Friday, November 16, 2012

The Sinogram

I love sinograms. They are rare and fascinating mixtures of science and art, and often are quite beautiful. One should be able to look at a sinogram and intuitively picture the two-dimensional image. Unfortunately, I rarely can do this, except for the most simple examples.

Russ Hobbie and I define the sinogram in the 4th edition of Intermediate Physics for Medicine and Biology. We explain how to calculate the projection, F(θ, x'), from the image, f(x,y). This transformation and its inverse—determining f(x,y) from F(θ,x')—is at the heart of many imaging algorithms, such as those used in computed tomography.
The process of calculating F(θ, x') from f(x, y) is sometimes called the Radon transformation. When F(θ, x') is plotted with x’ on the horizontal axis, θ on the vertical axis, and F as the brightness or height on a third perpendicular axis, the resulting picture is called a sinogram. For example, the projection of f(x, y) = δ(x − x0)δ(y − y0) is F(θ, x') = δ(x' − (x0 cos θ + y0 sin θ)). A plot of this object and its sinogram is shown in Fig. 12.17.
Figure 12.17 does indeed contain a sinogram, but a very simple one: the sinogram of a point is just a sine wave. The reader is asked to produce a somewhat more complicated sinogram in homework Problem 29.
Problem 29 An object consists of three δ functions at (0, 2), (√3,−1), and (−√3,−1). Draw the sinogram of the object.
This sinogram consists of three braided sine waves. I like this example, because its simple enough that you the reader should be able to reason out the structure of the sinogram by imagining the projection in your head, but it is complicated enough that its not trivial.

When preparing the 4th edition of Intermediate Physics for Medicine and Biology, I derived a couple new homework problems (Chapter 12, Problems 23 and 24) for which the inverse transformation can be solved analytically. I think these are useful exercises that build intuition with the Fourier transform method of reconstructing an image (see Fig. 12.20, top path). It occurs to me now, however, that while these problems do provide insight and practice for the mathematically inclined reader, they also offer the opportunity to further illustrate the sinogram. So this week I made the figures below, showing the image f(x,y) on the left and the corresponding sinogram F(θ,x') on the right, for the functions in Problems 23 and 24.
An object (left) and its sinogram (right) corresponding to Chapter 12, Homework Problem 23 in Intermediate Physics for Medicine and Biology.
Problem 23.
An object (left) and its sinogram (right) corresponding to Chapter 12, Homework Problem 24 in Intermediate Physics for Medicine and Biology.
Problem 24.
Let us try to interpret these pictures qualitatively. The vertical axis in the sinogram (right panel) indicates the angle, specifying the direction of the projection (the direction that the x-rays come from, to use CT terminology). The bottom of the θ axis is an angle of zero indicating x-rays are incident on the image from the bottom, the middle of the θ axis is x-rays incident from the left, and the top of the θ axis is x-rays incident from the top (see Fig. 12.12). Some authors extend the θ axis so it ranges from 0 to 360°, but to me that seems unnecessary since having the x-rays come from one side or the opposite side does not matter; it provides no new information. Its best if you, dear reader, pause now and stare at these sinograms until you understand how they relate to the image. If you really want to build your intuition, cover the left panel, and try to predict what the hidden image looks like from just the right panel. Or, solve homework Problems 30 and 31 in Chapter 12, and then plot both the image and its sinogram like I do above.

This website has some nice examples of sinograms. For instance, a sinogram of a line is just a point. Think about it and sketch some projections to convince yourself this is correct. Also this website shows a sinogram of a square located away from the center of the image (it looks like the sinogram above for Fig. 23, but with interesting bright curves tenuously weaving throughout the sinogram arising from the corners of the square). Finally, the website shows the sinogram of an image known as a Shepp-Logan head phantom. (Warning, the website displays its sinograms rotated by 90° compared to the way Russ and I plot them; it plots the angle along the horizontal axis.) The video shown below provides additional insight into the construction of the sinogram for the Shepp-Logan head phantom.

Here is one of my favorite images: a detailed image of a brain, and its lovely sinogram. If you can do the inverse transformation of this complicated sinogram in your head, you’re a better medical physicist than I am. 

An image of a brain (left) and its sinogram (right).
An image of a brain, and its sinogram,
adapted from Wikipedia.


  1. Nice post, thanks.

    Do(es) your Oakland (course)es cover this far into IPMB? I only got a one semester teaser circa 2nd Edition which covered through chapter 9. I've got to find a way to do the entire book as a two semester sequence.

    Come on Brad, don't hog the stuffing. Surely as an author you could get a list of institutions (instructors) that use IPMB in their curriculum, no? Where can we go in person or online to take a course based on all of IPMB?

    Now that's a problem we'd like you to solve as we wish you Happy Thanksgiving and thanks for IPMB.

  2. But I love the stuffing......

    At OU my Biological Physics class (PHY 325) covers chapters 1-10 (but we skip Chpt 9), and my Medical Physics class (PHY 326) covers chapters 11-18. Of course, we skip some topics within chapters, but that is the outline.

    I will go through some of my old emails, and see if I can figure out other institutions that use IPMB for their class. I continue to toy with the idea of doing an online version of those two classes, but I am not seriously planning on doing that soon.

    Russ and I hope the book is clear enough that someone who reads the book and works the problems could get 99% of the learning without actually taking the class. You could always email one of us with questions.

  3. The following schools have contact Russ or I at one time or the other and said they were at least considering using IPMB for a class:

    University of Texas Health Science Center at San Antonio

    University of Pennsylvania

    King Abdulaziz University, Saudi Arabia

    Pukyong National University in Korea

    Hillsdale College

    University of Central Oklahoma

    DRCC - IFGW – UNICAMP, Brazil

    Miami Univ of Ohio

    Université de Strasbourg, France

  4. Thanks for the follow-up. My thought was an author might have access to the publishers' info on who is buying the text and from that a list of institutions that have adopted it for use.

    Part of the beauty of IPMB is that it is indeed a solid book for self-study. It's transformative for a devoted student. For the person no longer concerned about credits and credentials, a formal course simply "helps get the person to the gym" for the transformative exercise of the mind.

    A final on my wish list is for an accompanying set of programs, written in something practical like Matlab, which could be used as a ready-made toolset. A Matlab work kit based on the text and problems would be an awesome extension of the work and would enable an enormous advancement of quality research.

    I never availed myself of the Pascal routines that were part of my 2nd edition, but would definitely use ready-made routines in Matlab or Mathematica--packages that are ubiquitous and that have a developed set of input/output/export functionality.

    I think an amateur could do wonders with that type of tool-kit.