Integral of the Bessel Function

Have you ever been reading a book, making good progress with everything making sense, and then you suddenly stop at say “wait… what?”. That happened to me recently as I was reading Homework Problem 31 in Chapter 12 of Intermediate Physics for Medicine and Biology. (Wait…what? I’m a coauthor of IPMB! How could there be any surprises for me?) The problem is about calculating the two-dimensional Fourier transform of 1/r, and it supplies the following Bessel function identity 

The function J₀ is a Bessel function of the first kind of order zero. What surprised me is that if you let x = kr, you get that the integral of the Bessel function is one,

Really? Here’s a plot of J₀(x).

It oscillates like crazy and the envelope of those oscillations falls off very slowly. In fact, an asymptotic expansion for J₀ at large x is

The leading factor of 1/√x decays so slowly that its integral from zero to infinity does not converge. Yet, when you include the cosine so the function oscillates, the integral does converge. Here’s a plot of

The integral approaches one at large x, but very slowly. So, the expression given in the problem is correct, but I sure wouldn’t want to do any numerical calculations using it, where I had to truncate the endpoint of the integral to something less than infinity. That would be a mess!

Here’s another interesting fact. Bessel functions come in many orders—J₀, J₁, J₂, etc.—and they all integrate to one.

Who’s responsible for these strangely-behaved functions? They’re named after the German astronomer Friedrich Bessel but they were first defined by the Swiss mathematician Daniel Bernoulli (1700–1782), a member of the brilliant Bernoulli family. The Bernoulli equation, mentioned in Chapter 1 of IPMB, is also named for Daniel Bernoulli. 

There was a time when I was in graduate school that I was obsessed with Bessel functions, especially modified Bessel functions that don’t oscillate. I’m not so preoccupied by them now, but they remain my favorite of the many special functions encountered in physics.

The Preface as Poetry

Alexander Pope

I mentioned before in this blog that I’m reading Will and Ariel Durant’s eleven-volume masterpiece The Story of Civilization. Recently, in Volume Nine about The Age of Voltaire, they discussed Alexander Pope, the eighteenth century English poet who, among other things, translated the Iliad into English poetry using iambic pentameter heroic couplets. Thus inspired, I decided to translate the preface of Intermediate Physics for Medicine and Biology from prose to poetry. I’d tell you to “Enjoy!” but I’m not sure that’ll be possible for such doggerel.

In the preface to the third edition, 

Russell Hobbie set out on a mission. 

For the two years before seventy three 

He audited the University

Of Minnesota’s medical courses. 

He found a lack of physics discourses. 

An intermediate physics class would 

For these students be so useful and good. 

This book is the result of when he taught 

Such a course that students needed a lot. 

He hoped that even those physics teachers 

Scared of bio would value its features. 

And doctors would find it a good reference 

With the physics concepts never too dense. 

Because the book was used by those whose tools 

From math were not vast he set down four rules: 

Calculus would be used without regret, 

And reviewed in detail when not seen yet. 

Readers should know the vocabulary 

But it’s told from the start, it’s not scary. 

He did not skip steps in derivations, 

And shunned any weirdo math notations. 

Hobbie added someone to help him write 

The Fourth Edition, and they did not fight. 

They wrote a new chapter, but made sure that 

The new edition did not get too fat. 

They added more than one homework problem, 

And a solution manual for them. 

Chapter One reviews biomechanics, 

Stress and strain, also fluid dynamics. 

Then Two discusses the exponential 

A math function that’s truly essential.

Next Three deals with temperature and heat, 

Biothermodynamics it does treat. 

Diffusion’s the topic of Chapter Four 

A random walk, and drift, and so much more. 

Chapter Five describes flow across membranes, 

And osmosis from ions that it contains. 

Six covers bioelectricity, 

And a model by Hodgkin and Huxley. 

Chapter Seven contains the EKG, 

And defibrillation is really key. 

Chapter Eight deals with all things magnetic, 

Interesting, but not too poetic. 

Nine begins with the model of Donnan, 

Then Nernst-Planck, Debye-Huckel, and so on. 

Chapter Ten has lots of mathematics, 

With feedback and chaos and dynamics. 

Eleven derives Fourier transforms.

Least squares and correlations, it brainstorms. 

Next Twelve analyzes tomography, 

Which allows you to image in 3D!

Chapter Thirteen considers ultrasound, 

A topic that you’ll find really profound. 

Next Chapter Fourteen summarizes light, 

Mix red and green and blue and you get white!

Fifteen considers photons and matter: 

Photoelectric and Compton Scatter. 

Chapter Sixteen is about the x-ray, 

Where the unit for dose is named the gray. 

Seventeen is about technetium, 

Nuclear medicine, and even then some. 

Then Eighteen is not about anything 

But magnetic resonance imaging. 

Biophysics is a subject quite broad, 

Of which we are always completely awed. 

Our book has grown and become large enough, 

To fit molecular stuff would be tough. 

We would like to get any corrections,

Or even hear about your suggestions. 

And last we thank our long-suffering wives, 

We know that this book disrupted their lives.

The Age of Voltaire,
by Will and Ariel Durant.

 The Iliad, translated by Alexander Pope (the heroic couplets start at 2:20).

https://www.youtube.com/watch?v=28RNGOCIzYI

 

Numerical Recipes is Online

Numerical Recipes,
by Press, Teukolsky, Vetterling, and Flannery.

In Intermediate Physics for Medicine and Biology, Russ Hobbie and I often refer to Numerical Recipes: The Art of Scientific Computing, by William Press, Saul Teukolsky, William Vetterling, and Brian Flannery. We usually cite the second edition of the book with programs written in C (1995), but the copy on my bookshelf is the second edition using Fortran 77 (1992). For those of you who don’t own a copy of this wonderful book, did you know you can read it online?

The address of the Numerical Recipes website is easy to remember: numerical.recipes. There you will find free copies of the second editions of Numerical Recipes for Fortran 77, Fortran 90, C, and C++ (2002). If you want easy, quick access to the third edition (2007), you will have to pay a fee. But if you are willing to put up with brief delays and annoying messages (which the authors call “nags”), you also can read the third edition for free.

The text below is from the Preface to the third edition.

“I was just going to say, when I was interrupted...” begins Oliver Wendell Holmes in the second series of his famous essays, The Autocrat of the Breakfast Table. The interruption referred to was a gap of 25 years. In our case, as the autocrats of Numerical Recipes, the gap between our second and third editions has been “only” 15 years. Scientific computing has changed enormously in that time.

The first edition of Numerical Recipes was roughly coincident with the first commercial success of the personal computer. The second edition came at about the time that the Internet, as we know it today, was created. Now, as we launch the third edition, the practice of science and engineering, and thus scientific computing, has been profoundly altered by the mature Internet and Web. It is no longer difficult to find somebody’s algorithm, and usually free code, for almost any conceivable scientific application. The critical questions have instead become, “How does it work?” and “Is it any good?” Correspondingly, the second edition of Numerical Recipes has come to be valued more and more for its text explanations, concise mathematical derivations, critical judgments, and advice, and less for its code implementations per se.

Recognizing the change, we have expanded and improved the text in many places in this edition and added many completely new sections. We seriously considered leaving the code out entirely, or making it available only on the Web. However, in the end, we decided that without code, it wouldn’t be Numerical Recipes. That is, without code you, the reader, could never know whether our advice was in fact honest, implementable, and practical. Many discussions of algorithms in the literature and on the Web omit crucial details that can only be uncovered by actually coding (our job) or reading compilable code (your job). Also, we needed actual code to teach and illustrate the large number of lessons about object-oriented programming that are implicit and explicit in this edition.

Russ and I cited Numerical Recipes in IPMB when we discussed integration, least squares fitting, random number generators, partial differential equations, the fast Fourier transform, aliasing, the correlation function, the power spectral density, and bilinear interpolation. Over the years, in my own research I have consulted the book about other topics, including solving systems of linear equations, evaluation of special functions, and computational analysis of eigensystems.

I highly recommend Numerical Recipes to anyone doing numerical computing. I found the book to be indispensable.

Is Shot Noise Also White Noise?

In Chapters 9 and 11 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss shot noise.

9.8.1 Shot Noise

The first (and smallest) limitation [on our ability to measure current] is called shot noise. It is due to the fact that the charge is transported by ions that move randomly and independently through the channels....

11.16.2 Shot Noise

Chapter 9 also mentioned shot noise, which occurs because the charge carriers have a finite charge, so the number of them passing a given point in a circuit in a given time fluctuates about an average value. One can show that shot noise is also white noise [my italics].

Introduction to Membrane Noise,
by Louis DeFelice

How does one show that shot noise is white noise (independent of frequency)? I’m going to follow Lou DeFelice’s explanation in his book Introduction to Membrane Noise (cited in IPMB). I won’t give a rigorous proof. Instead, I’ll first state Campbell’s theorem (without proving it), and then show that the whiteness of shot noise is a consequence of that theorem.

Campbell’s Theorem

To start, I’ll quote DeFelice, but I will change the names of a few variables.

Suppose N impulses i(t) arrive randomly in the time interval T. The sum of these will result in a random noise signal I(t). This is shown qualitatively in Figure 78.1.

Below is my version of Fig. 78.1.

A diagram illustrating the sum of N impulses, i(t), each shown in red, arriving randomly in the time interval T. The blue curve represents their sum, I(t), and the green dashed line represents the average, <I(t)>. Adapted from Fig. 78.1 in Introduction to Membrane Noise by Louis DeFelice.

DeFelice shows that the average of I(t), which I’ll denote <I(t)>, is

Here he lets T and N both be large, but their ratio (the average rate that the impulses arrived) remains finite.

He then shows that the variance of I(t), called σ_I², is

Finally, he writes

In order to calculate the spectral density of I(t) from i(t) we need Rayleigh’s theorem [also known as Parseval’s theorem]…

where î(f) is the Fourier transform of i(t) [and f is the frequency].

He concludes that the spectral density S_I(f) is given by

These three results (for the average, the variance, and the spectral density) constitute Campbell’s theorem.

Shot Noise

Now, let’s analyze shot noise by using Campbell’s theorem assuming the impulse is a delta function (zero everywhere except at t = 0 where it’s infinite). Set i(t) = q δ(t), where q is the charge of each discrete charge carrier.

First, the average <I(t)> is simply Nq/T, or the total charge divided by the total time. 

Second, the variance is the integral of the delta function squared. When any function is multiplied by a delta function and then integrated over time, you get that function evaluated at time zero. So, the integral of the square of the delta function gives the delta function itself evaluated at zero, which is infinity. Yikes! The variance of shot noise is infinite.

Third, to get the spectral density of shot noise we need the Fourier transform of the delta function. 

The key point is that S_I(f) is independent of frequency; it’s white.

DeFelice ends with

This [the expression for the spectral density] is the formula for shot noise first derived by Schottky (1918, pp. 541-567) in 1918. Evidently, the variance defined as

is again infinite; this is a consequence of the infinitely small width of the delta function.

As DeFelice reminds us, shot noise is white because the delta function is infinitely narrow. As soon as you assume i(t) has some width (perhaps the time it takes for a charge to cross the membrane), the spectrum will fall off at high frequencies, the variance won’t be infinite (thank goodness!), and the noise won’t be white. The bottom line is that shot noise is white because the Fourier transform of a delta function is a constant.

Conclusion

Perhaps you’re thinking I haven’t helped you all that much. I merely changed your question from “why is shot noise white” to “how do I prove Campbell’s theorem.” You have a point. Maybe proving Campbell’s theorem can be the story of another post.

I met Lou DeFelice in 1984, when I was a graduate student at Vanderbilt University and he came to give a talk. In the summer of 1986, my PhD advisor John Wikswo and I traveled to Emory University to visit DeFelice and Robert DeHaan. During that trip, Wikswo and I were walking across the Emory campus when Wikswo decided he knew a short cut (he didn’t). He left the sidewalk and entered a forest, with me following behind him. After what seemed like half an hour of wandering through a thicket, we emerged from the woods at a back entrance to the Yerkes Primate Research Center. We’re lucky we weren’t arrested.

DeFelice joined the faculty at Vanderbilt in 1995, and we both worked there in the late 1990s. He was a physicist by training, but spent most of his career studying electrophysiology. Sadly, in 2016 he passed away.

The Central Slice Theorem: An Example

The central slice theorem is key to understanding tomography. In Intermediate Physics for Medicine and Biology, Russ Hobbie and I ask the reader to prove the central slice theorem in a homework problem. Proofs are useful for their generality, but I often understand a theorem better by working an example. In this post, I present a new homework problem that guides you through every step needed to verify the central slice theorem. This example contains a lot of math, but once you get past the calculation details you will find it provides much insight.

The central slice theorem states that taking a one-dimensional Fourier transform of a projection is equivalent to taking the two-dimensional Fourier transform and evaluating it along one direction in frequency space. Our “object” will be a mathematical function (representing, say, the x-ray attenuation coefficient as a function of position). Here is a summary of the process, cast as a homework problem.

Section 12.4 

Problem 21½. Verify the central slice theorem for the object

(a) Calculate the projection of the object using Eq. 12.29,

Then take a one-dimensional Fourier transform of the projection using Eq. 11.59,

 

(b) Calculate the two-dimensional Fourier transform of the object using Eq. 12.11a,

Then transform F̂(k_x,k_y ) to F̂(θ,k) by converting from Cartesian to polar coordinates in frequency space.

(c) Compare your answers to parts (a) and (b). Are they the same?

I’ll outline the solution to this problem, and leave it to the reader to fill in the missing steps. 

 

Fig. 12.12 from IPMB, showing how to do a projection.

The Projection 

Figure 12.12 shows that the projection is an integral of the object along various lines in the direction θ, as a function of displacement perpendicular to each line, x'. The integral becomes

Note that you must replace x and y by the rotated coordinates x' and y'

You can verify that x² + y²= x'² + y'².

After some algebra, you’re left with integrals involving e^−by'2 (Gaussian integrals) such as those analyzed in Appendix K of IPMB. The three you’ll need are

The resulting projection is

Think of the projection as a function of x', with the angle θ being a parameter.

 

The One-Dimensional Fourier Transform

The next step is to evaluate the one-dimensional Fourier transform of the projection

The variable k is the spatial frequency. This integral isn’t as difficult as it appears. The trick is to complete the square of the exponent

Then make a variable substitution u = x' + ik⁄2b. Finally, use those Gaussian integrals again. You get

This is our big result: the one-dimensional Fourier transform of the projection. Our next goal is to show that it’s equal to the two-dimensional Fourier transform of the object evaluated in the direction θ.

Two-Dimensional Fourier Transform

To calculate the two-dimensional Fourier transform, we must evaluate the double integral

The variables k_x and k_y are again spatial frequencies, and they make up a two-dimensional domain we call frequency space.

You can separate this double integral into the product of an integral over x and an integral over y. Solving these requires—guess what—a lot of algebra, completing the square, and Gaussian integrals. But the process is straightforward, and you get

Select One Direction in Frequency Space

If we want to focus on one direction in frequency space, we must convert to polar coordinates: k_x = k cosθ and k_y = k sinθ. The result is 

This is exactly the result we found before! In other words, we can take the one-dimensional Fourier transform of the projection, or the two-dimensional Fourier transform of the object evaluated in the direction θ in frequency space, and we get the same result. The central slice theorem works.

I admit, the steps I left out involve a lot of calculations, and not everyone enjoys math (why not?!). But in the end you verify the central slice theorem for a specific example. I hope this helps clarify the process, and provides insight into what the central slice theorem is telling us.

Projections and Filtered Projections of a Square

Chapter 12 of Intermediate Physics for Medicine and Biology describes tomography. Russ Hobbie and I write

The reconstruction problem can be stated as follows. A function f(x,y) exists in two dimensions. Measurements are made that give projections: the integrals of f(x,y) along various lines as a function of displacement perpendicular to each line. For example, integration parallel to the y axis gives a function of x,

as shown in Fig. 12.12. The scan is repeated at many different angles θ with the x axis, giving a set of functions F(θ, x'), where x' is the distance along the axis at angle θ with the x axis.

One example in IPMB is the projection of a simple object: the circular top-hat

The projection can be calculated analytically

It’s independent of θ; it looks the same in every direction.

Let’s consider a slightly more complicated object: the square top-hat

This projection can be found using high school geometry and trigonometry (evaluating it is equivalent to finding the length of lines passing through the square at different angles). I leave the details to you. If you get stuck, email me (roth@oakland.edu) and I’ll send a picture of some squares and triangles that explains it.

The plot below shows the projections at four angles. For θ = 0° the projection is a rectangle; for θ = 45° it’s a triangle, and for intermediate angles (θ = 15° and 30°) it’s a trapezoid. Unlike the circular top-hat, the projection of the square top-hat depends on the direction.

The projections of a square top-hat, at different angles.

What I just described is the forward problem of tomography: calculating the projections from the object. As Russ and I wrote, usually the measuring device records projections, so you don’t have to calculate them. The central goal of tomography is the inverse problem: calculating the object from the projections. One way to perform such a reconstruction is a two-step procedure known as filtered back projection: first high-pass filter the projections and then back project them. In a previous post, I went through this entire procedure analytically for a circular top-hat. Today, I go through the filtering process analytically, obtaining an expression for the filtered projection of a square top-hat. 

Here we go. I warn you, there’s lots of math. To perform the filtering, we first calculate the Fourier transform of the projection, C_F(θ,k). Because the top-hat is even, we can use the cosine transform

where k is the spatial frequency.

Next, place the expression for F(θ,x') into the integral and evaluate it. There’s plenty of book-keeping, but the projection is either constant or linear in x', so the integrals are straightforward. I leave the details to you; if you work it out yourself, you’ll be delighted to find that many terms cancel, leaving the simple result 

To high-pass filter C_F(θ,k), multiply it by |k|/2π to get the Fourier transform of the filtered projection C_G(θ,k)

Finally, take the inverse Fourier transform to obtain the filtered projection G(θ,x') 

Inserting our expression for C_G(θ,k), we find

This integral is not trivial, but with some help from WolframAlpha I found

where Ci is the cosine integral. I admit, this is a complicated expression. The cosine integral goes to zero for large argument, so the upper limit vanishes. It goes to negative infinity logarithmically at zero argument. We’re in luck, however, because the four cosine integrals conspire to cancel all the infinities, allowing us to obtain an analytical expression for the filtered projection

We did it! Below are plots of the filtered projections at four angles. 

The filtered projections of a square top-hat, at different angles.

The last thing to do is back project G(θ,x') to get the object f(x,y). Unfortunately, I see no hope of back-projecting this function analytically; it’s too complicated. If you can do it, let me know.

Why must we analyze all this math? Because solving a simple example analytically provides insight into filtered back projection. You can do tomography using canned computer code, but you won’t experience the process like you will by slogging through each step by hand. If you don’t buy that argument, then another reason for doing the math is: it’s fun!

A Portable Scanner for Magnetic Resonance Imaging of the Brain

A Portable Scanner for Magnetic
Resonance Imaging of the Brain
Cooley et al., Nat. Biomed. Eng., 2020

Chapter 18 of Intermediate Physics for Medicine and Biology describes magnetic resonance imaging. MRI machines are usually heavy, expensive devices installed in hospitals and clinics. A recent article by Clarissa Cooley and her colleagues in Nature Biomedical Engineering, however, describes a portable MRI scanner. The abstract states

Access to scanners for magnetic resonance imaging (MRI) is typically limited by cost and by infrastructure requirements. Here, we report the design and testing of a portable prototype scanner for brain MRI that uses a compact and lightweight permanent rare-earth magnet with a built-in readout field gradient. The 122-kg low-field (80 mT) magnet has a Halbach cylinder design that results in a minimal stray field and requires neither cryogenics nor external power. The built-in magnetic field gradient reduces the reliance on high-power gradient drivers, lowering the overall requirements for power and cooling, and reducing acoustic noise. Imperfections in the encoding fields are mitigated with a generalized iterative image reconstruction technique that leverages previous characterization of the field patterns. In healthy adult volunteers, the scanner can generate T1-weighted, T2-weighted and proton density-weighted brain images with a spatial resolution of 2.2 × 1.3 × 6.8 mm³. Future versions of the scanner could improve the accessibility of brain MRI at the point of care, particularly for critically ill patients.

Cooley et al.’s design has four attributes.

1. It’s designed for imaging the head only. Most critical care MRIs are of the brain, so focusing on imaging the head is not as limiting as you might think. By restricting the device to the head they are able to reduce the weight of their prototype to 230 kg (about 500 pounds); not something you could carry in your pocket, but light enough to be transported in an ambulance or wheeled on a cart. The power required, about 1.7 kW, is far less than for a traditional MRI device, so the portable scanner can be operated from a standard wall outlet.
2. The static magnetic field is produced by permanent magnets. Typical MRI scanners create a static field of a few Tesla using a superconductor, which must be kept cold. Cooley et al.’s device avoids cryogenics completely by using room-temperature, permanent neodymium magnets in a Halbach configuration, producing a static magnetic field of 0.08 T. The lower field strength reduces the signal-to-noise ratio, so advanced MRI techniques such as echo-planar, functional, or diffusion tensor imaging are not feasible. However, many emergency MRIs are used to diagnose traumatic brain injury and don’t rely on these more advanced techniques. The Halbach design results in a small magnetic field outside the scanner, which minimizes safety hazards associated with iron-containing objects being sucked into the scanner.
3. The readout gradient is static. In IPMB, Russ Hobbie and I describe how magnetic field gradients are used to map the Larmor frequency to position. Usually the readout gradient of an MRI pulse sequence is turned on and off as needed. By making this gradient static, Cooley and her collaborators eliminate the need for a power supply to drive it. Most MRI pulse sequences require gradients in three directions, and in Cooley et al.’s device the gradients in the other two directions must still be switched on and off in the traditional way. One side effect of the reduced gradient switching is that this MRI scanner is quieter than a traditional device. This may seem like a minor advantage, but try having your head imaged in a typical MRI scanner with its gradient switching causing a deafening racket.
4. Much of the signal analysis is switched from hardware to software. Because of nonlinearities in the gradient magnetic field, traditional Fourier transform algorithms to convert from spatial frequency to position produce artifacts, and iterative methods that correct for these errors are needed.

Cooley et al.’s article fascinated me because of its educational value; the challenges they face force readers to think carefully about the design parameters and limitations of traditional MRI. If you want to learn more about normal MRI scanners, read this article to see how researchers had to modify the traditional design to overcome its limitations. 

Low-cost MRI systems for brain imaging. by Clarissa Cooley.

https://www.youtube.com/watch?v=bZz3-lmWv4I