the horrible situation of the local gender gap with a single graph:

To date, we still have no women math professors here. As far as I know, there are only two female full-professors in mathematics in all Finland. Now, this is not a question of aiming to a perfect 50-50 situation, neither to push too hard women to take part, but I think this figures are worrying and should at least make us wonder.

Sofia and many women before her (Maria Agnesi, Mary Sommerfield, Hypatia, ...) managed to exploit their talent, pursue their dreams, and succeed because they had a male champion. Mentoring still stands as one of the acclaimed possible solutions to encourage women to follow their aspirations in science. With the ultimate goal being getting free from such constraint and having women completely free from social pressure and able to pursue a scientific career, I anyway encourage all my colleagues, women and men, to dedicate some of their time and effort to mentor girls, female students, and female colleagues, to share the privilege and build all together a more fair and even more productive work environment.

Bibliography

http://www.epigenesys.eu/en/science-and-you/women-in-science/739-sofia-kovalevskaya

http://www.mathematics-in-europe.eu/ru/2013-03-19-21-49-35/76-enjoy-maths/strick/774-sonya-kovalevskaja-january-15-1850-february-10-1891-by-heinz-kaus-strick-germany

http://en.wikipedia.org/wiki/Cauchy–Kovalevskaya_theorem

http://www.encyclopediaofmath.org/index.php/Cauchy–Kovalevskaya_theorem

http://en.wikipedia.org/wiki/Kovalevskaya_Top

http://scienceworld.wolfram.com/physics/KovalevskayaTop.html

Cooke, The Mathematics of Sonya Kovalevskaya, Springer-Verlag, 1984.

Sofya Kovalevskaya, A Russian Childhood, Springer-Verlag, 1978 edition.

Gantumur, The Cauchy-Kovalevskaya theorem, Math 580 lecture notes.

L'articolo Sofia Kovalevskaya: the girl who wanted something else sembra essere il primo su Paola Elefante.

www.fips.fi/dataset.php

The measurement data was collected and documented by K. Hämäläinen, L. Harhanen, A. Kallonen, A. Kujanpää, E. Niemi and S. Siltanen. Our group used the walnut data for testing several tomographic reconstruction algorithms, as you can see in the homepage. You are free to use the dataset for scientific purposes, but please take care to cite the authors.

Have fun reconstructing!

L'articolo The "Helsinki walnut" dataset sembra essere il primo su Paola Elefante.

In this post, I will describe how the human voice is produced and why this is useful to the approach of the Inverse Problems Group at University of Helsinki.

Our voice is produced by air flowing from our lungs through our vocal folds and vocal tract. Vocal folds - commonly known as vocal cords - are a part of the glottis and flap to produce a sound from the flowing air. They move extremely fast (hundreds of times per second) and "break" the air flow creating sound (*). The dimension of vocal folds characterises the pitch: women's vocal folds are usually smaller than men's. The pitch is related to the number of vibrations per second, as you can see in the following chart:

Here you can watch a stroboscopic video of vocal folds moving (the slow motion is an effect of the stroboscopic light):

Another characterisation of the voice pitch comes from the vocal tract, resulting to be generally longer in men. The vocal tract is quickly deformed while we talk, as you can notice in this real-time MRI video:

The vocal tract acts as a resonator and its shape changes the "buzzing sound" produced by the vocal folds into a voice sound as we hear it. For studying purposes, our group printed some approximation models of the vocal tract corresponding to five vowels:

In the next post I will (finally!) unveil how inverse problems meet human speech. Wait for it!

(*) To know more about this topic, check out "Principles of voice production" by I. R. Titze.

L'articolo Creating a voice: a challenging inverse problem (part 2) sembra essere il primo su Paola Elefante.

Maybe you have never heard of him, but Majorana was a gifted Italian physicist who had produced great theoretical results in particle physics and quantum mechanics. The discovery of neutrinos was bestowed on him and he speculated on the existence of Majorana fermions (evidence found in 2012), just to give you a hint of his merits. He studied Physics in Rome together with Segrè (Nobel prize in Physics in 1959) and Enrico Volterra.

He showed to be a bright science scholar already at the age of 5. During his university studies, he met Enrico Fermi, who showed him some of his current particle physics research, including a novel table summing up some particle potentials (the Fermi potential) he calculated. The day afterwards, Majorana went back to Fermi, asking him to show that table again, since the previous day he could only spot it for few seconds. He took a piece of paper from his pocket, with the same numbers on it: in the previous 24 hours he had calculated the same numbers and wanted to check that Fermi's table was correct.

He graduated at 23, under supervision of Fermi (of course!). Apparently he was no piece of cake: he was so surly and stiff that his colleagues named him "the great inquisitor". At some point he started working in isolation in his flat, warding off visitors and letters by writing on the envelope "Rejected for death of the receiver". After refusing professorship positions from Cambridge and Yale, he became full professor at University of Naples. In 1938 he suddenly disappeared after withdrawing a significant amount of money and sending some mysterious notes to his family and his closest friends. The money withdrawal is a clue fighting against the suicide hypothesis, together with some testimonies of his presence in Naples after the day of the disappearance.

Some say he had a mystical crisis and retired in a monastery for the rest of his life. The famous Italian writer Leonardo Sciascia wrote a book (*) about his disappearance. The most popular theory is that he was scared from his scientific discoveries and their possible social consequences. Let's not forget he lived during Mussolini's time and the dictator was very attentive towards discoveries that could help his war. The latest investigation of Rome's district place him in Venezuela during the 1950s. He took the name of Bini and lived there as an Italian immigrant. The proofs are a picture of Mr. Bini that has perfect correspondence with Majorana's facial traits, a testimony by another Italian immigrant who met him and a postcard from Quirino Majorana, Ettore's uncle and brilliant physicist himself, addressed to a certain W. G. Conklin and found in Bini's car. Unfortunately there are no other traces of Majorana after 1959 and a big part of the mystery remains. Why has he left? Why not coming back after the II World War?

If this story made you curious, you can read more in the book referenced below.

(*) I could not find an English edition, but if you can read Italian you can search for Sciascia's book. An English language biography can be found here.

L'articolo Ettore Majorana: is the mystery solved? sembra essere il primo su Paola Elefante.