An Analytical Example of Filtered Back Projection

One of my hobbies is to find tomography problems that can be solved analytically. I know this is artificial—all tomography for medical imaging uses numerical computation—but as a learning tool analytical analysis helps build insight. I have some nice analytical examples using the Fourier method to solve the tomography problem (see homework problems 26 and 27 in chapter 12 of Intermediate Physics for Medicine and Biology), but I don't have a complete analytical example to illustrate the filtered back projection method (see a previous post for a partial example). Russ Hobbie and I do include a numerical example in section 12.6 of IPMB. I have always wondered if I can do that example analytically. Guess what. I can! Well, almost.

Start with a top-hat function for your object

If we set x = 0, we can plot it as function of y.

The projection of this function is given in IPMB; Homework Problem 36 asks the reader to derive it.

Because the object looks the same from all directions, the projection is independent of the angle. Below is a plot of the projection as a function of x'. It is identical to the top panel of IPMB's Figure 12.22.

The next step is to filter the projection, which means we have to take its Fourier transform, multiply the transform by a high-pass filter, and then do the inverse Fourier transform. The Fourier transform of the projection is

This integral is not trivial, but Abramowitz and Stegun’s Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables contains (Page 360, Equation 9.1.20)

where J₁ is a first-order Bessel function (see Homework Problem 10). Because the projection is an even function, the sine part of the Fourier transform vanishes.

Filtering is easy; multiply by |k|/2π. The result is

To find the inverse Fourier transform, we need

This integral appears in Abramowitz and Stegun (Page 487, Equation 11.4.37)

After some simplification (which I leave to you), the filtered projection becomes

Below is a plot of the filtered projection, which you should compare with the middle panel of Fig. 12.22. It looks the same as the plot in IPMB, except in the numerical calculation there is some ringing near the discontinuity that is not present in the analytical solution

The final step is back projection. Because the projection is independent of the angle, we can calculate the back projection along any radial line, such as along the y axis

If |y| is less than a, the back projection is easy: you just get 1. Thus, the filtered back projection is the same as the object, as it should be. If |y| is greater than a, the result should be zero. This is where I get stuck; I cannot do the integral. If any reader can solve this integral (and presumably show that it gives zero), I would greatly appreciate hearing about it. Below is a plot of the result; the part in red is what I have not proven yet. Compare this plot with the bottom panel of Fig. 12.22.

What happens if you do the back projection without filtering? You end up with a blurry image of the object. I can solve this case analytically too. For |y| less than a, the back projection without filtering is

which is 4a times the complete elliptic integral of the second kind

For |y| greater than a, you get the more complicated expression

which is the incomplete elliptic integral of the second kind

The trickiest part of the calculation is determining the upper limit on the integral, which arises because for some angles the projection is zero (you run into the same situation in homework problem 35, which I highly recommend). Readers who are on the ball may worry that the elliptic integral is tabulated only for kappa less than one, but there are ways around this (see Abramowitz and Stegun, Page 593, Equation 17.4.16). When I plot the result, I get

which looks like Fig. 12.23 in IPMB.

So, now you have an analytical example that illustrates the entire process of filtered back projection. It even shows what happens if you forget to filter before back projecting. For people like me, the Bessel functions and elliptic integrals in this calculation are a source of joy and beauty. I know that for others they may be less appealing. To each his own.

I’ll rely on you readers to fill in the one missing step: show that the filtered back projection is zero outside the top hat.

Five Generations

A five generation picture.

When my first daughter Stephanie was born, we included her in this photo of five generations. From left to right are my maternal grandmother, my great-grandmother (born 1889), my daughter Stephanie (born 1988), me, and my mom. My great grandmother lived to be over 100 years old. I remember playing poker with her when I was young; she generally won and kept the money!

All five editions of
Intermediate Physics for Medicine and Biology.

Recently I took another five-generation photo. There now exist five generations (editions) of Intermediate Physics for Medicine and Biology. My office is one of the few places you can find all five on one bookshelf. I was coauthor on the fourth and fifth editions; the first three editions were authored by Russ Hobbie alone.

Suki with all five editions of
Intermediate Physics for Medicine and Biology.

The yellow book is the first edition of IPMB, published by John Wiley and Sons in 1978. The blue version with the yellow sine wave on the cover is the second edition, again published by Wiley in 1988. The green cover is the third edition, published by Springer with AIP Press in 1997. The blue fourth edition was published by Springer alone in 2007. Finally, the blue/purple fifth edition, again published by Springer, appeared in 2015. My dog Suki seems to like them all.

All five editions of
Intermediate Physics for Medicine and Biology.

I have a special fondness for the first edition, which I bought for a class taught by my PhD advisor John Wikswo at Vanderbilt University in the early 1980s (price: $31.95). That is where I learned much of my biological and medical physics. When Russ was preparing the second edition, he asked John and I to create some three-dimensional figures of the electrical potential and magnetic field of a nerve axon. There figures have appeared in each subsequent edition of IPMB, and are Figs. 7.13 and 8.14 in the fifth. My third edition is pretty beat up. It is the textbook I taught out of for several years after I arrived at Oakland University. The fourth and fifth editions I know best, as I helped write them (although Russ remains the primary force behind every edition).

All five editions of
Intermediate Physics for Medicine and Biology.

IPMB has changed over the years. The first seven chapters are the same in all versions, but Russ added chapters on charged membranes and biomagnetism in the second edition. The first edition’s chapter on signal analysis split into two in the second: one on one-dimensional signal analysis and another on two-dimensional images. The 4th edition picked up a chapter on ultrasound. The first edition’s chapter on x-rays fissioned into a chapter on how x-rays interact with tissue and a chapter on the medical uses of x-rays. Finally, the second edition introduced a chapter on magnetic resonance imaging. Early editions featured a figure on the cover. I particularly like the first edition’s electrocardiogram picture (Fig. 7.16 in the 5th edition). Russ and I planned on using a computed tomography illustration, Fig. 12.12, on the 4th edition cover, but Springer opted to use a generic cover with no figure.

Me holding all five editions of
Intermediate Physics for Medicine and Biology.

Working on revisions of IPMB has been a pleasure and an honor. But really, the five generations of IPMB is a tribute to Russ Hobbie and his vision of advancing the teaching of physics in medicine and biology, which he has pursued over nearly four decades. I hope you find the book as useful as I have.

Trivial Pursuit IPMB

Trivial Pursuit.

Trivial Pursuit is a popular and fun board game invented in the 1980s. While playing it, you learn many obscure facts (trivial, really).

When my daughter Kathy was in high school, she would sometimes test out of a subject by studying over the summer and then taking an exam. Occasionally I would help her study by skimming through her textbook and creating Trivial Pursuit-like questions. We would then play Trivial Pursuit using my questions instead of those from the game. I don’t know if it helped her learn, but she always passed those exams.

Readers of Intermediate Physics for Medicine and Biology may want a similar study aid to help them learn about biological and medical physics. Now they have it! At the book website you can download 100 game cards for Trivial Pursuit: IPMB. To play, you will need the game board, game pieces, and instructions of the original Trivial Pursuit, but you replace the game cards by the ones I wrote.

The game pieces for Trivial Pursuit.

In case you have never played, here are the rules in a nutshell. The board has a circle with spots of six colors. You roll a die and move your game piece around the circle, landing on the spots. Your opponent asks you a question about a topic determined by the color. If you answer correctly you roll again; if you are wrong your opponent rolls. There are special larger spots where a correct answer gets you get a little colored wedge. The first person to get all six colored wedges wins.

The original version of Trivial Pursuit had topics such as sports or literature. The Trivial Pursuit: IPMB topics are

• Numbers and Units (blue)
• People (pink)
• Anatomy and Physiology (yellow)
• Biological Physics (brown)
• Medical Physics (green)
• Mathematics (orange).

One challenge of an interdisciplinary subject like medical and biological physics is that you need a broad range of knowledge. I suspect mathematicians will find the math questions to be simple, but the biologists may find them difficult. Physicists may be unfamiliar with anatomy and physiology, and chemists may find all the topics hard. The beauty of the game is that it rewards a broad knowledge across disciplines.

A game card for Trivial Pursuit.

Many may find the People section most challenging. I suggest you only require the player to know the person’s last name, although the first name is also given on my game card. In Units and Numbers I generally only require numbers to be known approximately. The goal is to have an order-of-magnitude knowledge of biological parameters and physical constants. Many questions ask you to estimate the size of an object, like in Section 1.1 of IPMB. For the math and physics questions you may need a pencil and paper handy, because some of the questions contain equations. You can’t simply show your opponent the equation on the game card, because both the questions and answers are together. This is unlike the real Trivial Pursuit game cards, which had the answers on the back. Unfortunately, such two-sided cards are difficult to make.

I know the game is not perfect. Some questions are truly trivial and others ask for some esoteric fact that no one would be expected to remember. Some questions may have multiple answers of which only one is on the card. You can either print out the game cards (requiring 100 pieces of paper) or use a laptop or mobile device to view the pdf. I try to avoid repetitions, but with 100 game cards some may have slipped in inadvertently.

Trivial Pursuit.

I may try using Trivial Pursuit: IPMB next time I teach Biological Physics (PHY 325) or Medical Physics (PHY 326) here at Oakland University. It would be excellent for, say, the last day of class, or perhaps a day when I know many students will be absent (such as the Wednesday before Thanksgiving). It doesn’t teach important high-level skills, such as learning to use mathematical models to describe biology, or understanding how physics constrains the way organisms evolve. You can’t teach a complex and beautiful subject like tomography using Trivial Pursuit. But for learning a bunch of facts, the game is useful.

Enjoy!

Science Blogging

Science Blogging:
The Essential Guide,
by Wilcox, Brookshire, and Goldman.

After writing this blog for nine years, I decided it is time to figure out what I’m doing. So I read the book Science Blogging: The Essential Guide (Yale University Press, 2016), edited by Christie Wilcox, Bethany Brookshire, and Jason Goldman. Their preface concludes

By bringing together some of the most experienced voices from around the science blogosphere, we hope this book will have something to teach everyone. Whether you’re just getting started, have some blog posts under your belt, or are looking for fresh inspiration, you are not alone. The science communication community may seem overwhelming, but it’s friendly. Dive in and show us what you can do. Seriously. Tweet us and show us your stuff. And use our hashtag, #SciBlogGuide, and find us online at http://www.theopennotebook.com/science-blogging-essential-guide.

I enjoyed Science Blogging, but oddly I didn’t feel connected to what many of the authors discussed. What you are reading now is less a science blog and more an auxiliary resource for the textbook that Russ Hobbie and I wrote: Intermediate Physics for Medicine and Biology. My goal is to provide materials that help instructors use the book in their classes, and extend and update topics that readers of the book are interested in. I view this blog as being similar to the solutions manual and the errata: it augments the book. The closest Science Blogging came to my blog is in the last chapter, “From Science Blog to Book,” by Brian Switek. But his chapter was primarily about using a blog as a springboard to writing a book, and only at the end of his chapter did he add that “there’s no reason to stop blogging when your book comes out.” I did the opposite. My blog began after Russ and I published the 4th edition of IPMB, and my goal was to improve sales. Has it worked? It’s hard to say, because our sales have never been spectacular. I hope it has had some impact.

Matt Shipman’s chapter on “Metrics” inspired me to look into the statistics for my blog. The post with by far the most page views is Frank Netter, Medical Illustrator. While I liked that post, I don’t know why it has more than three times as many page views as the next most viewed entry. In fact, I see no correlation between the number of page views and what I consider quality or relevance.

Bethany Brookshire wrote a chapter on “Science Blogging and Money.” I like money as much as the next guy, but I don’t subject my dear readers to ads. Hobbieroth.blogspot.com is add-free. There is one exception: each blog post contains a reference to IPMB. I guess that is a sort of advertisement.

Several authors talked about building a following using Twitter. I don’t tweet, but should I? Do you want to hear about IPMB several times a day? I don’t think so. I’ll continue posting once a week; every Friday morning, like clockwork. By the way, what’s a hashtag? I always thought I was a hep cat, but I guess not.

My favorite chapter was Ed Yong’s essay about “Building an Audience for Your Blog.” Accumulating a large following is not my goal; I am more a citation man than a page view man. Yong’s advice is that “you have to have something worth writing about, and you have to write it well.” That sums it up nicely. I think that physics applied to medicine and biology is something worth writing about; I hope I write it well. Yong also writes “picture your ideal readers in your head: who are they?” While I hope anyone interested in biological physics or medical physics will find my blog useful, I don’t write it for such a broad audience. I write it for the students and teachers using Intermediate Physics for Medicine and Biology. And, I write it for myself. I hope you enjoy it. I do.