During the week I shared many images, slides, topics. I decided to collect them all, so that they do not get lost in the feed and may turn up useful for myself or someone else in the future. I will not write complete explanations, it will be more of a list of resources. Feel free to use whatever, but please be polite and quote the source (I will indicate the author or the original website for everything).

Mathematics and its applications

X-ray tomography

X-ray tomography is my area of research (see here for details about my current project). Here some resources:

• Wikipedia page on X-ray tomography
• X-ray tomography used in archeology: mummies and lost letters.
• Wikipedia pages on J. Radon and Radon transform.
• VT-cube, a device that reduces X-ray irradiation in dental imaging.
• Denoysing 4D cardiac tomography images.
• Lecture notes on tomography.
• How tomographic data is collected.
• Reference: Smith et al. 1977. "A finite set of radiographs tells nothing at all".
• Compare FBP with other reconstruction algorithms for sparse data.
• An application of dynamic CT: coronary angiography. About 10 million tests carried every year in US.
• Another research group working in dynamic CT.
• Public CT real dataset by University of Helsinki.

Other Inverse Problems

Wikipedia page on Inverse Problems.

Image enhancement and inpainting.

• Inpainting Wikipedia page.
• Image restoration: deblurring & co.
• Our group will soon start working in image enhancement.

Invisibility cloak. How Harry Potter meets mathematics: link 1, link 2.

Creating synthetic voice. I talked about this problem here. A video where my advisor explains the project. His slides: I love this talk, every time it has great success. Speech problems affect many people, 6 to 8 millions in US only.

Seismic tomography. Seismic tomography exploits seismic wave and advanced topology to infere about the inner structure of our planet.

• Wikipedia page on seismic tomography.
• Our research in seismic tomography.

Electrical Impedance Tomography. EIT is a widely used medical imaging technique, employed especially in breathing monitoring.

• Wikipedia page on EIT.
• Our group's research on EIT.

Gravitational Lensing. Wikipedia page: measuring space bodies' properties from their interaction with remote light.

Forest monitoring. The Tampere Inverse Problems group uses inverse problems techniques to monitor and model forests, contributing to fire prevention and paper industry estimations: video.

Other applications

(see also Computer Science section)

Cryptography. Internet transactions require sensible information to be properly masked. For instance, when you buy online with your credit card, data transmission is protected by cryptography techniques (ex. RSA). Here some links:

• RSA Wikipedia page.
• Prime numbers are fundamental for RSA.
• The standard protocol SSL uses RSA.

Big Data. Wikipedia page on Big Data.

Rollercoaster engineering. Rollercoaster loops are designed according to a mathematical curve called Euler spiral, to maximise the fun and minimise nausea. Euler spiral can be found also in highways, to make turning at high speed safer.

Biomathematics. A fun example of biomathematics modelling can be found in this zombie scenario paper. Biomathematics had a big role in containing the Ebola infection.

Finance. Likely the person dealing with your investments is a mathematician. Here a Wikipedia page.

Computer Science

• Mathematics in Google algorithms: article on The Guardian.
• Mathematics in animation movies.
• P=NP problem: link 1, a simple explanation, an article on New Yorker,

Women in Mathematics

• The Fields Medal was won by a woman for the first time in 2014.
• Agora, a movie about a female mathematician of antiquity, Hypatia.
• University of Nottingham playlist on women in mathematics.
• How to propose to a mathematician.
• Sofia Kovalevskaya. My post on her, Google's dedicate Doodle, a movie on her life.
• Book on Minorities in Mathematics.
• TV series: The Bletchey Circle.
• Sophie Germain, who studied with Lagrange.
• How gender stereotypes affect mathematics skills at early age.
• [not only about women] Childcare at conferences: a post on Tenure She Wrote, my vision, Geoscience conference that provides it, some family care grants: LMS - UK, Franklin Women, SMBE, AustMS.
• Representation of women in mathematics at conference: a practical guide.
• Some links on women in STEM.
• Emmy Noether, allowed to work (for free or under a male colleague's name!) in a time when women were not allowed in academic life. A video about her by Katy Mack.

Algebra and Number Theory

Goldbach's conjecture (Wikipedia page). Goldbach's conjecture is an open problem in number theory. It states that every even number greater than 2 is sum of two prime numbers. For instance: 8=5+3, 18=11+7, 22=11+11. This has been checked for huge numbers (), but to date there is no proof that it's true for all even numbers. Some links:

• Page with updates on the verification of the conjecture.
• Post on Terence Tao's blog on his progress.

Collatz's conjecture (Wikipedia page). The conjecture states: take any natural number n. If n is even, compute n/2. If n is odd, compute 3n+1. Iterate the procedure. You'll alway end up at 1.

Fermat's Last Theorem (Wikipedia page). Stated by Fermat in 1635 and proven only in 1995 by Andrew Wiles, it claims that, given any n greater than 2, there is no integer solution to the equation . For instance given a cube, you cannot split it in the sum of two other cubes. Fermat wrote on the margin of a book "I have discovered a truly remarkable proof of this theorem which this margin is too small to contain.". Here Wiles recalls the moment when he understood his proof was complete (after 7 years of work). If you find the topic interesting, don't miss this great popular book by Simon Singh.

Induction. A technique to prove statements for natural numbers is induction. It works this way: (1) I want to prove statement A is true for all natural numbers, (2) I check A's true for n=1, (3) I assume A's true for a generic n and based on that, I prove A's true for n+1.

Proof of infinity of prime numbers. Euclid proved that prime numbers are infinitely many by a reduction ad absurdum proof. One assumes something, through logic gets to something absurd and concludes the initial assumption must have been false in the first place.

Perfect numbers. A perfect number is an integer that is equal to the sum of its divisors (except the number itself). An example is 6, whose divisors are 1,2,3,6, and 1+2+3=6. A cool property is that the sum of the reciprocals of perfect numbers' divisors is always equal to 2: . So far only even perfect numbers have been found and the question if odd perfect numbers exist is open.

Weird numbers. Some weird names for numbers: sexy primes, friendly numbers, amicable numbers, sociable numbers, twin primes.

Pi. Pi is one the most celebrated numbers ever, even having its own anniversary. Even Star Trek talks about it and Kate Bush wrote a song! Pi is irrational, meaning it cannot be expressed as a fraction. An equivalent formulation is that its infinitely many digits have no pattern, so in principle you can easily find any string of numbers in there. Another property of Pi is that it cannot be expressed as zero of an polynomial with rational coefficients ( is irrational but a zero of instead), making it trascendental. The most decimal places of Pi memorised is 70,000, and was achieved by Rajveer Meena in 2015.

Geometry and Topology

Art and mathematics.

[add Mobius scarf and bacon: downloaded. Add Bathsheba Grossman.]

I mentioned two weird geometry constructions (manifolds) in particular: Möbius' strip and Klein's bottle. They are non-orientable. Möbius's strip has not "over" or "below", Klein's bottle has no "inside" or "outside.

Non-euclidean geometries. Eucliden geometry, the "classical one", is based on some axioms. At some point some mathematicians wondered if they were all necessary and experimented with creating new geometry settings.

Here the Wikipedia page on the topic. Also, non-euclidean geometries relate to Einstein's General Relativity Theory.

The Seven Bridged of Königsberg. This problem was solved by Euler in 1736 and led to the birth of graph theory. Graph theory is used when solving or creating mazes. Another related famous problems is the Four Colour Theorem: given any map, you can use only four colours to fill all countries so that any two adjacent countries have different color. The latter was proved by Kenneth Appel and Wolfgang Haken in 1976. They reduced to 1'936 possible cases and then checked them all with a computer. It was the first time computing was used in a formal proof.

The happy ending problem. Below I state the problem visually. George Szekeres went even further and proved you can always draw the polygon you want, if you draw enough initial points (for a convex pentagon: you need 9 points). There is a conjecture stating you need points to be sure to find a convex polygon with sides. The question was named this way because it led to the long-lasting love of Esther and George Szekeres. They were married 70 years and died within one hour from each other. Mathematics can make you fall in love sometime...

Homeomorphisms. One of the funniest tools in topology. Here a video showing why donuts are mugs after all. Below another weird transformation: from a square to a (almost) torus.

Manifolds. Another powerful maps are diffeomorphisms, a level on top of homeomorphisms. Manifolds are geometrical objects that are locally similar to for some n. For instance a sphere is a manifold of dimension 2, small regions are comparable to planar regions. If you consider a sphere, you imagine it embedded in tridimensional space, but truth is you don't know the third dimension. All you know is your 2-dimensional "curved" life on the sphere. This is why you need new tools to define differentiation, that is usually linear and "flat". Manifolds find many real-life applications: for instance seismic tomography (see above) or robotics.

Analysis and Calculus

Harmonic series.  Their name comes from music. The basic question is: can we get infinity by summing up infinitely many infinitesimal quantities? Turns out it depends, as you can see from the figures. Relating to harmonic series is the problem of book stacking.

Integration. The theory of integration started several centuries ago, when people questioned how to calculate "difficult areas", but it was rigorously stated by Riemann in the XIX century. Below you find my explanation of his theory and a visual solution about integrability of monotone functions.

Pompeiu problem. Formulated in the past century, it's still open. Here's my favourite reference on the topic. See the problem below.

Banach-Tarski paradox. With a very formal proof, Banach and Tarski showed it is in principle possible to cut a sphere in pieces and recombine the pieces to get two spheres identical to the first. The trick is to decompose the sphere in non-measurable sets, that is something really artificial and odd.

History of Mathematics

Math duels ("disfide"). Around 1500, mathematicians used to challenge each other to math duels, to prove who was smarter and more able to solve problems. A famous series of such episodes concerned the formula to solve cubic equations, that is equations of the form . It all started with Tartaglia, an Italian mathematician nicknamed like that for his stammering. Born in a poor family, he had been a soldier, a topographer, and was a talented self-taught mathematician. At the time, mathematicians rarely published their results and only shared them with their students and maybe family, so that they would have an ace in their hole when it came to being hired by some university. At some point, a mathematician contemporary to Tartaglia, Antonio Maria del Fiore, started bragging about knowing the formula to solve cubic equations, which he learned from his teacher Scipione Del Ferro. Tartaglia was an ambitious researcher and found the formula independently. He then accepted a "math duel" from del Fiore. Tartaglia annihilated del Fiore and became immediately famous. Cardano, famous mathematician of the time, invited him to Milan. Tartaglia told him the "secret formula", under the oath he would not reveal it. However, he still hesitated to publish it. Years later, Cardano found out that Del Ferro - then long gone - had discovered the formula previously and independently, thus he felt relieved from his promise to Tartaglia. He then allowed his student Ferrari to publish their research and improvements on Tartaglia's formula. The result was a duel, in which Tartaglia lost due to his stammering and lack of confidence in public speaking. Luckily history gave him credit and the solving formula is named also after him.

Evariste Galois. One of the most brilliant mathematicians ever existed, he died very young in a duel (a real one!). Here his biography. His results were crucial to Wiles to prove Fermat Last Theorem.

The stolen theorem. One of the most popular calculus theorems is believed to have been commissioned by De L'Hopital to Johann Bernoulli.

Riemann's hypothesis. A great popular book to know more about prime numbers and a legendary problem in analytical number theory is The Music of Primes by Marcus Du Sautoy.

The Newton-Leibniz controversy. The two illustrious contemporary scientists fought to be recognised as inventor of calculus. Historians now believe they both were right and invented calculus independently, but at the time Newton's influence allowed him to win the argument and Leibniz died in disgrace. This rivalry was quoted in the popular show The Big Bang Theory.

Paul Erdos. Some say he was the greatest mathematician of the past century. He collaborated with more than 500 people, writing more than 1,525 papers, in many different areas and topics! Erdos belonged to an Hungarian Jewish family that lived during Nazism. His father died in the Holocaust and his mother survived in hiding. He didn't have a home: he kept travelling around the world, stopping at conferences or hosted by colleagues. He would knock in the middle of the night at the door of some colleague and say he was ready to solve some problem. No one would dare to turn him down and throw the chance away, 'cause his productivity and genius were out of ordinary. He received 15 honorary doctorates.

Alexander Grothendieck. I wrote about him when he passed last year. Someone translated some of his lectures in English. You can read more on his life here.

Riddles

The hanging picture problem. Hang a picture to the wall using two nails and a string. Fix the string in such a way that if one (any) of the two nails is removed, the picture falls down.

Solution: name by variables the following actions: means "to pass the string over nail 1 clockwise",   means "to pass the string over nail 1 counterclockwise". Similarly for and nail 2. We can combine actions through a sort of multiplications. Passing the string over nail 1 in one sense and then the other would be , so we understand that corresponds to "nothing is done", meaning "the picture falls down". Also, observe (physically) that the actions are not commutative: try and check that it is not equivalent to do only . Also, removing for instance nail 1 corresponds to ignore all actions as and . Hence our problem translates to: write a product so that by removing all occurrences of $x, x^{-1}$ or you get 1. Since our product is not commutative, one solution will be: . Through abstraction you can easily generalise this trick to any number of nails!

A reference link on the topology behind this.

The prisoners. Three prisoners meet a guard in a room. The guard says: "I have hats with me, two of which are black and the others are white". The prisoners ask: "How many hats there are all together?", the guard replies: "It's a secret!". The guard picks three of the hats and puts them on the prisoners' heads. All prisoners see the others' hats color, but not their own. The guard asks them, one after another: "What is the color of your hat?". The first prisoner looks around and replies: "I don't know". The second prisoner looks around and replies: "I don't know". What does the third prisoner reply and what is his hat's color?

Solution: denote B=black and W=white. Because of the replies of prisoner 1 and 2, we can rule out the combinations: W B B and B W B. We are left with the following cases: W W B, B W W, W W W, B B W, W B W. All cases except one show that the third prisoner's hat must be W. Consider the case W W B. The second prisoner hears the first's reply, so he understand he cannot see two black hats, leaving only the possibilities of him seeing two white hats or one black and one white. When his turn comes, he sees the third wearing black, so he concludes his own must be white and replies so. Since he says he doesn't know, we must rule this case out and conclude the third prisoner has white hat and says so.

Palindromes. Find the smallest 3-digit palindrome number that is divisible by 18.

Solution: our number, denote by ABA, must be divisible by 2 and 9. A number is divisible by 9 if the sum of its digits is divisible by 9, hence A+B+A=2A+B is divisible by 9. ABA must be also even, that leaves us the following possibilities:

A=2 -> 2A+B=4+B -> B must be 5

A=4 -> B=1

A=6 -> B=6

A=8 -> B=2

The smallest is 252.

The Monty Hall problem. This is a counter-intuitive problem of probability. Even Paul Erdos, considered the best mathematician of past century, did not believe the solution until he saw computer simulations! The problem asks: you are a guest of a TV programme. The presenter shows you three closed doors: behind one there's the car of your dreams, behind the other two there are goats. He lets you pick a door (say, n.1), but before opening it, he opens one of the other two (say, n.3) and shows a goat behind it. He then asks you: "Do you want to stick with your choice or change to door n.2?". What's the best strategy? The solution says the best strategy is changing your initial choice. Here some material to understand this: link 1, link 2, link 3.

True statements. How many statements are true?

At most 0 statements in this block are true.

At most 1 statement in this block is true.

At most 2 statements in this block are true.

At most 3 statements in this block are true.

Solution: only the last two statements are true. If you assume no statement is true, then the first is true, which is a contradiction. If you assume only one is true, the last three are true, again a contradiction.

The 12 ball problem. You have 12 balls, looking exactly the same, but one is an odd weight (you do not know if it's lighter or heavier). You have a scale (see figure below) and at most three chances to use it. How do you find the odd ball? Solution is in the gallery below.

Sticks. Use 6 identical sticks to build 4 identical triangles.

Solution: click here.

Miscellanea

• How to draw a perfect ellipse.
• The golden ratio to be found in architecture and nature.
• Visualising Pythagoras' theorem.
• Communication of science: SoapBoxScience.
• The Mathematics of Sudoku. 17 is the minimum number of clues to have a unique solution. The "most difficult scheme" was created by a Finn.
• Magic square. In antiquity, they were believed to hold mystical power. Euler's work on magic squares (translated to English).
• Bayesian approach to Inverse Problems: link 1, link 2.
• Gödel incompleteness theorems. Proved by a young Gödel in 1931, they threw mathematicians in panic (still do). A funny look into the topic.
• Mathematical models for traffic jams.
• Free version of the book "A Mathematician's Apology" by the great number theorist G. H. Hardy.

L'articolo Tweeting for Real Scientists: aftermath. 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.

Today I gave a talk to an audience of master students about the state of my research on dynamic X-ray tomography (find the talk here) and I found many of them puzzled by the fact we often do not test our algorithms with real medical data. One even asked me why I would not contact a doctor and ask for a medical scan to use for my tests. I tried to explain that research is not that simple.

Creating such contact would require that:

• I find someone who is interested to engage in a complete new, challenging and long project.
• That someone is easy to communicate with, even if we speak different technical languages.
• I would have complete freedom (not to mention experience!) to work without thinking of the number or frequency of publications.
• Finally, I should have free access to anonymised medical data.

Also, students do not have a clear idea of how long research takes. I do not blame them, it was a shock to me as well. I used to think that you magically get an idea, you contact experts and - BOOYAH - here is something new in the name of science and progress. Half an year, at most. It blew my mind away when I was explained that even just starting a collaboration in industrial mathematics can take a year, during which people communicate and explain to each other a problem from their own perspective. I have the feeling that I was not the only one thinking in such naive way, since almost all Ph. D. students I know have spent their first year panicking for their lack of results and understanding of what was going on, feeling like impostors.

I feel lucky because I have two advisors who listen to what I have to say but are not afraid to tell me "No, I don't think this is the right way to go for you"; in this sense, I have a good degree of freedom, for someone in my position of small expertise. On the other hand, truth is that whenever you apply for an academic position you get evaluated on the number and quality of your publications, not to mention that you are supposed to finish your Ph. D. studies in about 4 years, so you have to be humble with your expectations. I am not arguing against anything here, just explaining the facts.

Nevertheless, I cannot help dreaming of starting my own projects, winning Nobel Prizes and maybe conquering the world ;P... But I guess you have to never stop dreaming and believing if you work in research.

L'articolo Research idealism VS real world sembra essere il primo su Paola Elefante.

Stem cells and cancer

Stem cells are defined through three properties: (1) they can regenerate and repair themselves, (2) they are unspecialised but (3) can generate specialised cells. Even though stem cells are relatively few in all tissues (**), they play the very important role of "repairing" organs. During such process of division, there is a chance something goes wrong and a defective cell is generated.

Tomasetti - Vogelstein's study

The authors prove for many organs that there is a strong correlation (Spearman's rho 0.81) between number of stem cells divisions and lifetime risk. To underline the result, they show the following example:

Consequences

If further investigation confirmed this study, early diagnosis of cancer would result fundamental to save people's lives. As I discussed previously in the case of breast cancer, X-rays used in CT scans are considered one of the environmental factors causing cancer. Early diagnosis may translate into relatively frequent scans, hence low radiation dose becomes a central issue. The research group I participate in, develops algorithms to reproduce nice imaging outcomes with few measurement data (that is, lower radiation doses).

I really hope to read soon updates on this exciting research!

(*) To check other sources about the study: Time, The Guardian,

(**) Tomasetti and Vogelstein collected such datas in this appendix.

Featured image is courtesy of The Telegraph.

L'articolo What if getting cancer is mostly about bad luck? sembra essere il primo su Paola Elefante.

The battle we didn't choose: my wife's battle with breast cancer (*)

Angelo Merendino is a photographer based in Cleveland. Angelo and his wife Jennifer's story is moving: love at first sight (at least from Angelo's side :)), soon culminated to a wedding in Central Park. After only 5 months, Jennifer finds out she has breast cancer. While they fought side by side, Angelo collected everyday amazing pictures of their daily difficult battle. Usually I do not get so impressed by pictures, but in this case I could really feel a hint of the everyday obstacles they had to face and I was deeply moved.

X-ray CT scans are a valuable alternative to mammography, as diagnostic methods to detect breast cancer and cancer in general. Problem is, several studies (**) highlighted a correlation between CT scans and risk of getting cancer. Irradiating a patient with X-rays increases their risk of getting cancer by a small percentage. At the end of the day, such small percentage is a someone out there.

This is why we focus our research on sparse tomography. Sparse indicates a situation where we have constraints and can get only few measurements. We could be able to get just few projections (for instance to limit the irradiation) or just from a limited angle instead of all around the person (or, in general, the object of study). The classical approach of FBP (filtered back projection) works poorly when you have a limited amount of data. We have developed and are still developing several alternative reconstruction algorithm to improve this. The algorithm I am currently working is an implementation of a modified version of the level set method. As an example of what such reconstruction technique can do with respect to FBP, see the following pictures (***):

You can clearly see how much of an improvement this is. 10 is a very low number of projections compared to "usual", therefore small irradiation. During my recents work travelling, I met several medical physicists and found out there is still a long way to go to share such new techniques outside the mathematical world, even thought such research has been going on for about a decade.

My hope for the near future is to be able to develop new mathematical reconstruction techniques and to be able to spread them among the medical imaging community, in the hope of experimenting with real life situations. Stories like the one of Angelo and Jennifer are the fuel that keeps me going and my inner motivation. Thank you Angelo for having the courage and strength to share your story, as you can see it has a wide reach.

(*) The featured image of this post is part of Angelo Merendino's collection. Please check out also his foundation The Love You Share, dedicated to assisting breast cancer patients and their families.

(**) See for instance this study by American Medical Association (2009) and  this paper on JAMA (2007).

(***) You can learn more by reading Limited data X-Ray tomography using nonlinear evolution equations by V. Kolehmainen, M. Lassas, S. Siltanen.

L'articolo Mathematics and breast cancer prevention sembra essere il primo su Paola Elefante.