 any time if you have questions don't wait for the end of the talk this will not really work very well. What you're seeing here in the background is a thing you have seen many times before it's a piece of aluminum foil crumpled together like you will be throwing it away in a second like chocolate paper or something like that and you see all this riches and intricate structures in here only because I've cut these pieces of aluminum foil this little ball in the middle using extra tomography so you're seeing a thin slice cut out of the middle of such a ball let me show you all of the slices so we have the ball in little rods now we're moving up the rod you're starting to get inside this ball you can see layer by layer how this unfolds here now the clue is gone and you can see it is all this riches coming in and going out I think first of all this is actually aesthetically pleasing picture I mean I first time saw that I thought this is nice because I've never seen the worlds around me this way and there's a good reason I've never done that but there's also interesting physics to be done with that you know you have an extra experiment here talking about riches and crumbling stuff together so this would allow you to have insight from the inside to go inside such a sample we can actually quantify how much more insight we can get out of the tomography everyone knows that the picture is worth a thousand words so if we move on and go to something which is 3d actually we have thousand times to one point a point half power which gives about 30,000 words so we can really have a lot of more inside but just doing a 3d image of something instead of doing a slice what you see atop is the system which is very similar to the one you were actually doing in the hands-on session it's a container containing in this case glass beads and again it's just one slice cut through the whole sample of glass beads you put all those slice together you get a stack and that stack actually gives you that stack gives you then a chance to analyze it and find a particle position like here in 3d and then do things like the vulnerable tessellation that mark has been talking about only do it in 3d okay the basic outline of this talk is the following I'm going to first talk to you a little bit about x-ray imaging a lot of it might be known to some of you from from other courses but I think it's good to have back the basics altogether to understand how do we do a good x-ray image then how does actually tomography work we have these 2d projection images how do we create a 3d image out of those projection images then the probably most important thing how do you get started using it in your own research day to day work and finally I have a couple of special topics I'm not sure that after time to go all of them but there's some extra hints which might be helpful if you want to get started on things you all know that x-rays is just another form of electromagnetic waves so we're just moving up a little bit in the frequency or down the wavelength compared to our vision spectrum which is over here and so we expect them to actually be rather similar to what we know already except that we don't have sensors for them and we can't we don't have actually sources for x-rays the reasons of the for a long time didn't have so let me put it that way the reason for that is that x-rays cannot be created by blackbody radiation almost all light sources mankind invented first we're actually making something really really hot and then it emits x it emits blackbody radiation and that blackbody radiation for some of something is 5000 Kelvin warm is right into the visible range so you can use a fire you can use a little metal wire which is heated up electrons you create visible light now to move over to x-rays you would have to heat up something to hundreds of thousands of Kelvin or something like that so there's no way we could have done that by the usual way we learned to actually create light so we had to come up with new ways of creating x-ray source and that's what about a hundred years ago when Mr. Röntgen invented this x-ray tube so what you have here is an evacuated glass tube and you have an filament over here which is heated up which is heated up and by the seated up it emits thermal electrons so just moving around that cathode over there and then we put on a high voltage supply which accelerates our electrons towards the R-node and while they go over to the R-node they acquire kinetic energy which is propelled to their charge times the voltage we apply so they have a really high kinetic energy they bump into the n-node and while they bump in there they create x-rays which we then can use to eliminate our samples now this bumping is very hand-waving we can do is a little bit more precise there's actually two different mechanisms which create x-rays world is called bremsstrahlung what you have is an electron coming in seeing the electrons often the atoms into material and getting then decelerated or changing its course or maybe even completely stopped every time you decelerate and charge you actually create x-ray you create photons you create my electromagnetic waves and because the way the exact way how the electron can be decelerated inside this atomic orbitals can be very different you're actually not getting only one specific energy you get a broad spectrum of energies depending on the exact details of that collision there's a second way of creating x-rays which is the characteristic emission so the electron actually bumps out one electron which is already in shell somewhere there once the electron is missing the atom is not happy and new electron is falling down and because it's going between very distinct energy levels it's actually sending out a photon with a very precise way of putting that two mechanisms together you get a spectrum which typically looks something like that so here you have a photon flux that's photons per area and time and down here you have the energy of the photons coming out of this is in this case it's a tungsten anode and you have four different curves this four different curves correspond to four different acceleration voltages and you can see that either of these curves 80 hundred twenty hundred forty kilo volts ends exactly at that energy level you can't create any photons with more energy and what you actually have put into the electrons while accelerating them towards the target you cannot put get more high energy photons out of that you can also see that the characteristic radiation might be very strong in a very small energy band but overall the most photons are actually the bremsstrahlung so you have to worry much more about what is going on there and going on in this little characteristics radiation bands okay that's the way you can create x-rays one thing you need to know when doing so is that x-ray tubes are horribly inefficient about 99% of the energy you put in by accelerating the photons you actually just convert into heat only one percent comes out of extra photons that is a problem because this much you try normally to have a very small spot you want to have you have geometric optic you want to have a very focused small focus spot that's more focus spot you're dumping a lot of energy which means that if you're not careful you're actually just melting away your targets so you're limited naturally by the heat flux out of that focus spot which how strong of a photon flux you can create and I will show you in a second that photon flux is everything in x-ray tomography okay now you created the x-rays they come out of the tube they will start interacting with matter and it depends on what they find there the first thing to keep in mind is if you think about electromagnetic waves you normally think about something like sned's law some wave comes in change crosses the interface between two different materials these two different materials have two different influence index of a fraction and actually this the electromagnetic wave gets bent over that's how we make lenses for example well there's something you don't need to worry about for x-rays because the cof index of a fraction and is one minus delta and delta is a very very small number over basically all the photon range those 10 to minus 10 to the minus 7 for different materials so basically x-rays just goes straight on they're not getting bent they get absorbed they get scattered but they're not getting bent over so we have just purely geometric optic and don't need to worry about things like lens actually that's a problem because sometimes you would like to have a lens it's very very hard to build something like that so what we have is then directly geometric imaging so as I already said we typically have an x-ray tube where we created a very small spoke a spot a lot of x-rays and then they come out and it just goes straight on pass through some material they're interested in and end up on some camera over here and because of this cone beam geometry we have magnification because everything over here gets just expanded when it and defense finally up at the ccd screen and we can compute a magnification but just comparing the length so by moving the object closer to the target or closer to the camera we control the magnification without any lenses in there that's the classical x-ray imaging setup that's what you do and the medical doctor chest x-ray or anything dental x-ray and it's called a radiogram a single individual picture taken that way is a radiogram now the thing you need to also remember from your physical chemistry whatever entry class is that the actual absorption the interaction with the material happens because this is the scattering absorption is governed by this lumbar bear law so if this is your x-ray source is your x-ray beam coming out that's an intensity I zero over here certain number of photons per time and if it has passed the material it's hand gone down to I and if material of the thickness fit the x over here is the one thing which determines what how the image looks like for example again this image of the walnut the other thing which it determines is the attenuation coefficient different materials have different attenuation coefficients this attenuation coefficients only depend on the electrons of the atoms the x-rays are actually encountering there is no chemistry dependence for a perspective an x-rays it's called totally it doesn't really matter if you have like a cube of sugar or if you put about the same amount of carbon coal and hydrogen and oxygen atoms in the same path length as long as the x-ray encounters the same numbers of atoms it will see the same attenuation there's no chemistry at all involved in that type of imaging there's different ways of doing chemistry of x-ray but not in the imaging okay and then you can end up with the lumbar bear law so to bet these two internalization it's to get the two intensities into together you just have an exponential factor which goes with this attenuation coefficient and the length of the beam which is actually the material passing if you have different materials like with this walnut down here you have the flash inside the walnut and you have to shell what you do is you actually just sum up over the whole length of the path the attenuation coefficients times the length of the different materials which go over there and this summer they don't come back again as an integral if we go and analyze what is actually going on in tomography if you want to what is the extra attenuation of a certain material there's a very nice service by anist there's a database you can go another database you find all the attenuation coefficients beside other things for a very broad range of energies from one KV to 100 GEV that's much more than you want to know believe me and you can really search that database easily with this form and you can just download all the data you need to analyze to predict how strong the extra attenuation will be which you're observing now I've done that and I added it up for certain materials now what you see here is the extra attenuation coefficient as a function of the photon energies for four different materials glass water polyethylene and air so do you read that if you say okay I would have a photo of 100 KV then this would for glass for example give me something like 0.8 one over centimeter attenuation you can see several things over here first of all it's really depending on the energy and to understand this depends precisely would not have to talk about different mechanisms which do actually scattering and the attenuation inside the x-rays but that's actually beyond the scope of what we want to talk about here and as a user you don't really need to know that you just need to have that curve and you also see that you want to have contrast but contrast is better over here than over here and we will come back to some of that but if you typically have an object let's say out of glass or polyethylene in air the contrast between the plastic and the air is large is huge doesn't really depend so much on what photon energy you have you always will get a probably very nice image of that keep in mind also that the x-ray setup we will be using here just goes up to something about 550 kV so we can't really access that range over here but that's for example medical x-rays often work in that range 100 to 300 kV okay an important and very annoying thing is the following the x-ray spectrum changes while it passes the material as I already showed you the attenuation is much higher for low photon energies so they get filtered out much faster than for high energies that means if I go with my x-ray beam and forms example to have a spectrum here for my 120 kV tungsten anode tube if I measure the spectrum directly at the anode I get this red curve if I go for 75 centimeters of air I get the blue curve which is a little bit below I filled it out or something of it but not a lot but then if I go for example through what is that yellow should be something some millimeters of aluminum if they go for let's take the yellow curve is if I go to 2.5 millimeters of aluminum that could be for example a sidewall of your container you see that at a high energy range you almost not lost any of the photons but in a low energy range this is completely gone everything to 20 kV is completely out of your beam that means while penetrating food for the material the x-ray is changing the x-rays are changing in the spectrum and that change in spectrum is not accounted for for the Lambert-Behr law the Lambert-Behr law actually is only strictly correct with a constant coefficient of attenuation if you have only a monochromatic beam but because you have a polychromatic beam which changes while it's penetrating if you still want to keep the Lambert-Behr law it looks to you like your attenuation coefficient is actually becoming a function of the thickness you have this is called beam hardening so for example here's different materials is actually all glass BK7 glass so the three curves over here sixty hundred and forty kV that's different spectrum which we are the seven acceleration waters we had at the electrode so you get different spectra out of here that's the attenuation coefficient we've measured and you can see it's going down the figure the glass becomes and it's saturates against some value at some point you basically have removed all of the low energy photons out of your spectrum and the remaining photons more or less behave like the monochromatic but this takes a mile so Lambert-Behr's law is an approximation which is strictly with a constant attenuation coefficient only valid for monochromatic energies if you're not having monochromatic energies you have to take into account this beam hardening and I come back to that because this is a big nuisance in taking images and some of you will actually see that in the experiment if they have something in there which has for example some metal you will see artifacts which come exactly from this effect but then come back to that okay finally yeah so yeah this slide before that one you're talking about this peak this is no this this peak is just completely filtered out there is no photons of that energy left that's a characteristic radiation again so the alpha beta tells you which shell is falling toward other shell but these photons basically are all gone after you pass 2.5 millimeters over the minimum they've been completely filtered out the second peak is always there you just can't see it up here there's red and blue is still here on the top it's not like these peaks are shifting in energies it's just hard to see okay I should have pointed out the red and blue peak are right on top of the yellow and the green peak it's just going down in intensity but not as much as this peak okay let me come back to your question you're actually addressing the way of one of the ways to mitigate a problem and I have an extra slide for that further down and talk but you're addressing exactly the way how you can mitigate that okay so I will show you if I have the time if I show you a movie in about 10 minutes where we have tried we had a box of sand we shake that we try to compute a packing fraction and the numbers were wrong for a long time and we thought I was why how can the numbers be so wrong and then we said okay it's beam hardening and once you put in beam hardening in our analysis then the most became correct this is this is everywhere there's not an effect which is just limited to certain specific experiments or so okay once you have your x-rays passing your object you have to turn them into a signal well for light we all know we have CCD cameras down here that's a CCD camera but a CCD camera is basically blind to these high energies so what we do is we put a scintillator screen or directly in front of it on some distance that's to get some rare earth metals which have to property that they convert x-ray photons in photons of visible light and these photons of visible light are done just detected by a standard CCD camera there's other ways they're semiconductor detectors in the meantime and so on but that's still the most standard way a scintillator in front of a CCD camera to get a decent image keep in mind that this is a stochastic process so there's a finite probability q that a photon is actually detected it's not that every photon is detected and that probability q depends on the detector and the energy now if you would sit down on if you take a detector you have something standing in front of you to take x-rays and you don't change any of the geometry it just keep on taking images in a perfect world of course you would have a certain pixel certain intensity computed by Lambert Bayer's law but because both the creation of the x-ray photons and the actually detection of the x-ray photons are stochastic processes you actually get a fluctuation intensity so this is the intensity of two different pixels inside such a static arrangement and it fluctuates around the mean value actually if you go in and do all the math you find out that these photon energies are the photon counts are Poisson distributed and that's basically the case for almost all imaging problems it's just as I will show you in a second that for x-ray tomography noise is specifically detrimental and we have to keep it down as much as possible so if we know it's Poisson distributed we can also say the mean value is proportional to the detection probability times the photon flux so yeah and it's the photon flux coming in at that pixel so that's the mean value and the variance is in a Poisson distribution also given by q times n star so that means if I want to have a figure of merit a signal to noise ratio I divide my mean value the average gray value of that pixel by the standard deviation of the fluctuations around that mean value and use the knowledge that I have Poisson distribution I derive that the signal to noise ratio only depends on the square root of q times the mean photon flux q is something I cannot change I have already picked the camera so that's static the only thing I can change is the photon flux and that's why we never have enough photons we if you get 10 times more photons coming in we only prove a signal to noise ratio by factor of about three so that's a but that's a rather very standard thing in almost all of image creation it's just that in x-rays it specifically hits us in point later down the road so if you take x-ray if you if you have sample and you go to an x-ray machine and you want to take an extra image of it one of the first question is okay how much acceleration voltage do I actually use for that I mean I have you have this knob you can go between typically but an all machine you can go between 25 and 45 or 50 kv another machines you can go between 30 and 150 170 kv so what is the acceleration voltage use now if you go back to this diagram and say okay we have a photon energy which is down here in the low energy around we actually get a pretty good material contrast all of those lines are separated pretty well so if material has different things in there different different chemical positions we can probably distinguish them pretty well on the other side the overall attenuation is rather large on that side of the spectrum large attenuation means very few photons will reach our detector that means we have a bad signal to noise ratio so that's the one side of it if you go to the other side well all of those lines are now pretty close next to each other so the contrast between different materials really goes down and it's becoming much much harder to image something or to come post of simple things but we have a smaller attenuation so that means our photon code count goes up and we that's good for a signal to noise ratio so that's basically a trade-off we always have if you do an x-ray image and you have to figure out depending on what you have what is the best thing typical rule of thumb is high-contrast images of things like biologic specimen you're better off than paying the price of low photon energies if you have something like metal in there or thick layers of yeah metal or something you always go up to high photon energies and the rest you need to have experience how to do it there's a couple of tricks one of very nice triggers is one of your phases is liquid where you can dope this liquid phase with some salt which has a high order number like in this case cesium iodide is pretty far from the iteration so even small concentrations like half a mole of cesium iodide will make your liquid very well visible this is for example used over here there's a slice now this is from a tomography already not an x-ray image slice for tomography of sand which has been vetted by a liquid and you would normally have a hard time distinguishing between the liquid and the grains the different grains and there's liquid bridges in between them but because they added sink iodide to the liquid you can very nicely see where the water bridges between the different sand grains are so that's the way to to bump up your contrast so I'm mostly going to talk about x-ray tomography but let me point out that radiography is also a great tool you might if you have access to an x-ray system and you might even not want to go for the tomography because it's a time consuming you can only do static things but you have something which is dynamic extra radiograms can be much much faster so this is work which has been done in collaboration with the people from the Fraunhofer Institute by the way the Fraunhofer Institute which also built our city portable and you probably know it from your mp3 player so they are the guys who invented the mp3 protocol so they're doing really broad range of different things and this city stuff is one of them so there's a collaboration with Fraunhofer Norman Ullmann from there and Jonathan Coymer from our group what you see up here is an x-ray camera there's a small box of sand on some guarding on some rail which can be shaken here you see a top view of that sample which can be shaking this way and up here you see an x-ray tube which is then going to illuminate this sample box in here but if we do that well again this is our sandbox if take such a sandbox and shake it horizontally various things happen there's a convection roll going on in there and a lot of interesting dynamics but one thing which is rather peculiar has been found by Torsten Puschel some years ago is that if you pick one point in the height of that surface you just study that point and as a function of time so this is a time lapse of one of these points in the middle of the box you can see that the system seems to be breathing it's expanding and shrinking expanding and shrinking and the timescale of that breathing motion is on the order of two to three seconds it's very very different from any other timescale we have in the system the convection a timescale is different the shaking timescale is different and it's also completely different symmetry we shake it this way why would then the sample go up and down there's no real reason to understand that so that was something we're really asking ourselves can be understood what is the mechanism behind that why is that actually breathing there's no real surface pattern it's not like the radically shaking stuff it's really just so so there is a flow coming in as a convection roll going yet but over the surface stays rather flat there is no no undulations or wavelength you would just turn from that it's just the whole system goes up and goes down goes up and goes down okay so what we did is we took x-ray images and here it's just a grayscale image so you can see the box we take the images why we shake them so we synchronized the shaking motion in the x-ray camera so that's it therefore the box seems to be static standing there but actually we shake it with 75 hertz back and forth for a couple of millimeters and if we now analyze this gray value profile over here and if we know the limit bear law and if you know the beam hardening that was actually the necessary ingredient to get the numbers correct then we can compute the volume fraction of the sand average along the path through that box so just from these gray values you can see that in this gray values over here which we look at this blue box there's some stripes here which are brighter than the rest of it this brighter starts correspond actually to a dip there's less sand here there's a layer of sand which is more dilute than the remaining sand this layer of sand the loot sand comes from the fact that the stuff on top of it is actually moving back and forth and the sand below it is actually static and you get a shear band between these two areas and in this shear band sand get becomes more diluted and now if I start a movie and you can see what the dynamics of the system is so keep in mind it's synchronized so the box is really moving you can see that these bands of dilution form and then they travel upwards periodically again here you get the packing fraction out of it and we can show that once the band reaches the top surface or comes as close as it gets actually sand collapses down new band forms so this production of this breathing mode really depends on how many bands are actually in there and how expanded are they so there's a strict connection between that the red dots are actually tracer particles we have added in there so this is little particles which are higher extra attenuation and they show you this convection walls you can see if you see that they're actually going like that there's a conventional on the side over here but this conventional phenomenon is independent of this shear band phenomenon yeah no I take the box and shake it like that horizontally only constantly so the box is really going like this you're just not seeing it because it's synchronized with the extra there was 1.5 millimeters I think and the box is something like 10 centimeters to give you a skate it looks like there's kind of an inverse tornado an inverse at the bottom you mean like now I see it it's kind of like going up it's kind of like a tornado that's outside down okay I would be interested to talk you later about this tornado because I'm not sure I understand what you're seeing from here oh this one this one yeah so we get actually get another shear band over here which we don't understand because apparently that the motion which we're pretty sure is rather small we added with another image where we added more tracer particles so we have a rather good idea of this action motion of the ground and there is almost no motion down here so we don't really understand very very wide as a now at some point it vanishes it if it's close to the walls this this kids it no the volume fraction is changing you're correct but the thickness of the box is not changing the thickness the length is the same so the beam goes along the whole box the worm fraction will change right yeah but if you know an analytic expression how the volume fraction depends on the length you can actually compute it back so we first measured how the volume fraction we took another sample this other sample we measured how does the yeah we do both we need a kind of calibration measurement which is independent of that measurements otherwise we cannot get the numbers here okay well no there is these convection rules and the question was is these are these convection rules responsible for the shear bands so we needed to show that these shear bands are actually separate phenomenon from these convection laws that's why we had this other experiment which also I can show you I mean okay I don't have it here but if you're interested I can show the paper we can I can show you that exactly the point where the shear band is the tracer particles on top of the shear band they move much much faster than the tracer particles below the shear band that's all proof that it actually really is a shear band okay so that's it can radiograms we've not done anything tomography here and I'm seeing so okay let's let's talk about how does tomography actually work if we have talked about radiograms when I first work with tomography found this is kind of a black box so you buy this machine and this machine takes all those radiograms and then you get like here you have this walnut in there it takes an image it rotates it a little bit it takes another image it takes a whole stack of images over 360 degrees and the more images you take the better the qualities and it takes this whole stack of radiograms and gives you back a 3d volume so you have a 3d matrix x y set and every part that 3d matrix corresponds to the attenuation coefficient of the sector that space how is it actually doing that I mean it's they don't well you can read it up in books but the machine you buy has the software already together with you don't have to program that yourself anymore luckily just buy it yeah that's the 3d volume okay to explain how it works I'm going to talk now about 2d slice so I'm taking one slice of this is my sample I'm not just talking what happens in one slice like I have a line scan camera over here and I move this line scan camera around the sample so I have like just a single image over here so that's my x-ray source that's my slice for the object and that's what my line scan camera sees in fact I'm actually computing something which is called projection or projection integral so again if this is the setup over here I have this lumber bear law which says okay the intensity at a detector is the intensity coming in times the eight function minus the integral of all the new coefficients along the path which the photons have been taken now I can do a little bit of money algebraic manipulation and especially I turn around I have now I saw on top ID down here I take the logarithms the logarithm and then I find out that actually I from my intensities I can compute something which is the integral of the attenuation coefficients along the path so my photons so to speak compute me that integral at each point at each path they're going for and that is then called the projection integral or protection it's this P over here so it computes the integral over mu there is no muse over here the path coming in here is not going to see any of the materials over here the projection degree of zero and then it becomes larger the larger the actual attenuation coefficients along the path over here so we assume the divide areas have a higher attenuation than the gray areas and then we would become something like this projection integral so that's one measurement of my line scan camera after run just this little normalization over here the whole tomography probably kind of described the following way I have this first line scan integral then I rotated it by the guy and see so sorry size size I should have said that size deep is the the length along my my line scan camera so that's basically a pixel number if you want that that's the numbers of pics yeah I should sorry for that now I'm going to rotate my my setup I take another line integral another image of my line scan camera and I get another projection integral and now my P my projection agree not only depends on side the pixel number but it also depends on the angle which I have been taking it and I take more of those images again I get a series of these integrals and the problem I have to know if something is an inverse problem where I take all these projects and integrals and project them back into the volume and want to know the mu of x and y inside that volume where I actually get it from that's what tomography has to solve this is just something which is it's not the explanation it's just a way which is typically considered if we use this as a phantom so we assume this is our slice our particles these are glass particles in the container is a muse they're very high over here if I now take this all this projection integrals I always get like one line of intensity so if I have 180 or textation angles for example I get 180 lines and I put all those lines below each other I have a sense of position psi over here and rotation angle over here I get something just called the sinogram the sinogram is just a way of visualizing all this projection integrals into one nice picture instead of all this other individual curves so the intensity that's the projection integrals as an intensity that's the angle and that's the extent of position so this radiogram so the sinogram is now there the basis from what we were going to do the reconstruction let me remind you again what my credit told you on Monday morning you can do free transform in 2d a typical space of free modes looks like this this is at least the way mudlap represents it in the middle you have the so-called DC mode so that's just the average grade value of the image you're looking at after free transform and then surrounding it you have all the different waves you so this is just the three mode which fits in one time four or five seven whatever times into your image and in the horizontal line they're all aligned horizontally in the vertical line they're all around vertically and depending where you are the angle is different so wavelength decreases from the center outward so the largest wavelength is inside here and the smallest wavelength is on the outside that's the picture of the free space you need to keep in mind for the following discussion of course we have the complex variables here and a phase and amplitude and so on but that doesn't end anything about a wavelength picture now let's step back for a second and say okay let's assume we would already know the iteration coefficients in here and we would compute from our knowledge a free transform of that how would that look like well I just take the free transform equation that mark has been showing to you I have these two wave numbers you and we here's my attenuation coefficients and here's my functions with the two terms u times x and v times y now and that would be then the coefficients these would give me a different put in different use and we use and I get different amplitudes and that those amplitudes I would use to populate my free space over here if I would know it that way let's first for the sake of the argument assume we are only considered in the coefficients along that line you we ignore everything of the free space so we set a v to zero if we just to say that to say we really say okay so it's over here you and me so we only see coefficients along that way now we drop that second term here in our exponential functions we only left with the u times x term in here but this allows us to simplify the integral we can now put the attenuation with together with the integral over dy because the e function doesn't depend on y any longer it's only depending on x and we can take it out of the integral and that is however what is inside here is exactly the projection integral we had so what is there what I've written down here is nothing else than for this specific assumption that I said one of the lines 0 I get the free transform of my projection integral and that's the free slice theory I have not talked about one specific line this is valid for any possible line for any possible projection integral I could look at I can use any angle gamma for which I want and for each one of them I have the fact that the free transform of the projection integral populates me a certain line my Fourier space the full line which is parallel to it so what I can do now is I can take all my different lines of my sinogram free transform each of these lines and each of these lines will populate me my free space along one line going for the center so if I take this line for example this line over here would be that line so move down one more line I'm going to change the angle slightly gamma becomes slightly larger and so on and so off but in the end I have something like that I have all my different projection integral populate me but these are the free modes I've put into my free space by analyzing the free transform of all this projection integrants and this is not what I need exactly I have now some idea how the free modes look like but if I want to do a Fourier back transform from the free space to the real space I actually need something like that I need to know my coefficients on this rectangular grid these that's these are the even spaced out free modes or the gray dots over here are the evenly spaced out free modes which I need to know to the back transformation and I've gotten from my Fourier slice theorem which I just showed you I've gotten points along these diagonals that means I've gotten a much higher density of information around the center for the long wavelength and a much lower density of information the further I go out the higher the wavelength is the lower my information density becomes the fewer points are there and now doing my back transformation I have to interpolate all those gray points over here for much less information which I actually have and that's why actually free transform the back transformation rate of transformation requires a high pass filter I'm overvaluing the high frequency information I have less of it and I have to amplify it so to speak to fill in all my lack of information and that's why free transfer by tomography is so noise sensitive because the rate on transformation is first-line stream gives me information about what is going on inside there but it gives me more information about the low wave length modes and about the high wave length modes yeah somehow yes we not correlate but I'm the pink points over here that's the different amplitudes we get from so each line of these corresponds to one projection integral and we did a Fourier transform of this one projection integral the first life stream said oh once you've done the full transform you got some amplitudes these amplitudes are actually lying along this diagonal line over here so we filled in basically these I don't know this 15 20 points over here in our free space however if you want to do the back transformation because we in the end want to know what is the actual system is we cannot use that that's not on the grid we use we actually need to know the gray points over here on that grid so for example if you want to know this great point something we can do we can interpolate maybe this and this pink point to get the closest possible value to understand what the free transform of that point would be but this interpolation step becomes less and less good the further out we go because the less information we basically have that there's another question okay okay so it's coming up the whole tomography progress so to speak so we start by having some slice of some image over here we do a radon transform so we've heard all the projections we've get this projection integral we just sum them up all together and that's the result from what we actually did this down here this radon transformation that's actually the result which you measure by our tomography so this landscape camera has given us the synogram synogram now the black box inside our tomograph does two things it first takes the Fourier slice theorem and use the synogram to populate the Fourier space with all those lines and then it does an inverse Fourier transform to give us back to distribute spatial distribution of the menus now this is the idea of course the actual algorithm do something which is slightly to vastly different but overall that is what happens in the way of back transformation of the Fourier transform okay and just to sum that up because we have this little information on the bounders and the high frequency range that's why fully the tomography is so sensitive to noise and why it really matters how good our signal to noise ratio is much more than for other image modalities and here's an image example this is flint phantom and it just added two percent of Gaussian noise to the synogram and back transformed it and it's already hard to recognize the five percent you basically start losing all the features in there so this is much more sensitive than normal other many other methods okay this is the time to address the elephant in the room this is a school on hands on experiments I'm talking about x-ray tomography x-ray tomography even though we have a hands-on device is not what you typically would consider to be something which is tabletop let me give you a small answer why it still might be something you might want to consider first of all okay that's the machine I brought with us from the front of Institute it comes at a price tag of 55,000 euros which is already a pretty good price tag and given its limitations that for example you can go only about 10 millimeters in aluminium and 45 millimeters in plastic that limits you and also relatively small size of your sample that's a pretty strong limitation if you want to buy something which is more general for example what we had before in gutting is this nanotome for cheese sensing inspection nanotome because it actually can resolve two gold lines which are separated by 500 nanometers but only gold lines if you have a bad loss contrast it doesn't do anything in nanometer range then your price tag is in the 350 kilo euro range and that's definitely not tabletop anymore there's two there's two options two main options why I think you might still be able to use that technique one of them is that medical CT scanner and now ubiquitous and almost anywhere I mean if you look a little bit around you will find a place which has a medical CT scanner and these medical CT scanners they might not be as good in a space resolution and the others okay you only have like 0.3 millimeter of this voxel size by the way voxel is the freely equivalent of a pixel so little cube is a voxel that have voxel size of 300 millimetres that's much worse than what we have but you get other nice perks for example you have a speed of hundred four hundred twenty forty two milliseconds for one image that's much much faster than a nanotome can do so if you have if you find a way of getting access to medical x-ray you can do a lot of science with that and just as a proof of concept actually let me show you the results by rafshanarius and tamash brazzani they study a granular system there's a split bottom cell it's very similar to the taylor quetzel actually only don't fill it with liquids they fill it with granular media they fill it with these little rods over here which are little woodpecks so to speak and then they went to a medical facility and just put their container the split bottom cell on that table where the patient is normally lying and take images of all the pecs in there and then you're done you have your 3d data set and out coming downwards from there is image analysis which you can do with just any desktop pc and and they got nice results out of that so here is the alignment of the packs here this is the this shear zone if you she's moved a split bottom you can see that the packs and the shear zone align much more and you can make nice analogies with liquid crystals for example then there's another option you can build your own tomography so actually my own exposure first exposure in 2006 was with a self-built setup at a setup and a new and camera Timson has built that so what you see in the background that's the x-ray camera that's the x-ray tube there's a little rotation table over here and on top of that you see my fluidized bed which I have been using when I was there actually started running media inside a fluidized bed and that machine is in a lab clouded room and that was one of the first machines I know where physicists have actually built something like that you can do that also with less work for example this is a set up which has been in hand I've been handed a guest lab in the University of Chicago by an extraordinary undergrad actually at an asiatis probe is pronounced that sorry he built that as an undergrad build a whole x-ray setup not only the extra setup actually on top of the extra setup he built an instrument machine which can put a pressure which measure it very precisely on that and he's taking x-rays with that and he has written a very nice review scientific instrument paper which gives you all the ins and outs and details if you want to ever build an extra setup this is probably your first go to place for doing so if you're really interested find me I have more references but that's the most important one I would say I actually don't know I think the point is this is a dental x-ray source so that's something you should say that so one cheap way of doing is getting a dental x-ray source actually can be an old dental x-ray source which is not used so dental x-ray source also get lower and lower on energy and sometimes the doctors throw them out because they have a new system you can get some system like that you already have half of your almost half of your equipment I don't know what he paid for that but I would assume it if you buy a new one you're somewhere in a 30,000 euro or dollar range something like that but don't name it on a number on the other side the flat panel detector is what you need if you buy new ones you easily can spend also 20,000 dollars on that but there's another trick you can use scintillator screens and just a normal ccd high-speed camera or a normal ccd camera and to that show you in a second and you need a rotating table a rotating table comes starting a thousand dollars or something you get probably something where you buy so if you build yourself you can get the price tag definitely down in a 50k region again that's another set of thermographs actually in Dan Goldman's lab who's also a frequent lecturer here so he was actually studying and spilling tunnels inside of granular media and he had used also an x-ray source I think it's also a dental one on the rotation stage and he just used that he had a detector screen and he visualized that with a normal camera and then he could do thermographies and you can see here is three times the system of tunnels dug by the ends in there as a function of time and he also gives some information how he did that so yeah he said he's this phillips x-ray system and he has a high-speed camera phantom which is very critical and there's a there's different open source software to actually do the reconstruction Dan is using oscar but I'm no for sure there's at least one other very well developed system which you can use open access open source may arise so yes if you dedicate you can definitely build such a set of yourself you can do that with mudlap I've done it but you really want to have all the best and with us I would go for some of the two sets okay okay so you can either use Oscar or you can just go direct to Mike and then get his throat there's two other options I want to mention one of them is international collaboration so you might not have direct access to such as machine but the fact is that typically the data acquisition takes something like one two days and then you leave with a hard disk with hundred of gigabyte to terabyte of data and then you have about two years where you sit in front of your desktop and write programming code and analyze that so it's not too hard to sign find someone who has a tomography setup which lets you take data for one or two days in the terms of collaboration I think we'll have a special session on that later in this week or next week where you actually speak about that type of collaborations and there's a final option that is apply for beam tunnel synchrotrons we already had talked about this this morning it's also not really the spirit of the hands on school and I would also not advise it as a first way of doing tomography because synchrotron beam time is only given to people which can prove they can't do their science with a stationary machine and if you've not done any tomography proving that you cannot do it with a stationary machine will be really really hard this is something if you can prove that for example you need many many images very hard short exposure times or something like that then you can move to that but collaboration is definitely something to take into account okay I have about five minutes left let me make a couple of points in passing so one of them is I'm think there's something which really as a community not use enough and that's sharing data open data if we have taken this data for example from packings of sorry from packings of tumor of tetrahedra or ellipsoids and this took us a long time to take them and analyzing them and running software took even longer and now there is somebody sitting on our hard drive and no one else can ever use them we should as a community much more start to share our raw data for other people having different ideas and use repositories like try it Zenodo and the Sun is another one of those I think this should become almost obligatory for everyone using such type of equipment to actually later on give the data out the other thing is that's a very special remark for the hands-on lecture if you've not done it yet and you're coming bring your own sample you will have we will write your own code but you will also have a chance to take your own tomography with this machine so bring some small sample the size should be about like like I said this little box this little plastic box I have here could be a little bit smaller try to avoid metal plastic is better and this for example was a little wooden dog and now you can see one slice for a tomography we can very nicely see the fiber of the wood in here you can see that the furnish is something made of the color on the outside is made by some other material which must be much denser otherwise it wouldn't be actually being so much denser here in x-rays you can see a bit of the rope connecting those so you get a very nice inside of other things I not recommend bringing LA electronics like USB sticks which you still plan to use because we already killed integrated circuits of x-rays so don't put in your smartphone then I promise to speak about one of the reasons why beam hardening such a nuisance what you see here is a cardboard tube which fit with paper shavings from a document shredder and you can see in the radiogram very nicely there's a piece of a clip of a paper clip which has been used to clip that together the radiogram no big deal a little piece of metal it's dark you can see that let me show you the tomography reconstruction of that so we're going for the paper shreds and now you see over here there's something really bright and there's radiating lines coming out of that and these radiating lines is a typical beam hardening artifacts whenever you have something which is much higher in density the x-ray tomography reconstruction algorithm assumes basically that your mu is not changing a lot and it's not definitely not having beam hardening now if you have something in there which is beam hardening the construction goes weird and there's no real way to fix that in software I mean there's some algorithm to try to do that but honestly they're not really helping a lot at all so this is one of the reasons why for things like that you have to typically go to rather high energies or you have to use a trick which is called a filter and that's what you said in the beginning if I bring some extra material in here like a little bit of aluminium I already remove the low energy spectrum so what I do over here right before I actually go through my sample putting in this aluminium filter already took out all of those photons those photons never take place actually in the actual imaging but it's still aluminium disc I have put over here and that improves the beam hardening quite a bit this filtering and they actually doing this also if you look at the x-ray machine you will see you will see that right in front of the x-ray tube there's a little bit of an aluminium slice which is a slot piece which actually filters the air coming out and last three slides I want to show you I already said that you go somewhere you take images and you come back with gigabytes and you need months or years to actually analyze them the actual problem with x-rays is not so much the image acquisition because that's mostly turnkey by now for most problems at least at some places the actual problem is getting something out of it like here we have sample for example this little plastic tetrahedra of seven millimetres with a whole container of them we tap them we see how they compactify I can actually show you the movie so what we're interested is how do they change how they change the orientation you see now the more taps they give the system all those little tetrahedra change their position change your orientation we try to understand the mechanical stability of that sample to understand that we need to find the position and the orientation of all those little tetrahedra in there now as I already said this is a photograph of the sample this is a rendering that's just meaning which is putting it in an algorithm which sets everything to black or gray which is in trans transparent and the air is made transparent but nothing has been found here that is just a picture so to speak like that from the x-rays this is a geometric representation here we know from each tetrahedra the send-off mass and the different angles which give its rotation and going from here to here from here to here that's a full PhD for the expert photo you will actually see if you do the image analysis we have one step there which is called erosion which very easily separates the particles and what you do in the hands-on course you cannot do erosion over here because you face-to-face context tetrahedra can lie side-by-side an erosion will not get in between them and by not being able to get in between them it can also not separate them so you have to come up with new methods once you know how you do it it's actually rather simple and if you would have known in the beginning we could have done it probably in six months but figuring out all the ways not to do it and then finally find the way that's made the PhD actually and just to give you an idea that's actually PhD of Max Neudegger so what we actually ended up with doing is to follow me wrote some code so these are three slices through the cube of volume blue means there is some material gray means this is air and then we put in a probe body the probe body is this yellow tetrahedra over here and we rotate the trope body and we grow it and we shrink it around and try to maximize the overlap with the blue so you can see the projections of the tetrahedra is the orange over here and at that time the code was about running in real time so you can see okay it grows it has found this tetrahedra and we have an overlap of 99 percent we are happy it grows it found this tetrahedra it's happy now this guy has a problem has to rotate more than it thinks it takes it's a steepest ascender algorithm but at some point it figures out okay I have to rotate around now it's switching in and again with an overlap of 99.8% of the volume this way we identify all of these different tetrahedra in there detection rate of 99% or something like that but as I said this this is the hard part and that also means if you're interested in doing quantitative science with x-ray tomography you actually have to be able to group programming okay thank you for attention