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, one 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.
Any feedback regarding this page is welcome (firstname.lastname@example.org
). If I find this website is being utilized I will try to expand on the content.
Page version 1.0, updated March 10, 2014
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
). All gif images will play as animations when opened with modern internet browsers.
Users are welcome to use images for any non-commercial use
THE IMAGING PROCESS
The main steps in imaging are
(1) Acquire image data
| (2) Sort/store data
||(3) Reconstruct images
||(4) Utilize images (physician review) |
|Raw tomographic data can be acquired using CT, PET, SPECT
||Data stored (in computer) as either
as raw detector signal output (listmode), or sinograms (pictured above) which
are angle specific histrograms of detected events. In a sinogram, every detected event can be
sorted and stored using the angle and offset charactoristic to its detection.
Usually, sinograms are much smaller than listmode files, but less flexible for
||Images can be reconstructed
using analytic or iterative reconstruction. (this webpage focuses on analytic
Reconstructed (3d) images can be rendered and displayed in many useful ways:
2d slices, MIP images, 3d renderings,…
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)
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.
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 my machine manufacturer). The below gif is provided to help visualize the lines of response and how they correlate to the sinogram.
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.
The 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 RECONSTRUCATION (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. This 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”
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 transormation 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).
This concludes the tour of animations I made to help understand image reconstruction.
Alternative modality animation can be downloaded: