INTRODUCTION

IMAGE RECONSTRUCTION

Basic 3D image reconstruction does not need to be a complicated topic. The main principle of image reconstruction is this:

When multiple 2D projection images are acquired of an object from many angles, we can use mathematical tools to reconstruct a 3D representation of that object.

It is with this principle that we are able to acquire 3D images in medical imaging modalities: Computed Tomography (CT), Positron Emission Tomography (PET), Single Photon Emission Tomography (SPECT). There already exists on the internet many useful resources for understanding image reconstruction (Wikipedia, SPECT Reconstruction (Bruyant, JNM), …plus many others).

Since this page was created by a nuclear medicine specialist, the animations were designed with respect to emission tomography (PET, SPECT), yet the concepts are also appropriate for CT.

WHY THIS PAGE WAS CREATED

As an educator in medical imaging, I believe animations can play a large role in helping students or inquisitive minds understand the principles of tomography. After searching the web, I felt this resource was lacking, and that prompted my creation of this page.

If you find this page useful, we’d love to hear it. The feedback we receive helps us justify further development of the page/site.

Any comments/questions can be emailed to adam.kesner@fulbrightmail.org

WEBSITE UPDATES:

Page version 1.0, March 2014: Page launched.

Page version 1.1, December, 2016: gifmaker executable added (see bottom of page).

Page version 1.2, May 9th, 2016: backprojection gif added.

Page version 2.0, March, 2017: Page relocated to KesnersMedicalPhyscs.com (formerly hosted on University of Colorado website)

DOWNLOADABLE CONTENT

Any gif image can be downloaded by simply right clicking and choosing “save as”. In addition, several sets of images have been generated, and users may download zipped folder here (PET, CT, Phantom, Horse). All gif images will play as animations when opened with modern internet browsers.

A PowerPoint presentation is also available for download: Kesner-Haeggstroem Fundamentals of Medical Image Reconstruction Explained with Animations Lecture. It consists of 13 slides (including a title and final slide). The main topics covered in the presentation are: principles of sinograms/image data storage, forward projection, principles of PET acquisitions, and filtered back-projection. The slides set is free for download, distribution, and use.

To generate your own gifs, see bottom of page.

Users are welcome to use images for any non-commercial use.

THE IMAGING PROCESS

The main steps in imaging are

ANALYTIC VS ITERATIVE RECONSTRUCTION

There are two main types of mathematical algorithms for image reconstruction: analytic reconstruction (filtered back projection) and iterative reconstruction.

• Analytic reconstruction: on this website we focus on an image reconstruction technique called filtered back projection (FBP). The mathematics of FBP are based on the central slice theorem (link-Wikipedia), but are not discussed here.

• Iterative reconstruction: these algorithms involve a feedback process that permits sequential adjustments of an estimated image so that its virtual acquisition corresponds to the raw acquisition. They run by repeating (ITERATING) two distinct steps: (1) Expected projections are calculated by forward projecting data (using system matrix), and is based on activity distribution estimation from the previous iteration, and (2) the current image estimate is compared to the raw acquisition and updated so as to maximize the likelihood it is the “correct” image estimation.

FBP has been the reconstruction algorithm traditionally used for medical imaging. It is much faster, simpler, reproducible, and linear (performs uniformly across environments). Now that computing power is getting more accessible, many vendors are incorporating iterative reconstruction techniques into their systems. While iterative reconstruction is more complex, it has advantages in that it is capable of dealing with noise and other practical issues by incorporating their expected impact/uncertainty into the reconstruction process.This website shows animations of FBP reconstruction.

FILTERED BACK PROJECTION (FBP)

The main steps involved in a filtered back projection image acquisition include:

1. Forward projection (data acquired and forward projected into sinogram space)

2. Data is filtered (the filter in filtered back projection)

3. Filtered sinograms are back projected into image space (the back project in filtered back projection)

IMAGE ACQUISITION (FORWARD PROJECTION)

CT, SPECT

In CT and SPECT imaging, a sinogram is generated by rotating detectors around a patient, and storing the detected projection profiles at each angle in the sinogram, as depicted in the gif above. This gif specifically illustrates a SPECT acquisition, where the information about the biodistribution of a radioactive tracer is being emitted from within the patient (through photon emission) and virtually registered by the rotating detectors (an emission scan). A CT scan would work very similarly, with respect to image acquisition and reconstruction, except the photons reaching the detector would be coming from an x ray generating source at the other side of the patient (a transmission scan).Of course we can recall that a CT scan, a transmission scan, would give us information about the attenuation properties of the object being imaged – thus providing anatomical information. In contrast a SPECT scan, which is an emission scan, would tell us where a pharmaceutical is distributing throughout the body – thus providing functional information.

PET TOMOGRAPHY

For a PET acquisition, a patient is placed within a ring of detectors. Unlike SPECT, there are no roatating cameras or parts. However, sinograms are created much the same way. Virtual (emission) profiles per angle can be generated by mapping and sorting the detector pair events (based on a system martix provided by machine manufacturer). The below gif is provided to help visualize the lines of response and how they correlate to the sinogram.

PET ACQUISITION

PET images are generated through detection of the 511 Kev photons that arise during positron annihilation (the process of a positron combining with an electron, resulting in a transformation of particles with mass, to (massless) photons with energy. Data is collected by sorting each event into its appropriate location in sinogram space – each line of response has a corresponding angle and offset to indicate its location in the sinogram.

PET gif animation illustrates photon annihilation events taking place with a PET detector ring. As events are detected, they are recorded in the scan’s sinogram. Only sample events are illustrated, as the total true events would be too numerous to display.

IMAGE RECONSTRUCTION (BACKPROJECTION AND FILTERED BACK PROJECTION)

Once a sinogram is created (and stored in a computer), we can then use it to reconstruct a 3d image.

BACK PROJECTION (WITHOUT FILTERING)

Back projection is a process in which we “smear” the measured profile associated with each specific angle of acquisition, across the image space. Each projection profile in the sinogram (originally created and stored in sinogram at image acquisition), can be backprojected at the angle it was acquired at. The animation below illustrates how information stored at each angle in the sinogram is backprojected into image space.

The above animation illustrates how sinogram data correspond to image space backprojections. When we are reconstructing an image, we backproject more than a single angle at a time – specifically we backproject ALL the angles, and add their backprojections together in image space. The animation below shows how multiple angles in the sinogram can be backprojected together to create a backprojected image.

Backprojection alone is an inadequate image reconstruction strategy because we are left with a blurred representation of the image, as illustrated.

The blurring which takes place during back projection is referred to as “1/r blurring”

FILTERING

Back projection does not work as a useful image reconstruction method because of the blurring mentioned above. This blurring however can be corrected if we first filter the data. A useful/requisite ramp filter can be applied very quickly, as it is simply a multiplication function in the frequency domain (data can be transformed quickly using the Fourier transform). In imaging, where we are working with discrete/digital data, we can use the “fast Fourier transform” (see numerical recipes in C link) for speedy and accurate transformation of data to and from frequency space, thus allowing for very fast processing.

Several different types of filters can be used, but the basic most used filter is called a ramp filter, which corrects for the 1/r blurring effect that manifests during back projection. The negative of using the ramp filter alone is that it amplifies high frequency noise. This would not be a problem if our images were noiseless, but that is not the case in medical imaging. Other filters, which incorporate the ramp filter, can be used to alleviate the amplification of high frequency noise. (iterative reconstruction methods also use filters for optimizing properties).

FILTERING THEN BACKPROJECTION

Once sinograms are filtered, they can be back projected to recover an accurate representation of the original subject.

FILTERED BACK PROJECTION

The process of filtering sinogram data and then back projecting it is called filtered back projection reconstruction.

We can notice that to accurately reconstruct an image, we need to back project information from all 180 degrees of acquisition data. This can be exemplified with the following gif, in which images are created using only subsets of the data (split into angles).

And

SUMMARY

This concludes the tour of animations I made to help understand image reconstruction.

Alternative modality animation can be downloaded:

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NEW December 2014 – generate gifs with your own images!

Download gifmaking executable here (link)

COMPUTER REQUIREMENTS: RecondGifMaker3000.sav is an executible built in the IDL programming language. To run, host computer must have IDL or the (free) IDL virtual machine installed (http://www.exelisvis.com/).

INSTRUCTIONS – double click the ReconGifMaker3000.sav. The program will ask the user to point to a 200×200 pixel image saved in .jpg format. Gifs will then be generated based on user supplied .jpg. processing may require 1-3 minutes (user should see animation frames being created)

OUTPUT – gifs will be created in an output folder placed at the location of the image

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Any comments/suggestions for the page are welcome: adam.kesner@fulbrightmail.org

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That concludes this page. You made it to the end!. Here’s some final eye candy cutesy of Picasso (1949).