The New Witch-Hunt

(originally published on Y / 03 /2017)

The new threat to the civil development of the Western society is not capitalism, but radical liberalism, by hand of politically correctness. I am observing a worrying trend especially on social media, by which any opinion that vaguely contradicts the self-calling liberal loud voices, provokes waves of indignation, derision, rudeness, even bullying. Healthy debate is out of trend and the new concept of "respect" does not include discussion. The game is succeeding at polemicise on anything or find two-faced meanings that allow us to fill with indignation.

This is simply the modern version of witch-hunting, with torch and pitchfork held high against who threatens the modern "mono-thinking". Spits and public humiliation, anything is allowed, including strip of humanity and identity layers any virtual stranger. Feeling strong from their ideology, some people feel authorised to scream in the face of those who disagree.

To show an example, few weeks ago the story of a (Caucasian) girl who had slowly poisoned her (Afro-American) college roommate and bragged about it on Instagram went viral. I came across it when the picture of the accused, complete with full name, went viral with the caption "Be a nasty bigot and we will make you famous for it". I found myself wondering which one was the worst example of human nature, the original story or the long series of deplorable comments and threats which followed. A woman with the same name has to temporarily suspend all her social media accounts out of the verbal abuse she had endured.

I believe our ideas never allow us to rejoice of other people's misery, nor to verbally assault or superficially judge others. From those who claim to champion for diversity, I except that they do the same with ideas. I conclude by quoting Evelyn Beatrice Hall's famous words, "I disapprove of what you say, but I will defend to the death your right to say it".

L'articolo The New Witch-Hunt sembra essere il primo su Paola Elefante.

Why all this? Women are severely underrepresented in mathematics, especially in Scandinavian countries, even though they top charts for gender equality. Recently EWM collected data from the country coordinators to describe the single situations and filed a report. The situation in Nordic countries is too extreme to ignore:

L'articolo Women in Mathematics in Finland: Amal Attouchi sembra essere il primo su Paola Elefante.

I am proud to say one of the speakers was my advisor and inverse problems researcher Prof. Samuli Siltanen. His talk was titled "How to defuse a photobomb" and showed some mathematical techniques for image enhancement and manipulation, in particular inpainting. Samuli was claimed winner at the end! You can enjoy his talk right here:

You can browse a photogallery of the event at this link.

It was great to see a mathematics talk winning the hearts of the audience. Congratulations, Samuli!

L'articolo Science Slam Helsinki 2015: a triumph for Inverse Problems sembra essere il primo su Paola Elefante.

Today I will report how to crop multiple images on a Mac. Sometime in my presentation I describe a sequence of events by similar images, slide after slide. Of course it looks much better if you place the image at the same coordinates and at the same size in each slide, so that the changes is well-highligthed when you scroll down. This visual offers the benefit of being able to scroll freely up and down to show your point. One problem is the following: how to fix a series of images which you saved, say, from MatLab, in the same way to get a series of still similar images.

I searched for solutions online but apparently the latest version of Preview does not allow to simultaneously crop a series of images. You need a little more work, but here is what you need to do:

Step 1.

Select all your images, then right-click and select Open with > Preview.

Step 2.

Select: File > Print. Make sure the orientation is as you want it. Then select PDF > Save as PDF (look image below).

Now your images are all saved in a PDF, one for each page. This step was necessary because the latest version of Preview allows simultaneous cropping of PDF pages but not images.

Step 3.

Open your PDF again with Preview. Make sure you have thumbnails on the left side. If not, select:

Step 4.

Click View > Show Edit Toolbar. Use the Select tool to select the area you want to crop on one of the pages. Then select one of the thumbnails on the left and press ⌘+A to select all thumbnails. Finally, press Crop.

Step 5.

Now you need to split the pages of your PDF file. I used the online free tool Zamzar, but there are alternatives (Google will help).

And here you have your simultaneously cropped images. If you have a faster method, please feel free to share your solution in the comments. Hope this turns out useful to someone else as well :).

Featured image from http://got2belinux.com/?cat=1.

L'articolo Visual Tips: Cropping multiple images sembra essere il primo su Paola Elefante.

Baby-wise it went much better than I expected: she played, slept and watched cartoon and I was able to listen to almost all talks. I could even present my own during her afternoon nap (slides to be found here).

Science-wise, I heard about many interesting projects. Inverse Days is the annual meeting of the Inverse Problems research community in Finland. Since it's a great chance to get to know all ongoing projects in Finland, there are often guests from abroad as well. Since 2010, I have been missing only one meeting, because of my parental leave. The dominant theme this year has been EIT, on which many groups in Finland are working at the moment.

For me, the highlight has been a nice presentation by Lauri Harhanen from DTU. He started last May in the huge ongoing project HD-Tomo and he's working on an interesting optimization of gradient descent minimization methods, based on physical modelling of material a priori information in CT. When I will upgrade to real data, this could be a fruitful improvement of my framework.

During the conference, we also had the second meeting of Women in Inverse Problems, a small network started one year ago. We agreed that it would be nice to have some statistics (local or nation-wide) on presence of women in mathematics, and investigate on reasons why the disproportion is created. Let's hope we can follow up on this idea. At the moment our most active action is a small mailing list, where we share interesting events or discussion topics (drop me an email if you want to join, since it's closed).

It's been nice to be at Inverse Days once again and I also realised that this was my first talk at this event. See you soon, Lappeenranta!

L'articolo Live from Inverse Days 2015: baby on board 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.

The Doctoral School of the Department was kind enough to grant us some fundings this year, so that we could advertise the seminar more and organise some events with special speakers; I sincerely thank DOMAST for the trust!

After about one year and a half from the start of our management, the seminar gets the room full almost every week and I am very pleased to see new faces every time. We even had exceptional events that, to my opinion, contributed to increase the feeling of community among students of different levels (see the pics from the talk on mathematics and zombies here).

https://www.youtube.com/watch?v=bgzidIygynk&feature=youtu.be

The main purposes of this happening are to offer students a "safe place" where to train their presenting skills and to learn something together. Since people in the audience have different backgrounds, we encourage presenters to choose topics that are accessible to everyone. As Anssi and I are both interested in teaching, communication skills and techniques, in the current semester we advised speakers to privilege explanations on the blackboard than slides (credit to Anssi for the idea). Even though I am myself a fan of slides, I personally think this forced speakers to a slower pace and made the talks better, in general. It was a very interesting experiment! (**)

As we have been getting such good reaction from students, we decided to push further the seminar and we are now also trying to pursue three additional goals:

• try to connect the different departments of Kumpulan Kampus, showing how mathematics is used or needed in other fields,
• encourage networking with other Finnish universities (we had two speakers from Aalto University this spring),
• creating a bridge between university education and the working world, to make math students more aware of what they need "out there".

To this extent, I have been contacting people from other science departments, asking to give math-related presentation of their field of study/research (for instance, Oona Kupiainen-Määttä from Department of Physics will be speaking next Thursday). Personally, my hope is to encourage networking within campus, to create a lively and heterogeneous research/study environment.

Concerning the other idea, we noticed great enthusiasm in students when Prof. Samuli Siltanen presented some suggestions for facing the working world last January. We had to rush to an auditorium because the audience was definitely too wide for our usual little room! This large affluence and the questions after the talk made me think that maybe students wished to be more aware of what they needed to study and do to have an easier life when applying for jobs after their degree. For this reason, we invited a mathematician working at Varma Insurances, Jarno Ruokokoski, who was kind enough to accept promptly. I wish to thank Jarno and Varma for making this possible.

I invite all interested students to participate, since presence will be an important feedback indicator for us for future events!

As a conclusion, I wished to explain all these aspects of the Students' Seminar to show our commitment and purposes. I am really grateful to all speakers who I harassed in the past year and a half and who proved open and generous to give talks, and to the Department for supporting the activity. I strongly hope to provide a good platform for students at different levels to learn new things and to train themselves to become good communicators.

(*) The current webpage of the seminar. Since it changes every semester, in general you can reach it through this webpage, then clicking "Opiskeljioiden seminaari".

(**) I care to underline that this was a mere suggestion. We leave speakers free to choose their own way of presenting and the language they want to talk in (Finnish or English). There is no point in making a speaker uncomfortable!

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